A Simplified Network Model for Travel Time Reliability...

Research ArticleA Simplified Network Model for Travel Time ReliabilityAnalysis in a Road Network

Kenetsu Uchida and Teppei Kato

Graduate School of Engineering Hokkaido University Hokkaido Japan

Correspondence should be addressed to Kenetsu Uchida uchidaenghokudaiacjp

Received 1 September 2017 Revised 30 October 2017 Accepted 7 November 2017 Published 26 November 2017

Academic Editor Antonio Comi

Copyright copy 2017 Kenetsu Uchida and Teppei Kato This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper proposes a simplified network model which analyzes travel time reliability in a road network A risk-averse driver isassumed in the simplified model The risk-averse driver chooses a path by taking into account both a path travel time varianceand a mean path travel time The uncertainty addressed in this model is that of traffic flows (ie stochastic demand flows) In thesimplified network model the path travel time variance is not calculated by considering all travel time covariance between twolinks in the network The path travel time variance is calculated by considering all travel time covariance between two adjacentlinks in the network Numerical experiments are carried out to illustrate the applicability and validity of the proposed model Theexperiments introduce the path choice behavior of a risk-neutral driver and several types of risk-averse drivers It is shown thatthe mean link flows calculated by introducing the risk-neutral driver differ as a whole from those calculated by introducing severaltypes of risk-averse drivers It is also shown that the mean link flows calculated by the simplified network model are almost thesame as the flows calculated by using the exact path travel time variance

1 Introduction

Conventional frameworks for analyzing and modeling trans-portation systems have been confined to average representa-tions of the network state (eg average link flow or averagetravel flow) For instance in the traditional traffic assignmentmodel one can obtain a deterministic prediction of a futureflow on a certain link in the network based on averageorigin-destination (O-D) flows link capacities and a formof proportional path choice model (either deterministic userequilibrium (DUE) or stochastic user equilibrium (SUE))This represents a deterministic view of the environment andthemodelerrsquos postulation that the variability or uncertainty inthe system is not influential in system design and evaluation

There has been a growing concern over the uncertainty oftravel time in transport systems and its effect on the reliabilityof transport services [1] Research on network reliability hasbegun to address this problem [2ndash5] From the travelerrsquosperspective the issue of travel time reliability has been amajor concern Travelersmay experience excessive variabilityof travel time from day to day on the same trips [6ndash8]

In transport modeling some advances have been madetoward incorporating traffic flow uncertainties into the net-work modeling framework (ie developing a stochastic net-work model) Watling [9] proposed a second-order networkequilibrium model that explicitly considers random pathchoice behavior His model in contrast to the conventionalSUE model uses path choice probability as predicted bya SUE model to define stochastic path flows that followa multinomial distribution The path flows derived usingthe traditional SUE model are in fact the expected flowsof this multinomial distribution In this model the driverschoose their paths so as tominimize their perceived long-runexpected travel costs With a nonlinear travel cost functionthis long-run expected travel cost will differ from the equi-librium cost computed by the conventional SUE model [10]

Clark and Watling [6] extended this stochastic networkmodel to the case with a Poisson distribution of O-Dflows Similarly Nakayama and Takayama [11] proposed astochastic network model with random path choice behaviorbut using a binomial distribution of O-D flowsThesemodelsfully represent stochasticuncertain path choice behavior

HindawiJournal of Advanced TransportationVolume 2017 Article ID 4941535 17 pageshttpsdoiorg10115520174941535

2 Journal of Advanced Transportation

with uncertain flows Lo et al [12] extended their originalmodel to consider the concept of travel time budget in pathchoice decision-making Shao et al [13] adopted a similarpostulation of the central limit theorem to derive the normaldistribution of the path travel time but with O-D flowdistribution (normal distribution) Szeto and Solayappan [14]proposed a nonlinear complementarity problem formulationfor the risk-aversive stochastic transit assignment problemSumalee et al [15] address stochastic flow and stochasticcapacity in a multimodal network Uchida et al [16] addressnetwork design problem in a stochastic multimodal networkmodel Uchida [17] proposed a model which simultaneouslyestimates the value of travel time and of travel time reliabilitybased on the risk-averse driverrsquos path choice behavior Uchida[18] proposed a network equilibrium model which estimatestravel time reliability from the observed link flows in thenetwork Kato and Uchida [19] proposed a benefit estimationmethod that considers travel time reliability

In the context of the advancements in theoretical studieson travel time variability in road networks transporta-tion benefit-cost analysis (BCA) considering travel timevariability (httpssitesgooglecomsitebenefitcostanalysisbenefitstravel-time-reliability [20]) is now becoming a bigconcern The traditional DUE traffic assignment model hasbeen widely used to estimate the value or benefits of a policyprogram or project considering no travel time variability Ifwe consider the value of travel time variability in estimatingthe benefit of a policy the DUE traffic assignment modelcannot be applied since it does not address the stochasticnature of a travel time variability Therefore a plausiblenetwork equilibrium model which is well established interms of theory and practice is needed For BCA in whichtravel time variability is considered a measure of travel timevariability which is discussed in the next section needsto be determined Once the travel time variability measureis determined then a network equilibrium model whichcombines risk-averse driverrsquos path choice behavior with thegeneralized travel time which is defined as a mean traveltime plus a travel time variability measure multiplied by acalibration parameter is developed If we put more weight onthe accuracy than the practicality of a network equilibriummodel then the validity of BCA may increase however thecosts required for calculating BCA may increase and viceversa Therefore the modeler has to consider the trade-offbetween accuracy and practicalityThe objective of this studyis to propose a simplified and plausible network equilibriummodel which takes into account both the risk-averse driverrsquospath choice behavior and the travel time variability Thedifference between this study and the other studies that theauthors have presented is that we propose a model thatcan be applied to a large network problem The stochasticnetworkmodels that the authors have developed require pathenumeration in the network However the enumeration ofall possible paths is difficult in the case of a large networkTherefore a stochastic networkmodel for travel time reliabil-ity analysis that solves a large network problem is demanded

This paper starts by examining several measures oftravel time variability in the next section Based on thediscussion provided in the next section we will employ a

travel time variance as a measure of travel time variability inthis study Then link and path travel time under stochasticdemand flows are formulated in Section 3 In Section 4two network equilibrium models under stochastic demandflows are formulated considering a risk-averse driverrsquos pathchoice behavior in a road networkThe firstmodel introducespath travel time variance which is calculated considering alltravel time covariances between two links in the networkThe generalized travel time in this model is not additive sincethe generalized path travel time is not equal to the sum ofthe generalized link travel times related to that path Thesecond model which we propose in this study is a simplifiedversion of the first model In this model the path traveltime variance is not calculated by considering all travel timecovariance between two links in the networkThe path traveltime variance is calculated by considering all travel timecovariance between two adjacent links in the network Thegeneralized path travel time in this model is additive It isshown that a unique solution is provided by the simplifiednetwork model Numerical experiments are carried out toillustrate the applicability and validity of the proposedmodelFinally concluding remarks are provided in Section 6

2 Measures of Travel Time Variability

We will briefly review how to obtain a measure of traveltime variability based on the expected utility maximizationprinciple Vickrey [21] considered a separable or additiveutility function which is a sum of utilities obtained from timespent at an origin and time spent at a destination of a tripUsing such formulation of utility it is possible to considera driver who chooses a departure time optimally in orderto maximize expected utility when facing uncertain traveltime Noland and Small [22] Bates et al [23] Fosgerau andKarlstrom [24] Fosgerau and Engelson [25] and Engelson[26] have shown how measures of travel time variability canbe derived from the driversrsquo scheduling preferences

A popular formulation of scheduling preferences is the120572-120573-120574 preference inwhich themarginal utility of time (MUT)at the origin is constant and that at the destination is astep function [27 28] By assuming an exponential traveltime distribution or a uniform travel time distributionNoland and Small [22] derived the scheduling utility whichis linear in (120583 120590) where 120583 and 120590 are the mean and standarddeviation (SD) of the stochastic travel time Fosgerau andKarlstrom [24] generalized this result to any travel timedistributions Fosgerau and Engelson [25] considered thevalue of travel time reliability under scheduling preferencesthat were defined in terms of linear MUTs being at the originand at the destination They found that the scheduling utilitywas linear in (120583 1205832 1205902) and that this result was independentof the shape of a travel time distribution

Engelson [26] derived the scheduling utility for thetwo cases when the MUTs at both the origin and thedestination are either quadratic or exponential in form anddemonstrated special cases when the scheduling utility isadditiveThe necessary condition when the scheduling utilityis additive is that theMUT at the origin is a positive constant

Journal of Advanced Transportation 3

Engelson and Fosgerau [29] derived a measure of traveltime variability for travelers equipped with scheduling pref-erences defined in terms of MUT and who chose departuretime optimality In the case of ℎ(119905) = ℎ0 and 119908(119905) = ℎ0 +(120574120573) sdot (119890120573sdot(119905minus1199050) minus 1) which are respectively MUT at theorigin and that at the destination evaluated at clock time 119905(the parameters in theMUTs satisfy the following conditionsℎ0 gt 0 120573 ge 0 120574 gt 0 and ℎ(1199050) = 119908(1199050)) the schedulingutility is

119906 (119905 119879) = int1199050119905ℎ0119889120591

+ 119864[int119905+1198791199050

(ℎ0 + 120574120573 sdot 119890120573sdot(120591minus1199050) minus 120574120573)119889120591]= ℎ0 sdot 120583 minus 120574120573 sdot (119905 minus 1199050 + 120583) + 1205741205732sdot (119890120573sdot(119905minus1199050) sdot 119864 [119890120573sdot119879] minus 1)

(1)

where 119879 is an independently distributed random travel timewith a mean value of 120583 and an SD of 120590 and 1199050 is the time suchthat the individual prefers being at the origin before this timeand at the destination after this time From the first-ordercondition the scheduling utility is derived as

119906 (119879) = ℎ0 sdot 120583 + 1205741205732 sdot ln119864 [119890120573sdot(119879minus120583)] (2)

where ln119864[119890120573sdot(119879minus120583)] is the cumulant-generating function(CGF) of the centralized travel time distribution A limitingcase of 120573 rarr 0 yields

lim120573rarr0

119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 (3)

If the travel time distribution has compact support then thescheduling utility is finite for any 120573 and can be presented asthe convergent Taylor series

119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 + 120574 sdot infinsum119899=3

120573119899minus2 sdot 119896119899119899 (4)

where 119896119899 is the cumulant of order 119899 of the travel timedistribution If the travel time follows a normal distributionhowever the normal distribution does not have compactsupport and for any 120573 the scheduling utility is

119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 (5)

A special case of this problem with constant ℎ(119905) equiv 120572 and thetwo-valued function 119908(119905) = 120572 minus 120573 for 119905 lt 1199050 and 119908(119905) =120572 + 120573 for 119905 ge 1199050 leads to the 120572-120573-120574 preferences model Itwas reported that (2) and (3) have advantages over the 120572-120573-120574preferences model First they do not depend on the shapeof the travel time distribution The second is additivity withrespect to parts of trip with independent travel time whichis an important property to analyze travel time reliability ina road network Travel time variance as a measure of travel

time variability may be criticized for not taking into accountthe skewness of the travel time distribution [30] The CGFdepends on the skewness (11989631205903 = 119864[(119879 minus 120583)3]1205903) of thetravel time distribution for nonzero 120573 However the traveltime covariance between two links is not taken into accountin a CGF in which independent link travel time is assumedThe effect of link travel time covariance terms on the pathtravel time variance becomes larger than that of the skewnessas the number of links in a path increases For example if apath is comprised of 119899 links the number of link travel timecovariance terms taken into account in calculating the pathtravel time variance is 1198991198622 We recognize that the travel timecovariance between two links is a more important factor inanalyzing travel time reliability in a road network than theskewness of the travel time distribution

