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OPERATIONS RESEARCHVol. 66, No. 6, November–December 2018, pp. 1641–1660

http://pubsonline.informs.org/journal/opre/ ISSN 0030-364X (print), ISSN 1526-5463 (online)

Static Routing in Stochastic Scheduling: Performance Guaranteesand Asymptotic OptimalitySantiago R. Balseiro,a David B. Brown,b Chen Chenb

aGraduate School of Business, Columbia University, New York, New York 10027; b Fuqua School of Business, Duke University, Durham,North Carolina 27708Contact: [email protected], http://orcid.org/0000-0002-0012-3292 (SRB); [email protected],

http://orcid.org/0000-0002-5458-9098 (DBB); [email protected], http://orcid.org/0000-0001-8455-2977 (CC)

Received: July 22, 2016Revised: June 15, 2017Accepted: February 16, 2018Published Online in Articles in Advance:December 5, 2018

Subject Classifications:Production/scheduling: sequencing, stochastic;production/scheduling: approximations/heuristic,dynamic programmingArea of Review: Optimization

https://doi.org/10.1287/opre.2018.1749

Copyright: © 2018 INFORMS

Abstract. We study the problem of scheduling a set of J jobs onMmachines with stochasticjob processing times when no preemptions are allowed and with a weighted sum ofexpected completion times objective. Our model allows for “unrelated” machines: thedistributions of processing times may vary across both jobs and machines. We study staticrouting policies, which assign (or “route”) each job to a particular machine at the start of theproblem and then sequence jobs on each machine according to the weighted shortestexpected processing time rule.We discuss how to obtain a good routing of jobs tomachinesby solving a convex quadratic optimization problem that has J ×M variables and dependsonly on the job processing distributions through their expected values. Our main result isan additive performance bound on the suboptimality of this static routing policy relative toan optimal adaptive, nonanticipative scheduling policy. This result implies that such staticrouting policies are asymptotically optimal as the number of jobs grows large. In the specialcase of “uniformly related”machines—that is, machines differing only in their speeds—weobtain a similar but slightly sharper result for a static routing policy that routes jobs tomachines proportionally to machine speeds. We also study the impact that dependence inprocessing times across jobs can have on the suboptimality of the static routing policy. Themain novelty in our work is deriving lower bounds on the performance of an optimaladaptive, nonanticipative scheduling policy; we do this through the use of an informationrelaxation in which all processing times are revealed before scheduling jobs and a penaltythat appropriately compensates for this additional information.

Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2018.1749.

Keywords: stochastic scheduling • unrelated machines • dynamic programming • information relaxation duality • asymptotic optimality

1. IntroductionThere is a rich and well-developed literature onscheduling problems in the operations research andcomputer science communities, and many varietiesof scheduling problems arise in a broad range ofapplications. In this paper, we study the problem ofnonpreemptively scheduling a set of J jobs on a set ofM unrelated machines when job processing times arestochastic. Each job j has a positive weight wj andmust be processed nonpreemptively by one machine.Machines operate in parallel, and at any given time,a machine can process at most one job. Processingtimes for a job depend on the job as well as themachine that processes the job, and the processingtime of a job is not fully known until a job is com-pleted. The objective is to minimize the weightedsum of expected completion times; following thenotation of Graham et al. (1979), this problem is writ-ten as R ||E[∑J

j�1 wjCj], where R indicates unrelatedmachines, and Cj denotes the completion time ofjob j.

Hoogeveen et al. (2001) study the deterministicversion of this problem and show that no polynomialtime approximation scheme exists if 3≠13. In thestochastic case, the problem may be formulated asa stochastic dynamic program in which we minimizeover nonanticipative policies, but the effort in com-puting an optimal policy scales exponentially in thenumber of jobs. Furthermore, optimal policies may becomplex: for example, Uetz (2003) shows that de-liberately idling machines can be optimal even whenmachines are identical. Given the difficulty of theproblem, we typically can only consider suboptimal,heuristic policies. In this paper, we study a class of staticpolicies and analyze their performance relative to anoptimal nonanticipative policy, with an aim of showingthat these policies are asymptotically optimal in theregime of many jobs.When processing times are deterministic, the problem

reduces to two sets of decisions: first, to which machinesto assign or route jobs and, second, how to sequence jobsoptimally on each machine. For deterministic scheduling

1641

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on one machine, Smith (1956) shows that the optimalsequencing of jobs is in decreasing order of the ratio ofweight to processing time. Rothkopf (1966) extends thisresult to the single-machine stochastic scheduling prob-lem 1 ||E[∑J

j�1 wjCj], showing that sequencing jobs indecreasing order of the ratio of weight to expectedprocessing time is optimal. This sequencing is referredto as the weighted shortest expected processing time(WSEPT) rule. Motivated by these insights, we considerstatic routing policies that at the start of the problemroute each job to a given machine, then sequence thejobs routed to eachmachine according toWSEPT. Tofindan optimal static routing policy, the task then reduces tooptimizing over the routing.

The objective of the resulting optimization problemis quadratic but not convex. For the deterministic ver-sion of this problem, Sethuraman and Squillante (1999)and Skutella (2001) independently propose a convexrelaxation to efficiently compute a routing and showedthat the resulting solution performs within a factor of3/2 of optimal. Inspired by these papers, we also useconvex relaxations to optimize over static routing policiesfor the stochastic problem. This leads to convex qua-dratic optimization problems with the same form as inSethuraman and Squillante (1999) and Skutella (2001)with job processing times set to their expected values.Our main result is establishing an additive performanceguarantee on the suboptimality of these static routingpolicies. A corollary to this result is that, as the numberof jobs grows large relative to the number of machines,these static routing policies are asymptotically optimal,provided job weights, expected processing times, andthe coefficients of variation of processing times areuniformly bounded.

Themain challenge is establishing a good lower boundon the performance of an optimal nonanticipative,adaptive scheduling policy. The result in this paper isqualitatively similar to an additive guarantee developedby Mohring et al. (1999) for the case when processingtimes are identical across machines (in particular, seecorollary 4.1 of that paper) but extended to the casewith unrelated machines. The key technology used byMohring et al. (1999) to develop lower bounds is poly-matroid inequalities that describe the performance spaceof expected completion times achievable by feasiblepolicies. This builds on similar inequalities in Schulz(1996) and Hall et al. (1997) for deterministic sched-uling on identical machines; a key step in establishingthese inequalities is effectively an aggregation of jobsacross machines, which appears to rely critically onjob times being the same across machines. As we showin Section 5, bounds based on such relaxations of theperformance space can be generalized to the special case ofuniformly related machines (i.e., machines differing onlyin their speeds), but it is unclear how to generalize thisapproach to the more general case of unrelated machines.

By contrast, we derive lower bounds on the per-formance of an optimal scheduling policy by usinga duality approach based on information relaxations.This approach involves relaxing some or all of thenonanticipativity constraints and imposing a penaltythat punishes violations of the nonanticipativity con-straints. The theory behind this approach is developedin Brown et al. (2010) and builds on earlier work on“martingale duality methods,” independently de-veloped by Rogers (2002) and Haugh and Kogan(2004) for pricing American options. A penalty issaid to be dual feasible if it does not penalize anynonanticipative policy in expectation. Brown et al.(2010) establish weak and strong duality results: withany information relaxation and any dual feasiblepenalty, we obtain a bound on the performance of anoptimal policy and there exists an “ideal” dual feasiblepenalty such that the bound is tight.In this paper, we focus exclusively on perfect infor-

mation relaxations in which all job processing times arerevealed before making any scheduling decisions. Theperfect information problem then reduces to solving adeterministic scheduling problem for a given realiza-tion of jobprocessing times.Absent a penalty, the resultinglower bound may be quite weak, especially when jobprocessing times are highly uncertain. As a result, pen-alties are needed, and we develop dual feasible penaltiesthat improve the lower bound. An attractive feature ofthis approach is that the problem reduces to the analysisof deterministic schedulingproblems in every sample path,and we can leverage ideas from deterministic schedulingin this analysis—namely, the convex relaxations developedby Sethuraman and Squillante (1999) and Skutella (2001).This allows us to “close the loop” and relate the resultinglower bound on performance of an optimal adaptive,nonanticipative policy to the performance of the staticrouting policy.The rest of this paper is organized as follows. Section 1.1

reviews some relevant literature. In Section 2, we for-mulate the problem, and in Section 3, we discuss staticrouting policies. Section 4 presents the main perfor-mance analysis results and an overview of the keysteps of the proof. Section 5 studies the special caseof uniformly related machines, and we show how toobtain a slightly stronger result using a very simplestatic routing policy that routes jobs proportionally tomachine speeds. In Section 6, we discuss the impactthat dependence in processing times on jobs can haveon the suboptimality of static routing policies. InSection 7, we demonstrate the performance of thestatic routing policies and the lower bounds we de-velop on randomly generated examples with up to1,000 jobs. Section 8 concludes. The main proofs are inAppendix A; additional proofs and derivations are inOnline Appendices B, C, and D.

Balseiro, Brown, and Chen: Static Routing in Stochastic Scheduling1642 Operations Research, 2018, vol. 66, no. 6, pp. 1641–1660, © 2018 INFORMS

1.1. Literature ReviewOur paper has connections to several streams of papersthat analyze different forms of scheduling problemsusing different techniques.

1.1.1. Time-Indexed Linear Programs. In time-indexedformulations, the planning horizon is divided intodiscrete time periods, and binary variables are used toindicate whether a job is processed in a machine ina given time period. Linear programming (LP) re-laxations of time-index formulations provide stronglower bounds, which have been extensively used todevelop approximation algorithms for deterministicscheduling problems (see, e.g., Hall et al. 1997 andSchulz and Skutella 2002). In recent, related work,Skutella et al. (2016) study static routing policies forstochastic scheduling on unrelatedmachines. The staticrouting policies they consider are based on a noveltime-indexed LP relaxation. This time-indexed LP re-laxation is powerful in that the approach allows forextensions such as release dates before which jobscannot be processed. Moreover, Skutella et al. (2016)derive strong constant factor approximation results forversions of the problem both with and without releasedates. However, the approach requires a discretizationof the time dimension, a potentially large number ofvariables, and a full input of all cumulative distribu-tions of job processing times along the discretization.By contrast, the static routing policies we consider,although more limited in scope (e.g., we do not handlerelease dates), are simple and intuitive, require solvingonly a convex quadratic problem with J ×M variables,and use only the job weights and the expected values ofjob processing times as inputs. Moreover, our focus ison the asymptotic regime with many jobs, and wedevelop an additive performance bound with that goalin mind. Given that the policies that we study appearto be simple—especially compared with an optimaladaptive, nonanticipative policy—and use little infor-mation about processing time distributions, it is perhapssurprising that these policies can perform well.

