Lattice path approach for busy period density of GIa/Gb/1GIa/Gb/1 queues using C2C2 Coxian distributions

In this paper busy period analysis of non-Markovian queueing system GIa/Gb/1GIa/Gb/1, starting initially with i0i0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C2C2, 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,ab,a and b   are co-prime. Distributions having rational Laplace–Stieltjes transform and square coefficient of variation lying in [1/2,∞)[1/2,∞) form a very wide class of distributions. As any distribution of this class can be approximated by a C2C2, the use of C2C2, therefore, has led us to achieve results applicable to almost any real life queueing system GIa/Gb/1GIa/Gb/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.