Hjorth et al [31] analyzed the stated preference databy applying the scheduling preferences model that assumesMUTs at the origin and at the destination They have shownthat the value of travel time variability can be proportional tothe variance of travel time This result can partially supportthe use of travel time variance as a measure of travel timevariability

The additivity of the scheduling utility is a convenientproperty for a network equilibriummodel from the practicalviewpoint (calculation cost efficiency ease of handling thenetwork equilibriummodel etc) If we employ an SD relatedmeasure of the travel time variability (SD percentile valueetc) unrealistic driversrsquo path choice behavior as shown nextmay be generated

The following example is cited from Cominetti andTorrico [32] The generalized travel time of a random traveltime 119879 is given by 119888 (119879) = 120583 + sdot 120590 (6)

where we assume = 1 without loss of generality Weconsider then the traffic situation shown in Figure 1 in whicha road network consists of three nodes and three links andthe stochastic link travel time is shown In the network linktravel time is denoted by 119879119894 (119894 = 1 3) where 119894 is linknumber which follows the normal distribution119873(120583 1205902)witha mean of 120583 and a variance of 1205902 From the link travel timesshown in the figure we obtain 119888(1198791) = 12 119888(1198792) = 10 + radic5119888(1198791 + 1198793) = 21 + radic3 and 119888(1198792 + 1198793) = 20 + radic7 Fromthese four generalized travel times the following results canbe obtained The minimum generalized travel time betweennodes 1 and 2 is 12 and the minimum path consists of link 1The minimum generalized travel time between nodes 1 and 3is min(119888(1198791 + 1198793) 119888(1198792 + 1198793)) = 20 +radic7 and the minimumpath consists of links 2 and 3This example shows that a driverin a network with an origin node 1 and a destination node 2will choose the path comprising link 1 if heshe prefers smallergeneralized travel time However the same driver in thenetworkwhose origin and destination nodes are respectively1 and 3 will choose the path comprising two links 2 and3 However the path choice criterion is clear and the pathchoice behavior of the driver is somehow unrealistic Eventhough the mean deviation is used instead of SD such anunrealistic case can occur since the generalized path travel


1 2 3

T1 = N(11 1)

T3 = N(10 2)

T2 = N(10 5)

Figure 1 A paradoxical path choice

time is not equal to the sum of the generalized travel timesof the links that comprise that path in the case of meandeviation

We examined the convenience and importance of theadditivity of the generalized path travel timewhen addressingit in a network problem In the following we formulate somenetwork equilibrium models in which travel time varianceis employed as a measure of the travel time variability inreference to (5) According to (5) the generalized travel timeis given by

119888 = 119906ℎ0 = 120583 + 120596 sdot 1205902 (7)

where 120596 = 1205742 sdot ℎ0 (8)

3 Link and Path Travel Times underStochastic Flows

31 Notation The notations below are used in this paper119860 Set of links in the network119868 Set of O-D pairs in the network119869119894 Set of paths between O-D pair 119894120575119886119895 Variable that equals 1 if link 119886 is part of path 119895 andequals 0 otherwise119881119886 Stochastic flow of link 119886V119886 Mean flow of link 119886V119886119887 Mean flow that passes through both links 119886 and 119887119888119886 Capacity of link 119886119865119894119895 Stochastic flow of path 119895 between O-D pair 119894119891119894119895 Mean flow of path 119895 between O-D pair 119894119876119894 Stochastic flow for O-D pair 119894119902119894 Mean flow for O-D pair 119894119876 Stochastic total O-D flow119902 Total mean O-D flow119901119894119895 Path choice probability for O-D pair 119894 choosingpath 119895119901119894 Proportion of mean flow for O-D pair 119894 119902119894 to totalmean flow 119902119888V119894 Coefficient of variation of random flow 119876119894

Ξ119894119895 Stochastic travel time of path 119895 which serves O-Dpair 119894119888119894119895 Generalized travel time of path 119895which servesO-Dpair 119894

32 Stochastic Traffic Flows An O-D flow 119876119894 is assumedto be a random variable with a mean of 119864[119876119894] = 119902119894 and avariance of var[119876119894] = (119888V119894 sdot 119902119894)2 where 119888V119894 is the coefficientof variation of the random flow119876119894 Following Lam et al [33]the stochastic flow on path 119895 isin 119869119894 119865119894119895 is then given by

119865119894119895 = 119901119894119895 sdot 119876119894 forall119894 isin 119868 forall119895 isin 119869119894 (9)

119865119894119895 is a random variable with a mean of 119891119894119895 = 119901119894119895 sdot 119902119894 ge 0and a covariance of cov[119865119894119895 119865119894119896] = 119901119894119895 sdot 119901119894119896 sdot var[119876119894] where119901119894119895(119895 isin 119869119894) is path choice probability which can be determinedby a path choice model (DUE SUE etc) The following flowconservation law holds for each O-D pair

sum119895isin119869119894

119891119894119895 = 119902119894 forall119894 isin 119868 (10)

The variance of 119865119894119895 is given by

var [119865119894119895] = var [119901119894119895 sdot 119876119894] = (119901119894119895)2 sdot var [119876119894]= (119888V119894 sdot 119891119894119895)2 forall119894 isin 119868 forall119895 isin 119869119894 (11)

The conservation of the path flow variance in relation to theO-D flow variance holds (Appendix A) The stochastic flowof link 119886 119881119886 is given by

119881119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119865119894119895 forall119886 isin 119860 (12)

The mean and covariance of the stochastic link flow are then

V119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119891119894119895 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119901119894119895 sdot 119902119894forall119886 isin 119860 (13)

cov [119881119886 119881119887] = var[[sum119894isin119868sum119895isin119869119894 120575119886119895 sdot 120575119887119895 sdot 119865119894119895]] forall119886 119887 isin 119860 (14)

where sum119894isin119868sum119895isin119869119894 120575119886119895 sdot 120575119887119895 sdot 119865119894119895 is the sum of all stochastic pathflows that pass through both links 119886 and 11988733 Stochastic Link Travel Time and Stochastic Path TravelTime In this study link travel time is represented by thefollowing BPR function [34]

119905119886 (V119886) = 1199050119886 sdot (1 + 120581 sdot (V119886119888119886 )120582) forall119886 isin 119860 (15)


where 1199050119886 is free flow travel time of link 119886 and 120581(ge 0) and120582(ge 1)are calibration parameters By substituting V119886 in (15) with 119881119886we obtain 119905119886 (119881119886) = 1199050119886 + 120581119886 sdot (119881119886)120582 forall119886 isin 119860 (16)

where 120581119886 = 1199050119886 sdot 120581(119888119886)120582Next we will show how to calculate both a mean value

and variance of the stochastic link travel time shown by (16)By performing an119898th-order (119898 ge 1)Taylor expansion to (16)at 119881119886 = V119886 we obtain

119905119886 (119881119886) = 119898sum119896=0

119887119896119886 sdot (119881119886 minus V119886)119896 forall119886 isin 119860 (17)

where 119887119896119886 is the coefficient of the 119896th term of the Taylorexpansion given by

119887119896119886 = 1119896 sdot 120597(119896)119905119886 (119881119886)120597119881(119896)1198861003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886

= 119905119886 (V119886) if 119896 = 0120581119886 sdot prod119896119897=1 (120582 minus 119897 + 1)119896 sdot V120582minus119896119886 otherwise

(18)

The mean link travel time is then calculated as

119864 [119905119886 (119881119886)] = 119898sum119896=0

119887119896119886 sdot 119864 [(119881119886 minus V119886)119896] forall119886 isin 119860 (19)

The travel time covariance between two links is

cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=0

119898sum119897=0

119887119896119886 sdot 119887119897119887sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]minus 119864 [119905119886 (119881119886)] sdot 119864 [119905119887 (119881119887)]forall119886 119887 isin 119860

(20)

As shown in Clark and Watling [6] (19) and (20) can becalculated by applying a method proposed by Isserlis [35]given the moments of 119881119886 by assuming that the link flowfollows normal distribution [17 18 33] This assumption wassupported by Rakha et al [36] which demonstrated thatthe normality assumption may be sufficient from a practicalstandpoint given its computational simplicity Equation (19)can be calculated as follows

119864 [119905119886 (119881119886)] = 1198870119886 + 1198982sum119896=1

119896prod119897=1

(2119897 minus 1) sdot 1198872119896119886 sdot (var [119881119886])119896 119898 even number

1198870119886 + (119898minus1)2sum119896=1

119896prod119897=1

(2119897 minus 1) sdot 1198872119896119886 sdot (var [119881119886])119896 119898 odd numberforall119886 isin 119860 (21)

The results of119898 = 4 are provided in Appendix BWe now assume that the coefficient of each O-D flow

takes a specific value By applying this assumption to (14) (ie119888V119894 = 119888V forall119894 isin 119868) we obtaincov [119881119886 119881119887] = var[[sum119894isin119868sum119895isin119869119894 120575119886119895 sdot 120575119887119895 sdot 119865119894119895]] = (119888V sdot V119886119887)2

forall119886 119887 isin 119860 (22)

where

V119886119887 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 120575119887119895 sdot 119891119894119895 forall119886 119887 isin 119860 (23)

V119886119887 in (23) is the mean flow that passes through both links 119886and 119887 If 119886 = 119887 in (22) thenwe obtain var[119881119886] = (119888V sdotV119886)2 forall119886 isinA

In fact this assumption can be justified if we regard totalO-D flow in the network 119876 = sum119894isin119868119876119894 as a random variablewith amean of119864[119876] = 119902(= sum119894isin119868 119902119894) and a variance of var[119876] =(119888Vsdot119902)2 where119876119894 = 119901119894 sdot119876119901119894 = 119902119894119902 is the proportion of theO-D flow 119902119894 to total O-D flow 119902 In this case all O-D flows are

statistically dependent on each otherThe conservation of theO-D flow variance in relation to the total O-D flow varianceholds (Appendix C)

By substituting (22) into (19) and (20) we obtain

119864 [119905119886 (119881119886)] = 119905119886 (V119886) + sum119896=1

119896prod119897=1

(2119897 minus 1) sdot 2119896119886sdot (V119886)120582 forall119886 isin 119860 (24)

cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=1

119888119898119896 sdot (V119886)120582minus119896 sdot (V119887)120582minus119896sdot (V119886119887)2119896 forall119886 119887 isin 119860 (25)

where

119896119886 = 120581119886 sdot (119888V)119896 sdot prod119896119897=1 (120582 minus 119897 + 1)119896 (26)

119888119898119896 in (25) is the coefficient for the 119896th term The results of119898 = 4 are provided in Appendix DNote that it is shown from (24) and (25) that the mean

variance or covariance of link travel time is expressed by


using onlymean link flow(s)with some given parameters andit will be shown that both119864[119905119886(119881119886)] and cov[119905119886(119881119886) 119905119887(119881119887)] areincreasing functions with respect to V119886 and V119886119887 respectivelyIt will be shown that these two mathematical properties areconvenient for developing a network equilibriummodelThemost dominant reason for these two properties is that thecoefficients of variation of all O-D flows are assumed tobe the same Thanks to this assumption any moments forthe stochastic link flow can be calculated by using its meanvalue Also as far as the Taylor series expansion providesgood approximation the method presented in this study canbe applied to any functional forms Since the Taylor seriesexpansion can approximate well the function of 119891(119909) =1(1 minus 119909) for 0 lt 119909 le 1 the proposed method canbe applied to the Davidson type link cost function From(25) cov[119905119886(119881119886) 119905119887(119881119887)] gt 0 if and only if V119886119887 gt 0 From(23) if V119886119887 gt 0 then V119886 gt 0 and V119887 gt 0 however theinverse relationship does not always hold for any two linksin the network (ie even though V119886 gt 0 and V119887 gt 0 V119886119887can be zero) These two mathematical properties show thatthe travel time covariance of two links cov[119905119886(119881119886) 119905119887(119881119887)]is greater than zero if and only if V119886119887 is greater than zeroand that V119886 and V119886 can influence the travel time covarianceof two links cov[119905119886(119881119886) 119905119887(119881119887)] if and only if V119886119887 is greaterthan zero Therefore a calculation of V119886119887 is important tocalculate cov[119905119886(119881119886) 119905119887(119881119887)] in the network As discussed inthe next section the calculation of V119886119887 for two adjacent linksin the network is easily implemented since there is no needto enumerate a path set in the network In contrast thecalculation of V119886119887 for two unconnected links in the networkbecomes more difficult than that for two adjacent links sincethat may need to enumerate a path set in the network