1.1.2. Fluid Relaxations for High-Multiplicity Scheduling.The high-multiplicity version of a scheduling problemassumes that jobs can be partitioned into a small numberof typeswith all jobs being identical within a type.Whenthe number of jobs within each type is large, it has beenshown that many scheduling problems can be wellapproximated using fluid relaxations in which discretejobs are replaced with a flow of continuous fluid. Thisapproximation builds on the observation that whenevery machine processes many jobs, the contribution ofeach job is infinitesimal, and the sum of realized pro-cessing times tends to concentrate around its mean.

Bertsimas et al. (2003) study a deterministic job-shopscheduling problem with the objective of minimizing

the total holding cost. They develop an approximationalgorithm based on appropriately rounding an optimalsolution of a fluid relaxation and show that this algo-rithm is within an additive amount from optimal and isasymptotically optimalwhen the number of jobs is large.Fluid models have been applied in a number of sto-chastic problems as well, such as multiclass queueingnetworks (e.g., Maglaras 2000 andNazarathy andWeiss2009) and stochastic processing networks (e.g., Dai andLin 2005, 2008).Through the use of a fluid relaxation, it may well be

possible to show that simple heuristic policies, such asthe static routing policy considered in our paper, areasymptotically optimal when the number of jobs islarge. This involves showing that, using the functionallaw of large numbers, the system behavior under anadmissible policy converges almost surely to a solutionof the fluid problem. Asymptotic optimality then fol-lows because the proposed policy is optimal for thefluid problem. Typically, most papers restrict attentionto work-conserving policies that are independent of thesize of the problem (see, e.g., theorem 4.1 in Dai 1995).Such policies are not necessarily optimal in our setting:as discussed, Uetz (2003) shows that deliberately idlingmachines can be optimal even when machines are iden-tical. Additionally, the fluid problem typically does notprovide a bound for thefinite, discrete stochastic system,and it only establishes optimality in the limit (an ex-ception is Bertsimas et al. 2003, who provide an additiveperformance guarantee based on their fluid relaxationanalysis for a deterministic job shop scheduling problem).An advantage of the perfect information relaxation

approach is that it provides an additive performancebound on the suboptimality of the static routing policyrelative to an optimal adaptive, nonanticipative sched-uling policy (which might not be work-conserving). Thisbound holds for problems of all sizes and not only in thelimit, which, in turn, provides an explicit rate of con-vergence to optimality.

1.1.3. Primal-Dual Schemas and Dual Fitting. Our ap-proach is in spirit similar to the primal-dual schemaand the closely related dual-fitting approach used todesign approximation algorithms in many combina-torial and stochastic optimization problems (see, e.g.,Bar-Yehuda and Even 1981, Agrawal et al. 1995, andJain and Vazirani 2001). In such an approach, onetypically uses a candidate primal feasible solution todesign a dual feasible solution to a relaxation (typically,an LP relaxation) of the original problem; the objectivevalue of this dual feasible solution then providesa bound that ideally can be related to the performanceof the candidate solution.The idea of “dualfitting” is closely related and involves

constructing a good dual feasible solution by prop-erly scaling an initial candidate dual solution. Both

Balseiro, Brown, and Chen: Static Routing in Stochastic SchedulingOperations Research, 2018, vol. 66, no. 6, pp. 1641–1660, © 2018 INFORMS 1643

Anand et al. (2012) and Gupta et al. (2017) use a dual-fitting approach to study variations of scheduling prob-lems. Anand et al. (2012) consider several deterministicscheduling problems with different objectives and pre-emption, whereas Gupta et al. (2017) study an onlineversion of the stochastic and nonpreemptive schedulingproblem on unrelated machines with a weighted sum ofexpected completion times objective inwhich jobs arriveover time. Applying a dual-fitting approach to the time-indexed LP, both of these papers derive constant-factorguarantees for the heuristic policies that they study.

Our approach can be viewed as a primal-dual schemein that we use our “primal feasible solution” (the staticrouting policy) to generate dual variables that play a keyrole in the performance bound. Specifically, to obtain aperformance bound, we first consider the (penalized)perfect information relaxation, which relaxes the non-anticipativity constraints. This problem is a determin-istic scheduling problem for every realization of jobprocessing times. We then consider a convex relaxationof this problem and further consider a Lagrangian dualof this relaxation that dualizes the constraints that requirethat each job be assigned to exactly one machine. Thisprocedure provides a lower bound for any dual fea-sible penalty and any value of the Lagrange multi-pliers. By properly designing the penalty and settingthe Lagrange multipliers to be equal to those from theconvex relaxation used to generate the static routingpolicy, we show that the optimal value of the penalizedperfect information problem is “close” to the performanceof our static routing policy in every sample path.

1.1.4. Information Relaxations. The use of informationrelaxations to this point has largely been as a methodfor computing numerical bounds on specific large-scaleproblem instances; by contrast, here we use the approachas a proof technique to derive analytical bounds. Thisfollows recent work by Balseiro and Brown (2018); werefer the reader to that paper for a more thorough lit-erature review on information relaxations. Balseiro andBrown (2018) study stochastic scheduling on identicalparallel machines (in addition to some other problems)and recover some of the results from Mohring et al.(1999). Specifically, Balseiro and Brown (2018) studythe WSEPT priority policy by combining performancespace relaxations and information relaxations; thisproof technique does not extend to the unrelatedmachinecase. In addition to studying a more general stochasticscheduling problem than that of Balseiro and Brown(2018), our static routing policy is different from theWSEPT priority policy they study, our penalty is quitedifferent from the one considered there (namely, wehave a routing component that is critical for our results;see Section 4.3), and our analysis of the information re-laxations relies on quadratic programming andLagrangian

duality, whereas their analysis relies on linear program-ming relaxations of the performance space.

1.1.5. Offline Analysis. Another line of work studiesthe performance of heuristic policies relative to the per-formance of an optimal “offline” algorithm that knowsall processing times in advance (i.e., what we call theperfect information relaxation without penalty; see, e.g.,Scharbrodt et al. 2006, and Souza and Steger 2006). Forexample, Souza and Steger (2006) show that theexpected competitive ratio of the WSEPT list prioritypolicy—given by the expected ratio of the performance ofWSEPT to that of the perfect information relaxation—isat most 3 when job processing times are identical acrossmachines and exponentially distributed. In general,no nonanticipative policy can achieve asymptotic opti-mality with respect to the (unpenalized) perfect infor-mation benchmark. Thus, as inMohring et al. (1999) andSkutella et al. (2016) and other papers, we measure theperformance of heuristic policies relative to the perfor-mance of an optimal adaptive, nonanticipative policy.

2. Problem FormulationWe consider the problem of scheduling a set of jobs onunrelated machines with the objective of minimizingthe weighted sum of expected completion times whenno preemptions are allowed. Job processing times arestochastic, and the realized processing times of each jobare learned only after completion of the job. Each jobmust be processed by one machine. Machines maywork in parallel, but at any given time, a machine canprocess at most one job. We let ) � {1, . . . , J} denotea set of jobs to be scheduled on a set of machines } �{1, . . . , M}. We denote the processing time of job j∈)on machine m ∈} by pjm, which is a random variablewith positive expected value (denoted by E[pjm]) andfinite variance (denoted by Var[pjm]). In Remark 4.1, wediscuss how our results extend when each job can onlybe processed by a subset of machines. With the ex-ception of Section 6, we assume processing times areindependent across jobs. For our main results in Section 4,we make no assumptions about the dependence ofprocessing times across machines for each job. Becausepreemptions are not allowed, jobs are independent,and each job is assigned to exactly onemachine in everysample path, the dependence structure acrossmachinesfor each job has no impact on the expected performanceof any nonanticipative scheduling policy.We let Cj denote the completion time of job j∈): that

is, Cj denotes the sum of the time until the processingof j starts and the realized processing time of j. Eachjob has a positive weight wj, and the objective is tominimize E[∑j∈) wjCj]. We let Π denote the set ofnonanticipative, adaptive policies. In this model, timeis continuous, and decision epochs correspond to times


when some machine is available; a policy π ∈Π de-termines the next job to be processed at each decisionepoch based on the information available at that time.We let Cπ

j denote the completion time of job j underpolicy π ∈Π. We can write the problem as

V∗ � minπ∈Π E

∑

j∈)wjCπ

j

[ ],

where V∗ denotes the performance of an optimal non-anticipative adaptive policy.

3. Static Routing PoliciesAn optimal policy will generally assign jobs to ma-chines dynamically based on the realizations of pro-cessing times. In general, solving the associatedstochastic dynamic program (DP) for an optimalpolicy requires tracking the state of every job (un-completed, completed, or assigned to a particularmachine) as well as the elapsed processing times for alljobs currently in process; we provide a description ofthe DP in Online Appendix C. Given the difficulty offinding an optimal policy even for problems with fewjobs, we instead consider a static routing policy that,at the start of the problem, assigns each job j∈)to machine m ∈} with probability xjm, independentlyacross jobs, and then it sequences the jobs assigned toeach machine m ∈} in decreasing order of weight perexpected processing time rjm ≜wj/E[pjm]. This sequencingis referred to as the WSEPT rule. As mentioned in theintroduction, Rothkopf (1966) shows that the WSEPTrule minimizes the expected total weighted completiontime on each single machine. Thus, for a given routingof jobs to machines, the WSEPT rule is the optimalsequencing.

We let-�{x�(xjm)j∈),m∈}∈RJ×M+ :

∑m∈}xjm�1,∀j∈)}

denote the set of all routing matrices and let VR(x)denote the expected performance of the static routingpolicy when jobs are routed according to routing matrixx∈-. Formachinem∈},3m denotes the total order thatsorts jobs by the WSEPT rule: that is, jobs i,j∈) satisfythat i3m j if rim>rjm (if rim� rjm, we use the conventionthat i3m j if and only if i< j). Using linearity of expec-tations and the fact that jobs are routed independently,we can then write VR(x) as

VR(x) �∑

j∈)wj

∑

m∈}xjm E[pjm] +

∑

i3mjximE[pim]

( ).

