Analysis of an infinite-buffer batch-size-dependent service queue with discrete-time Markovian arrival process: D-$MAP/G_n^{(a,b)}/1$

TL;DR

This paper models an infinite-buffer discrete-time queue with batch arrivals and size-dependent service times, deriving joint distributions and performance metrics using advanced mathematical techniques for applications in digital communication systems.

Contribution

It introduces a comprehensive mathematical framework for analyzing batch-service queues with size-dependent service times using the supplementary variable technique.

Findings

Derived the joint distribution of queue and server content.

Established relationships between distributions at different epochs.

Provided numerical examples illustrating the system's performance measures.

Abstract

Discrete-time queueing models find huge applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems, computer networks, etc. 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 epoch, which 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 a few numerical examples.