For notational simplicity in the rest of the paper119864[119905119886(119881119886)] var[119905119886(119881119886)] and cov[119905119886(119881119886) 119905119887(119881119887)] are denoted by119886(V119886)1205902119886(V119886) and120590119886119887(V119886119887 V119886 V119887) respectivelyThe travel timeof path 119895 which serves O-D pair 119894 (Ξ119894119895) is given by

Ξ119894119895 = sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895 forall119894 isin 119868 forall119895 isin 119869119894 (27)

The mean path travel time and path travel time variance arerespectively given by

119864 [Ξ119894119895] = 119864[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895] = sum119886isin119860

119886 (V119886) sdot 120575119886119895forall119894 isin 119868 forall119895 isin 119869119894

var [Ξ119894119895] = var[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895]= sum119886isin119860

sum119887isin119860

120575119886119895 sdot 120575119887119895 sdot 120590119886119887 (V119886119887 V119886 V119887)= sum119886isin119860

1205902119886 (V119886) sdot 120575119886119895 + 2 sum119886isin119860

sum119887( =119886)isin119860

120575119886119895 sdot 120575119887119895sdot 120590119886119887 (V119886119887 V119886 V119887) forall119894 isin 119868 forall119895 isin 119869119894

(28)

The path travel time covariance is

cov [Ξ119894119895 Ξ119894119896]= cov[sum

119886isinA120575119886119895 sdot 119905119886 (119881119886) sum

119887isinA120575119887119895 sdot 119905119887 (119881119887)]

= sum119886isinA

sum119887isinA

120575119886119895 sdot 120575119887119896 sdot 120590119886119887 (V119886119887 V119886 V119887)forall119894 isin 119868 forall119895 isin 119869119894 forall119896 isin 119869119894

(29)

For calculation methods of the mean travel time and traveltime variance when each link flow in the network followsa lognormal distribution the reader is referred to Tani andUchida [37] in which each link capacity in the network is alsoassumed to follow a lognormal distribution

4 Network Equilibrium Model underStochastic Flows

41 DUE Principle A risk-averse driver may take intoaccount both a mean travel time and travel time variabilityin hisher path choice decision Travel time variance isemployed as a measure of the travel time variability in thisstudyThe generalized travel time of path 119895which serves O-Dpair 119894 (Ξ119894119895) is defined as

119888119894119895 = 119864 [Ξ119894119895] + 120596 sdot var [Ξ119894119895] (30)

where 120596 ge 0 is a relative weight assigned to var[Ξ119894119895] Therisk-averse driver assumed in this study chooses the pathwith lower path travel time variance if the mean path traveltimes of all the alternative paths are the same Such risk-aversedriverrsquos path choice problem based on the DUE principle canbe formulated as follows119888lowast119894119895 = 120587119894 if 119891lowast119894119895 gt 0119888lowast119894119895 ge 120587119894 if 119891lowast119894119895 = 0forall119894 isin 119868 forall119895 isin 119869119894

(31)

subject to (10) (13) and (23) where 120587119894 is the minimumgeneralized travel time of O-D pair 119894 The superscript lowast isused to denote the variables that are obtained at equilibriumIt is known that this problem is equivalent to the followingnonlinear complementary problem (NCP)119891lowast119894119895 sdot (119888lowast119894119895 minus 120587119894) = 0119888lowast119894119895 minus 120587119894 ge 0119891lowast119894119895 ge 0 forall119894 isin 119868 forall119895 isin 119869119894

(32)

subject to (10) (13) and (23) The equilibrium path flows canbe obtained by solving the following variational inequality(VI) problem [38]


Find flowast isin Ω119891 such thatsum119894isin119868

sum119895isin119869119894

(119891119894119895 minus 119891lowast119894119895 ) sdot 119888lowast119894119895 ge 0 forallf isin Ω119891 (33)

where Ω119891 = f | sum119895isin119869119894 119891119894119895 = 119902119894 forall119894 isin 119868 119891119894119895 ge 0 forall119894 isin 119868 forall119895 isin119869119894 and f = (119891119894119895)119894isin119868119895isin119869119894 There are efficient solution algorithms for solving the

DUE traffic assignment problem However most of suchalgorithms cannot be applied to solve the VI problem shownabove in which the path travel time variance is nonadditivedue to the link travel time covariance between two links inthe network Therefore path-based solution algorithms [39]which in general require enumeration of a path set need tobe applied in order to solve the VI problem However theenumeration of all possible paths is almost impossible forthe case of a large network Therefore in the next sectionwe will propose a simplified network model in which onlythe covariance terms between two adjacent links in thenetwork are taken into account in calculating the path travelvariance by considering practicality Thus an efficient link-based algorithm for DUE traffic assignment problem can beapplied to the simplified network model

42 Simplification Paths enumeration may be required forsolving the VI problem presented in the previous sectionEnumeration of all noncyclic paths in a large road network isimpossible From a practical standpoint it may be reasonableto enumerate several paths for each O-D pair (eg two orthree paths for each O-D pair) However different solutionscan be obtained depending on paths enumerated and thatmay be a troublesome issue in estimating the benefit of aproject

If we assume that the link travel time follows an indepen-dent distribution then path travel time variance is the sum ofthe link travel time variance related to that path In this casethe path choice problem can be formulated as the followingconvex programming problem

min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)

subject to (10) (13) and (23) where 119892119886(V119886) = 119886(V119886) + 120596 sdot1205902119886(V119886) This problem has the same mathematical structureas the standard DUE traffic assignment model and thus canbe solved easily by applying standard link-based algorithms(MSA (Method of Successive Averages) the Frank-Wolfealgorithm etc) [40]However ignoring all travel time covari-ance in (28) may bring about unrealistic solutions

Rakha et al [36] presented extensive evidence of asignificant correlation between travel times of two adjacentlinks in the network Although they analyzed travel timevariability over vehicles this evidence may support traveltime variability over days By utilizing this evidence we nowtake into account all travel time covariance between twoadjacent links in the network when calculating the path traveltime variance The corresponding path choice problem canbe then formulated as the following link-based VI problemsimplified network model (SNM)

Find klowast isin ΩV and klowast isin ΩV such that

sum119886isinA

((V119886 minus Vlowast119886 ) sdot 119892119886 (Vlowast119886 ) + sum119887isin120579(119886)

(V119886119887 minus Vlowast119886119887)sdot 119892119886119887 (Vlowast119886119887 Vlowast119886 Vlowast119887 )) ge 0 forallk isin ΩV forallk isin ΩV (35)

where119892119886119887 (V119886119887 V119886 V119887) = 2 sdot 120596 sdot 120590119886119887 (V119886119887 V119886 V119887)ΩV = k | V119886 = sum

119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119891119894119895 forallf isin Ω119891 forall119886 isin 119860 ΩV = k | V119886119887 = sum

119894isin119868

sum119895isin119869119894

120575119886119895 sdot 120575119886119895 sdot 119891119894119895 forallf isin Ω119891 forall119886isin 119860 forall119887 isin 120579 (119886)

k = (V119886)119886isin119860 k = (V119886119887)119886isin119860119887isin120579(119886)

(36)

120579(119886) is the set of linkswhich are adjacent to link 119886 in front of itSNM includes no stochastic variable although SNM analyzestravel time reliability in the network

To solve SNM one can apply a network representationwhich may be used when addressing intersection delays inwhich dummy links connecting between all adjacent linksin the network are added to the original network Consideran original network that consists of a set of links and a setof nodes It is assumed that each node in the network alsohas an identical number Consider then a directed link inthe original network We can find each link of which originnode number is the same as the destination node of the linkBy using this relationship we can construct the augmentednetwork (eg right-hand side of Figure 2) that correspondsto the original network (eg left-hand side of Figure 2) byinserting a directed dummy link between these two linksIn the augmented network the generalized travel time oflink 119886 forall119886 isin 119860 is 119892119886(V119886) and that of dummy link 119886119887 forall119886 isin119860 forall119887 isin 120579(119886) is 119892119886119887(V119886119887 V119886 V119887) Once the augmented networkis constructed V119886119887 is easily calculated as the flow of link 119886119887 Byapplying this network representation SNMcan be solved by adiagonalization method in which V119886 and V119887 in 119892119886119887(V119886119887 V119886 V119887)are regarded as constant terms [41ndash45] Also MSA can beapplied for solving SNM Due to the mathematical structureof 119892119886119887(V119886119887 V119886 V119887) however SNM has the same mathematicalproperty as an asymmetric DUE traffic assignment problemwhich can have multiple solutionsTherefore SNMmay havemultiple solutions Next we will examine the uniqueness ofthe solution of SNM

If both generalized link travel time functions119892119886(V119886) forall119886 isin119860 and 119892119886119887(V119886119887 V119886 V119887) forall119886 isin 119860 forall119887 isin 120579(119886) in the aug-mented network are monotone functions SNM has a unique


ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)

Figure 2 Network representation for addressing intersection delays

solution Or equivalently if the Jacobian matrix nablag(k k)where g = ((119892119886)119886isin119860 (119892119886119887)119886isin119860119887isin120579(119886)) is positive-definite SNM has a unique solution The Jacobian matrix nablag(k k) is given

by the following lower triangle matrix

nablag (k k) =((((((((((((((

1198891198921119889V10 d 00 0 119889119892|119860|119889V|119860|1205971198921120579(1)120597V1 sdot sdot sdot 1205971198921120579(1)120597V120579(1) sdot sdot sdot 0 1205971198921120579(1)120597V1120579(1) d0 sdot sdot sdot 120597119892|119860|120579(|119860|)120597V120579(|119860|) sdot sdot sdot 120597119892|119860|120579(|119860|)120597V|119860| 0 sdot sdot sdot 120597119892|119860|120579(|119860|)120597V|119860|120579(|119860|)

))))))))))))))

(37)

If every eigenvalue of a Jacobian matrix is positive theJacobian matrix is positive-definite The eigenvalues of alower triangle matrix are equal to the values of diagonalelements of thematrix In SNM the following two conditionshold for each link in the augmented network (see Appendix Efor the proofs)120597119892119886119887 (V119886119887 V119886 V119887)120597V119886119887 gt 0 forall119886 119887 isin 119860120597119892119886 (V119886)120597V119886 gt 0 forall119886 isin 119860 (38)

Therefore nablag(k k) is positive-definite and thus SNM has aunique solution If the link travel time covariance for any twolinks in the network is taken into consideration in calculatingpath travel time variance the uniqueness of the solution is notobtained

Since SNM calculates the stochastic variables at theequilibrium the stability as well as the uniqueness of thesolution is guaranteed in SNM Even in a problem in a staticcontext the stability and uniqueness of the solution can beanalyzed by applying theory in a dynamic contextThis is truefor our static problem

5 Numerical Experiments

51 Settings In this section we carry out numerical exper-iments for illustrating the application and validity of SNMWe adopt the network of Nguyen and Dupuis [46] with 4 O-D pairs 25 paths and 19 directed links (Figure 3) The linksequences of the paths are shown in Table 1

