A Simple and Extended Computational Analysis of M/Gj(a,b)/1 and M/Gj(a,b)/1/(B + b) Queues Using Roots

Abstract

A recent article on bulk-service queues gives a generating function for the steady-state probabilities of the embedded Markov chain for a single-server infinite-space system in which, the customers arrive according to a Poisson process and are served in batches with quorum a and capacity b, and the service time follows a general distribution dependent on batch size. This system is equivalent to the bulk queueing model M/Gj(a,b)/1, whose general solution requires finding the roots of the denominator of the underlying generating function. The article claims that the use of roots may result in numerical inaccuracies, especially for large values of b. Hence it only solves for the finite-space model M/Gj(a,b)/1/(B + b) using (B+1) simultaneous linear equations. We present a simple way to obtain the probability distribution of queue length at post-departure epochs for the infinite-space model M/Gj(a,b)/1 using roots, then an alternative method to solve the finite-space queue M/Gj(a,b)/1/(B+b). We derive, for the first time, closed-form formulas for the queue-length distribution of models with deterministic service time for both infinite (M/Dj(a,b)/1) and finite-space (M/Dj(a,b)/1/(B+b)) systems. We also show that the queue-length distribution of (M/Dj(a,b)/1 can be approximated by a Poisson distribution when the traffic intensity p is low. Numerical results are both tabulated and graphed.