Stochastic comparisons for single‐server queues

Carl M. Harrissp and N. U. Prabhu

Naval Research Logistics (NRL), 1987, vol. 34, issue 4, 555-567

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.

Date: 1987
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https://doi.org/10.1002/1520-6750(198708)34:43.0.CO;2-7

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