We employed the mean link travel time and travel timecovariance shown in (24) and (25) that are calculated byassuming 119898 = 4 respectively Parameters for the BPRfunction 120581 and 120582 are 262 and 5 respectively The coefficientof variation of total O-D flow 119888V is 01 The other parametersused in (24) and (25) are shown in Table 2

In this study we prepared four experimental cases bychanging the effect of the path travel time variance on thedriverrsquos path choice behavior in the network In the first casewe assume a risk-neutral driver who chooses a path basedonly on mean path travel time in the network Thereforethe relative weight assigned to path travel time variance120596 in (30) is 0 in this case The other three cases assumethree types of risk-averse driversThe relative weight assignedto var[Ξ119894119895] is assumed as 03 in these three cases In thesecond case we assume that there is no travel time correlationbetween two different links in the network In the third caseSNM is employed (ie all travel time covariance betweentwo adjacent links in the network is introduced to calculatethe path travel time variance) In the fourth case all traveltime covariance between two different links in the network isintroduced to calculate the path travel time variance In thelatter three cases the path travel time variance is calculatedas

var [Ξ119894119895] = sum119886isin119860

sum119887isin119861

cov [119905119886 (119881119886) 119905119887 (119881119887)] sdot 120575119886119895 sdot 120575119887119895forall119894 isin 119868 forall119895 isin 119869119894 (39)

by using both the relative weight assigned to path travel timevariance 120596 and the set of link(s) 119861 shown in Table 3


6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin

Figure 3 The Nguyen and Dupuis network

Table 1 Paths and link sequences

O-D O-D pair Path Link seq

(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16

Table 2 Link travel time parameters

Link Free-flow travel time Capacity1199050119886 119888119886(1) 10 1500(2) 10 1500(3) 10 1500(4) 20 1500(5) 10 1500(6) 10 1500(7) 10 1500(8) 10 1500(9) 10 1500(10) 10 1500(11) 10 1500(12) 10 1500(13) 20 1500(14) 10 1500(15) 10 1500(16) 10 1500(17) 10 1500(18) 40 1500(19) 10 1500

For solving the first three cases we employed MSA as alink-based solution algorithmThe final case can be solved byminimizing the gap function for NCP [17 47 48] In fact thepaths set shown in Table 1 is prepared only for the final case


Table 3 Assumptions of each case

Case Relative weight (120596) Set of link(s) (119861)2 03 1198863 03 120579 (119886)4 03 119860

Table 4 Mean path flows

O-D Path Case 1 Case 2 Case 3 Case 4

(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97

Only case 4 cannot be solved byMSAwhich does not requirepaths enumeration and thus is applicable to a large networkOn the other hand it is difficult to solve the problem of alarge network using the method based on the gap functionthat requires paths enumeration

52 Results Table 4 shows the mean path flows for eachcase For the first three cases since we applied MSA to solvecorresponding path choice problems the path flows for eachcase were not uniquely determined Even so presenting pathflows for the first three cases may be useful to understandroughly how path flows change according to different expres-sions of path travel time variance Also by presenting thepath flows for cases 1ndash3 to which MSA was applied it is easyto understand that the generalized travel times for the pathsbetween each O-D pair that are used by the drivers are thesame although both the mean travel time and travel timevariance of a path between theO-D pair can be different from

Table 5 Generalized path travel time


(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851

the others In contrast the path flows for case 4were uniquelydetermined since we applied a path-based algorithm whensolving its path choice problem In all cases if a path flow iszero such path flow is denoted by bold figures in Table 4 sothat we can know that the corresponding path is not chosenby the drivers in the network Table 5 shows generalized pathtravel times for all cases which are calculated by using boththe mean path travel times shown in Table 6 and the pathtravel time variance shown in Table 7 It is observed that thepaths chosen by the drivers have the minimum generalizedtravel time and that the paths which are not chosen bythe drivers have generalized travel times which are equal toor greater than the minimum generalized travel time Thegeneralized path travel times for unused paths are denotedby bold figures in Table 5 It is shown in Tables 4 6 and 8that although the mean path flows of each case are differentfrom the other cases mean link flows are similar among allcases Since mean travel time of a path is calculated by usingmean link flows therefore similar mean path travel times areobtained among the four cases

For O-D pair 1 the flows of paths 4 5 and 8 in cases 2ndash4are smaller than those in case 1 whereas the flows of path 1in cases 2ndash4 are larger than that in case 1 These differencescan be explained as follows The travel time variances of path1 in cases 2ndash4 which are denoted by bold figures in Table 7are much smaller than those of paths 2ndash8 although the mean


Table 6 Mean path travel time


(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712

travel time of path 1 is longer than those of paths 2ndash8 whichare denoted by bold figures in Table 6 However path 1 haslonger mean travel time than paths 2ndash8 and that path is morereliable in terms of travel time variance than paths 2ndash8 incases 2ndash4 In total the path choice probabilities of path 1 incases 2ndash4 are larger than that in case 1 In contrast since thedrivers in case 1 choose their paths based only onmean traveltimes the flow of path 1 is larger than those in cases 2ndash4Thispath choice switch can be observed in the other O-D pairs(eg from path 10 to path 9 for O-D pair 2 from paths 17 and19 to paths 16 and 18 for O-D pair 3 and from paths 20 and24 to path 25 for O-D pair 4

We can find from Table 6 a tendency of the mean pathtravel time in case 1 to be greater than those in cases 2ndash4Surprisingly this tendency holds for all paths in the networkexcept for path 1 Obviously this tendency was derived fromthe introduction of travel time variance to the driversrsquo pathchoice behavior We will then check how total mean traveltime in the network is shortened by the introduction of traveltime variance An index for the total mean travel time forcases 119899 isin 1 4 TTT119899 can be given by

TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)

By using both the mean link flows shown in Table 8 andthe mean link travel time shown in Table 9 TTT119899 forall119899 isin1 4 are calculated as 2847times105 2789times105 2797times105

Table 7 Path travel time variance


(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465

Table 8 Mean link flows

Link Case 1 Case 2 Case 3 Case 4(1) 904 914 896 890(2) 1096 1086 1104 1110(3) 1024 1036 1040 1044(4) 976 964 960 956(5) 1010 1017 1021 1028(6) 918 933 914 906(7) 1215 1151 1157 1155(8) 392 295 325 342(9) 514 363 383 387(10) 701 788 774 768(11) 1013 1021 1026 1028(12) 837 873 855 846(13) 1057 1024 1019 1016(14) 1229 1167 1181 1188(15) 987 979 974 972(16) 943 976 981 984(17) 597 428 461 469(18) 499 658 643 641(19) 1057 1024 1019 1016

and 2794 times 105 respectively An introduction of travel timevariance to the path choice behavior model may bring about


Table 9 Mean link travel time


an efficient use of the road network in terms of the meantravel time This effect is similar to the one obtained byapplying the system optimal principle (ie Wardroprsquos secondprinciple) From the mean link flows shown in Table 8 it isobserved that the mean flows of links 5 9 11 12 16 and 18in cases 2ndash4 are larger than those in case 1 whereas those oflinks 4 7 8 13 14 15 and 17 in cases 2ndash4 are smaller than thosein case 1 These changes happened by the path flow changesin cases 1ndash4 which we have discussed

As discussed in the previous section if we do notconsider the effect of travel time reliability on the driverrsquospath choice behavior in the network unrealistic results maybe obtained However the introduction of all travel timecovariance between two different links in the network tothe calculation of the path travel time variance is difficultto implement Therefore we proposed SNM in which alltravel time covariance between two adjacent links in thenetwork is considered in calculating the path travel timevariance Our concern is now how far the results obtainedin cases 1ndash3 are from the results obtained in case 4 Theresults obtained in case 4 can be regarded as the exactsolution since it is obtained by adopting the path travel timevariance denoted by (28) Table 10 shows the coefficientsof correlation of mean link flows among four cases Thebold figures in the table show the coefficients of correlationbetween case 4 and the other three cases One can see thatSNM (case 3) reproduces almost the same mean link flowsas case 4 since the coefficient of correlation between case 4and case 3 is 1 whereas the other coefficients of correlation areless than 1 Note that the coefficient of correlation betweencase 4 and case 1 is the smallest Almost the same resultswere obtained by calculating the coefficients of correlationof mean link travel times These results may bring about two

Table 10 Coefficients of correlation of mean link flows

Case 1 Case 2 Case 3 Case 4Case 1 1000Case 2 0957 1000Case 3 0973 0995 1000Case 4 0969 0997 1000 1000

practical implications that all travel time covariance betweentwo different links in the network may not be required incalculatingmean link flows Instead all travel time covariancebetween two adjacent links in the network is required incalculating mean link flows and that calculation of all traveltime covariance between two adjacent links in the network iseasy even in a large network

6 Conclusions

In this study we proposed a simplified network model thatis SNM for travel time reliability analysis The uncertaintyaddressed in this model is that of O-D flows In this modelthe generalized path travel time is a linear combination ofmean path travel time and path travel time variance In cal-culating the path travel time variance we consider all traveltime covariance between two adjacent links in the networkin SNM A risk-averse driver in the network is assumedBy applying a network representation used for addressingintersection delays SNMcanbe solved by applying a standardlink-based algorithm The other property of SNM whichneeds to be emphasized here is that its formulation requiresonlymean network flowsThis propertymay be important forpractitioners since once the coefficient of variation of totalO-D flow is determined one can apply SNM to real roadnetwork analysis for which network data sets for a conven-tional network model (eg a deterministicstochastic userequilibrium traffic assignment model) are already prepared

Numerical experiments are carried out for illustrating theapplications and validity of SNM The experiments assumedfour types of drivers in the network The first type of driveris a risk-neutral driver who chooses a path based only onmean path travel time The other three types of drivers arerisk-averse drivers who choose their paths based on bothmean path travel time and path travel time variance Thesecond type of driverrsquos path travel time variance is calculatedby assuming statistically independent link travel time Thethird type of driverrsquos path travel time variance is calculated byconsidering all travel time covariance between two adjacentlinks in the network The fourth type of driverrsquos path traveltime variances is calculated by considering all travel timecovariance between two different links in the network Thepath travel time variance of the fourth type of driver is theexact one in the network It is shown thatmean network flowsobtained by assuming the risk-neutral driver differ as a wholefrom those obtained by assuming the risk-averse driversThese differences in mean network flows are generated bythe effects of travel time reliability on path choice behaviorby the driver It is also shown that mean link flows obtainedby assuming the third type of driver that is mean link flows


calculated by SNM are almost the same as the mean linkflows calculated by assuming the fourth type of driver In apractical sense it may be difficult to calculate network flowsin a large road network by assuming the fourth type of driverIn contrast SNMcan be easily applied to a large road networkin calculating network flows

In this study we recognize that the introduction of pathtravel time variance considering all travel time covariancebetween two links in a road network to the generalized pathtravel time is more important in expressing the driverrsquos routechoice behavior than the introduction of the skewness of linktravel time to the generalized path travel time ThereforeSNM introduces the path travel time variance consideringall travel time covariance between two adjacent links in thenetwork to the generalized path travel time It is interestingto see how the path choice probabilities which are calculatedby assuming both the statistically independent link traveltime and the skewness of link travel time differ from theprobabilities calculated by SNM In light of this there isa need for a network equilibrium model that introduces adriverrsquos path choice preference based on (2) A deterministicpath choice model based on Wardroprsquos first principle isemployed in this study in order to express the driverrsquospath choice behavior in the network The introduction ofstochastic models for example logit-basedmodels or probit-based models to SNM is needed in order to express thedriverrsquos perception error on the generalized travel timeThesetwo challenges are our future tasks

Appendix

A The Conservation ofthe Path Flow Variance in Relation tothe O-D Flow Variance

O-D flow variance is calculated as the sum of correspondingpath flow variance as follows

var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894

1199011198941198951sdot 1199011198941198952 sdot var [119876119894]= (119888V119894 sdot 119902119894)2 sdot (sum

