ADVANCED MANAGEMENT SCIENCE (MSC551D) INSTRUCTOR: Dr. Jose Ramon Albert Statistical Research ...

ADVANCEDADVANCEDMANAGEMENT MANAGEMENT

SCIENCE (MSC551D)SCIENCE (MSC551D)

INSTRUCTOR: INSTRUCTOR: Dr. Jose Ramon AlbertDr. Jose Ramon Albert

Statistical Research & Training Center Statistical Research & Training Center www.srtc.gov.phwww.srtc.gov.ph

Email: [email protected]: [email protected]

QUEUING MODELSQUEUING MODELS

3

Advanced Management Science

INTRODUCTIONINTRODUCTION The study of queues (or waiting lines) is one The study of queues (or waiting lines) is one

of the oldest and widely used quantitative of the oldest and widely used quantitative methods. methods.

Queues are a common everyday experienceQueues are a common everyday experience• banks/supermarkets - waiting for service banks/supermarkets - waiting for service • computers - waiting for a response computers - waiting for a response • failure situations - waiting for a failure to occur e.g. in a failure situations - waiting for a failure to occur e.g. in a

piece of machinery piece of machinery • business/industrial service systemsbusiness/industrial service systems• social service systems, e.g. judicial systemsocial service systems, e.g. judicial system• public transport - waiting for a train, LRT/MRT, jeepney public transport - waiting for a train, LRT/MRT, jeepney

or a bus or a bus

4


INTRODUCTIONINTRODUCTION Queues form because resources are limited.Queues form because resources are limited. It makes It makes economic senseeconomic sense to have queues. to have queues.

How many buses or trains would be needed How many buses or trains would be needed if queues were to be avoided/eliminated? if queues were to be avoided/eliminated? How many employees would a bank have to How many employees would a bank have to hire if the bank needs to avoid queuing?hire if the bank needs to avoid queuing?

In designing queuing systems we need to In designing queuing systems we need to aim for a balance between service to aim for a balance between service to customers (short queues implying many customers (short queues implying many servers) and economic considerations (not servers) and economic considerations (not too many servers). too many servers).

5


INTRODUCTIONINTRODUCTION Decisions regarding the amount of capacity Decisions regarding the amount of capacity

to provide are needed. to provide are needed. It is often impossible to predict accurately It is often impossible to predict accurately

when units will arrive to seek service and/or when units will arrive to seek service and/or how much time will be required to provide how much time will be required to provide that service. This makes decision making that service. This makes decision making difficult. difficult.

Queuing theory contributes vital information Queuing theory contributes vital information required for such decisions by predicting required for such decisions by predicting various operating characteristics of the various operating characteristics of the waiting line such as the average waiting time. waiting line such as the average waiting time.

6


INTRODUCTIONINTRODUCTIONIIn essence all queuing systems can n essence all queuing systems can be broken down into individual sub-be broken down into individual sub-systems consisting of systems consisting of entitiesentities queuing for some queuing for some activity.activity.

Typically we can talk of this Typically we can talk of this individual sub-system as dealing individual sub-system as dealing with with customerscustomers queuing for queuing for serviceservice. .

7


INTRODUCTIONINTRODUCTIONTo analyze a sub-system we need To analyze a sub-system we need information on:information on:

Arrival processArrival process Service mechanism (including Calling Service mechanism (including Calling

Popn)Popn) Queuing characteristicsQueuing characteristics

Queue Server

CustomerArrivals

Calling Population(Input Source)

Servedcustomers

Queuing system

8


INTRODUCTIONINTRODUCTION

QueueServer

Arrivals

SINGLE CHANNELSINGLE PHASE SYSTEM

9



Queue Type 1 Server

Arrivals

SINGLE CHANNELMULTIPHASE SYSTEM

Type 2 Server

10



Queue

Server 1

Arrivals

MULTICHANNELSINGLE PHASE SYSTEM

Server 3

Server 2

11



Queue

Type 1Server 1

Arrivals

MULTICHANNELMULTI PHASE SYSTEM

Type 1Server 2

Type 2Server 2

Type 2Server 1

12


Arrival ProcessArrival Process how big is the size of the how big is the size of the calling calling

populationpopulation (the total number of customers (the total number of customers that arrive in the system) that arrive in the system)

how customers arrive e.g. singly or in groups how customers arrive e.g. singly or in groups (batch or bulk arrivals) (batch or bulk arrivals)

how the arrivals are distributed in time, e.g. how the arrivals are distributed in time, e.g. what is the probability distribution of time what is the probability distribution of time between successive arrivals (the between successive arrivals (the interarrival time distributioninterarrival time distribution). ).

