Necessary and Sufficient Conditions for Delay Moments in FIFO Multiserver Queues with an Application Comparing s Slow Servers with One Fast One

Abstract

In this paper we establish the weakest known sufficient conditions for the existence of stationary delay moments in FIFO GI/GI/s queues, for s≥2. These conditions involve not only the service time distribution, as in the classic Kiefer and Wolfowitz conditions, but also the interplay of the traffic intensity and the number of servers in the queue. We then prove the necessity of our conditions for a large class of service times having finite first, but infinite αth, moment for some finite α. Such service time distributions include many, but not all of, the class of heavy-tailed distributions: The Pareto and Cauchy are members; the Weibull is not. Our results are then applied to provide one answer to the classic question: When are s slow servers (operating at rate 1/s) better than one fast server (operating at rate 1)? We consider this question with respect to the rate of decay of the tail of stationary customer delay. In a system characterized by service times that have finite mean but lack some higher moments, such as are often used to model telecommunications traffic, for s greater than a traffic-related constant, the answer is always. Our results help to quantify the benefits of extra servers, while also pointing the way towards the derivation of bounds and asymptotics for the stationary delay distribution of multiserver queues having these types of service times. Such queues are attracting a great deal of academic research, motivated by their practical use modeling telecommunications systems.

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