Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-MAP/G_n^(a,b)/1

Abstract

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer's epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Stationary queue and server content distribution of a vacation queue with N -policy and set up time

“…Nandy and Pradhan 10,11 have discussed infinite-buffer queues with the inclusion of batch-size-dependent service and also provided a complete joint distribution at different epochs. Gupta et al 12 have analyzed a discrete-time queue with correlated arrival in which they have provided a complete results of joint distribution at different epochs. In above five research articles, no one has incorporated 𝑁policy and set up time together in their works.…”