Queueing system with server breakdowns and individual customer abandonment

ABSTRACT

We consider a single server queueing system with a finite capacity buffer in which the server is subject to breakdowns and repairs. At the time of a breakdown, the customers, if any, present in the system will either decide to stay or abandon according to a probabilistic mechanism. This decision is made by every customer present in the system independent of the others. We assume that the arrivals of customers occur according to a Markovian arrival process (MAP). The shocks that create breakdowns of the servers are also assumed to follow a MAP independent of the other MAP. The service and the repair times of the server are assumed to follow (possibly) different phase type (PH) distributions. An arriving customer finding the buffer full will be lost. Further, an arriving customer finding the server under repair may decide to balk or join the queue at that instant. This queueing system, which generalizes some of the well-known queueing models in the literature, is studied as a GI/ M/ 1- type with a finite state space. The model is analyzed in steady-state and illustrative numerical examples including an optimization problem of interest are presented.

KEYWORDS:

• Unreliable server
• Markovian arrival process
• $\mathit{PH}$ distribution
• structured multidimensional Markov chain

Disclosure statement

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

Additional information

Notes on contributors

A.N. Dudin

Alexander Dudin has got PhD degree in Probability Theory and Mathematical Statistics in 1982 from Vilnius University and Doctor of Science degree in 1992 from Tomsk University. He is Head of Laboratory of Applied Probabilistic Analysis in Belarusian State University. He is author of more than 450 publications including 5 books and more than 130 papers in top level Journals. He coedited more than ten volumes of the Springer series. In 2013, he received Scopus Award Belarus for outstanding contribution to the field of Mathematics. He is the Chairman of Belarusian Winter Workshops in Queueing Theory which are held since 1985 and the Chairman of IPC of the conference named after A.F. Terpugov since 2014. He was invited for lecturing and research to USA, UK, Germany, France, the Netherlands, Japan, Korea, India, Russia, China, Italy, and Sweden.

S.R. Chakravarthy

Srinivas Chakravarthy, a Distinguished Faculty (Kettering's Faculty and Alumni Honor Wall), retired as a Professor in the Departments of Industrial and Manufacturing Engineering & Mathematics at Kettering University, Flint, Michigan. Srinivas Chakravarthy’s research interests are in the areas of algorithmic probability, queuing, reliability, inventory, and simulation. He has published more than 135 papers in leading journals and his recent contributions include a 2-volume book on Matrix-Analytic Methods in Queues, published by ISTE-Wiley, 2022.

S.A. Dudin

Sergei Dudin was graduated from Belarusian State University in 2007. In 2010, he earned PhD degree in Belarusian State University in System Analysis, Control and Information Processing and works currently as leading scientific researcher of Research Laboratory of Applied Probabilistic Analysis in Belarusian State University. His main fields of interests are queueing systems with correlated arrival flows and controlled tandem models. He published the monograph and over 100 papers including more than 70 papers cited in Scopus.

O.S. Dudina

Olga S. Dudina was graduated from Belarusian State University in 2007. In 2010, she earned PhD degree in Belarusian State University in Probability Theory and Mathematical Statistics. Her PhD dissertation got Award as the best dissertation of the year in Republic of Belarus in Natural Sciences. She works as leading scientific researcher of Research Laboratory of Applied Probabilistic Analysis in Belarusian State University. Her main fields of interests are queueing tandem queueing models with correlated arrival flows, non-markovian queueing systems. She published the monograph and over 100 papers including more than 65 papers cited in Scopus.