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Asymptotically Optimal Routing and Servive Rate Allocation in a Multiserver Queueing System

Author

Listed:
  • J. George Shanthikumar

    (University of California at Berkeley, Berkeley, California)

  • Susan H. Xu

    (The Pennsylvania State University, University Park, Pennsylvania)

Abstract

We consider a single stage queueing system with c heterogeneous servers. Customers arrive at this system according to a renewal process with mean 1/λ and squared coefficient of variation (scv) C a 2 . An incoming customer is routed to server i with probability θ i , ∑ i =1 c θ i = 1. The service times at server i are i.i.d random variables with mean 1/μ i , and scv C s i 2 . The holding cost rate of queue i is h i per customer, i = 1, 2, …, c . The problems of interest are twofold: (a) for a fixed service rate allocation μ i , ∑ i =1 c μ i = μ, find the routing probabilities, θ i *, ∑ i =1 c θ i * = 1, that minimize the average total holding cost; and (b) for fixed routing probabilities θ i , ∑ i =1 c θ i , and total service rate μ, find the service rate allocation μ i * = μδ i *, ∑ i =1 c δ i * = 1, that minimizes the average total holding cost of the system. For each problem, we characterize the optimal policy under heavy traffic conditions. We also derive the routing probabilities, θ̂ i (proportions δ̂ i ), i = 1, …, c , that are strongly asymptotically optimal. That is, the difference between the average total holding costs under θ̂ i , i = 1, …, c , and θ i *, i = 1, …, c (δ̂ i , i = 1, …, c , and δ i *, i = 1, …, c ) is bounded by a fixed constant independent of the routing probabilities (proportions) and the arrival rate. In addition, we discuss the necessity and sufficiency of the accurate knowledge of the means and scvs of the interarrival and service times m obtaining asymptotically optimal policies.

Suggested Citation

  • J. George Shanthikumar & Susan H. Xu, 1997. "Asymptotically Optimal Routing and Servive Rate Allocation in a Multiserver Queueing System," Operations Research, INFORMS, vol. 45(3), pages 464-469, June.
  • Handle: RePEc:inm:oropre:v:45:y:1997:i:3:p:464-469
    DOI: 10.1287/opre.45.3.464
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    Citations

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    Cited by:

    1. Amir Ahmadi-Javid & Mohammadreza Fathi, 2022. "Design of multi-service systems with facilities functioning as open Jackson queueing networks: application to online shopping stores," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 44(4), pages 1255-1286, December.
    2. Cathy H. Xia & George J. Shanthikumar & Peter W. Glynn, 2000. "On the Asymptotic Optimality of the SPT Rule for the Flow Shop Average Completion Time Problem," Operations Research, INFORMS, vol. 48(4), pages 615-622, August.
    3. Kimberly M. Wasserman & Tava Lennon Olsen, 2001. "On Mutually Interfering Parallel Servers Subject to External Disturbances," Operations Research, INFORMS, vol. 49(5), pages 700-709, October.

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