• Typical assumption : exponential distributionTypical assumption : exponential distribution

13


The Exponential Distn (w/ parameter

P arrival time < t( ) 1 - e

e = the mathematical constant 2.71828

- t

= the population mean of arrivals

t = any value of the continuous random variable e.g. Drivers Arriving at a Toll Bridge

Customers Arriving at an ATM Machine

14


Describes time Describes time or distanceor distance betweenbetween eventsevents• Used for queuesUsed for queues

Density functionDensity function

ParametersParameters

f(t) = e - t

f(t)

t

= 2.0

= 0.5

The Exponential Distn The Exponential Distn (w/ parameter (w/ parameter

15


The Exponential Distn The Exponential Distn (w/ parameter (w/ parameter

f(t) = f(t) = exp{- exp{- t} t} where t>0where t>0 The mean and variance of the exp distn The mean and variance of the exp distn

with parameter with parameter are 1/ are 1/; 1/; 1/22

The distn is the only memoryless distn, i.e. The distn is the only memoryless distn, i.e. P{ T > s+t | T > t} = P{T > s}P{ T > s+t | T > t} = P{T > s}

Related to the Poisson distn. Related to the Poisson distn. If T ~ exp distn with parameter If T ~ exp distn with parameter then X(t) then X(t) has a Poisson distn with mean has a Poisson distn with mean tt

16


The Poisson DistnThe Poisson Distn

Poisson ProcessPoisson Process::• Discrete EventsDiscrete Events in an in an ‘Interval’‘Interval’ The Probability ofThe Probability of One SuccessOne Success in in Interval isInterval is Stable Stable The Probability of The Probability of More than One More than One SuccessSuccess in this Interval is in this Interval is 00• Probability of Success is Probability of Success is IndependentIndependent from Interval tofrom Interval toIntervalInterval

# Customers Arriving in 15 min.# Customers Arriving in 15 min.# Defects Per Case of Light # Defects Per Case of Light

Bulbs. Bulbs.

P X x

x

x

( |

!

e-

17


P(x ) = probability of x successes given = expected (mean) number of ‘successes’e = 2.71828 (base of natural logs)x = number of ‘successes’ per unit

P xx

x( )

!

e

e.g. Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6.

P(X) = e-3.6

3.64!

4= .1912

The Poisson Distn

18


= 0.5

= 6

Mean

Standard Deviation

ii

N

i

E X

x P x

( )

( )1

0.2.4.6

0 1 2 3 4 5X

P(X)

0.2.4.6

0 2 4 6 8 10X

P(X)

The Poisson Distn

19


Arrival ProcessArrival Process whether there is a finite population whether there is a finite population

of customers or (effectively) an of customers or (effectively) an infinite number that could arrive in a infinite number that could arrive in a certain time interval. certain time interval.

• The simplest arrival process is one where we The simplest arrival process is one where we have completely regular arrivals (i.e. the same have completely regular arrivals (i.e. the same constant time interval between successive constant time interval between successive arrivals). arrivals).

20


Arrival ProcessArrival Process• A A PoissonPoisson stream of arrivals corresponds to stream of arrivals corresponds to

arrivals at random. Here, arrivals at random. Here, inter-arrivalinter-arrival time of time of customers are independent, customers are independent, exponentially exponentially distributeddistributed. The Poisson stream is described by . The Poisson stream is described by a single parameter - the average arrival rate a single parameter - the average arrival rate . .

• Other important arrival processes are Other important arrival processes are scheduled arrivals; batch arrivals; and time scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate dependent arrival rates (i.e. the arrival rate varies according to the time of day). varies according to the time of day).

21


Service MechanismService Mechanism a description of the resources needed for a description of the resources needed for

service to begin service to begin how long the service will take (the how long the service will take (the service service

time distributiontime distribution) ) the number of servers available the number of servers available whether the servers are in series (each whether the servers are in series (each

server has a separate queue) or in parallel server has a separate queue) or in parallel (one queue for all servers) (one queue for all servers)

whether preemption is allowed (a server can whether preemption is allowed (a server can stop processing a customer to deal with stop processing a customer to deal with another "emergency" customer) another "emergency" customer)

22


Service MechanismService Mechanism Assuming that the service times for Assuming that the service times for

customers are independent and do not customers are independent and do not depend upon the arrival process is depend upon the arrival process is common. Another common assumption common. Another common assumption about service times is that they are about service times is that they are exponentially distributed. exponentially distributed.

