Lattice path approach for busy period density of [formula omitted] queues using [formula omitted] Coxian distributions

Abstract

In this paper busy period analysis of non-Markovian queueing system GI a / G b / 1 , starting initially with i 0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C 2 , that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Arrivals occur in batches of size a and services occur in batches of size b , a and b are co-prime. Distributions having rational Laplace–Stieltjes transform and square coefficient of variation lying in [ 1 / 2 , ∞ ) form a very wide class of distributions. As any distribution of this class can be approximated by a C 2 , the use of C 2 , therefore, has led us to achieve results applicable to almost any real life queueing system GI a / G b / 1 occurring in computer systems, communication systems, manufacturing systems, etc. Numerical computations have been performed for different sets of values of the parameters involved using software Mathematica and presented graphically.