An M/G/1 queueing model with k sequential heterogeneous service steps and vacations in the transient state

ABSTRACT

This article studies a transient state $M/G/1$ queueing model with $k$ sequential heterogeneous service phase and vacations. According to a Poisson process, entrants, each arrival is serviced sequentially in k phases in the order of arrival. The service times follow a general distribution function. Upon completion of the service, the server could take a vacation with the probability $\theta$ and remain in the system to provide service to other customers with the probability $1-\theta$. For this model, first, we obtain the Laplace transform of the probability generating functions. The probability generating functions of system sizes are obtained for a special case in the transient state. The mean system size is obtained by using numerical analysis. Then, the sensitivity of the model is considered.

KEYWORDS:

• $M/G/1$ queue
• $k$ phase of heterogeneous service
• Bernoulli vacation
• Transient behaviour
• Laplace transform

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Ali Mohammadi

Ali Mohammadi is PHD student in the Department of Statistics, Faculty of Statistics, Mathematics and Computer, Allameh Tabataba'i University. He is currently a member of the faculty of the Islamic Azad University Gachsaran Branch.

Mohammad Reza Salehi Rad

Mohammad Reza Salehi Rad is currently an associate Professor in the Department of Statistics, Faculty of Statistics, Mathematics and Computer, Allameh Tabataba'i University. He received his PhD degree in Ferdowsi university of Mashhad, Iran. His research interests include stochastic processes, queueing models, time series and optimization.