23


Queue CharacteristicsQueue Characteristics Is the queue Is the queue infinite infinite or or finite?finite?

• The standard model assumes an infinite queue The standard model assumes an infinite queue even for situations where there is a finite upper even for situations where there is a finite upper bound on the no. of permissible customers (as bound on the no. of permissible customers (as this bound may be rather large)this bound may be rather large)

The The queue discipline:queue discipline: How, from the set of How, from the set of customers waiting for service, do we choose customers waiting for service, do we choose the one to be served next the one to be served next • FIFO (first-in first-out) FIFO (first-in first-out) • LIFO (last-in first-out) LIFO (last-in first-out) • randomly randomly

24


Queue CharacteristicsQueue Characteristics Do we have: Do we have:

• balkingbalking (customers deciding not to join the (customers deciding not to join the queue if it is too long) queue if it is too long)

• renegingreneging (customers leave the queue if (customers leave the queue if they have waited too long for service) they have waited too long for service)

• jockeyingjockeying (customers switch between (customers switch between queues if they think they will get served queues if they think they will get served faster by so doing) faster by so doing)

25


Queue CharacteristicsQueue Characteristics Changing the queue discipline (the rule Changing the queue discipline (the rule

by which we select the next customer by which we select the next customer to be served) can often reduce to be served) can often reduce congestion. congestion.

Often the queue discipline "choose the Often the queue discipline "choose the customer with the lowest service time" customer with the lowest service time" results in the smallest value for the results in the smallest value for the time (on average) a customer spends time (on average) a customer spends queuing. queuing.

26


Typical Questions of InterestTypical Questions of Interest How long does a customer expect to wait in How long does a customer expect to wait in

the queue before they are served, and how the queue before they are served, and how long will they have to wait before the service long will they have to wait before the service is complete? is complete?

What is the probability of a customer having What is the probability of a customer having to wait longer than a given time interval to wait longer than a given time interval before they are served? before they are served?

What is the average length of the queue? What is the average length of the queue?

27


Typical Questions of InterestTypical Questions of Interest What is the probability that the queue will What is the probability that the queue will

exceed a certain length? exceed a certain length? What is the expected utilization of the server What is the expected utilization of the server

and the expected time period during which and the expected time period during which he will be fully occupied (remember servers he will be fully occupied (remember servers cost us money so we need to keep them cost us money so we need to keep them busy). In fact if we can assign costs to busy). In fact if we can assign costs to factors such as customer waiting time and factors such as customer waiting time and server idle time then we can investigate how server idle time then we can investigate how to design a system at minimum total cost. to design a system at minimum total cost.

28


Typical ProblemsTypical Problems Is it worthwhile to invest effort in reducing Is it worthwhile to invest effort in reducing

the service time? the service time? How many servers should be employed? How many servers should be employed? Should priorities for certain types of Should priorities for certain types of

customers be introduced? customers be introduced? Is the waiting area for customers adequate? Is the waiting area for customers adequate?

29


Basic Approaches to Solving Basic Approaches to Solving Problems/QuestionsProblems/Questions

analytic methods or queuing theory analytic methods or queuing theory (formula based)(formula based)this is only available for relatively simple this is only available for relatively simple queuing systemsqueuing systems

simulation (computer based)simulation (computer based)Complex queuing systems are almost always Complex queuing systems are almost always analyzedanalyzedusing using discrete eventdiscrete event simulation. simulation.

