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

¹ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
² Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur, Nagpur 440010, India
³ Department of Mathematics, Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Hyderabad 500078, India
⁴ Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston, ON K7K7B4, Canada

^* Corresponding author: umesh@maths.iitkgp.ac.in

Received: 19 August 2020
Accepted: 5 April 2021

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.