Reversibility of Tandem Blocking Queueing Systems

Abstract

This paper is concerned with queueing systems of several service stations in series in which each station may consist of multi-servers. An infinite number of customers always waits in front of the first station, and each customer passes through all of the stations in sequence. There is only a finite number of waiting positions between any two adjacent stations. The service time for a customer at any station is assumed to be a random variable, the distribution of which may depend on the station. In this mode of operation the servers at any station will at any time be busy, idle, or blocked. This blocking system is said to be C-reversible if the capacity remains invariant under reversal of the system. The reversed system is obtained by reversing the original stations' order, that is, every customer in the reversed system passes through the original stations in the reverse order. It has already been proved that C-reversibility holds for any blocking system in which each station consists of either a single server of nondeterministic service times or multi-servers of deterministic service times, and that the blocking system has a stronger property than C-reversibility. In this paper we show that two-station blocking systems with multi-server stations of nondeterministic service times are C-reversible, but this property can no longer be extended to three or more station blocking systems with multi-server stations of nondeterministic service times. We also show for the case of multi-server stations of nondeterministic service times that the stronger property which involves invariance of distribution does not hold even for two-station blocking systems.

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