Studia Scientiarium Mathematicarum Hungarica 34. (1998)

4. szám - Sen K.-Gupta R.: Time dependent analysis of T-policy M/M/1 queues - a new approach

Studia Scientiarum Mathematicarum Hungarica 34 (1998), 453-473 TIME DEPENDENT ANALYSIS OF T-POLICY M/M/l QUEUES — A NEW APPROACH KANWAR SEN and RITU GUPTA Abstract This paper demonstrates a simple and elegant lattice path combinatoric technique for computing transient probabilities concerning M/M/l queueing models. Through this lattice path approach time dependent analysis of T-policy M/M/l queue is presented. The transient probabilities computed herein are free from modified Bessel function and are amenable to pragmatic probabilistic interpretations. As a special case the results for ordinary M/M/l queues are checked. 1. Introduction Consider a T-policy M/M/l queueing model which activates the server T time units after the end of a busy period to determine if customers are present. If no customers are found when the server scans the queue, it is turned off, and the system is scanned again after an interval of length T. This procedure is repeated until the server finds at least one customer waiting, after which the server is kept in active state until the system becomes empty. This model can also be viewed as one where the server takes a sequence of vacations each of duration T, at the end of busy period (see Doshi [4]). Henceforth T-policy M/M/l queueing model will be referred to as M/M/l(T). Different aspects of T-policy queues were studied by Heyman [8] (see also Teghem [21], Takagi [20]). However, little effort was made to find the tran­sient solution of this model (see [20]). As opposed to classical method which entails formulation of tedious unwieldy difference-differential equations, in this paper the lattice path approach — a new combinatorial technique is adopted for studying transient behaviour of M/M/l(T) queues. Starting initially with k (^ 0) units, the probability of i arrivals and j departures up to time t is found for the M/M/l (T) queue. This probability in turn leads to the probability of the number of units in the system up to time t. Over the years combinatorial techniques have been successfully employed in solving queueing problems (refer to Takács [18], [19]). Recently, using lattice path combinatorics Mohanty and Panny [15], Böhm and Mohanty [2], Kanwar 1991 Mathematics Subject Classification. Primary 60K25; Secondary 60J15. Key words and phrases. T-policy, vacation period, busy period, M/M/l(T) model, discretized M/M/1{T) model, slot, lattice path, transient probability. 0081-6906/98/$ 5.00 ©1998 Akadémiai Kiadó, Budapest

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