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Stochastic comparisons for single‐server queues

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  • Carl M. Harrissp
  • N. U. Prabhu

Abstract

We consider single‐server queueing systems with the queue discipline “first come, first served,” interarrival times {uk, k ≥ l}, and service times {uk, k ≥ l}, where the {uk} and {uk} are independent sequences of non‐negative random variables that are independently but not necessarily identically distributed. Let Xk = uk − uk (k ≥ 1), S0 0, Sn = X1 + X2 … + Xn(n≥1). It is known that the (possibly nonhomogeneous) random walk {Sn} determines the behavior of the system. In this paper we make stochastic comparisons of two such systems σ1,σ2 whose basic random variables X (1)k and X (1)k are stochastically ordered. The corresponding random walks are also similarly ordered, and this leads to stochastic comparisons of idle times, duration of busy period and busy cycles, number of customers served during a busy period, and output from the system. In the classical case of identical distributions of {uk} and {uk} we obtain further comparisons. Our results are for the transient behavior of the systems, not merely for steady state.

Suggested Citation

  • Carl M. Harrissp & N. U. Prabhu, 1987. "Stochastic comparisons for single‐server queues," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(4), pages 555-567, August.
    • Handle: RePEc:wly:navres:v:34:y:1987:i:4:p:555-567
    DOI: 10.1002/1520-6750(198708)34:43.0.CO;2-7
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    1. Douglas R. Miller, 1979. "Almost Sure Comparisons of Renewal Processes and Poisson Processes, with Application to Reliability Theory," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 406-413, November.
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