M^x/G/1 retrial vacation queue for multi-optional services, phase repair and reneging

Abstract

The present paper deals with a batch arrival general retrial queue with multi-optional services, vacation and impatient customers undergoing reneging. The incoming customer (or a batch of customers) on finding the server in busy or in non-working condition forms a virtual pool of customers called orbit otherwise undergoes service if the server is free. The service is provided at two levels; first essential service or second optional service depending upon customer’s choice to opt for it or not. Moreover, the server may go for vacation if he finds no customer waiting for service and returns back to the service center again as soon as the customer approaches for the service. The server is unreliable as such subject to breakdown during the service; as soon as the server fails, it is immediately sent for repair so as to restore its functionality as before failure. The repairman assigned for the repair of the server also takes some setup time before commencing the repair process. By using supplementary variables technique and probability generating function method, the explicit expressions for the queue size distribution and other performance measures have been derived. The sensitivity of different parameters on various performance indices has been examined numerically by taking an illustration. Also, the neuro fuzzy approach has been used to approximate the analytical results.

Keywords:

• Retrial queue
• ANFIS
• bulk arrival
• multi-optional services
• phase repair
• vacation
• supplementary variables

Acknowledgements

The authors are thankful to MHRD for providing financial grant to carry out the research work.

Disclosure statement

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of this article.

Notes on contributors

Dr. Madhu Jain is the faculty member in the Department of Mathematics, I.I.T. Roorkee. She has been a gold medalist of Agra University at M. Phil. level. There are more than 360 research publications in refereed International/National journals and more than 20 books to her credit in addition to two reference books. She was recipient of the Young Scientific award and SERC visiting fellow of DST, and Career award of UGC. She has successfully completed six sponsored major research projects of DST, UGC and CSIR. Her current research interest includes Performance Modeling, Soft Computing, Bio-informatics, Reliability Engineering and Queueing Theory. Twenty Eight candidates have received their Ph.D. degrees under her supervision. She has visited various reputed Universities/Institutes in USA, Canada, UK, Germany, France, Holland and Belgium. She has participated and presented her research work in more than 35 International and 80 National Conferences/Seminars.

Amita Bhagat has completed her Ph.D from the Department of Mathematics, I.I.T. Roorkee under the supervision of Dr. Madhu Jain. She has been a gold medalist at both B.Sc and M.Sc level of her institute. She is a recipient of MHRD scholarship after passing GATE during her research. Her research interest includes Performance Modeling and Queueing Theory. She has to her credit 9 research publications in reputed international journals and conference proceedings.