30


Queuing NotationsQueuing NotationsIt is common to use to use the symbols: It is common to use to use the symbols: lambdalambda to be the mean (or average) to be the mean (or average)

number of arrivals per time period, i.e. number of arrivals per time period, i.e. the mean arrival rate the mean arrival rate

mumu µ to be the mean (or average) µ to be the mean (or average) number of customers served per time number of customers served per time period, i.e. the mean service rate period, i.e. the mean service rate

31


Queuing NotationsQueuing NotationsStandard notation system A/B/C/D/E to Standard notation system A/B/C/D/E to classify queuing systems where: classify queuing systems where:

A represents the prob’ty distribution for the arrival A represents the prob’ty distribution for the arrival process process

B represents the probability distribution for the B represents the probability distribution for the service process service process

C represents the number of channels (servers) C represents the number of channels (servers) D represents the maximum number of customers D represents the maximum number of customers

allowed in the queuing system (either being served or allowed in the queuing system (either being served or waiting for service) waiting for service)

E represents the maximum number of customers in E represents the maximum number of customers in total total

32


Queuing NotationsQueuing NotationsCommon options for A and B are: Common options for A and B are: M for an exponential interarrival (Poisson M for an exponential interarrival (Poisson

arrival distribution) or an exponential service arrival distribution) or an exponential service time distribution time distribution

D for a deterministic or constant value (called D for a deterministic or constant value (called also a degenerate distribution)also a degenerate distribution)

EEkk for an Erlang distribution for an Erlang distribution G for a general distribution (but with a known G for a general distribution (but with a known

mean and variance) mean and variance) If D and E are not specified then it isIf D and E are not specified then it isassumed that they are infinite.assumed that they are infinite.

33


Single Server QueueSingle Server QueueThe M/M/1 queuing system,The M/M/1 queuing system, the simplest queuing systemthe simplest queuing system has a Poisson arrival distribution, an has a Poisson arrival distribution, an

exponential service time exponential service time distribution and a single channel distribution and a single channel (one server). (one server).

Since D and E are not specified, Since D and E are not specified, then it is assumed that they are then it is assumed that they are infinite.infinite.

34


Single Server Queue ExampleSingle Server Queue Example

Suppose we have a single server in a Suppose we have a single server in a shop and customers arrive in the shop shop and customers arrive in the shop with a Poisson arrival distribution at a with a Poisson arrival distribution at a mean rate of mean rate of lambdalambda=24 customers =24 customers per hour, i.e. on average one customer per hour, i.e. on average one customer appears every 1/appears every 1/lambdalambda = 1/24 = 1/24 hours=2.5 minutes. hours=2.5 minutes.

Interarrival times have an exponential Interarrival times have an exponential distribution with an average distribution with an average interarrival time of 2.5 minutes. interarrival time of 2.5 minutes.

35



The server has an exponential service The server has an exponential service time distribution with a mean service time distribution with a mean service rate of 30 customers per hour, i.e. the rate of 30 customers per hour, i.e. the service rate µ=30 customers per hour. service rate µ=30 customers per hour.

As we have a Poisson arrival rate, an As we have a Poisson arrival rate, an exponential service time, and a single exponential service time, and a single server we have a M/M/1 queue in terms server we have a M/M/1 queue in terms of the standard notation.of the standard notation.

36


Formulas for Single-Server ModelFormulas for Single-Server Model

LL = =

- -

Average number of Average number of customers in the systemcustomers in the system

Probability that no customers Probability that no customers are in the system (either in the are in the system (either in the queue or being served)queue or being served)

PP00 = 1 - = 1 -

Probability of exactly Probability of exactly nn customers in the systemcustomers in the system

PPnn = • = • PP00

nn

= 1 -= 1 -

nn

Average number of Average number of customers in the waiting linecustomers in the waiting line LLqq = =

(( - - ))

37


Formulas for Single-Server ModelFormulas for Single-Server Model

==

Probability that the server Probability that the server is busy and the customer is busy and the customer has to waithas to wait

Average time a customer Average time a customer spends in the queuing systemspends in the queuing system WW = = = = 11

--

LL

Probability that the server Probability that the server is idle and a customer can is idle and a customer can be servedbe served

= 1 - = 1 -

= 1 - == 1 - = P P00

Average time a customer Average time a customer spends waiting in line to spends waiting in line to be servedbe served

WWqq = = (( - - ))

38


A Single-Server ModelA Single-Server ModelGiven Given = 24 per hour, = 24 per hour, = 30 customers per hour = 30 customers per hour

Probability of no Probability of no customers in the customers in the systemsystem

PP00 = 1 - = 1 - = = 1 - = 1 - = 0.200.20

24243030

LL = = = 4 = = = 4Average number Average number of customers in of customers in the systemthe system

--

242430 - 2430 - 24

Average number Average number of customers of customers waiting in linewaiting in line

LLqq = = = 3.2 = = = 3.2(24)(24)22

30(30 - 24)30(30 - 24)22

(( - - ))

