The Gated-Service M/GI/1 Queue with Single Vacations and Its Application to Batch-Service Queues

TL;DR

This paper analyzes a gated-service M/GI/1 queue with single vacations, deriving key transforms for queue length and delay, and applies findings to a batch-service queue with size-dependent service times.

Contribution

It provides new analytical results for the stationary distribution and delay in a gated-service queue with vacations, and extends to batch-service applications.

Findings

Derived the probability generating function of queue length.

Obtained the Laplace-Stieltjes transform of system delay.

Analyzed the joint transform of busy cycle length and served customers.

Abstract

In this paper, we consider the M/GI/1 queue with single vacations under the gated service discipline. We obtain the probability generating function of the stationary queue length, the Laplace-Stieltjes transform of the system delay distribution in steady state, and the joint transform of the busy cycle length and the number of customers served in the busy cycle. Furthermore, as an application, we consider a batch-service M/G/1 queue, where service times depend on the number of customers in batch.