119895isin119869119894

119901119894119895)2 = (119888V119894 sdot 119902119894)2= var [119876119894] forall119894 isin 119868

(A1)

B The Results of (19) and (20) for 119898 = 4Mean and variancecovariance of link travel time obtainedby performing the fourth-order Taylor expansion to (16) arerespectively given by

119864 [119905119886 (119881119886)] = 1198870119886 + 1198872119886 sdot 119864 [(119881119886 minus V119886)2] + 1198874119886sdot 119864 [(119881119886 minus V119886)4] = 1198870119886 + 1198872119886 sdot var [119881119886] + 3 sdot 1198874119886sdot (var [119881119886])2 cov [119905119886 (119881119886) 119905119887 (119881119887)] = 1198870119886 sdot 1198870119887 + 1198870119886 sdot 1198872119887sdot 119864 [(119881119887 minus V119887)2] + 1198870119886 sdot 1198874119887 sdot 119864 [(119881119887 minus V119887)4] + 1198871119886sdot 1198871119887 sdot 119864 [(119881119886 minus V119886) sdot (119881119887 minus V119887)] + 1198871119886 sdot 1198873119887sdot 119864 [(119881119886 minus V119886) sdot (119881119887 minus V119887)3] + 1198872119886 sdot 1198870119887sdot 119864 [(119881119886 minus V119886)2] + 1198872119886 sdot 1198872119887 sdot 119864 [(119881119886 minus V119886)2sdot (119881119887 minus V119887)2] + 1198872119886 sdot 1198874119887 sdot 119864 [(119881119886 minus V119886)2sdot (119881119887 minus V119887)4] + 1198873119886 sdot 1198871119887 sdot 119864 [(119881119886 minus V119886)3 sdot (119881119887 minus V119887)]+ 1198873119886 sdot 1198873119887 sdot 119864 [(119881119886 minus V119886)3 sdot (119881119887 minus V119887)3] + 1198874119886 sdot 1198870119887sdot 119864 [(119881119886 minus V119886)4] + 1198874119886 sdot 1198872119887 sdot 119864 [(119881119886 minus V119886)4sdot (119881119887 minus V119887)2] + 1198874119886 sdot 1198874119887 sdot 119864 [(119881119886 minus V119886)4sdot (119881119887 minus V119887)4] minus (1198870119886 + 1198872119886 sdot 119864 [(119881119886 minus V119886)2] + 1198874119886sdot 119864 [(119881119886 minus V119886)4]) sdot (1198870119887 + 1198872119887 sdot 119864 [(119881119887 minus V119887)2] + 1198874119887sdot 119864 [(119881119887 minus V119887)4]) = 1198871119886 sdot 1198871119887 sdot cov [119881119886 119881119887] + 3 sdot 1198871119886sdot 1198873119887 sdot var [119881119887] sdot cov [119881119886 119881119887] + 2 sdot 1198872119886 sdot 1198872119887sdot (cov [119881119886 119881119887])2 + 12 sdot 1198872119886 sdot 1198874119887 sdot var [119881119887]sdot (cov [119881119886 119881119887])2 + 3 sdot 1198873119886 sdot 1198871119887 sdot var [119881119886]sdot cov [119881119886 119881119887] + 1198873119886 sdot 1198873119887 sdot (9 sdot var [119881119886] sdot var [119881119887]sdot cov [119881119886 119881119887] + 6 sdot (cov [119881119886 119881119887])3) + 12 sdot 1198874119886 sdot 1198872119887sdot var [119881119886] sdot (cov [119881119886 119881119887])2 + 1198874119886 sdot 1198874119887 sdot (3 sdot 8sdot (cov [119881119886 119881119887])4 + 3 sdot 24 sdot var [119881119886] sdot var [119881119887]sdot (cov [119881119886 119881119887])2)

(B1)

In the above calculations we applied the following momentcalculations [35]119864 [(119881119886 minus V119886) sdot (119881119887 minus V119887)3] = 3 sdot var [119881119887] sdot cov [119881119886 119881119887] 119864 [(119881119886 minus V119886)2 sdot (119881119887 minus V119887)2] = var [119881119886] sdot var [119881119887] + 2sdot (cov [119881119886 119881119887])2 119864 [(119881119886 minus V119886)2 sdot (119881119887 minus V119887)4] = 3 sdot (var [119881119887])2sdot var [119881119886] + 12 sdot var [119881119887] sdot (cov [119881119886 119881119887])2


119864 [(119881119886 minus V119886)3 sdot (119881119887 minus V119887)3] = 9 sdot var [119881119886] sdot var [119881119887]sdot cov [119881119886 119881119887] + 6 sdot (cov [119881119886 119881119887])3 119864 [(119881119886 minus V119886)4] = 3 sdot (var [119881119886])2 119864 [(119881119886 minus V119886)4 sdot (119881119887 minus V119887)4] = 3 sdot (8 sdot (cov [119881119886 119881119887])4+ 24 sdot var [119881119886] sdot var [119881119887] sdot (cov [119881119886 119881119887])2 + 3sdot (var [119881119886])2 sdot (var [119881119887])2) (B2)

A case of 119886 = 119887 in cov[119905119886(119881119886) 119905119887(119881119887)] yieldsvar [119905119886 (119881119886)]= (1198871119886)2 sdot var [119881119886] + (2 sdot 3 sdot 1198871119886 sdot 1198873119886 + 2 sdot (1198872119886)2)sdot (var [119881119886])2+ (2 sdot 12 sdot 1198872119886 sdot 1198874119886 + 9 sdot (1198873119886)2 + 6 sdot (1198873119886)2)sdot (var [119881119886])3 + (3 sdot 8 + 3 sdot 24) sdot (1198874119886)2sdot (var [119881119886])4

(B3)

C The Conservation of the O-D Flow Variancein Relation to the Total O-D Flow Varianceunder the Stochastic Total O-D Flow

Total O-D flow variance is calculated as the sum of O-D flowvariance as follows

var[sum119894isin119868

119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868

(119888V sdot 119901119894 sdot 119902)2 sdot (sum119895isin119869119894

119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868

(119888V sdot 119901119894 sdot 119902)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942 sdot (119888V sdot 119902)2 = (119888V sdot 119902)2

sdot (sum119894isin119868

(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)

D The Results of (24) and (25) for 119898 = 4By using (23) the mean link travel time and link travel timecovariance can be respectively calculated as119864 [119905119886 (119881119886)] = 1198870119886 + 1198872119886 sdot var [119881119886] + 3 sdot 1198874119886sdot (var [119881119886])2 = 1198870119886 + 1198872119886 sdot (119888V sdot V119886)2 + 3 sdot 1198874119886 sdot (119888Vsdot V119886)4 = 1198870119886 + 2119886 sdot (V119886)120582 + 3 sdot 4119886 sdot (V119886)120582

cov [119905119886(119881119886) 119905119887(119881119887)] = 1198871119886 sdot 1198871119887 sdot cov [119881119886 119881119887] + 3 sdot 1198871119886sdot 1198873119887 sdot var [119881119887] sdot cov [119881119886 119881119887] + 2 sdot 1198872119886 sdot 1198872119887sdot (cov [119881119886 119881119887])2 + 12 sdot 1198872119886 sdot 1198874119887 sdot var [119881119887]sdot (cov [119881119886 119881119887])2 + 3 sdot 1198873119886 sdot 1198871119887 sdot var [119881119886]sdot cov [119881119886 119881119887] + 1198873119886 sdot 1198873119887 sdot (9 sdot var [119881119886] sdot var [119881119887]sdot cov [119881119886 119881119887] + 6 sdot (cov [119881119886 119881119887])3) + 12 sdot 1198874119886 sdot 1198872119887sdot var [119881119887] sdot (cov [119881119886 119881119887])2 + 1198874119886 sdot 1198874119887 sdot (3 sdot 8sdot (cov [119881119886 119881119887])4 + 3 sdot 24 sdot var [119881119886] sdot var [119881119887]sdot (cov [119881119886 119881119887])2) = 1119886 sdot 1119887 sdot (119888V sdot V119886119887)2 + 3 sdot 1119886sdot 3119887 sdot (119888V sdot V119887)2 sdot (119888V sdot V119886119887)2 + 2 sdot 2119886 sdot 2119887 sdot (119888Vsdot V119886119887)4 + 12 sdot 2119886 sdot 4119887 sdot (119888V sdot V119887)2 sdot (119888V sdot V119886119887)4 + 3sdot 3119886 sdot 1119887 sdot (119888V sdot V119886)2 sdot (119888V sdot V119886119887)2 + 3119886 sdot 3119887 sdot (9sdot (119888V sdot V119886)2 sdot (119888V sdot V119887)2 sdot (119888V sdot V119886119887)2 + 6 sdot (119888V sdot V119886119887)6)+ 12 sdot 4119886 sdot 2119887 sdot (119888V sdot V119886)2 sdot (119888V sdot V119886119887)4 + 4119886 sdot 4119887sdot (3 sdot 8 sdot (119888V sdot V119886119887)8 + 3 sdot 24 sdot (119888V sdot V119886)2 sdot (119888V sdot V119887)2sdot (119888V sdot V119886119887)4) = 1119886 sdot 1119887 sdot (V119886)120582minus1 sdot (V119886)120582minus1 sdot (V119886119887)2+ 3 sdot 1119886 sdot 3119887 sdot (V119886)120582minus1 sdot (V119886)120582minus1 sdot (V119886119887)2 + 2 sdot 2119886sdot 2119887 sdot (V119886)120582minus2 sdot (V119886)120582minus2 sdot (V119886119887)4 + 12 sdot 2119886 sdot 4119887sdot (V119886)120582minus2 sdot (V119886)120582minus2 sdot (V119886119887)4 + 3 sdot 3119886 sdot 1119887 sdot (V119886)120582minus1sdot (V119886)120582minus1 sdot (V119886119887)2 + 9 sdot 3119886 sdot 3119887 sdot (V119886)120582minus1 sdot (V119886)120582minus1sdot (V119886119887)2 + 6 sdot 3119886 sdot 3119887 sdot (V119886)120582minus3 sdot (V119886)120582minus3 sdot (V119886119887)6


+ 12 sdot 4119886 sdot 2119887 sdot (V119886)120582minus2 sdot (V119886)120582minus2 sdot (V119886119887)4 + 3 sdot 8sdot 4119886 sdot 4119887 sdot (V119886)120582minus4 sdot (V119886)120582minus4 sdot (V119886119887)8 + 3 sdot 24 sdot 4119886sdot 4119887 sdot (V119886)120582minus2 sdot (V119886)120582minus2 sdot (V119886119887)4 = 11988841 sdot (V119886)120582minus1sdot (V119887)120582minus1 sdot (V119886119887)2 + 11988842 sdot (V119886)120582minus2 sdot (V119887)120582minus2 sdot (V119886119887)4+ 11988843 sdot (V119886)120582minus3 sdot (V119887)120582minus3 sdot (V119886119887)6 + 11988844 sdot (V119886)120582minus4sdot (V119887)120582minus4 sdot (V119886119887)8 (D1)

where11988841 = 1119886 sdot 1119887 + 3 sdot 1119886 sdot 3119887 + 3 sdot 3119886 sdot 1119887 + 9 sdot 3119886 sdot 311988711988842 = 2 sdot 2119886 sdot 2119887 + 12 sdot 2119886 sdot 4119887 + 12 sdot 4119886 sdot 2119887 + 3 sdot 24sdot 4119886 sdot 411988711988843 = 6 sdot 3119886 sdot 311988711988844 = 3 sdot 8 sdot 4119886 sdot 4119887(D2)