We note that, given any stochastic (i.e., noninteger)routing matrix x∈-, it is straightforward to obtaina deterministic routing matrix whose performance isno worse than VR(x) using the method of conditionalprobabilities (see, e.g., Motwani and Raghavan 1995).

We will show that with a “smart” choice of therouting matrix, the static routing policy is asymptoti-cally optimal as the number of jobs grows large. Wewill make this precise in Section 4. We next show howto obtain such a routing matrix by solving a convexquadratic optimization problem.

3.1. Optimal Static RoutingTo obtain the optimal static routing matrix, we can, intheory, optimize over the routing matrix directly; thatis, we can solve

minx∈-

VR(x). (1)

The expected performance of a static routing policy,VR(x), is not convex in x because the quadratic form isnot positive semidefinite.1 As a result, we will insteadsolve a convex relaxation of (1).Sethuraman and Squillante (1999) and Skutella (2001)

develop convex relaxations for the deterministic versionof this problem. This approach involves adding a suffi-ciently large diagonal to the quadratic form to make theobjective convex and subtracting a commensurate linearterm with the net effect of providing a lower bound onthe problem. We adapt this method to obtain a convexrelaxation toVR(x). In this result and in later discussions,it will be helpful to denote byRm � (rmij ) ∈RJ×J thematrixsuch that

rmij �rim i � j,rim j3m i,rjm i3m j.

⎧⎪⎪⎪⎨⎪⎪⎪⎩In addition, we denote the vector of expected pro-cessing times on machine m by E[pm] � (E[pjm])j∈)∈RJ ,and we denote by diag(E[pm]) ∈RJ×J the diagonalmatrix whose diagonal entries are E[pm].

Lemma 3.1. Let cm � wjE[pjm]( )j∈)∈RJ and Dm �

diag(E[pm])Rmdiag(E[pm]) ∈RJ×J , and consider the opti-mization problem

ZR ≜ minx∈-

∑

m∈}12c′mxm + 1

2x′mDmxm, (2)

where xm � (xjm)j∈) ∈ [0, 1]J . Then (2) is a convex optimi-zation problem; moreover, ZR ≤VR(x) for all routing ma-trices x∈-.Problem (2) is a convex quadratic optimization

problemwith J ×M variables and linear constraints thatimpose the requirement that all jobs must be assignedto one machine. Problem (2) is equivalent to the convexrelaxations in Sethuraman and Squillante (1999) andSkutella (2001) when job processing times all equaltheir expected values. The objective of (2) can bewritten


asVR(x)−1/2∑j∈)wj∑

m∈}E[pjm]xjm(1−xjm), which showsthat the lower bound of Lemma 3.1 is tight when theoptimal solution of (2) is integral. We let x∗ denote anoptimal solution of (2). Using the previous expression, wecan relate the expected performance of the associatedstatic routing policy (which we denote simply by VR) tothe optimal objective of (2) as

VR ≜VR(x∗) � ZR + 12

∑

j∈)wj

∑

m∈}x∗jm(1 − x∗jm)E[pjm]. (3)

In all discussions that follow, unless otherwise stated,when we say “the static routing policy,” this should beunderstood to mean a static routing policy induced byan optimal solution of (2), whichwill be ourmain focus.By contrast, an “optimal static routing policy” shouldbe understood to mean a static routing policy inducedby an optimal solution of (1).

4. Performance AnalysisIn this section, we provide our main results.

Theorem 4.1. Any static routing policy obtained from anoptimal solution of (2) satisfies

V∗ ≤VR ≤V∗ + 12

∑

j∈)wj

(M − 1M

maxm∈}

E[pjm]

+maxm∈}

Var[pjm]E[pjm]

). (4)

In particular, if Var[pjm]/E[pjm]2 ≤Δ holds for all jobs j∈)and machines m ∈}, then the static routing policy satisfies

V∗ ≤VR ≤V∗ + M − 12M

+ Δ

2

( )∑j∈)

wj maxm∈}

E[pjm].

We prove Theorem 4.1 by bounding the performanceVR of the static routing policy from above in terms ofZR and bounding the optimal performance V∗ frombelow by using a penalized perfect information re-laxation. By properly choosing the penalty in the in-formation relaxation, we can connect the lower boundtoZR. The analysis leads to a gap between these boundsthat is linear in the number of jobs.

We note that in the case of identical machines—thatis, when pjm does not vary with m for every jobj∈)—(4) leads to

VR − V∗ ≤ 12

∑

j∈)wjE[pj]

M − 1M

+ Var[pj]

E[pj]2( )

.

This is nearly identical to the suboptimality gap estab-lished for the WSEPT priority policy in corollary 4.1 ofMohring et al. (1999), with the difference that thecoefficient on the second term is slightly weaker

(1/2 versus 1/2 − 1/2M) than the result in corollary 4.1of Mohring et al. (1999). In Section 5, we establish aslightly stronger result for a different static routingpolicy in the case of uniformly related machines; thisresult reduces to the same gap as in corollary 4.1 ofMohring et al. (1999) when machines are identical.If job processing times are deterministic, then a static

routing policy is optimal. Note that in this case thevariance term in (4) is zero. Thus we can interpret thefirst term in the gap in (4) as the performance lossresulting from employing a convex relaxation to finda good routing. If processing times are stochastic,however, the scheduling problem is more challenging,as scheduling policies may be adaptive to realizedprocessing times. Thus, we can interpret the secondterm in the gap in (4) as the performance loss due torestricting to static policies, which, by definition, ig-nore the evolution of job processing information overtime.If we fix the number of machines M and consider

scaling the number of jobs J, then the optimal perfor-mance V∗ scales quadratically with J, because on av-erage, each of the J jobs has to wait for Ω(J) jobs beforebeing processed. From Theorem 4.1, the gap betweenthe performance of the static routing policy and V∗,however, only scales as O(J), provided job weights,mean processing times, and coefficients of variation ofprocessing times are uniformly bounded as J grows.Theorem 4.1 thus implies that the static routing policyis asymptotically optimal under this scaling. This can,in fact, be strengthened: asymptotic optimality holds aslong as the number of jobs grows faster than thenumber of machines.

Corollary 4.2. Suppose w ≤wj ≤ w, p ≤E[pjm]≤ p, andVar[pjm]/E[pjm]2 ≤Δ for all jobs j∈) and machines m ∈},for some positive constants w , w, p , p, and Δ. If

M/J�→J→∞ 0, then

VR − V∗

V∗ →J→∞

0.

In the next two subsections, we provide further detailson the proof of Theorem 4.1; we now describe somehigh-level intuition as to why we might expect thestatic routing policy from problem (2) to perform wellwhen there are many jobs relative to machines. First, aswe will show shortly, the performance loss of applyinga convex relaxation to find a good static routing relativeto the optimal static routing is not substantial whenthere are many jobs. The question then becomes, whyshould we expect an optimal static routing policy toperform well at all in this regime? Intuitively, an op-timal static policy attempts to optimally balance ex-pected job workload statically across machines whenjobs are sequenced at each machine by WSEPT. By


committing jobs to machines at the outset, the optimalstatic policy may suffer relative to an optimal, adaptivepolicy, which balances workloads dynamically depend-ing on machine availability. When every machine has toprocess many jobs, however, the expectations used in thestatic routing policy’s objective become good approx-imations of the realized workload, as the total pro-cessing time on each machine will tend to concentratearound its mean. Thus, the benefit of dynamicallyreacting to the realized workload is small in relativeterms, and we might expect the optimal static routingpolicy to perform well, as it optimizes an objectivefunction that provides a very good approximation ofthe problem.

It is perhaps interesting to note that, although thisintuition for the asymptotic optimality result rests onaveraging (sums of independent variables concentrat-ing around their mean), we do not invoke any con-centration inequalities in establishing the performancebounds.

Remark 4.1 (Infeasible Job-Machine Assignments). Theupper bound in (4) involves taking, for each job, themaximum of the expected processing times across ma-chines. In some applications, machines may be special-ized and simply unable to process certain jobs—thatis, E[pjm] � ∞ for some (j,m) pairs. In such problems,(4) would be a useless bound. A simple modification,however, corrects for this. Specifically, we let }j ⊆}denote the set of machines that are feasible to job j, andwe include the constraints that xjm � 0 for all m∉}j forall j∈) in the definition of the set - of feasible routingmatrices. Because an optimal nonanticipative policywould never use such assignments, these constraints arevalid in all boundswe derive. Themodified version of (4)is then

V∗ ≤VR ≤V∗ + 12

∑

j∈)wj

(M − 1M

maxm∈}j

E[pjm]

+ maxm∈}j

Var[pjm]E[pjm]

).

Asymptotic optimality as in Corollary 4.2 followsunder the weaker requirement that the conditions onthe processing times’ means and coefficients of varia-tion hold for all feasible job-machine pairs.

Overview of Proof. Theorem 4.1 follows from four keysteps:

1. We bound from above the performance VR of thestatic routing policy in terms of the optimal value ZR ofthe convex relaxation plus a term that is linear in thenumber of jobs (Proposition 4.3).

2. We bound from below the performance V∗ ofan optimal adaptive, nonanticipative policy using a

penalized perfect information relaxation. The penal-ized perfect information relaxation involves relaxingthe nonanticipativity constraints (i.e., revealing alljob processing times in advance) and adding to theobjective a penalty that punishes violations of thenonanticipativity constraints. The penalty we use hastwo components:

(a) a sequencing penalty, which penalizes the per-fect information scheduler for prioritizing, on eachmachine, jobs that have a high ratio of weight to re-alized processing time; and

(b) a routing penalty, which penalizes the perfectinformation scheduler for routing jobs to machines thathave short realized processing time.

We show that with these penalties, the penalizedperfect information problem, in every sample path, re-duces to a deterministic scheduling problem inwhich it isoptimal to sequence jobs routed to each machineaccording to WSEPT, plus a term that is linear in the job-machine assignments and depends on the routing pen-alty (Lemma 4.4).

3. We further bound from below the objective valueof the penalized perfect information problems in everysample path in terms of ZR. To do this, we considera convex relaxation of the resulting deterministicscheduling problem and then dualize the job assignmentconstraints in this relaxation. Let ν∗j denote the optimaldual variables for the job assignment constraints in theconvex problem (2) used to generate the static routingpolicy. By using ν∗j in the routing penalty as well as in thedual variables of this relaxation, we can show that thepenalized perfect information problem, in every samplepath, is bounded from below by ZR less a term that islinear in the number of jobs (Proposition 4.5). We obtaina lower bound on V∗ by taking expectations of this lowerbound.