39


A Single-Server ModelA Single-Server ModelGiven Given = 24 per hour, = 24 per hour, = 30 customers per hour = 30 customers per hour

Average time in the Average time in the system per customer system per customer WW = = = 0.167 hour = = = 0.167 hour11

--

1130 - 2430 - 24

Average time waiting Average time waiting in line per customer in line per customer WWqq = = = 0.133 = = = 0.133

((--))2424

30(30 - 24)30(30 - 24)

Probability that the Probability that the server will be busy and server will be busy and the customer must waitthe customer must wait

= = = 0.80= = = 0.80

24243030

Probability the Probability the server will be idleserver will be idle 1 - 1 - = 1 - 0.80 = 0.20 = 1 - 0.80 = 0.20

40



41



42


Single Server Queue ExampleSingle Server Queue ExampleNote the first line of the output says that Note the first line of the output says that the results are from a formula. For this the results are from a formula. For this very simple queuing system, there are very simple queuing system, there are exact formulae that give these statistics exact formulae that give these statistics in the output under the assumption that in the output under the assumption that the system has reached a the system has reached a steady state steady state - that is that the system has been - that is that the system has been running long enough so as to settle running long enough so as to settle down into some kind of equilibrium down into some kind of equilibrium position.position.

43


Single Server Queue ExampleSingle Server Queue ExampleIf customers are served faster than they If customers are served faster than they arrive, i.e., arrive, i.e.,

1.1. System utilizationSystem utilization or or traffic intensitytraffic intensity U= U= (arrival rate)/(departure rate) (arrival rate)/(departure rate) ==Traffic intensity is a measure of the congestion Traffic intensity is a measure of the congestion of the system. If it is near to zero there is very of the system. If it is near to zero there is very little queuing and in general as the traffic little queuing and in general as the traffic intensity increases (to near 1) the amount of intensity increases (to near 1) the amount of queuing increases. For the system we have queuing increases. For the system we have considered above the arrival rate is 24 and the considered above the arrival rate is 24 and the departure rate is 30 so the traffic intensity is departure rate is 30 so the traffic intensity is 24/30 = 0.8024/30 = 0.80

44



2.2. Average number of customers in the Average number of customers in the queuing system (customers being queuing system (customers being serviced and in the waiting line) is serviced and in the waiting line) is

LL= = Here, 24/(30-24) = 4.0Here, 24/(30-24) = 4.0

3.3. Average number of customers in the Average number of customers in the waiting waiting line is line is

LLqq= (U) = (U) Here, (0.8) 24/(30-24) = 3.2Here, (0.8) 24/(30-24) = 3.2

45



4.4. Average time a customer spends in the Average time a customer spends in the queuing system isqueuing system is

WW= L/= L/Here, 4.0/24=0.1667Here, 4.0/24=0.1667

5.5. Average time a customer spends in the Average time a customer spends in the waiting line is waiting line is

WWqq= L= Lqq/ / Here, (3.2)/24=0.1333Here, (3.2)/24=0.1333

Note: The relationship LNote: The relationship L= = W W is called is called Little’s formula. (John D. C. Little, 1961)Little’s formula. (John D. C. Little, 1961)

46



6.6. Probability server is idle (i.e. an arriving Probability server is idle (i.e. an arriving customer can be served) iscustomer can be served) is

II= (1- = (1- //Here, 1-24/30=0.20Here, 1-24/30=0.20Note this is the same as the probty that Note this is the same as the probty that no customers are in the queuing system. no customers are in the queuing system.

7.7. Probability an arriving customer has to Probability an arriving customer has to waitwait

1-I1-IHere, 1-0.2 =0.8Here, 1-0.2 =0.8

47



48


With QM for WindowsWith QM for Windows

49


Soln with QM for WindowsSoln with QM for Windows

50



51



52


Faster Server or More Servers?Faster Server or More Servers?Consider the M/M/1situation, which would Consider the M/M/1situation, which would we prefer:we prefer:

one server working twice as fast; or one server working twice as fast; or two servers each working at the original two servers each working at the original

rate? rate? The simple answer is that we can analyze The simple answer is that we can analyze this problem using the WINQSB. For the this problem using the WINQSB. For the first situation one server working twice as first situation one server working twice as fast corresponds to a service rate µ=60 fast corresponds to a service rate µ=60 customers per minute yielding the customers per minute yielding the following performance.following performance.