A condition of 119886 = 119887 in cov[119905119886(119881119886) 119905119887(119881119887)] yieldsvar [119905119886 (119881119886)] = (11988841 + 11988842 + 11988843 + 11988844 ) sdot (V119886)2120582 forall119886 isin 119860 (D3)

where 11988841 = (1119886 + 33119886)2 11988842 = 2 sdot (2119886 + 6 sdot 4119886)2 11988843 = 6 sdot (3119886)2 11988844 = 3 sdot 8 sdot (4119886)2 (D4)

E The Proofs for (38)

Firstly we will prove 120597119886(V119886)120597V119886 gt 0 By differentiating bothsides of (19) with respect to V119886 we obtain120597119864 [119905119886 (119881119886)]120597V119886 = 119898sum

119896=0

120597119887119896119886120597V119886 sdot 119864 [(119881119886 minus V119886)119896]100381610038161003816100381610038161003816119881119886=V119886+ 119898sum119896=0

119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1

119887119896119886 sdot 119896 sdot 119864 [(119881119886 minus V119886)119896minus1]100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886 + 1198871119886 gt 0 forall119886 isin 119860

(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)

Therefore 120597119886 (V119886)120597V119886 = 120597119864 [119905119886 (119881119886)]120597V119886 gt 0 (E3)

Next we will prove 1205971205902119886(V119886)120597V119886 gt 0 By assuming 119881119886 = 119881119887 in(20) and by differentiating both sides of (20) with respect toV119886 we obtain120597 var [119905119886 (119881119886)]120597V119886

= 120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119886 sdot 119864 [(119881119886 minus V119886)119896+119897] minus (119864 [119905119886 (119881119886)])2120597V119886= 119898sum119896=0

119898sum119897=0

120597119887119896119886120597V119886 sdot 119887119897119886 sdot 119864 [(119881119886 minus V119886)119896+119897]100381610038161003816100381610038161003816119881119886=V119886 + 119898sum119896=0 119898sum119897=0 119887119896119886sdot 119887119897119886120597V119886 sdot 119864 [(119881119886 minus V119886)119896+119897]100381610038161003816100381610038161003816119881119886=V119886 + 119898sum119896=0 119898sum119897=0 (119896 + 119897) sdot 119887119896119886 sdot 119887119897119886sdot 119864 [(119881119886 minus V119886)119896+119897minus1]100381610038161003816100381610038161003816119881119886=V119886 minus 2 sdot 119864 [119905119886 (119881119886)]sdot 120597119864 [119905119886 (119881119886)]120597119881119886 1003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886 = 1205971198870119886120597V119886 sdot 1198870119886 + 1198870119886 sdot 1205971198870119886120597V119886 + 2 sdot 1198870119886sdot 1198871119886 minus 21198870119886 sdot 1198871119886 = 2 sdot 1205971198870119886120597V119886 sdot 1198870119886 gt 0 forall119886 isin 119860

(E4)

Therefore 1205971205902119886 (V119886)120597V119886 = 120597 var [119905119886 (119881119886)]120597V119886 gt 0 (E5)

Finally we will prove 120597120590119886119887(V119886119887 V119886 V119887)120597V119886119887 gt 0 By differenti-ating both sides of (20) with respect to V119886119887 we obtain

120597 cov [119905119886 (119881119886) 119905119887 (119881119887)]120597V119886119887 = 120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119887 sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897] minus 119864 [119905119886 (119881119886)] sdot 119864 [119905119887 (119881119887)]120597V119886119887= 120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119887 sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]120597V119886119887


= 120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119887 sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]120597V119886 sdot 120597V119886120597V119886119887+ 120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119887 sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]120597V119887 sdot 120597V119887120597V119886119887

(E6)

It is shown that cov[119905119886(119881119886) 119905119887(119881119887)] gt 0 if V119886119887 gt 0 andcov[119905119886(119881119886) 119905119887(119881119887)] = 0 otherwiseThus we will consider onlythe condition of V119886119887 gt 0 that is equivalent to 120597V119886120597V119886119887 =120597V119887120597V119886119887 = 1 in the following discussion The first term ofthe right-hand side of the equation above can be calculatedas follows120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119887 sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]120597V119886

= 119898sum119896=0

119898sum119897=0

120597119887119896119886120597V119886 sdot 119887119897119887sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]100381610038161003816100381610038161003816119881119886=V119886 119881119887=V119887+ 119898sum119896=1

119898sum119897=0

119896 sdot 119887119896119886 sdot 119887119897119887sdot 119864 [(119881119886 minus V119886)119896minus1 sdot (119881119887 minus V119887)119897]100381610038161003816100381610038161003816119881119886=V119886 119881119887=V119887= 1205971198870119886120597V119886 sdot 1198870119887 + 1198871119886 sdot 1198870119887 gt 0

(E7)

In the same manner the following relationship is alsoobtained120597sum119898119896=0sum119898119897=0 119887119896119886 sdot 119887119897119887 sdot 119864 [(119881119886 minus V119886)119896 sdot (119881119887 minus V119887)119897]120597V119887 gt 0 (E8)

Therefore120597120590119886119887 (V119886119887 V119886 V119887)120597V119886119887 = 120597cov [119905119886 (119881119886) 119905119887 (119881119887)]120597V119886119887 gt 0 (E9)

By using the results shown above the following two condi-tions are obtained120597119892119886119887 (V119886119887 V119886 V119887)120597V119886119887 = 2 sdot 120596 sdot 120597120590119886119887 (V119886119887 V119886 V119887)120597V119886119887 gt 0

forall119886 119887 isin 119860120597119892119886 (V119886)120597V119886 = 120597119886 (V119886)120597V119886 + 120596 sdot 1205971205902119886 (V119886)120597V119886 gt 0forall119886 isin 119860

(E10)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

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[2] Y Asakura ldquoEvaluation of network reliability using stochasticuser equilibriumrdquo Journal of Advanced Transportation vol 33no 2 pp 147ndash158 1999

[3] M G H Bell ldquoA game theory approach to measuring theperformance reliability of transport networksrdquo TransportationResearch Part B Methodological vol 34 no 6 pp 533ndash5452000

[4] A ChenH YangHK Lo andWH Tang ldquoCapacity reliabilityof a road network an assessment methodology and numericalresultsrdquo Transportation Research Part B Methodological vol 36no 3 pp 225ndash252 2002

[5] A Sumalee and F Kurauchi ldquoNetwork capacity reliabilityanalysis considering traffic regulation after a major disasterrdquoNetworks and Spatial Economics vol 6 no 3-4 pp 205ndash2192006

[6] S Clark and D Watling ldquoModelling network travel timereliability under stochastic demandrdquo Transportation ResearchPart B Methodological vol 39 no 2 pp 119ndash140 2005

[7] H X Liu X He and W Recker ldquoEstimation of the time-dependency of values of travel time and its reliability from loopdetector datardquo Transportation Research Part B Methodologicalvol 41 no 4 pp 448ndash461 2007

[8] R B Noland and J W Polak ldquoTravel time variability a reviewof theoretical and empirical issuesrdquo Transport Reviews vol 22no 1 pp 39ndash54 2002

[9] D Watling ldquoA second order stochastic network equilibriummodel I Theoretical foundationrdquo Transportation Science vol36 no 2 pp 149ndash166 2002

[10] E Cascetta ldquoA stochastic process approach to the analysis oftemporal dynamics in transportation networksrdquo TransportationResearch Part B Methodological vol 23 no 1 pp 1ndash17 1989

[11] S Nakayama and J Takayama inProceedings of the 82ndAnnualMeeting of the Transportation Research BoardThe 82nd AnnualMeeting of the Transportation Research Board WashingtonDC USA 2003

[12] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquo Transportation Research Part B Methodological vol40 no 9 pp 792ndash806 2006

[13] H Shao W H Lam and M L Tam ldquoA reliability-basedstochastic traffic assignment model for network with multiple


user classes under uncertainty in demandrdquoNetworks and SpatialEconomics vol 6 no 3-4 pp 173ndash204 2006

[14] W Y Szeto and M Solayappan ldquoReliability-based tran-sit assignment for congested stochastic transit networksrdquoComputer-Aided Civil and Infrastructure Engineering vol 26no 4 pp 311ndash326 2010

[15] A Sumalee K Uchida and W H K Lam ldquoStochastic multi-modal transport network under demand uncertainties andadverse weather conditionrdquo Transportation Research Part CEmerging Technologies vol 19 no 2 pp 338ndash350 2011

[16] K Uchida A Sumalee and HW Ho ldquoA stochastic multimodalreliable network design problem under adverse weather condi-tionsrdquo Journal of Advanced Transportation vol 49 no 1 pp 73ndash95 2014

[17] K Uchida ldquoEstimating the value of travel time and of traveltime reliability in road networksrdquo Transportation Research PartB Methodological vol 66 pp 129ndash147 2014

[18] K Uchida ldquoTravel Time Reliability Estimation Model UsingObservedLink Flows in aRoadNetworkrdquoComputer-AidedCiviland Infrastructure Engineering vol 30 no 6 pp 449ndash463 2015

[19] T Kato and K Uchida ldquoA study on benefit estimation thatconsiders the values of travel time and travel time reliability inroad networksrdquo Transportmetrica A Transport Science pp 1ndash212017

[20] G C de Jong and M C J Bliemer ldquoOn including travel timereliability of road traffic in appraisalrdquo Transportation ResearchPart A Policy and Practice vol 73 pp 80ndash95 2015

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[22] R B Noland and K A Small ldquoTravel-time uncertaintydeparture time choice and the cost of morning commutesrdquoTransportation Research Record no 1493 pp 150ndash158 1995

[23] J Bates J Polak P Jones and A Cook ldquoThe valuation ofreliability for personal travelrdquo Transportation Research Part ELogistics and Transportation Review vol 37 no 2-3 pp 191ndash2292001

[24] M Fosgerau and A Karlstrom ldquoThe value of reliabilityrdquoTransportation Research Part B Methodological vol 44 no 1pp 38ndash49 2010

[25] M Fosgerau and L Engelson ldquoThe value of travel time vari-ancerdquo Transportation Research Part B Methodological vol 45no 1 pp 1ndash8 2011

[26] L Engelson ldquoProperties of expected travel cost function withuncertain travel timerdquo Transportation Research Record no2254 pp 151ndash159 2011

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[28] K A Small ldquoThe scheduling of consumer activities work tripsrdquoTheAmerican Economic Review vol 72 no 3 pp 467ndash479 1982

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[30] J W C van Lint H J van Zuylen and H Tu ldquoTravel timeunreliability on freeways why measures based on variance tellonly half the storyrdquo Transportation Research Part A Policy andPractice vol 42 no 1 pp 258ndash277 2008

[31] K Hjorth M Borjesson L Engelson and M Fosgerau ldquoEsti-mating exponential scheduling preferencesrdquo TransportationResearch Part B Methodological vol 81 2015

[32] R Cominetti and A Torrico ldquoAdditive consistency of riskmeasures and its application to risk-averse routing in networksrdquo2013 httpsarxivorgabs13124193

[33] W H K Lam H Shao and A Sumalee ldquoModeling impactsof adverse weather conditions on a road network with uncer-tainties in demand and supplyrdquo Transportation Research Part BMethodological vol 42 no 10 pp 890ndash910 2008

[34] Bureau of Public Roads Traffic Assignment Manual USDepartment of Commerce Urban PlanningDivisionWashing-ton DC USA Bureau of Public Roads

[35] L Isserlis ldquoOn a formula for the product-moment coefficient ofany order of a normal frequency distribution in any number ofvariablesrdquo Biometrika vol 12 no 1-2 pp 134ndash139 1918

[36] H Rakha I El-Shawarby M Arafeh and F Dion ldquoEstimatingpath travel-time reliabilityrdquo in Proceedings of the ITSC 20062006 IEEE Intelligent Transportation Systems Conference pp236ndash241 September 2006

