A simple derivation of the waiting-time distribution (in the queue) for the bulk-service $$M/G^{(a,b)}/1$$ queueing system

This paper deals with a Poisson input infinite-buffer single-server queue, where the arrivals occur in singles and the server serves the customers in batches. The server serves customers in batches of maximum size “b” with a minimum threshold size “a”. The service time of each batch follows general distribution (including heavy-tailed distribution) independent of each other as well as of the arrival process. The probability generating function (pgf) of the queue-length distributions at an arbitrary epoch as well as at a post-departure epoch of a batch have been derived using the embedded Markov chain and the argument of the rate-in and rate-out principle. The Laplace-Stieltjes transform (LST) of the actual waiting-time distribution (in the queue) of a random customer has also been derived using functional relation between pgf’s. The proposed analysis is based on the roots of the characteristic equation associated with the LST of the waiting-time distribution (in the queue) of a random customer. Using LSTs, the closed-form expressions for the probability density functions and for an arbitrary number of moments of the waiting-time distributions have been presented. We have also done numerical implementation of this procedure for the case of a bulk service infinite-buffer queueing model, and obtained the probability density function for waiting-time distribution of a random customer in the queue.