53


Faster ServerFaster Server

54


Two Servers (M/M/2)Two Servers (M/M/2)

55


Two Servers (M/M/2)Two Servers (M/M/2)

56


Faster Server or More Servers?Faster Server or More Servers?It can be seen that with one server It can be seen that with one server working twice as fast, customers working twice as fast, customers spend less time in the system on spend less time in the system on average, but have to wait longer for average, but have to wait longer for service and also have a higher service and also have a higher probability of having to wait for probability of having to wait for service.service.

57


Faster Server or More Servers?Faster Server or More Servers?Of the figures in the outputs for the fastOf the figures in the outputs for the fastM/M/1 and the M/M/2, some are identical.M/M/1 and the M/M/2, some are identical.Extracting key figures which are differentExtracting key figures which are differentwe have:we have:

Fast M/M/1Fast M/M/1 M/M/2M/M/2Ave.time in systemAve.time in system .0278.0278 .0397.0397Ave time in queueAve time in queue .0111.0111 .0063.0063Probty of having to Probty of having to 40%40% 22.8571% 22.8571% wait for servicewait for service

58


Extending the example: Extending the example: M/M/2 with costsM/M/2 with costs

Consider the M/M/2 system we had Consider the M/M/2 system we had but now the service rate is at 40 but now the service rate is at 40 customers served per hour. We have customers served per hour. We have also entered a queue capacity also entered a queue capacity (waiting space) of 3 - i.e. if all servers (waiting space) of 3 - i.e. if all servers are occupied and 3 customers are are occupied and 3 customers are waiting when a new customer waiting when a new customer appears then they go away - this is appears then they go away - this is known as known as balkingbalking. .

59


Extending the example: Extending the example: M/M/2 with costs M/M/2 with costs

We have also added cost information We have also added cost information relating to the server and customers:relating to the server and customers:

each hour a server is idle costs us 10 each hour a server is idle costs us 10 dollars dollars

each hour a customer waits for a each hour a customer waits for a server costs us 20 dollarsserver costs us 20 dollars

each customer who is balked (goes each customer who is balked (goes away without being served) costs us away without being served) costs us 100 dollars 100 dollars

60


M/M/2/5 with CostsM/M/2/5 with Costs

61


M/M/2/5 with CostsM/M/2/5 with Costs

62


Simulation on M/M/2/5Simulation on M/M/2/5

63


Simulation Performance on Simulation Performance on M/M/2/5M/M/2/5

64


Constant Service TimesConstant Service Times

Constant service times occur with Constant service times occur with machinery and automated machinery and automated equipmentequipment

Constant service times Constant service times are a special case are a special case of the single-server of the single-server model with general model with general or undefined service timesor undefined service times

65


Operating Characteristics for Operating Characteristics for Constant Service TimesConstant Service Times

PP00 = 1 - = 1 -Probability that no customersProbability that no customersare in systemare in system

Average number of Average number of customers in systemcustomers in system LL = = LLqq + +

Average number of Average number of customers in queuecustomers in queue LLqq = = 22

22(( - - ))

66


Operating Characteristics for Operating Characteristics for Constant Service TimesConstant Service Times

= = Probability that the Probability that the server is busyserver is busy

Average time customer Average time customer spends in the systemspends in the system WW = = WWqq + +

11

Average time customer Average time customer spends in queuespends in queue WWqq = =

LLqq

67


LLqq = = = 1.14 cars waiting = = = 1.14 cars waiting(10)(10)22

2(13.3)(13.3 - 10)2(13.3)(13.3 - 10)

Constant Service TimesConstant Service Times

Automated car wash with service time = 4.5 minAutomated car wash with service time = 4.5 minCars arrive at rate Cars arrive at rate = 10/hour (Poisson) = 10/hour (Poisson) = 60/4.5 = 13.3/hour= 60/4.5 = 13.3/hour

WWqq = = 1.14/10 = .114 hour or 6.84 minutes = = 1.14/10 = .114 hour or 6.84 minutesLLqq

22

22(( - - ))

68


Finite Queue LengthFinite Queue Length A physical limit exists on length of queueA physical limit exists on length of queue MM = maximum number in queue = maximum number in queue Service rate does not have to exceed arrival Service rate does not have to exceed arrival

rate (rate () to obtain steady-state ) to obtain steady-state conditionsconditions