4. By combining the upper bound from step 1 andthe lower bound from step 3, we obtain an upperbound on the suboptimality gap of the static routingpolicy, i.e., an upper bound on VR − V∗, that is linear inthe number of jobs. Under the given conditions on thejob weights and processing distributions, this resultimplies asymptotic optimality of the static routing policyas the number of jobs grows large.We now move to the details of the proof.

4.1. Upper BoundWe first provide an upper bound for the performanceVR of the static routing policy, which of course alsobounds the optimal performance V∗ from above.

Proposition 4.3. Any static routing policy obtained bysolving (2) satisfies

ZR ≤VR ≤ZR +M − 12M

∑

j∈)wj max

m∈}j

E[pjm].


Proof.From (3),

VR � ZR + 12

∑

j∈)wj

∑

m∈}x∗jm(1 − x∗jm)E[pjm]

≤ZR + 12

∑

j∈)wj max

m ∈}j

E[pjm]∑

m∈}x∗jm(1 − x∗jm).

Fixing job j, because each x∗jm is nonnegative and∑

m∈} x∗jm � 1, we have∑

m∈} x∗jm(1 − x∗jm)≤ M−1M . As a

result,

ZR ≤VR ≤ZR +M − 12M

∑

j∈)wj max

m∈}j

E[pjm]. □

Proposition 4.3 states that the gap between ZR andthe performance VR of the static routing policy onlygrows as O(J), provided job weights and expectedprocessing times are uniformly bounded. Because bothZR and VR grow as Θ(J2), Proposition 4.3 thus impliesZR will be very close to the actual performance VR

when J is large.Moreover, becauseZR ≤ minx∈-

VR(x)≤ VR,when J is large, the performance of the static routingpolicy will be very close to the performance of an optimalstatic routing policy.

4.2. Lower BoundIn this section, we develop a lower bound on V∗ byusing a penalized perfect information relaxation.Specifically, we consider a decision maker who hasaccess to all realizations of the processing times p �{pjm}j∈),m∈} before making scheduling decisions. Givena sample path p ∈RJ×M, we can calculate the optimalvalue for the sample path in “hindsight” by solvinga deterministic scheduling problem. The expectedvalue with perfect information on processing timesprovides a lower bound on V∗. Unfortunately, thislower bound is not tight in general because the de-cision maker with perfect information on processingtimes can “cheat” (i.e., use information not availableto any nonanticipative scheduling policy) (a) byrouting jobs to machines that have correspondinglyshort realized processing times and (b) then, for eachmachine, by sequencing jobs with a higher ratio ofweight to realized processing time first. To improvethe lower bound, we will impose an appropriatepenalty that creates incentives for the perfect in-formation scheduler to not cheat in the routing andsequencing decisions. In Section 4.3, we show thatthis penalty is essential for obtaining good lowerbounds.

FollowingBrownet al. (2010),we say apenaltyYπ is dualfeasible if Yπ does not penalize, in expectation, any non-anticipative policy; that is, if for allπ ∈Π,E[Yπ] � 0 holds.2

The stochastic scheduling problem including any dualfeasible penalty Yπ can be written as

V∗Y � min

π∈Π E∑

j∈)wjCπ

j + Yπ

[ ].

Because the penalty is dual feasible, V∗Y � V∗. We let

VH(p) denote the optimal (deterministic) value of thepenalized perfect information problem for sample pathp∈R J×M, where H stands for hindsight. We then haveVH ≜E[VH(p)] ≤V∗

Y � V∗. A dual problem then is tochoose a dual feasible penalty so as to maximize thelower bound VH. As shown in Brown et al. (2010),strong duality holds, and we can, in theory, constructan ideal penalty such thatVH � V∗ by using the optimalvalue function to the stochastic scheduling problem.Because the optimal value function is difficult to char-acterize in general, we will instead consider simplerpenalties that will allow us to relate the lower boundVH

to the performance of the static routing policy.Specifically, to compensate for possible cheating at

the routing and sequencing steps in the perfect in-formation problem, we consider a penalty of the formYπ � Yπ

S + YπR, where Yπ

S is a sequencing penalty and YπR

is a routing penalty. The goal of the sequencing penaltyis to create incentives for the perfect informationscheduler to sequence jobs on every machine accordingto WSEPT in every sample path; the goal of the routingpenalty is to create incentives for the perfect in-formation scheduler to route jobs similarly to an op-timal solution x∗ of (2) in every sample path. We willalso ensure that both penalties are dual feasible.We set the sequencing penalty Yπ

S to be

YπS ≜

∑

j∈)

∑

m∈}rjmSπjm pjm − E[pjm]

( ),

where Sπjm denotes the starting time of job j on machinem with the convention that Sπjm � 0 if job j is notassigned to machine m. First, note that Yπ

S is dualfeasible: this follows from the fact that, for everynonanticipative policy π ∈Π, the processing time pjm ofjob j on machine m is independent of job j’s startingtime Sπjm. To understand why the sequencing penaltycan be helpful, imagine a given job j being routed tomachine m in the perfect information problem. If pjm isshort (compared with its expected value), to minimizethe total weighted completion time, the perfect in-formation scheduler prefers to sequence j earlier; on theother hand, pjm − E[pjm]< 0, so the sequencing penaltycreates an incentive to start j later (i.e., to delay thesequencing of j). If pjm is long relative to its expectedvalue, the reverse reasoning applies. With the weightsrjm on each term in the sequencing penalty, it turnsout that these competing incentives perfectly cancel,and it is optimal in the perfect information problem to


sequence all jobs on each machine according to WSEPTin every sample path. We show this formally below.

For the routing penalty YπR, we consider the form

YπR ≜

∑

j∈)

∑

m∈}xπjm λjm E[pjm] − pjm

( ) + γjm(E[p2jm] − p2

jm)[ ]

,

where xπjm indicates whether job j∈) is assigned tomachine m ∈} under policy π ∈Π, and λ �λjm( )

j∈),m∈}∈RJ×M and γ � (γjm)j∈),m∈} ∈RJ×M are two

constant matrices that we will specify shortly. First, YπR

is dual feasible: this follows from the fact that, for everynonanticipative policy π ∈Π, the decision about whetherto assign job j tomachinem is independent of pjm.We canview the routing penalty as creating incentives for theperfect information scheduler to route jobs to machinesthat have relatively large (compared with their expectedvalues) realized processing times. Note that Yπ

R is in-dependent of the sequencing of jobs on each machine.

Because both YπS and Yπ

R are dual feasible, the fullpenalty Yπ � Yπ

S + YπR is also dual feasible. The penal-

ized perfect information problem can be viewed asa combination of an assignment problem together witha deterministic scheduling problem. In this determin-istic scheduling problem, weights are proportional tothe realized processing times, and it is optimal to se-quence jobs according to the WSEPT ordering 3m onevery machine. We now formalize this.

Lemma 4.4. The penalized perfect information problemwith Yπ � Yπ

S + YπR can be written as

VH(p) � minx∈-

∑

j∈)

∑

m∈}rjmpjmxjm pjm +

∑

i3mjximpim

( )+ xjm[(λjm + pjmrjm)(E[pjm] − pjm)+ γjm(E[p2

jm] − p2jm)].

Proof. We write the completion time as Cj � ∑m∈}Cjm

with the assumption that Cjm � 0 if job j is not assignedto machine m. Using the fact that the starting time isgiven by Sjm � Cjm − pjmxjm, together with rjm �wj/E[pjm], we can write the objective of the penalizedperfect information problem as∑

j∈)wjCj + YS + YR �

∑

j∈)

∑

m∈}rjmpjmCjm

+∑

j∈)

∑

m∈}xjmpjmrjm(E[pjm] − pjm) + YR.

The first term corresponds to a sequencing cost thatcoincides with the objective value of the deterministicscheduling problem R ||∑J

j�1wjmCj with weights wjm �rjmpjm, and the second and third terms can be inter-preted as a routing cost that penalizes the perfect

information scheduler for assigning jobs to machinesin which the realized processing time is below themean. Note that the routing cost is independent ofthe order in which jobs are sequenced in a machine;thus once an assignment is fixed, the decision makershould sequence jobs so as to minimize the sequencingcost. The ratio of weights to realized processing timesin this modified scheduling problem is given bywjm/pjm � rjmpjm/pjm � rjm: thus, the optimal orderingis independent of the realized processing times andcoincides with the WSEPT rule. As a result, the pe-nalized perfect information problem can be written asin the statement. □

Aswe dowith the static routing policy, we will workwith convex relaxations of VH(p) that are easier toanalyze. Following the steps of Lemma 3.1 applied tothe deterministic problem VH(p), we have

VH(p)≥ZH(p)≜ minx∈-

∑

m∈}a′mxm + 1

2x′mQmxm,

where am � (12 p

2jmrjm + (λjm + pjmrjm)(E[pjm] − pjm) +

γjm(E[p2jm] − p2jm))j∈) ∈ RJ and Qm � diag(pm)Rmdiag

(pm) ∈ R J×J . Note that thematrixQm is analogous to thematrix Dm in (2) that we use to compute the staticrouting policy, with expected job processing timesreplaced by their realized values. The objective inZH(p)is convex, as the matrices Qm are positive semidefiniteby Lemma 3.1. Let ZH ≜ E[ZH(p)]. From the abovediscussion, we have V∗ ≥ VH ≥ ZH.We now turn to the issue of how to tune the routing

penalty to obtain a good lower bound onV∗. Recall thatthe goal of the routing penalty is to create incentives forthe perfect information scheduler to route jobs insimilar fashion to x∗, which is an optimal solution to (2).Because (2) is a convex optimization problem, we canfind an optimal Lagrange multiplier ν∗j on the assign-ment constraint for each job; ν∗j represents the marginalcost of assigning a job to any machine at optimality in(2). By choosing λjm in the routing penalty to be pro-portional to ν∗j , we align the marginal cost of routingjobs in the perfect information problemwith problem (2),which we use to obtain a static routing policy. As wenow show, this allows us to establish a strong rela-tionship between ZH(p) and ZR, the optimal value of (2),in every sample path.

