A generalised Little's law and its applications for a discrete-time G/D/1 queue with correlated arrivals

Abstract

The discrete-time G/D/1 queues with serially correlated batch arrivals and unit service times have wide applications in modern telecommunication systems. Despite the rich literature in their performance analysis, no simple formula on the relation between system size and sojourn time is known. We show that for this specific type of queues, the Little's law can be extended to higher moments. The benefit of this generalised result is that once the moments of either performance measure are available, those of the other will be obtained simultaneously. This result is applied to a particular example of OO-G/D/1 system, where the mean, variance, and skewness of the sojourn delay are derived in closed-form. Numerical examples are given to examine how the correlation influences these performance measures.

Keywords:

• queueing
• telecommunications
• discrete-time G/D/1 queue
• correlated arrivals
• Little's law
• higher moments

Acknowledgements

The authors would like to thank Prof. Yi-Ching Yao (Academia Sinica, Taiwan) and Prof. Ronald W. Wolff (University of California at Berkeley) for the helpful discussions on this work.