M/M/3/3 and M/M/4/4 retrial queues

Abstract

This paper studies M/M/$c$/$c$ retrial queues, where $c$ servers areall identical. In the retrial queues, an arriving customer is servedimmediately if it finds an idle server upon arrival, otherwise thecustomer tries to enter the system after an exponentially distributedtime independently of other customers. As is well known, it is achallenging problem to obtain an analytical solution for the stationaryjoint distribution of the numbers of retrial customers and busyservers in the M/M/$c$/$c$ retrial queue especially for $c \ge3$. Under some technical assumptions, a few analytical solutions havebeen presented for $c \ge 3$. This paper derives analytical solutionsfor M/M/3/3 and M/M/4/4 retrial queues without such technicalassumptions. Through many numerical examples, we show that the derivedanalytical solutions can be computed by a numerically stablealgorithm.

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