Proposition 4.5. Let ν∗j denote an optimal Lagrangemultiplier of the constraint

∑m∈} xjm � 1 in (2). If γjm �

−12 rjm and λjm � ν∗j/E[pjm] + 1

2 wj, then for every samplepath p,

VH(p)≥ZH(p)≥ZR − 12

∑

j∈)wj max

m∈}j

Var[pjm]E[pjm]

. (5)


Note that the scaling of ν∗j by 1/E[pjm] is simply amatching of units: ν∗j is measured in units of objective(weighted time), whereas λjm is in units of objective pertime. We also note that the second term 1/2wj in λjm

and the choice of γjm are not critical for the qualitativelyimportant aspect of the result, which is obtaining a gapin (5) that is linear in the number of jobs. These termsare merely for the purposes of cleanly expressing thisgap in terms of the coefficients of variation of the jobprocessing times.

Because the penalty is dual feasible, taking expec-tations over (5), we immediately obtain that

V∗ ≥VH ≥ZH ≥ZR − 12

∑

j∈)wj max

m∈}Var[pjm]E[pjm]

.

Proposition 4.5 and Proposition 4.3 thus imply Theo-rem 4.1 and show that the gap between the optimalperformance V∗ and the performance of the staticrouting policy VR only grows as O(J).

In the proof of Proposition 4.5, we bound ZH(p)from below in terms of ZR by dualizing the assignmentconstraints in ZH(p) with Lagrange multipliers ν∗j . Theproblem then decouples over machines. Ideally, onewould like to show that, for each machine, the optimalstatic routing vector x∗m is optimal for the perfect in-formation problem in every sample path. Instead, weshow that by properly perturbing the linear term inthe objective of the perfect information problem(which leads to the gap in (5)) and by setting λjm andγjm as above, the optimal routing of job j to machine mcan be shown to be xHjm � x∗jmE[pjm]/pjm. This guaran-tees that the optimal objective value of the staticrouting policy ZR coincides with the objective valueof the dualized and perturbed perfect informationproblem. This follows because the quadratic formssatisfy x∗m′Dmx∗m � xHm′QmxHm (and a similar equalityholds for the linear terms). The full proof ofProposition 4.5 and the proof of Corollary 4.2 aregiven in Appendix A.

4.3. The Importance of PenaltiesBoth the routing and sequencing penalties are essentialin obtaining good lower bounds on the performanceof an optimal, nonanticipative scheduling policy: wepresent two examples illustrating that in the absenceof either of these penalties, the perfect informationrelaxation can be weak (and, in particular, does notprovide an asymptotically tight bound). The first ex-ample excludes the sequencing penalty, and the secondexample excludes the routing penalty.

Example 4.1. (Role of the Sequencing Penalty). Consideran example with M � 1, and J identical jobs of weight 1and processing times pj having a Bernoulli distribution

with mean 0.5. For this single-machine schedulingproblem, the static routing policy is optimal, andwe have

VR � V∗ �∑J

j�1

(E[pj] +

∑

i<jE[pi]

)� J2 + J

4.

Because there is only onemachine, the routing penalty hasno effect in this example. Without a sequencing penalty,however, a scheduler with perfect information will cheatby scheduling first all jobs with zero realized processingtimes. Let N1 be the number of jobs with realized pro-cessing times of 1. The random variableN1 has a binomialdistribution with J trials and success probability 0.5. Theexpected value of the perfect information problem is

VH � EN1(N1 + 1)

2

[ ]� J2 + 3J

8.

For J large, this lower bound is off from V∗ by nearlya factor of 2.

Example 4.2. (Role of the Routing Penalty). Consider anexample with M � 2 and J identical jobs of weight 1.The processing times of each job on the first machineare deterministic and equal to pj1 � 0.5; on the secondmachine, each pj2 is given by a Bernoulli distributionwith mean 0.5. The static routing policy simply routeseach job randomly according to expected processingtimes, and thus xj1 � xj2 � 0.5 for each job j. This leads to

VR �∑J

j�1

∑2

m�112

E[pjm] +∑

i<j

E[pjm]2

( )� J2 + 3J

8.

Without a routing penalty, a scheduler with perfectinformation will cheat by routing all jobs that have zerorealized processing times on the second machine (therandommachine) to that machine. This leads to a weaklower bound even with a sequencing penalty. We nowshow this formally. Without the routing penalty, theobjective of VH(p) (refer to Lemma 4.4) is

∑

j∈)wjCj + YS �

∑

j∈)

∑

m∈}rjmpjmCjm

+∑

j∈)

∑

m∈}xjmpjmrjm(E[pjm] − pjm)

≤∑

j∈)

∑

m∈}rjmpjmCjm,

where the inequality follows from the fact thatE[pjm] � pjm for the first machine, and for the secondmachine, either pjm�0 or pjm�1, and in either case,pjm(E[pjm] − pjm)≤0.LetN1 be the number of jobs with realized processing

times of 1 on the random machine. The random vari-able N1 has a binomial distribution with J trials andsuccess probability 0.5. Clearly, it is optimal to route allother J −N1 jobs to the random machine because they


have zero realized job processing times there. Theremaining N1 jobs have “penalized” weights (from thesequencing penalty) rj2pj2 � 2 and realized processingtimes of one on the random machine and unit weightsand processing times of 0.5 on the deterministic ma-chine. It is feasible for the perfect information schedulerto route 1/5 of theN1 jobs to the randommachine and theremaining 4/5 of theN1 jobs to the deterministic machine(in fact, this can be shown to be asymptotically optimalwhen J is large). With this routing in every sample pathand the bound above on

∑j∈)wjCj + YS, we have

VH � E VH(p)[ ] � E min

x∈-∑

j∈)wjCj + YS

( )[ ]

≤E∑

j∈)

∑

m∈}rjmpjmCjm

[ ]� E

N21 + 2N1

5

[ ]� J2 + 5J

20.

For J large, the perfect information bound with asequencing penalty but no routing penalty in this ex-ample is thus off from V∗ by at least a factor of 5/2.

5. Uniformly Related MachinesIn this section, we consider the case of uniformly relatedmachines. Here, machines differ only in their speeds: theprocessing time of job j on machine m is pjm � pj/sm,where sm > 0 is the speed ofmachinem. Although this isa special case of the problem considered in Theorem 4.1,using our penalized perfect information relaxationapproach, we can establish slightly stronger perfor-mance guarantees for a simpler static routing policy. InSection 5.1, we show that similar bounds can also beobtained using LP-based performance space boundswhen machines are uniformly related.

Specifically, we consider the static routing policy in-duced by xjm � sm/

∑m′∈}sm′ , which we refer to as speed

proportional routing. This routing policy balances jobworkload across machines by routing jobs proportionallyto machine speeds. We note that solving problem (2)when machines are uniformly related does not lead tospeedproportional routing, so this policy is different fromthe static routing policy studied in Section 4. In the fol-lowing,we let S denote the sumof the speeds and assumethat machines are indexed in descending order of theirspeeds (i.e., s1 ≥ s2 ≥⋯≥ sM). We use VS to denote theexpected performance of speed proportional routing.

Theorem 5.1. If machines are uniformly related, thenspeed-proportional routing satisfies

V∗ ≤VS ≤V∗ + 12

∑

j∈)wjE[pj]

{2M − 1

S− 1κj

( )+ 1

κj− 1S

( )Var[pj]

E[pj]2}, (6)

where κj � sM if Var[pj]/E[pj]2 > 1 and κj � s1 ifVar[pj]/E[pj]2 ≤ 1.IfM � 1, then from (6) it follows that V∗ � VS, which

means the performance guarantee in Theorem 5.1 issharp. If all speeds are equal, then the problem reducesto that of identical parallel machines; without loss ofgenerality, we can set sm � 1 in this case, and (6) thenreduces to

V∗ ≤VS ≤V∗ +M − 12M

∑

j∈)wj E[pj] + Var[pj]

E[pj]

( ).

This is the same suboptimality gap as in corollary 4.1in Mohring et al. (1999) for the identical parallelmachine problem, although, as mentioned earlier,Mohring et al. (1999) study a different policy. It is alsoperhaps interesting to note that when Var[pj]/E[pj]2 ≤ 1for all jobs j∈) (uniform, exponential, or Erlang dis-tributions, for example), the gap in (6) does not dependon the slowest machine other than through the sum ofspeeds.In the proof of Theorem 5.1, we show that speed

proportional routing is optimal for a perturbed ver-sion of the penalized perfect information problem inwhich we slightly modify the linear terms of theobjective. In the case of uniformly related machines,the routing penalty is not critical to obtain thequalitative result, and the routing penalty is in-troduced for the purpose of cleanly expressing thisgap in terms of the coefficients of variation of the jobprocessing times. Intuitively, because the processingtimes of a job are perfectly correlated across machines,the perfect information scheduler cannot cheat byrouting jobs to machines that have correspondinglyshort realized processing times. This dependence ofprocessing times across machines allows us to pro-vide a bound for the speed proportional routingpolicy that is in some cases sharper to the one given inTheorem 4.1. The proof of Theorem 5.1 is given inAppendix A.4.

5.1. LP-Based Performance Space Bounds forUniformly Related Machines

For stochastic scheduling on identical parallel ma-chines, Mohring et al. (1999) provide a polyhedralrelaxation of the performance space and show that thispolyhedron is a polymatroid; these results build onsimilar relaxations developed by Schulz (1996) fordeterministic scheduling problems. This relaxationcritically relies on job processing times being the sameacross machines: the intuition is that M identical ma-chines working in parallel tend to process jobs ata similar rate as a single machine that isM times faster.Schulz (1996) derives similar polyhedral relaxations ofthe performance space for deterministic scheduling on


uniformly related machines (see lemma 9 of Schulz1996); we are not aware of similar results that extend tothe more general case of stochastic scheduling on un-related machines.

In this subsection, we show that for the case ofuniformly related machines, we can, in fact, derivepolyhedral representations of the performance spacethat lead to similar guarantees (albeit with slightlyweaker constants) as in Theorem 5.1. This represen-tation leverages the facts that (i) when machines areuniformly related, the processing times of jobs ondifferent machines will be proportional, and (ii) theprocessing time of any one job can always be boundedfrom below by the time it would take on the fastestmachine. Given this intuition, it is not clear that suchresults could be extended to the case of unrelatedmachines.

We have the following result, which parallels theo-rem 3.1 in Mohring et al. (1999) in the case of uniformlyrelated machines.

