Modeling and analysis of a discrete-time group-arrival and batch-size-dependent service queue with single and multiple vacation

TL;DR

This paper analyzes a discrete-time queueing system with batch arrivals and variable service times, considering single and multiple vacation policies, deriving key distributions and applying phase-type distributions for transmission error management.

Contribution

It introduces a comprehensive analysis of a batch-service queue with vacation policies, deriving the joint distributions and incorporating phase-type distributions for realistic modeling.

Findings

Derived the bivariate probability generating function of queue length and server content.

Obtained joint distributions at arbitrary and pre-arrival times.

Numerical illustrations demonstrate the impact of phase-type service times.

Abstract

Discrete-time queueing system has widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the probabilities in terms of roots of the characteristic equation. We also find the joint distribution at arbitrary and pre-arrival slot. Discrete phase type distribution plays a noteworthy role in order to regulate the high transmission error through a particular channel. In view of this, numerical illustrations include the service time distribution to follow the discrete phase type distribution.