# Steady-state analysis of single exponential vacation in a $PH/MSP/1/\infty$ queue using roots

## TL;DR

This paper analyzes a single-server queue with phase-type arrivals, Markovian service, and exponential vacations, deriving steady-state distributions and performance measures using root-based methods, with approximations and numerical insights.

## Contribution

It introduces a root-based analytical approach for steady-state analysis of PH/MSP/1 queues with vacations, including new approximations and numerical evaluations.

## Key findings

- Derived explicit steady-state distributions and performance metrics.
- Developed heavy- and light-traffic approximations.
- Provided numerical results illustrating parameter effects.

## Abstract

We consider an infinite-buffer single-server queue where inter-arrival times are phase-type ($PH$), the service is provided according to Markovian service process $(MSP)$, and the server may take single, exponentially distributed vacations when the queue is empty. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function (VGF) of system-length distribution at a pre-arrival epoch. Also, we obtain the steady-state system-length distribution at an arbitrary epoch along with some important performance measures such as the mean number of customers in the system and the mean system sojourn time of a customer. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. At the end, we present numerical results in the form of tables to show the effect of model parameters on the performance measures.