Proposition 5.2. If machines are uniformly related,then for every feasible, nonanticipative policy, theexpected completion times E[Cj] of jobs satisfy, for all!⊆),

∑

j∈!E[pj]E[Cj]≥ 1

2S

(∑j ∈!

E[pj])2

− 12

1sM

− 1S

( )∑j∈!

Var[pj]

− 12sM

− 1s1

( )∑j∈!

E[pj]2. (7)

As we have assumed above, if processing times sat-isfyVar[pj]/E[pj]2 ≤Δ for all jobs j∈), then (7) imply that

∑

j∈!E[pj]E[Cj]≥ 1

2S

( ∑j ∈!

E[pj])2 −

(1

2sM− 1s1

( )+Δ

21sM

− 1S

( ))∑j∈!

E[pj]2, (8)

holds for all feasible, nonanticipative policies and forall !∈). We thus obtain a lower bound on V∗ bysolving the LP min

∑j∈)wjE[Cj] subject to (8)

{ }treat-

ing E[Cj] as decision variables. It is straightforwardto show that the set function on the right-hand sideof (8) is supermodular, and thus a solution to thisproblem is given by the greedy algorithm of Edmonds(1971). We assume without loss of generality thatw1

E[p1]≥ w2

E[p2]≥⋯≥ wJ

E[pJ]and let CLP denote an optimal

solution to this problem. Because a greedy solution isoptimal, it follows that

CLPj � 1

S

∑j

k�1E[pk] − 1

2sM+ 12S

− 1s1

( )+ Δ

21sM

− 1S

( )( )E[pj].

Because∑

j∈)wjCLPj ≤V∗, this leads to a bound on the

suboptimality gap of the speed proportional routingpolicy:

VS − V∗ ≤VS −∑

j∈)wjCLP

j � 12

1sM

+ 2M − 1S

− 2s1

( )(+ 1

sM− 1S

( )Δ

)∑j∈)

wjE[pj]. (9)

The bound in (9) is qualitatively similar to that ofTheorem 5.1 and also implies asymptotic optimality ofspeed proportional routing in the case of uniformlyrelated machines, although the gap in Theorem 5.1 hasslightly stronger constants.One might wonder whether such an analysis with

performance space bounds generalizes to the case ofunrelated machines. The challenge is that the LP-basedbounds on the performance space are, by design, ag-nostic to the specifics of how policies might assign jobsto machines. Although this may seem like a limitationof this approach, in the case of uniformly relatedmachines, we can nonetheless obtain good (i.e., asymp-totically tight) bounds with this approach by leveragingthe fact that processing times are proportional across themachines.We do not know, however, how to generalizethese LP-based bounds on the performance space in theabsence of any particular structure for processing timesacross machines.

6. Dependent JobsThe analysis above assumes, consistent with much ofthe literature on stochastic scheduling, that processingtimes are independent across jobs.When job processingtimes are dependent, static policiesmay perform poorly:good scheduling policiesmay need to be adaptive, as theelapsed time for jobs in process (or completed) mayconvey valuable information about the distributionsfor unscheduled jobs, which could in turn influencescheduling decisions.In this section, we study the impact that dependence

in processing time distributions can have on the sub-optimality of the static routing policy, and we providesufficient conditions for asymptotic optimality of thestatic routing policy. At a high level, we use the sameproof technique involving penalized perfect informa-tion relaxations, but now, more care is required in thedefinition of penalties: for example, with dependentjobs, the start time of a job may no longer be inde-pendent of the processing time of the job, and thusthe routing and sequencing penalties as defined inSection 4.2 need not be dual feasible. In terms of theresults that follow, the conditions that we derive forasymptotic optimality of static routing to prevail in thecase of dependent jobs are strong conditions, andwewillshow by example that these conditions are necessary.


We will describe dependence across jobs in the fol-lowing way. For a feasible, nonanticipative schedulingpolicy π, we let ^π

t denote the σ-algebra describinginformation about processing times at time t, and again,Sπjm denotes the start time of job j on machine m underpolicy π.

Assumption 6.1. There exist constants αj and βj such that

|E[pjm |^πt ] − E[pjm] | ≤αj, (10)

|E[p2jm |^πt ] − E[p2jm] | ≤ βj, (11)

hold a.s. for every j∈), for every m ∈}, for all π ∈Π, andfor all t≤Sπjm.

Assumption 6.1 expresses limits on the amount anyfeasible, nonanticipative scheduling policy can “learn”about each job before it is scheduled from the pro-cessing times of jobs that are currently in process orcompleted. These limits are expressed in terms ofmaximum absolute deviations αj and βj in the first andsecond moments, respectively, of the job processingtime distribution relative to its unconditional (i.e., att � 0) value. Note that the static routing policy (and itsperformance) depend only on processing time distri-butions through their expected values, so the de-pendence structure does not affectVR—but the optimalvalue V∗ may be affected by the dependence structure.As in the case of independent jobs, we make no as-sumptions about the dependence structure of pro-cessing times across machines for each job.

In Online Appendix B.1, we provide a bound(Theorem B.1) on the suboptimality of the performanceof the static routing policy. This result is analogous toTheorem 4.1 in the case of independent jobs but in-cludes additional terms that depend on αj and βj: theseterms can be interpreted as bounds on the performanceloss of the static routing policy as a result of ignoringthe dependence across jobs. If the dependence acrossjobs is sufficiently weak, thenwemight expect the staticrouting policy to remain asymptotically optimal as thenumber of jobs grows. The following result makes thisprecise; this result follows from Theorem B.1 by similararguments as those in Corollary 4.2.

Corollary 6.2. Suppose the conditions on weights andprocessing times in Corollary 4.2 hold. Let α≜

∑j∈)αj/J and

β≜∑

j∈)βj/J. If αM32 �→J→∞ 0, βM/J �→J→∞ 0, and

M/J �→J→∞ 0, then

VR − V∗

V∗ �→J→∞0.

For a fixed number of machines, the key conditionthat implies asymptotic optimality of static routingpolicy is that the average (over jobs) of αj go to zeroas the number of jobs grows large: in other words, it

is sufficient that the “average dependence,” as measuredin terms of absolute deviations of the expected jobprocessing times in (10), vanishes. The condition onthe second moments is much weaker; for example, itsuffices that the conditional variance of processingtimes for each job be uniformly bounded for the resultto hold. We now show by example that the conditionon expected processing times is in fact necessary forthe static routing policy to be asymptotically optimal.

Example 6.1. (Jobs with Limited Dependence). Consideran example with M � 1 and J jobs and all job weightsare 1. We assume that the number of jobs J is even. Theprocessing time pj of each job j on the machine followsa Bernoulli distribution with mean 0.5. We assume thatthe processing times of every pair of adjacent odd- andeven-numbered jobs p2i−1 and p2i with i∈ {1, 2, . . . , J2} aredependent but are independent from all the other jobs.Specifically, the joint distribution of (p2i−1, p2i) is

P(p2i−1, p2i) p2i � 0 p2i � 1

p2i−1 � 0 qi 12 − qi

p2i−1 � 1 12 − qi qi

where qi ∈ [0, 12]. We have P(p2i � 1 | p2i−1 � 0) � 1 − 2qiand P(p2i � 1 | p2i−1 � 1) � 2qi. If qi � 1

4 for all job pairs(p2i−1, p2i), all jobs are independent, and from Section 4,we know that the static routing policy is asymptoticallyoptimal as J goes to infinity. If qi > 1

4, adjacent jobs arepositively correlated. In this example we assume thatqi ≥ 1

4 for all job pairs (p2i−1, p2i) and that jobs are sortedin decreasing order of qi.Because the (unconditional) expected processing

time of each job is the same (0.5), the static routingpolicy simply chooses an arbitrary order of jobs andprocesses jobs in this order. Thus,

VR �(∑J

j�1E[pj] +

∑

i<jE[pi]

)� J2 + J

4.

Now we consider a simple heuristic policy that isadaptive and exploits the correlation among adjacentjobs. Every time an odd-numbered job 2i − 1 is com-pleted, if p2i−1 � 0, then we continue to process job 2i;otherwise, if p2i−1 � 1, we move job 2i to the end of thesequence and process the next odd job 2i + 1 instead,because job 2i is more likely to be long as well. Whenall the odd-numbered jobs are completed, we processthe remaining even-numbered jobs in order. DenotebyVA the expected performance of this heuristic adaptivepolicy. It can be shown (see Online Appendix B.2) that

VA ≤ 516

J2 + 18J + (1 − J

2)∑J2

i�1qi.


Let q � 2J

∑J2i�1 qi be the average of {qi}

J2i�1. Because

VR − VA ≥ q− 14

4J2 − q− 1

4

2J (12)

and VA ≥V∗, the static routing policy is not asymp-totically optimal as J increases whenever q> 1

4.

Note that in this example, E[| pj |^t] − E[pj]|≤ αj �2(qj − 1

4

)for all jobs j and times t≤ Sπj ; thus α � 2

(q − 1

4

). In

this sense, the assumption on α in Corollary 6.2 is nec-essary (even with a fixed number of machines) for thestatic routing policy to be asymptotically optimal.

Example 6.1 demonstrates that even a very smallamount of dependence across jobs can lead to the staticrouting policy being asymptotically suboptimal. Evenif only a small, fixed fraction of all the job pairs areweakly dependent (e.g., qj � 1

4 + ε for some ε> 0), andall other jobs are independent, the static routing policywill not be asymptotically optimal. On the other hand,even if not asymptotically optimal, the static routingpolicy may perform well if the dependence (as mea-sured by the αj) is small, and Theorem B.1 providesa performance guarantee.

7. Numerical ExamplesAlthough the preceding analysis provides a guaranteeon the static routing policy and shows that this policyis asymptotically optimal, it is also helpful to havemethods that produce bounds in specific examples. Inparticular, we have observed in numerical examplesthat the lower bounds we study through the use ofpenalized perfect information relaxations often lead tovery good lower bounds on the performance of theoptimal scheduling policy.