[37] R Tani and K Uchida ldquoA stochastic user equilibrium assign-ment model under stochastic demand and supply followinglognormal distributionsrdquo Proceedings of Eastern Asia Society forTransportation Studies 2017

[38] M J Smith ldquoThe existence uniqueness and stability of trafficequilibriardquo Transportation Research Part B Methodological vol13 no 4 pp 295ndash304 1979

[39] D Bernstein and S A Gabriel ldquoSolving the nonadditive trafficequilibrium problemrdquo inNetwork Optimization P M PardalosD W Hearn and W W Hagar Eds vol 450 of Lecture Notesin Econom and Math Systems pp 72ndash102 Springer New YorkNY USA 1997

[40] Y Sheffi ldquoUrban transportation networks equilibrium analy-sis with mathematical programming methodsrdquo TransportationResearch Part A General vol 20 no 1 pp 76-77 1986

[41] M Abdulaal and L J Leblanc ldquoMethods for combining modalsplit and equilibrium assignment modelsrdquo Transportation Sci-ence vol 13 no 4 pp 292ndash314 1979

[42] M Florian and H Spiess ldquoThe convergence of diagonalizationalgorithms for asymmetric network equilibrium problemsrdquoTransportation Research Part B Methodological vol 16 no 6pp 477ndash483 1982

[43] S Dafermos ldquoConvergence of a network decomposition algo-rithm for the traffic equilibriummodelrdquo inProceedings of the 8thInternational Symposium on Transportation and Traffic Theorypp 143ndash156 1981

[44] S Dafermos ldquoRelaxation algorithms for the general asymmetrictraffic equilibrium problemrdquo Transportation Science vol 16 no2 pp 231ndash240 1982

[45] S Dafermos ldquoAn iterative scheme for variational inequalitiesrdquoMathematical Programming vol 26 no 1 pp 40ndash47 1983

[46] S Nguyen and C Dupuis ldquoAn efficient method for computingtraffic equilibria in networks with asymmetric transportationcostsrdquo Transportation Science vol 18 no 2 pp 185ndash202 1984

[47] H K Lo and A Chen ldquoTraffic equilibrium problemwith route-specific costs Formulation and algorithmsrdquo TransportationResearch Part B Methodological vol 34 no 6 pp 493ndash5132000

[48] D Watling ldquoUser equilibrium traffic network assignment withstochastic travel times and late arrival penaltyrdquo EuropeanJournal of Operational Research vol 175 no 3 pp 1539ndash15562006

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



with uncertain flows Lo et al [12] extended their originalmodel to consider the concept of travel time budget in pathchoice decision-making Shao et al [13] adopted a similarpostulation of the central limit theorem to derive the normaldistribution of the path travel time but with O-D flowdistribution (normal distribution) Szeto and Solayappan [14]proposed a nonlinear complementarity problem formulationfor the risk-aversive stochastic transit assignment problemSumalee et al [15] address stochastic flow and stochasticcapacity in a multimodal network Uchida et al [16] addressnetwork design problem in a stochastic multimodal networkmodel Uchida [17] proposed a model which simultaneouslyestimates the value of travel time and of travel time reliabilitybased on the risk-averse driverrsquos path choice behavior Uchida[18] proposed a network equilibrium model which estimatestravel time reliability from the observed link flows in thenetwork Kato and Uchida [19] proposed a benefit estimationmethod that considers travel time reliability

In the context of the advancements in theoretical studieson travel time variability in road networks transporta-tion benefit-cost analysis (BCA) considering travel timevariability (httpssitesgooglecomsitebenefitcostanalysisbenefitstravel-time-reliability [20]) is now becoming a bigconcern The traditional DUE traffic assignment model hasbeen widely used to estimate the value or benefits of a policyprogram or project considering no travel time variability Ifwe consider the value of travel time variability in estimatingthe benefit of a policy the DUE traffic assignment modelcannot be applied since it does not address the stochasticnature of a travel time variability Therefore a plausiblenetwork equilibrium model which is well established interms of theory and practice is needed For BCA in whichtravel time variability is considered a measure of travel timevariability which is discussed in the next section needsto be determined Once the travel time variability measureis determined then a network equilibrium model whichcombines risk-averse driverrsquos path choice behavior with thegeneralized travel time which is defined as a mean traveltime plus a travel time variability measure multiplied by acalibration parameter is developed If we put more weight onthe accuracy than the practicality of a network equilibriummodel then the validity of BCA may increase however thecosts required for calculating BCA may increase and viceversa Therefore the modeler has to consider the trade-offbetween accuracy and practicalityThe objective of this studyis to propose a simplified and plausible network equilibriummodel which takes into account both the risk-averse driverrsquospath choice behavior and the travel time variability Thedifference between this study and the other studies that theauthors have presented is that we propose a model thatcan be applied to a large network problem The stochasticnetworkmodels that the authors have developed require pathenumeration in the network However the enumeration ofall possible paths is difficult in the case of a large networkTherefore a stochastic networkmodel for travel time reliabil-ity analysis that solves a large network problem is demanded

This paper starts by examining several measures oftravel time variability in the next section Based on thediscussion provided in the next section we will employ a

travel time variance as a measure of travel time variability inthis study Then link and path travel time under stochasticdemand flows are formulated in Section 3 In Section 4two network equilibrium models under stochastic demandflows are formulated considering a risk-averse driverrsquos pathchoice behavior in a road networkThe firstmodel introducespath travel time variance which is calculated considering alltravel time covariances between two links in the networkThe generalized travel time in this model is not additive sincethe generalized path travel time is not equal to the sum ofthe generalized link travel times related to that path Thesecond model which we propose in this study is a simplifiedversion of the first model In this model the path traveltime variance is not calculated by considering all travel timecovariance between two links in the networkThe path traveltime variance is calculated by considering all travel timecovariance between two adjacent links in the network Thegeneralized path travel time in this model is additive It isshown that a unique solution is provided by the simplifiednetwork model Numerical experiments are carried out toillustrate the applicability and validity of the proposedmodelFinally concluding remarks are provided in Section 6

2 Measures of Travel Time Variability

We will briefly review how to obtain a measure of traveltime variability based on the expected utility maximizationprinciple Vickrey [21] considered a separable or additiveutility function which is a sum of utilities obtained from timespent at an origin and time spent at a destination of a tripUsing such formulation of utility it is possible to considera driver who chooses a departure time optimally in orderto maximize expected utility when facing uncertain traveltime Noland and Small [22] Bates et al [23] Fosgerau andKarlstrom [24] Fosgerau and Engelson [25] and Engelson[26] have shown how measures of travel time variability canbe derived from the driversrsquo scheduling preferences

A popular formulation of scheduling preferences is the120572-120573-120574 preference inwhich themarginal utility of time (MUT)at the origin is constant and that at the destination is astep function [27 28] By assuming an exponential traveltime distribution or a uniform travel time distributionNoland and Small [22] derived the scheduling utility whichis linear in (120583 120590) where 120583 and 120590 are the mean and standarddeviation (SD) of the stochastic travel time Fosgerau andKarlstrom [24] generalized this result to any travel timedistributions Fosgerau and Engelson [25] considered thevalue of travel time reliability under scheduling preferencesthat were defined in terms of linear MUTs being at the originand at the destination They found that the scheduling utilitywas linear in (120583 1205832 1205902) and that this result was independentof the shape of a travel time distribution

Engelson [26] derived the scheduling utility for thetwo cases when the MUTs at both the origin and thedestination are either quadratic or exponential in form anddemonstrated special cases when the scheduling utility isadditiveThe necessary condition when the scheduling utilityis additive is that theMUT at the origin is a positive constant



119906 (119905 119879) = int1199050119905ℎ0119889120591

+ 119864[int119905+1198791199050


(1)




lim120573rarr0

119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 (3)



120573119899minus2 sdot 119896119899119899 (4)


119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 (5)








1 2 3

T1 = N(11 1)

T3 = N(10 2)

T2 = N(10 5)




119888 = 119906ℎ0 = 120583 + 120596 sdot 1205902 (7)

where 120596 = 1205742 sdot ℎ0 (8)







sum119895isin119869119894

119891119894119895 = 119902119894 forall119894 isin 119868 (10)




119881119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119865119894119895 forall119886 isin 119860 (12)


V119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119891119894119895 = sum119894isin119868

sum119895isin119869119894









119905119886 (119881119886) = 119898sum119896=0



119887119896119886 = 1119896 sdot 120597(119896)119905119886 (119881119886)120597119881(119896)1198861003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886


(18)


119864 [119905119886 (119881119886)] = 119898sum119896=0



cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=0

119898sum119897=0


(20)


119864 [119905119886 (119881119886)] = 1198870119886 + 1198982sum119896=1

119896prod119897=1


1198870119886 + (119898minus1)2sum119896=1

119896prod119897=1




forall119886 119887 isin 119860 (22)

where

V119886119887 = sum119894isin119868

sum119895isin119869119894






119864 [119905119886 (119881119886)] = 119905119886 (V119886) + sum119896=1

119896prod119897=1


cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=1


where







Ξ119894119895 = sum119886isin119860



119864 [Ξ119894119895] = 119864[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895] = sum119886isin119860


var [Ξ119894119895] = var[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895]= sum119886isin119860

sum119887isin119860


1205902119886 (V119886) sdot 120575119886119895 + 2 sum119886isin119860

sum119887( =119886)isin119860


(28)


cov [Ξ119894119895 Ξ119894119896]= cov[sum

119886isinA120575119886119895 sdot 119905119886 (119881119886) sum

119887isinA120575119887119895 sdot 119905119887 (119881119887)]

= sum119886isinA

sum119887isinA


(29)




119888119894119895 = 119864 [Ξ119894119895] + 120596 sdot var [Ξ119894119895] (30)


(31)


(32)




sum119895isin119869119894






min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)




sum119886isinA




119894isin119868

sum119895isin119869119894


119894isin119868

sum119895isin119869119894



(36)





ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119906 (119905 119879) = int1199050119905ℎ0119889120591

+ 119864[int119905+1198791199050


(1)




lim120573rarr0

119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 (3)



120573119899minus2 sdot 119896119899119899 (4)


119906 (119879) = ℎ0 sdot 120583 + 1205742 sdot 1205902 (5)








1 2 3

T1 = N(11 1)

T3 = N(10 2)

T2 = N(10 5)




119888 = 119906ℎ0 = 120583 + 120596 sdot 1205902 (7)

where 120596 = 1205742 sdot ℎ0 (8)







sum119895isin119869119894

119891119894119895 = 119902119894 forall119894 isin 119868 (10)




119881119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119865119894119895 forall119886 isin 119860 (12)


V119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119891119894119895 = sum119894isin119868

sum119895isin119869119894









119905119886 (119881119886) = 119898sum119896=0



119887119896119886 = 1119896 sdot 120597(119896)119905119886 (119881119886)120597119881(119896)1198861003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886


(18)


119864 [119905119886 (119881119886)] = 119898sum119896=0



cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=0

119898sum119897=0


(20)


119864 [119905119886 (119881119886)] = 1198870119886 + 1198982sum119896=1

119896prod119897=1


1198870119886 + (119898minus1)2sum119896=1

119896prod119897=1




forall119886 119887 isin 119860 (22)

where

V119886119887 = sum119894isin119868

sum119895isin119869119894






119864 [119905119886 (119881119886)] = 119905119886 (V119886) + sum119896=1

119896prod119897=1


cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=1


where







Ξ119894119895 = sum119886isin119860



119864 [Ξ119894119895] = 119864[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895] = sum119886isin119860


var [Ξ119894119895] = var[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895]= sum119886isin119860

sum119887isin119860


1205902119886 (V119886) sdot 120575119886119895 + 2 sum119886isin119860

sum119887( =119886)isin119860


(28)


cov [Ξ119894119895 Ξ119894119896]= cov[sum

119886isinA120575119886119895 sdot 119905119886 (119881119886) sum

119887isinA120575119887119895 sdot 119905119887 (119881119887)]

