N-Policy for State-Dependent Batch Arrival Queueing System with l-Stage Service and Modified Bernoulli Schedule Vacation

Abstract

This paper deals with a batch arrival queueing system with modified Bernoulli vacation under N-policy, where the server starts the service of the customers only when the queue size becomes at least N (> 1) otherwise remains idle. The customers arrive in batches to the system in Poisson fashion but may also balk in case of long queue, when server is in working state. The customer needs l-stage of service in succession i.e. the first stage service (FSS) is followed by the second stage service (SSS), the second stage service followed by third stage service (TSS) and so on up to l-stages of service. It is assumed that after completion of l^h stage service either the server goes on vacation with probability p or decides to stay in the system with probability (1 − p) to give the service to the next customer. We determine the queue size distribution and other performance indices by using the generating function method to solve the governing equations constructed after introducing the supplementary variables. We propose a method to find the optimal value of the threshold parameter to minimize the total expected cost.

Key Words:

• Bernoulli schedule
• generating function
• M^X/G/1 queue
• N-policy
• queue size
• state dependent rate
• supplementary variable technique
• vacation time

Additional information

Notes on contributors

Madhu Jain

Dr. Madhu Jain is a Professor of Mathematics, Dr. B. R. Ambedkar University, Agra and presently faculty in the Department of Mathematics, IIT Roorkee, Roorkee, India. She received her M.Sc., M.Phil., Ph.D. and D.Sc. degrees in Mathematics from University of Agra. She has been a Gold Medalist of Agra University at M.Phil. level. There are more than 180 research publications in refereed International/National Journals and more than 15 books to her credit. She was recipient of the young scientific award, SERC visiting follow ship of Department of Science and Technology (India) and Career award of University Grants Commission (India). She has successfully completed six sponsored major research projects of Department of Science and Technology (India), University Grants Commission (India) and Council of Scientific and Industrial Research (India). Her current research interest includes Performance Modelling, Soft Computing, Bio-informatics, Reliability, Engineering and Queueing Theory. Twenty five candidates have received their Ph.D. degrees under her supervision. She has visited more than 25 reputed Universities/Institutes in USA, Canada, UK, Germany, France, Holland, and Belgium. She has participated and presented her research works in more than 30 Internationals and 75 Nationals Conferences/Seminars.

Praveen Kumar Agrawal

Praveen Kumar Agrawal is a research scholar in the Department of Mathematics at Institute of Basic Science, Khandari, Dr. B. R. Ambedkar University, Agra, India. After completing his Master’s degree in first division, he received his M. Phil. degree in Mathematics from Dr. B. R. Ambedkar University, Agra, India. His research interest includes Queueing Theory and Reliability Analysis. He has participated and presented his research work in 11 International/National Conferences/Seminars.