To demonstrate this, we examine the performance ofthe static routing policy and the penalized perfect in-formation relaxation lower bound on a set of randomlygenerated examples. We consider instances with num-ber of jobs J increasing linearly from 50 to 1000with steplength 50. We generate job weights as wj ~U[0.5, 1] andexpected processing times as E[pjm] ~U[0.5, 1], all in-dependent and identically distributed. We considerfour cases:

1. M � 4, pjm ~U[0, 2E[pjm]],2. M � 4, pjm ~Exponential(1/E[pjm]),3. M � ��

J√

, pjm ~U[0, 2E[pjm]], and4. M � ��

J√

, pjm ~Exponential(1/E[pjm]).In cases 1 and 2,M is fixed to be 4; in cases 3 and 4,M

scales with J asM � ��J

√(specifically, we roundM to the

integer closest to��J

√). For consistency across distributions,

for each value of J, all four cases share the same wj,cases 1 and 2 share the same E[pjm], and cases 3 and 4share the same E[pjm].

In each case and for each fixed J, we calculate (a) theperformance of the static routing policy VR, (b) thepenalized perfect information relaxation lower bound

ZH, and (c) the perfect information relaxation lowerbound without penalty ZH

0 . As the static routing policydepends only on expected processing times as does theassociated performance, VR is the same for each value ofJ in cases 1 and 2, as well as for each value of J in cases 3and 4. The values ZH � E[ZH(p)] and ZH

0 � E[ZH0 (p)]

are estimated with 100 sample paths; this led to verylow standard errors in the results. (We also calculatedbut do not report here bounds with penalized infor-mation relaxations that omitted either the sequencingpenalty or the routing penalty. These lower boundswith only one component of the penalty were not as-ymptotically optimal and were often quite weak—insome cases, these bounds were worse than those withno penalty. Following Section 4.3, this further under-scores the importance of having both the sequencingand routing penalties.)All bounds are calculated using MATLAB on a desk-

top PC, with theMOSEK optimization toolbox used tosolve the convex quadratic optimization problems.Solving (2) or optimizing ZH(p) in one sample path isof comparable difficulty. On the smallest examples(J � 50,M � 4), these optimization problems take about0.1 seconds to solve; on the largest examples (J � 1000,M � 32), these optimization problems take about 100seconds to solve. Considering the complexity of solvingfor an optimal policy, these run times appear quitemodest by comparison, and we could likely reduce thetotal computational effort somewhat in a variety ofways (e.g., variance reduction techniques for thelower bounds, solution techniques that take advan-tage of the special assignment structure of the opti-mization problems).Figure 1 reports the simulation results for cases 1

and 2, and Figure 2 reports the simulation results forcases 3 and 4. In all these cases, the perfect informationboundswith no penalty are quite weak, and the relativegaps using these bounds do not appear to convergeto zero. The penalized perfect information bounds,however, perform very well and clearly convey theasymptotic optimality of the static routing policy (inFigure 1, (a) and (b), in particular, the penalized perfectinformation bound is almost indistinguishable from theperformance of the policy). The performance gaps aresomewhat larger with exponential distributed jobsand when the number of machines scales with thenumber of jobs. We suspect these larger gaps arelargely the result of the lower bound being somewhatworse in these cases. With more machines, there aremore opportunities for “cheating” in the penalized perfectinformation problem. Similarly, exponential processingtimes are considerably more uncertain than uniformprocessing times, and thus knowing processing timesin advance is more valuable in this case.


Overall, the gaps are quite small, and in many of theexamples we can conclude that the static routing policyis at worst within a few percentage points from opti-mality. Although we have derived analytical boundson the suboptimality of static routing policies, it isencouraging to observe that static routing policiesperform well on these specific examples.

8. ConclusionWe have studied the problem of stochastic schedul-ing on unrelated machines with a weighted sumof expected completion times objective. Our main

contribution is to bound the loss in using a simple staticrouting policy relative to an optimal adaptive, non-anticipative policy; the result implies that the staticrouting policies we consider approach optimality inthe asymptotic regime of many jobs. Moreover, thestatic routing policies we study can be obtainedby solving a convex quadratic optimization problemwith J ×M variables that depends only on the jobprocessing distributions through their expected values.In the special case of uniformly related machines, wealso have established a similar but slightly sharperresult for a simple static routing policy; this policy

Figure 1. (Color online) Simulation Results with M � 4

Notes. Jobs are uniformly distributed in (a) and (c) and are exponentially distributed in (b) and (d). Standard errors ofZH andZH0 are not included

in the plots, but they are negligible relative to the cost. The penalized perfect information bound is almost indistinguishable from theperformance of the policy in (a) and (b).


involves no optimization and simply routes jobs pro-portionally to machine speeds. Finally, we also study theimpact that dependence across job processing timescan have on the suboptimality on the static routingpolicy, and we derive some conditions on the de-pendence under which static routing is still asymp-totically optimal; these conditions are relatively strongand, as we show by example, necessary. The maintechnique underlying our results has been the use ofperfect information relaxations with a properly designedpenalty.

There are a number of interesting directions toconsider moving forward. First, although the problemof stochastic scheduling on unrelated machines that westudy is general in many ways, there are, of course,

many important variations that we do not consider,such as release dates, job precedence constraints, anddifferent objectives (e.g., expected makespan). Thestatic routing policies we study clearly require somemodifications to handle such variations, and in fact,very different policies may be necessary. Nonetheless,we believe the machinery we use for developinglower bounds on optimal performance may be usefulfor other stochastic scheduling problems with dif-ferent features. It would also be interesting to seewhether the information relaxation duality approachwould be useful in other scheduling problems (e.g.,stochastic job shop models). Second, in some practicalapplications of the model, it may be the case that theproblem is too large to solve to optimality yet too

Figure 2. (Color online) Simulation Results with M � ��J

√

Notes. Jobs are uniformly distributed in (a) and (c) and are exponentially distributed in (b) and (d). Standard errors ofZH andZH0 are not included

in the plots, but they are negligible relative to the cost.


small for the static routing policies we studied hereto be nearly optimal. In such situations, it would beinteresting to consider the use of more sophisticatedpolicies and information relaxation lower boundswith more sophisticated penalties. Finally, the per-formance bounds we derived can be a useful buildingblock in analyzing broader “design”-type questions,such as the problem of selecting the best set of ma-chines to process a fixed set of jobs to minimizea combined objective of machine cost plus expectedcompletion time. The implications our results have forvariations on this theme (e.g., scheduling problemsthat include job selection as a decision) may also beinteresting to consider.

AcknowledgmentsThe authors thank their colleague Jim Smith, the area editor,an anonymous associate editor, and two anonymous refereesfor providing a number of helpful comments that greatlyimproved the paper.

Appendix A. ProofsA.1. Proof of Lemma 3.1. This result is established in bothSethuraman and Squillante (1999) and Skutella (2001); weprovide a proof here for completeness. Let Dm � (d

mij )∈RJ×J be

the matrix such that

dmij �

0 if i � j,wiE[pjm], if j3m i,wjE[pim], if i3m j.

⎧⎪⎪⎪⎨⎪⎪⎪⎩The objective in (1) can be written as

∑

m∈}c′mxm + 1

2x′mDmxm �

∑

m∈}c′mxm − 1

2x′mdiag(cm)xm

+ 12x′m Dm + diag(cm)

( )xm ≥

∑

m∈}12c′mxm

+ 12x′m Dm + diag(cm)

( )xm,

where we use the fact that x2jm ≤ xjm because the variablestake values in the interval [0, 1]. It is not hard to check thatthe matrix Dm � Dm + diag(cm) is positive semidefinite,which implies that problem (2) is a convex optimizationproblem.

A.2. Proof of Proposition 4.5. Define the Lagrangian func-tion of problem (2) to be

L(x, ν) � ν′1 +∑M

m�112x′mDmxm + cm

2− ν

( )′xm,

where ν ∈RJ , xm ∈RJ , and x � (xm)m∈} ∈RJ×M. Becauseproblem (2) is convex and all its constraints are linear, strongduality holds. Moreover, let x∗ ∈RJ×M be an optimal solutionto problem (2), and let ν∗ ∈RJ be an optimal Lagrangemultiplier of the assignment constraint

∑m∈}xm � 1 in (2).

Because the optimal objective value is finite, proposition 5.2.1 in

Bertsekas (1999) implies that the optimal solution x∗ minimizesthe Lagrangian evaluated at ν∗, or equivalently,

ZR � L(x∗,ν∗) � minx≥0 L(x, ν∗). (A.1)

Recall that

ZH(p) � minx∈-

∑

m∈}a′mxm + 1

2x′mQmxm,

with vectors am�(12p2jmrjm+ λjm+pjmrjm( )E[pjm]−pjm( )+γjm(E[p2jm]−

p2jm)j∈)∈RJ and matrices Qm�diag(pm) Rmdiag(pm)∈RJ×J . Forease of analysis in the future, we instead consider the followingproblem:

ZH(p) � min

x∈-∑

m∈}a′mxm + 1

2x′mQmxm,

with vectors am� ((λjm−12wj) E[pjm]−pjm

( )+12wjpjm)j∈). Setting

γjm� −12rjm, a direct relationship between ZH(p) and Z

H(p) is

given by

ZH (p) − ZH(p)≤

∑

j∈)maxm∈}

(ajm − ajm) � 12

∑

j∈)maxm∈}

rjmVar[pjm].

(A.2)

We now bound ZH(p) from below. Letw � (wj)j∈) ∈RJ .We set

λm to be

λm � diag E[pm]( )−1

ν∗ + 12w.

As a result, we have

am − ν∗ � −diag(pm)(λm −w). (A.3)

Define new variables xm such that if pjm > 0, then pjmxjm �E[pjm]xjm; otherwise, xjm can be an arbitrary number. Therefore,for every sample path,

ZH(p)≥(i)ν∗′1 +

∑M

m�1minxm≥0

12x′mQmxm + (am − ν∗)′xm

�(ii)ν∗′1 +∑M

m�1minxm≥0

12x′mQmxm − (λm −w)′diag(pm)xm

� ν∗′1 +∑M

m�1minxm≥0

12x′mdiag(pm)Rmdiag(pm)xm

−(λm −w)′diag(pm)xm

≥(iii)ν∗′1 +∑M

m�1minxm≥0

12x′mdiag(E[pm])Rmdiag(E[pm])xm

−(λm −w)′diag(E[pm])xm

� ν∗′1 +∑M

m�1minxm≥0

12x′

mDmxm + cm

2− ν∗

( )′xm

�(iv)ZR, (A.4)

where (i) follows from weak duality, (ii) from (A.3), (iii) fromthe change of variables, and (iv) from (A.1) because the


optimal solution x∗ minimizes the Lagrangian evaluated at ν∗.Combining (A.2) and (A.4), we have for every sample path

VH(p)≥ZH(p)≥ZR − 12

∑

j∈)wj max

m∈}Var[pjm]E[pjm]

.

