Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traac with Complete Resource Pooling: Asymptotic Optimality of a Continuous Review Threshold Policy 1

rights reserved. Personal scientiic non-commercial use only for individuals with permission from the authors (williams@math.ucsd.edu). Abstract This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to innnite capacity buuers and two (non-identical) servers working in parallel. One server can only process jobs from one buuer, whereas the other server can process jobs from either buuer. The service time distribution may depend on the buuer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisses both a heavy traac condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traac scaling) queue length. We rst review the analytic solution of the Brownian control problem (formal heavy traac approximation) for this system. We \interpret" this solution by proposing a \continuous review" threshold control policy for use in the original parallel server system. We show that this policy is asymptotically optimal in the heavy traac limit and the limiting cost is the same as the optimal cost in the Brownian control problem. The techniques developed here are expected to be useful for analyzing the performance of threshold-type policies in more complex multiserver systems. Short title: Dynamic scheduling of two parallel servers.

[1]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[2]  D. Iglehart,et al.  The Equivalence of Functional Central Limit Theorems for Counting Processes and Associated Partial Sums , 1971 .

[3]  J. Harrison,et al.  Brownian motion and stochastic flow systems , 1986 .

[4]  J. Michael Harrison,et al.  Brownian Models of Queueing Networks with Heterogeneous Customer Populations , 1988 .

[5]  Lawrence M. Wein,et al.  Scheduling networks of queues: Heavy traffic analysis of a simple open network , 1989, Queueing Syst. Theory Appl..

[6]  Lawrence M. Wein,et al.  Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Network with Controllable Inputs , 1990, Oper. Res..

[7]  Lawrence M. Wein,et al.  Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Closed Network , 1990, Oper. Res..

[8]  C. N. Laws,et al.  Dynamic Scheduling of a Four-Station Queueing Network , 1990, Probability in the Engineering and Informational Sciences.

[9]  L. F. Martins,et al.  Routing and singular control for queueing networks in heavy traffic , 1990 .

[10]  C. Laws.  Resource pooling in queueing networks with dynamic routing , 1992, Advances in Applied Probability.

[11]  F. P. Kelly,et al.  Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling , 1993, Queueing Syst. Theory Appl..

[12]  Lawrence M. Wein,et al.  Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Closed Network , 1993, Oper. Res..

[13]  J. Quadrat.  Numerical methods for stochastic control problems in continuous time , 1994 .

[14]  William C. Jordan,et al.  Principles on the benefits of manufacturing process flexibility , 1995 .

[15]  J. V. Mieghem.  Dynamic Scheduling with Convex Delay Costs: The Generalized CU Rule , 1995 .

[16]  L. F. Martins,et al.  Heavy Traffic Convergence of a Controlled, Multiclass Queueing System , 1996 .

[17]  L. F. Martins,et al.  Heavy Traffic Analysis of a Controlled Multiclass Queueing Network via Weak Convergence Methods , 1996 .

[18]  Jan A. Van Mieghem,et al.  Dynamic Control of Brownian Networks: State Space Collapse and Equivalent Workload Formulations , 1997 .

[19]  Anatolii A. Puhalskii,et al.  A critically loaded multirate link with trunk reservation , 1998, Queueing Syst. Theory Appl..

[20]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[21]  Ruth J. Williams,et al.  An invariance principle for semimartingale reflecting Brownian motions in an orthant , 1998, Queueing Syst. Theory Appl..

[22]  Isi Mitrani,et al.  Probabilistic Modelling , 1998 .

[23]  Ruth J. Williams,et al.  Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse , 1998, Queueing Syst. Theory Appl..

[24]  J. Harrison.  Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete-review policies , 1998 .

[25]  James Williams.  On Dynamic Scheduling of a Parallel Server System With Complete Resource Pooling , 1999 .

[26]  J. Michael Harrison,et al.  Heavy traffic resource pooling in parallel‐server systems , 1999, Queueing Syst. Theory Appl..

[27]  Two-Server Closed Networks in Heavy TraÆ : Di usion Limits and Asymptoti Optimality , 1999 .

[28]  H. Kushner,et al.  Optimal control of assignment of jobs to processors under heavy traffic , 2000 .

[29]  J. Harrison.  Brownian models of open processing networks: canonical representation of workload , 2000 .