Combinatorial approach to Markovian queueing models

Abstract

Some of the known results in transient behaviour of the Markovian queueing models such as the distribution of X(r)-number of units in the system at time t, joint distribution of length of the busy period and the number of units served, and the maximum length during a busy period have been derived using the combinatoric analysis of lattice paths representing the queueing process thus unifying the earlier results on transient solution of Markovian queueing models. AMS Subject Classification: Primary 60k25; secondary 6OCO5. Considerable attention has been paid to obtain the transient solution for the system size and the distribution for the length of a busy period of an M/M/l queue- ing system. A number of methods have been put forward and most of them require setting up of differential difference equations. Champernowne (1956) has used the combinatorial method for the first time for the same problem. Takacs (1962, 1967) adopted combinatorial approach which involved the use of celebrated ballot theorem to study sequences of inter-arrival and service times. Mohanty (1979) in- vestigated some results treating queueing system as a random walk on the lattice in the plane and then used generating functions. Neuts (1964) developed a new ap- proach avoiding the use of generating function and Rouche's theorem. Recently Mohanty and Panny (1990) obtained transient results by employing the technique of first discretizing the continuous time model and then representing the same by a random walk path. In this paper, we have rederived transient solutions by first discretizing the model and then using the combinatorics through the so called lattice path method. The classical transient results for M/M/l queueing system provided little insight into the behaviour of the queueing system through a fixed operation