PP00 = =Probability that no Probability that no customers are in systemcustomers are in system

1 - 1 - //1 - (1 - (//))MM + 1 + 1

Probability of exactly Probability of exactly nn customers in systemcustomers in system

PPnn = ( = (PP00) for ) for nn ≤ ≤ MMnn

LL = - = -Average number of Average number of customers in systemcustomers in system

//1 - 1 - //

((MM + 1)( + 1)(//))MM + 1 + 1

1 - (1 - (//))MM + 1 + 1

69


Finite Queue LengthFinite Queue Length

Let Let PPMM = probability a customer will not join system = probability a customer will not join system

Average time customerAverage time customerspends in systemspends in system WW = =

LL(1 - (1 -

PPMM))

LLqq = = L L - - (1- (1- PPMM))

Average number of Average number of customers in queuecustomers in queue

Average time customer Average time customer spends in queuespends in queue WWqq = = W W -- 11

70


Finite QueueFinite Queue

Quick Lube has waiting space for only 3 cars.Quick Lube has waiting space for only 3 cars. = 20, = 20, = 30, = 30, MM = 4 cars (1 in service + 3 waiting) = 4 cars (1 in service + 3 waiting)

Probability that Probability that no cars are in no cars are in the systemthe system

PP00 = = = 0.38 = = = 0.381 - 20/301 - 20/301 - (20/30)1 - (20/30)55

1 - 1 - //1 - (1 - (//))M + 1M + 1

PPnn = ( = (PP00) = (0.38) = 0.076) = (0.38) = 0.076Probability of Probability of exactly 4 cars in exactly 4 cars in the systemthe system

20203030

44

nn==MM

LL = - = 1.24 = - = 1.24Average number Average number of cars in the of cars in the systemsystem

//1 - 1 - //

((MM + 1)( + 1)(//))MM + 1 + 1

1 - (1 - (//))MM + 1 + 1

71


Finite QueueFinite Queue

Average time a carAverage time a carspends in the systemspends in the system W W = = 0.067 hr = = 0.067 hr LL

(1 - (1 -

PPMM))

LLqq = = L L - = 0.62- = 0.62(1- (1- PPMM))

Average number of Average number of cars in the queuecars in the queue

Quick Lube has waiting space for only 3 cars.Quick Lube has waiting space for only 3 cars. = 20, = 20, = 30, = 30, MM = 4 cars (1 in service + 3 waiting) = 4 cars (1 in service + 3 waiting)

Average time a carAverage time a carspends in the queuespends in the queue WWqq = = W W - = 0.033 hr- = 0.033 hr11

72


Finite Calling PopulationFinite Calling Population

Arrivals originate from a finite (countable) populationArrivals originate from a finite (countable) population NN = population size = population size

Probability of exactly Probability of exactly n n customers in systemcustomers in system PPnn = = PP00 where where nn = 1, 2, ..., = 1, 2, ..., NN

nn

NN!!((NN - - nn)!)!

Average number of Average number of customers in queuecustomers in queue LLqq = = NN - (1- - (1- PP00))

++

Probability that no Probability that no customers are in systemcustomers are in system

PP00 = =

nn = 0 = 0

NN!!((NN - - nn)!)!

NN nn

11

73



Arrivals originate from a finite (countable) populationArrivals originate from a finite (countable) population NN = population size = population size

WWqq = = LLqq

((NN - - LL) ) Average time customer Average time customer spends in queuespends in queue

LL = = LLqq + (1 - + (1 - PP00))Average number of Average number of customers in systemcustomers in system

WW = = WWqq + +Average time customerAverage time customerspends in systemspends in system

11

74


20 machines which operate an average of 200 hrs 20 machines which operate an average of 200 hrs before breaking down (before breaking down ( = 1/200 hr = 0.005/hr) = 1/200 hr = 0.005/hr)Mean repair time = 3.6 hrs (Mean repair time = 3.6 hrs ( = 1/3.6 hr = 0.2778/hr) = 1/3.6 hr = 0.2778/hr)

Probability that no Probability that no machines are in the systemmachines are in the system PP00 = 0.652= 0.652

Average number of Average number of machines in the queuemachines in the queue LLqq = 0.169= 0.169

Average number of Average number of machines in systemmachines in system LL = 0.169 + (1 - 0.652) = .520 = 0.169 + (1 - 0.652) = .520


