Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traffic with Resource Pooling: Asy

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

[2]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[3]  J. Harrison.  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]  William P. Peterson.  A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types , 1991, Math. Oper. Res..

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

[12]  Lawrence M. Wein,et al.  Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network with Controllable Inputs , 2011, Oper. Res..

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

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

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

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

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

[18]  Maury Bramson,et al.  Convergence to equilibria for fluid models of FIFO queueing networks , 1996, Queueing Syst. Theory Appl..

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

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

[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]  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]  Sunil Kumar.  Two-server closed networks in heavy traffic: diffusion limits and asymptotic optimality , 2000 .

[30]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

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