A.3. Proof of Corollary 4.2. Clearly, the ratio is non-negative. Here, we bound the ratio from above and thenshow that the upper bound converges to zero as the totalnumber of jobs J goes to infinity. From Theorem 4.1, the dif-ference between VR and V∗ can be bounded from above by

VR − V∗ ≤ M(Δ + 1) − 12M

wpJ. (A.5)

On the other hand, according to Proposition 4.5, theoptimal performance V∗ can be bounded from below by

V∗ ≥ZR − Δ

2

∑

j∈)wj max

m∈}E[pjm]≥ZR − Δ

2w pJ. (A.6)

Comparing with (2), ZR can further be bounded from belowby ZR, which is the optimal value of the following convexoptimization problem:

ZR ≜ minx∈-

∑

m∈}12c ′xm + 1

2x′m Dxm, (A.7)

with c � (wp)j∈) ∈RJ , and D � (d ij) ∈RJ×J the matrix suchthat d ij � w p . The optimal value of (A.7) provides a lowerbound on ZR because xjm ≥ 0, c j ≤ cjm, and d ij ≤ dmij . Becausethis new problem is symmetric across machines, the optimalsolution of (A.7) is xjm � 1

M (this can be verified easily bychecking the Karush–Kuhn–Tucker (KKT) conditions of(A.7)). As a result,

ZR ≥Z R � 12w p

J2

M+ J

( ). (A.8)

Combining (A.6) and (A.8), we obtain that

V∗ ≥ 12wp

J2

M− Δ

2w pJ. (A.9)

The right-hand side of (A.9) is positive when J is large enough.Finally, combining (A.5) and (A.9) together with the fact that

M/J �→J→∞ 0 yieldsVR − V∗

V∗ �→J→∞0.

A.4. Proof of Theorem 5.1. Here, we use the same formfor the sequencing and routing penalties as in the proofof Theorem 4.1, but we use different values for the parameters(λjm

)j∈),m∈} in the routing penalty.

Let c � (wjE[pj])j∈) ∈RJ and D � (dij)∈RJ×J be the matrixsuch that

dij �wiE[pi], if i � j,wiE[pj], if j3i,wjE[pi], if i3j.

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Then cm � 1smc andDm � 1

smD because machines are uniformly

related. Set the objective function ZR(x) to be

ZR(x) �

∑

m∈}12c′xm + 1

2x′mDmxm.

It is easy to check that the performance of the static routingpolicy with routing x ∈- satisfies

VR(x) � ZR(x) +∑

m∈}12(cm − c)′xm

+ 12

∑

j∈)wj

∑

m∈}xjm(1 − xjm)E[pjm]. (A.10)

Now consider the optimization problem:

ZR � minx∈-

ZR(x). (A.11)

Because problem (A.11) is convex and all its constraints arelinear, according to the KKT conditions, there exists vectorsx∗m ∈RJ , ν∗ ∈RJ , and µ∗

m ∈RJ such that∑

m∈}x∗m � 1,

x∗m ≥ 0,

µ∗m ≥ 0,

Dmx∗m + c2� ν∗ + µ∗

m,

x∗m′µ∗m � 0,

with ν∗ and µ∗m being the duals corresponding to the nor-

malization and nonnegativity constraints of (A.11), respec-tively. Furthermore, x∗m is an optimal solution of (A.11), and ν∗and µ∗

m comprise an optimal solution of the dual problem of(A.11). By checking the KKT conditions, it is easy to verify thatthe speed-proportional routing x∗jm � sm/S is an optimal so-lution of (A.11). The performance VS of speed-proportionalrouting satisfies VS � VR(x∗). According to (A.10)

VS � ZR + 2M − S − 12S

∑

j∈)wjE[pj]. (A.12)

Now set ZH � E[ZH(p)], where

ZH(p) � minx∈-

∑

m∈}a′mxm + 1

2x′mQmxm,

with vectors am�(12p2jmrjm+(λjm+pjmrjm)(E[pjm]−pjm)+γjm(E[p2jm]−p2jm)j∈)∈RJ and matrices Qm�diag(pm) Rmdiag(pm)∈RJ×J . As inSection 4.2, ZH is a lower bound on V∗. Consider the followingproblem:

ZH(p) � min

x∈-∑

m∈}a′mxm + 1

2x′mQmxm,

with vectors am � ((λjm−wj) E[pjm]−pjm( )+1

2cj)j∈). Setting γjm�

−12rjm, we have

ajm − ajm � 12(cj − cjm) + 1

2rjmVar[pjm]

� 12wjE[pj] 1 + 1

sm

Var[pj]

E[pj]2− 1

( )[ ].


Therefore, a direct relation betweenZH(p) and ZH(p) is given by

ZH(p) − ZH(p)≤

∑

j∈)maxm∈}

(ajm − ajm)

� 12

∑

j∈)wjE[pj]max

m∈}1 + 1

sm

Var[pj]

E[pj]2− 1

( )[ ].

Hence, letting ZH � E[ZH

(p)], we have

ZH − ZH ≤ 1

2

∑

j∈)wjE[pj]max

m∈}1 + 1

sm

Var[pj]

E[pj]2− 1

( )[ ]. (A.13)

Set the penalty λm to be λjm � wj. Similarly, it is easy to verifyby using the KKT conditions that the speed-proportional

routing x∗jm � sm/S is an optimal solution of ZH(p) for every

sample path p. As a result,

E[ZH(p)] � E

∑

m∈}a′mx

∗m + 1

2x∗m

′Qmx∗m

[ ]�

∑

m∈}12c′x∗m + 1

2x∗m

′Dmx∗m

+∑

m∈}12x∗m

′ E[Qm] −Dm( )x∗m� Z

R + 12

∑

m∈}

∑

j∈)

wjE[pj]sm

Var[pj]

E[pj]2(x∗jm)2

� ZR + 1

2S

∑

j∈)wjE[pj]

Var[pj]

E[pj]2. (A.14)

Let Δj ≜Var[pj]/E[pj]2. Combining (A.12), (A.13), and (A.14)we have

V∗ ≤VS ≤V∗ + 12

∑

j∈)wjE[pj]

{1S

2M − 1 − Δj( )

+maxm∈}

1sm

(Δj − 1)[ ]}

� V∗ + 12

∑

j∈)wjE[pj]

2M − 1S

− 1κj

( )+ 1

κj− 1S

( )Δj

{ },

where κj � sM if Δj > 1 and κj � s1 if Δj ≤ 1.

A.5. Proof of Proposition 5.2. We first develop some validinequalities for all feasible starting times in deterministic sched-uling on uniformly related machines; this result is analogous tolemma 9 in Schulz (1996).

Lemma A.1. If machines are uniformly related, and processingtimes are deterministic and denoted by pj, then the startingtimes Sj of jobs for all feasible scheduling policies satisfy, for all!⊆),

∑

j∈!pjSj ≥ − 1

21sM

− 1S

( )∑j∈!

p2j +12S

∑

i,j∈!,i≠jpipj

( ). (A.15)

Proof. Let!m ⊆! be the set of jobs processed on machinem.From Schulz et al. (1996), the completion times for all feasibleschedules on uniformly related machines satisfy

∑

j∈!pjCj �

∑M

m�1sm

∑

j ∈!m

pjsm

Cj ≥∑M

m�1sm2

(∑

j ∈!m

pjsm

( )2+

∑

j∈!m

pjsm

( )2)

�∑M

m�112sm

(∑

j∈!m

p2j +∑

j∈!m

pj

( )2),

(A.16)

where the inequality follows from the single machine case.Because Sj � Cj − pj

smfor all j ∈!m,

∑

j∈!pjSj�

∑

j∈!pjCj −

∑M

m�1

∑

j∈!m

p2jsm

≥(i)∑M

m�112sm

(∑

j∈!m

p2j +∑

j∈!m

pj

( )2)−∑M

m�11sm

∑

j ∈!m

p2j

� −∑M

m�112sm

∑

j ∈!m

p2j +∑M

m�112sm

∑

j ∈!m

pj

( )2

≥(ii) −∑M

m�112sm

∑

j ∈!m

p2j +12S

∑

j∈!pj

( )2

≥ − 12sM

∑

j∈!p2j +

12S

∑

j∈!pj

( )2

� −12

1sM

− 1S

( )∑j∈!

p2j +12S

∑

i,j∈!,i≠jpipj

( ),

where (i) is due to (A.16) and (ii) follows from the inequality∑M

m�1x2msm

≥ (∑Mm�1xm)2

∑Mm�1sm

for any nonnegative numbers {xm}Mm�1

and {sm}Mm�1, which can be proved easily by induction. □

Using Lemma A.1, we now prove Proposition 5.2. Be-cause pj and Sj are independent for any nonanticipativepolicy, and pj and pi are independent for all i≠ j, takingexpectations on both sides of (A.15) gives

∑

j∈!E[pj]E[Sj]≥ − 1

21sM

− 1S

( )∑j∈!

E[p2j ] +12S

∑

i,j∈!,i≠jE[pi]E[pj])

(

� 12S

∑

j∈!E[pj])2 − 1

21sM

− 1S

( )∑j∈!

Var[pj]

(− 12sM

∑

j∈!E[pj]2. (A.17)

Using the fact thatE[Cj]≥E[Sj] + E[pj]s1

and (A.17), we obtain (7).

Endnotes1Problem (1) can be formulated as a binary linear program.We showthis formulation in Online Appendix D.


2Brown et al. (2010) define dual feasible penalties more generallywith E[Yπ]≤ 0 for all π∈Π, although in most applications of theapproach, including here, a penalty has “no slack” in that equalityholds in this condition.

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Santiago R. Balseiro is an assistant professor in the De-cision, Risk, and Operations Division at the Graduate Schoolof Business, Columbia University. His primary research in-terests are in the areas of dynamic optimization, stochasticsystems, and game theory with applications in revenuemanagement and internet advertising.

David B. Brown is an associate professor in the DecisionSciences area at the Fuqua School of Business at Duke Uni-versity. His research focuses on developing and analyzingapproximation methods for large-scale stochastic optimiza-tion problems, with an emphasis on applications in opera-tions and finance.

Chen Chen is a PhD student in the Decision Sciences areaat the Fuqua School of Business at Duke University. He isinterested in the analysis and design of approximation al-gorithms for dynamic optimization problems.


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