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The longer queue model. (English) Zbl 1134.60395
Summary: Two queues forming two independent Poisson processes are served by one server with exponential service time. The server always works on the longer queue and, in case that they are of equal length, chooses either one with probability \({1\over 2}\). Let \(\pi_{ij}\) be the probability that the two queue lengths equal \(i\) and \(j\) at equilibrium and \(\Pi(z, w)= \sum\pi_{ij}z^i w^j\). We determine \(\Pi(z, w)\) and derive from this asymptotic formulas for \(\pi_{ij}\) as \(i,j\to\infty\). These asymptotic formulas are used to study the interdependence of the queue lengths. In particular, we obtain limit laws for the queue lengths conditioned on each other.

MSC:
60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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