75


20 machines which operate an average of 200 hrs 20 machines which operate an average of 200 hrs before breaking down (before breaking down ( = 1/200 hr = 0.005/hr) = 1/200 hr = 0.005/hr)Mean repair time = 3.6 hrs (Mean repair time = 3.6 hrs ( = 1/3.6 hr = 0.2778/hr) = 1/3.6 hr = 0.2778/hr)


Average time machineAverage time machinespends in queuespends in queue WWqq = 1.74 hrs = 1.74 hrs

Average time machineAverage time machinespends in systemspends in system WW = 5.33 hrs = 5.33 hrs

76


Capacity PerformanceCapacity Performance

WinQSB can automatically perform an analysis for us of how total cost varies with the number of servers

77


Capacity Performance Approx.Capacity Performance Approx.

78


General QueuingGeneral Queuing The screen on the The screen on the

right shows the right shows the possible input possible input parameters to the parameters to the package in the package in the case of a general case of a general queueing model queueing model (i.e. not a M/M/r (i.e. not a M/M/r system).system).

79


General QueuingGeneral Queuing Here we have a number of possible choices for Here we have a number of possible choices for

the service time distribution and the inter-arrival the service time distribution and the inter-arrival time distribution. In fact the package recognizes time distribution. In fact the package recognizes some 15 different distributions! Other items some 15 different distributions! Other items mentioned above are:mentioned above are:

service pressure coefficient - indicates how service pressure coefficient - indicates how servers speed up service when the system is servers speed up service when the system is busy, i.e. when all servers are busy the service busy, i.e. when all servers are busy the service rate is increased. If this coefficient is s and we rate is increased. If this coefficient is s and we have r servers each with service rate µ then the have r servers each with service rate µ then the service rate changes from µ to (n/r)service rate changes from µ to (n/r)ssµ when there µ when there are n customers in the system and n>=r. are n customers in the system and n>=r.

80


General QueuingGeneral Queuing arrival discourage coefficient - indicates how arrival discourage coefficient - indicates how

customer arrivals are discouraged when the customer arrivals are discouraged when the system is busy, i.e. when all servers are busy system is busy, i.e. when all servers are busy the arrival rate is decreased. If this coefficient the arrival rate is decreased. If this coefficient is s and we have r servers with the arrival rate is s and we have r servers with the arrival rate being being lambdalambda then the arrival rate changes then the arrival rate changes from from lambdalambda to (r/(n+1)) to (r/(n+1))s s lambdalambda when there when there are n customers in the system and n>=r. are n customers in the system and n>=r.

batch (bulk) size distribution - customers can batch (bulk) size distribution - customers can arrive together (in batches, also known as in arrive together (in batches, also known as in bulk) and this indicates the distribution of size bulk) and this indicates the distribution of size of such batches. of such batches.

81


General Queuing ExampleGeneral Queuing Example

82



83



84


General QueuingGeneral Queuing Clearly the longer we simulate, the more Clearly the longer we simulate, the more

confidence we may have in the confidence we may have in the statistics/probabilities obtained. statistics/probabilities obtained.

As before we can investigate how the As before we can investigate how the system might behave with more servers. system might behave with more servers. Simulating for 1000 hours (to reduce the Simulating for 1000 hours (to reduce the overall elapsed time required) and overall elapsed time required) and looking at just the total system cost per looking at just the total system cost per hour (item 22 in the above outputs) we hour (item 22 in the above outputs) we get the following results:get the following results:

85


General QueuingGeneral Queuing

No. of Servers (Total system cost)No. of Servers (Total system cost)1 (4452)1 (4452) 2 (3314) 2 (3314) 3 (2221) 3 (2221) 4 (1614)4 (1614) 5 (1257) 5 (1257) 6 ( 992) 6 ( 992) 7 (832) 7 (832) 8 (754)8 (754) 9 (718) 9 (718) 10 (772) 10 (772) 11(833) 11(833) 12(902) 12(902) Hence here the number of servers Hence here the number of servers associated with the minimum total associated with the minimum total system cost is 9system cost is 9

ADVANCED MANAGEMENT SCIENCE (MSC551D) INSTRUCTOR: Dr. Jose Ramon Albert Statistical Research ...

Documents

Transcript of ADVANCED MANAGEMENT SCIENCE (MSC551D) INSTRUCTOR: Dr. Jose Ramon Albert Statistical Research ...