It has been shown by Sharma and Tarabia  that a power series technique can be successfully applied to derive the transient solution for an empty M/M/1/N queueing system. In this paper, we further illustrate how this technique can be used to extend  solution to allow for an arbitrary number of initial customers in the system. Moreover, from this, other more commonly sought results such as the transient solution of a nonempty M/M/1/∞ queue can be computed easily. The emphasis in this paper is theoretical but numerical assessment of operational consequences is also given.
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"In general, such problems tend to be intractable or provide solutions so complicated as to be of little practical use. A technique expounded in   and later in   is known as the series approach or randomization. This approach has proved successful in modeling several queueing situations and random walks. "
[Show abstract][Hide abstract] ABSTRACT: Recently Tarabia and El-Baz [A.M.K. Tarabia, A.H. El-Baz, Transient
solution of a random walk with chemical rule, Physica A 382 (2007)
430-438] have obtained the transient distribution for an infinite random
walk moving on the integers -[infinity]<k<[infinity] of the real
line. In this paper, a similar technique is used to derive new elegant
explicit expressions for the first passage time and the transient state
distributions of a semi-infinite random walk having "chemical" rule and
in the presence of an absorbing barrier at state zero. The walker
starting initially at any arbitrary positive integer position i,i>0.
In random walk terminology, the busy period concerns the first passage
time to zero. This relation of these walks to queuing problems is
pointed out and the distributions of the queue length in the system and
the first passage time (busy period) are derived. As special cases of
our result, the Conolly et al. [B.W. Conolly, P.R. Parthasarathy, S.
Dharmaraja, A chemical queue, Math. Sci. 22 (1997) 83-91] solution and
the probability density function (PDF) of the busy period for the
M/M/1/[infinity] queue are easily obtained. Finally, numerical values
are given to illustrate the efficiency and effectiveness of the proposed
Full-text · Article · Mar 2009 · Journal of Computational and Applied Mathematics
[Show abstract][Hide abstract] ABSTRACT: Conolly et al. [Math. Scientist 22 (1997) 83-91] have obtained the transient distribution for a random walk moving on the integers -infinity < k <infinity of the real line. Their analysis is based on a generating function technique. In this paper, an alternative technique is used to derive elegant explicit expressions for the transient state distribution of an infinite random walk having "chemical" rule and starting initially at any arbitrary integer position (say i). As a special case of our result, Conolly et al.'s (1997) solution is easily obtained. Moreover, the transient solution of the infinite symmetric continuous random walk is also presented. Finally, numerical values testing the quality of our analytical results are illustrated.
No preview · Article · Aug 2007 · Physica A: Statistical Mechanics and its Applications
[Show abstract][Hide abstract] ABSTRACT: We find combinatorially the probability of having n customers in an M/M/1/c queueing system at an arbitrary time t when the arrival rate λ and the service rate µ are equal, including the case c = ∞. Our method uses path counting methods and finds a bijection between the paths of the type needed for the queueing model and paths of another type which are easy to count. The bijection involves some interesting geometric methods.
Preview · Article · Sep 2008 · International Journal of Mathematics in Operational Research