Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process: BMAP/Gn(a,b)/1

Abstract

Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This article analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service (a, b) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch using supplementary variable technique. The mathematical procedure for the complete extraction of distribution at departure epoch has been discussed and using those extracted probabilities, we achieve the queue and server content distribution at arbitrary epoch. Finally, numerical illustrations have been carried out in order to make a deep insight to the readers which contains deterministic as well as phase-type service time distributions.

Keywords:

• Batch-service
• BMAP
• Batch-size-dependent
• queueing
• server content

MATHEMATICS SUBJECT CLASSIFICATION:

• 60G05 ⋅ 60K25

Acknowledgements

The authors would like to thank the anonymous referee for their valuable comments and suggestions which led to the improvement and presentation of the article.