A Non-Markovian Bulk Queue with Multiple Vacations and Control Policy on Request for Re-Service

Abstract

The steady state behaviour of a single server non-Markovian queue with multiple vacations and controlled optional re-service is analysed. A batch of customers arrive according to Poisson with rate A, whereas the bulk service is rendered by a single server with a minimum batch size of ‘a and maximum of ‘b’. The service times follow a general distribution. At the completion of an essential service, the leaving batch of customers may request for a re-service with probability n. However, the re-service is rendered only when the number of customers waiting in the queue is less than ‘a’. If no request for re-service is made after the completion of an essential service and numbers of customers in the queue is less than ‘a, then the server will avail a vacation of a random length. When the server returns from vacation and if the queue length is still less than ‘a’ he avails another vacation so on until the server finds ‘a’ customers in the queue. After the completion of an essential service and numbers of customers in the queue is greater than ‘a then the server will continue the batch service with general bulk service rule. The probability generating function of queue size at a random epoch is obtained. Some important performance measures such as expected queue size, expected busy period and idle period are derived. Cost model is discussed with numerical illustration.

Key Words:

• Bulk service
• multiple vacations
• optional re-service
• supplementary variables

Additional information

Notes on contributors

S. Jeyakumar

S. Jeyakumar is currently Assistant Professor in the Department of Mathematics, Government Arts College, Coimbatore, India. He received his M.Sc.and M.Phil degrees in Mathematics from Bharathiar University, Coimbatore, India in 1988 and 1996 respectively and PGDCA from Bharathiar University in 1989 and Ph.D. Degree from Bharathiar University in 2007. His research interests include the Queueing theory, Graph theory and Computer Applications. He has publications in Applied Mathematical Modelling, International journal of Industrial, and Systems Engineering and International Journal of information and management sciences.

R. Arumuganathan

R. Arumuganathan is currently Professor in the Department of Mathematics and Computer Applications, PSG College of Technology, Coimbatore, India. He received his M.Sc. and M.Phil degrees in Mathematics from Madurai Kamarajar University, Madurai, India in 1985 and 1986 respectively and Ph.D. degree in Mathematics from Bharathiar University in 1998. His research interests include the Queueing theory, Number theory, abstract algebra and wavelet transforms. He has publications in Computers and Operations Research, Asia-pacific Journal of Operations Research, Applied Mathematical Modelling, International journal of Industrial and Systems Engineering and International Journal of information and management sciences and etc. He has given invited talks in several international conferences.