= sum119886isinA

sum119887isinA


(29)




119888119894119895 = 119864 [Ξ119894119895] + 120596 sdot var [Ξ119894119895] (30)


(31)


(32)




sum119895isin119869119894






min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)




sum119886isinA




119894isin119868

sum119895isin119869119894


119894isin119868

sum119895isin119869119894



(36)





ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3

T1 = N(11 1)

T3 = N(10 2)

T2 = N(10 5)




119888 = 119906ℎ0 = 120583 + 120596 sdot 1205902 (7)

where 120596 = 1205742 sdot ℎ0 (8)







sum119895isin119869119894

119891119894119895 = 119902119894 forall119894 isin 119868 (10)




119881119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119865119894119895 forall119886 isin 119860 (12)


V119886 = sum119894isin119868

sum119895isin119869119894

120575119886119895 sdot 119891119894119895 = sum119894isin119868

sum119895isin119869119894









119905119886 (119881119886) = 119898sum119896=0



119887119896119886 = 1119896 sdot 120597(119896)119905119886 (119881119886)120597119881(119896)1198861003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886


(18)


119864 [119905119886 (119881119886)] = 119898sum119896=0



cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=0

119898sum119897=0


(20)


119864 [119905119886 (119881119886)] = 1198870119886 + 1198982sum119896=1

119896prod119897=1


1198870119886 + (119898minus1)2sum119896=1

119896prod119897=1




forall119886 119887 isin 119860 (22)

where

V119886119887 = sum119894isin119868

sum119895isin119869119894






119864 [119905119886 (119881119886)] = 119905119886 (V119886) + sum119896=1

119896prod119897=1


cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=1


where







Ξ119894119895 = sum119886isin119860



119864 [Ξ119894119895] = 119864[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895] = sum119886isin119860


var [Ξ119894119895] = var[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895]= sum119886isin119860

sum119887isin119860


1205902119886 (V119886) sdot 120575119886119895 + 2 sum119886isin119860

sum119887( =119886)isin119860


(28)


cov [Ξ119894119895 Ξ119894119896]= cov[sum

119886isinA120575119886119895 sdot 119905119886 (119881119886) sum

119887isinA120575119887119895 sdot 119905119887 (119881119887)]

= sum119886isinA

sum119887isinA


(29)




119888119894119895 = 119864 [Ξ119894119895] + 120596 sdot var [Ξ119894119895] (30)


(31)


(32)




sum119895isin119869119894






min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)




sum119886isinA




119894isin119868

sum119895isin119869119894


119894isin119868

sum119895isin119869119894



(36)





ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119905119886 (119881119886) = 119898sum119896=0



119887119896119886 = 1119896 sdot 120597(119896)119905119886 (119881119886)120597119881(119896)1198861003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886


(18)


119864 [119905119886 (119881119886)] = 119898sum119896=0



cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=0

119898sum119897=0


(20)


119864 [119905119886 (119881119886)] = 1198870119886 + 1198982sum119896=1

119896prod119897=1


1198870119886 + (119898minus1)2sum119896=1

119896prod119897=1




forall119886 119887 isin 119860 (22)

where

V119886119887 = sum119894isin119868

sum119895isin119869119894






119864 [119905119886 (119881119886)] = 119905119886 (V119886) + sum119896=1

119896prod119897=1


cov [119905119886 (119881119886) 119905119887 (119881119887)] = 119898sum119896=1


where







Ξ119894119895 = sum119886isin119860



119864 [Ξ119894119895] = 119864[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895] = sum119886isin119860


var [Ξ119894119895] = var[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895]= sum119886isin119860

sum119887isin119860


1205902119886 (V119886) sdot 120575119886119895 + 2 sum119886isin119860

sum119887( =119886)isin119860


(28)


cov [Ξ119894119895 Ξ119894119896]= cov[sum

119886isinA120575119886119895 sdot 119905119886 (119881119886) sum

119887isinA120575119887119895 sdot 119905119887 (119881119887)]

= sum119886isinA

sum119887isinA


(29)




119888119894119895 = 119864 [Ξ119894119895] + 120596 sdot var [Ξ119894119895] (30)


(31)


(32)




sum119895isin119869119894






min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)




sum119886isinA




119894isin119868

sum119895isin119869119894


119894isin119868

sum119895isin119869119894



(36)





ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Ξ119894119895 = sum119886isin119860



119864 [Ξ119894119895] = 119864[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895] = sum119886isin119860


var [Ξ119894119895] = var[sum119886isin119860

119905119886 (119881119886) sdot 120575119886119895]= sum119886isin119860

sum119887isin119860


1205902119886 (V119886) sdot 120575119886119895 + 2 sum119886isin119860

sum119887( =119886)isin119860


(28)


cov [Ξ119894119895 Ξ119894119896]= cov[sum

119886isinA120575119886119895 sdot 119905119886 (119881119886) sum

119887isinA120575119887119895 sdot 119905119887 (119881119887)]

= sum119886isinA

sum119887isinA


(29)




119888119894119895 = 119864 [Ξ119894119895] + 120596 sdot var [Ξ119894119895] (30)


(31)


(32)




sum119895isin119869119894






min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)




sum119886isinA




119894isin119868

sum119895isin119869119894


119894isin119868

sum119895isin119869119894



(36)





ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











sum119895isin119869119894






min 119911 = sum119886isin119860

intV119886

0119892119886 (119908) 119889119908 (34)




sum119886isinA




119894isin119868

sum119895isin119869119894


119894isin119868

sum119895isin119869119894



(36)





ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ga(a) ga(a)

gb(b)gb(b)

gc(c)gc(c)

gab(ab)

gac(ac)




nablag (k k) =((((((((((((((


))))))))))))))

(37)








var [Ξ119894119895] = sum119886isin119860

sum119887isin119861




6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6

12

5

1 2

5

171

109 12

86

77

1114

10

89

215

11

4 3

4

13

13

319

16

18

Origin

Destination

Destination

Origin




(1) 1-2

(1) 2-18-11(2) 1-5-7-9-11(3) 1-5-7-10-15(4) 1-5-8-14-15(5) 1-6-12-14-15(6) 2-17-7-9-11(7) 2-17-7-10-15(8) 2-17-8-14-15

(2) 4-2

(9) 4-12-14-15(10) 3-5-7-9-11(11) 3-5-7-10-15(12) 3-5-8-14-15(13) 3-6-12-14-15

(3) 1-3

(14) 1-6-13-19(15) 1-5-7-10-16(16) 1-5-8-14-16(17) 1-6-12-14-16(18) 2-17-7-10-16(19) 2-17-8-14-16

(4) 4-3

(20) 4-13-19(21) 4-12-14-16(22) 3-6-13-19(23) 3-5-7-10-16(24) 3-5-8-14-16(25) 3-6-12-14-16









(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














(1)(1) 499 658 643 641(2) 81 58 0 75(3) 88 64 165 117(4) 45 47 0 0(5) 112 59 61 0(6) 22 76 88 0(7) 127 38 42 168(8) 25 0 0 0

(2)(9) 265 276 491 321(10) 411 229 294 313(11) 157 236 173 88(12) 16 126 0 127(13) 151 133 41 152

(3)(14) 317 413 304 268(15) 79 107 45 100(16) 36 58 296 210(17) 145 108 24 122(18) 248 255 330 296(19) 174 59 0 5

(4)(20) 591 491 441 481(21) 121 197 28 155(22) 149 120 273 268(23) 0 87 18 0(24) 96 4 30 0(25) 43 100 210 97





(1)(1) 705 759 775 800(2) 705 759 775 800(3) 705 759 775 800(4) 705 759 775 801(5) 705 759 775 802(6) 705 759 775 811(7) 705 759 775 800(8) 705 759 785 801

(2)(9) 725 791 813 851(10) 725 791 813 851(11) 725 791 813 851(12) 725 791 813 851(13) 725 791 813 851

(3)(14) 698 758 777 806(15) 698 758 777 806(16) 698 758 777 806(17) 698 758 777 806(18) 698 758 777 806(19) 698 758 787 806

(4)(20) 718 790 815 851(21) 718 790 815 851(22) 718 790 815 851(23) 718 790 815 852(24) 718 790 815 851(25) 718 790 815 851






(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(1)(1) 705 718 722 724(2) 705 685 686 687(3) 705 688 687 686(4) 705 682 685 688(5) 705 687 684 684(6) 705 677 685 687(7) 705 680 686 686(8) 705 675 684 688

(2)(9) 725 699 700 700(10) 725 706 711 713(11) 725 709 711 712(12) 725 703 709 714(13) 725 708 708 710

(3)(14) 698 680 672 668(15) 698 687 688 688(16) 698 682 686 690(17) 698 686 685 686(18) 698 680 687 688(19) 698 674 685 690

(4)(20) 718 692 688 684(21) 718 698 701 702(22) 718 701 696 694(23) 718 709 712 714(24) 718 703 710 716(25) 718 707 710 712



TTT119899 = sum119886isin119860

119886 (V119886) sdot V119886 (40)




(1)(1) 00 135 176 255(2) 00 248 296 378(3) 00 236 295 380(4) 00 256 301 378(5) 00 240 304 393(6) 00 273 300 413(7) 00 262 298 381(8) 00 281 336 378

(2)(9) 00 307 378 504(10) 00 285 343 460(11) 00 273 342 463(12) 00 292 348 458(13) 00 277 350 472

(3)(14) 00 261 351 462(15) 00 235 297 393(16) 00 255 303 388(17) 00 239 306 402(18) 00 261 300 394(19) 00 280 339 389

(4)(20) 00 328 426 557(21) 00 306 380 497(22) 00 298 398 525(23) 00 272 344 460(24) 00 291 350 452(25) 00 276 352 465












6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















6 Conclusions






Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Appendix



var[[sum119895isin119869119894

119865119894119895]] = sum119895isin119869119894

var [119865119894119895]+ 2 sum1198951isin119869119894

sum1198952( =1198951)isin119869119894

cov [1198651198941198951 1198651198941198952]= (119901119894119895)2 sdot var [119876119894] + 2 sum

1198951isin119869119894

sum1198952( =1198951)isin119869119894


119895isin119869119894


(A1)



(B1)





(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(B3)




119876119894] = sum119894isin119868

var [119876119894]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov [1198761198941 1198761198942] = sum119894isin119868

var[[sum119895isin119869119894

119865119894119895]]+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

cov[[ sum1198951isin1198691198941

11986511989411198951 sum1198952isin1198691198942

11986511989421198952]]= sum119894isin119868


119901119894119895)2+ 2 sum1198941isin119868

sum1198942( =1198941)isin119868

(1199011198941 sum1198951isin1198691198941

11990111989411198951)sdot (1199011198942 sum

1198952isin1198691198942

11990111989421198952) var [119876] = sum119894isin119868


sum1198942( =1198941)isin119868



(119901119894)2 + 2 sum1198941isin119868

sum1198942( =1198941)isin119868

1199011198941 sdot 1199011198942) = (119888V sdot 119902)2= var [119876]

(C1)










119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119896=0


119887119896119886 sdot 120597119864 [(119881119886 minus V119886)119896]1205971198811198861003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119881119886=V119886= 1205971198870119886120597V119886

+ 119898sum119896=1


(E1)

where 1205971198870119886120597V119886 = 120597119905119886 (V119886)120597V119886 1198871119886 = 120581119886 (E2)




119898sum119897=0


(E4)






(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(E6)


= 119898sum119896=0

119898sum119897=0


119898sum119897=0


(E7)





(E10)



References



















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









A Simplified Network Model for Travel Time Reliability...

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