A note on the waiting-time distribution in an infinite-buffer \(\text{GI}^{[X]}/ \text{C-MSP} / 1\) queueing system (Q1733133)

Summary: This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (\(\text{C-MSP}\)). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the ``\(\mathbf{R}\)'' matrix of the corresponding \(\mathrm{GI}/\mathrm{M}/ 1\)-type Markov chain embedded at a prearrival epoch of a batch.