Server Workload in an M/M/1 Queue with Bulk Arrivals and Special Delays

Affiliation(s)

Department of Mathematics & Statistics, University of Windsor, Windsor, Canada.

Department of Management Science, University of Windsor, Windsor, Canada.

Department of Mathematics & Statistics, University of Windsor, Windsor, Canada.

Department of Management Science, University of Windsor, Windsor, Canada.

ABSTRACT

We consider a variant of M/M/1 where customers arrive singly or in pairs. Each single and one member of each pair is called primary; the other member of each pair is called secondary. Each primary joins the queue upon arrival. Each secondary is delayed in a separate area, and joins the queue when “pushed” by the next arriving primary. Thus each secondary joins the queue followed immediately by the next primary. This arrival/delay mechanism appears to be new in queueing theory. Our goal is to obtain the steady-state probability density function (pdf) of the workload, and related quantities of interest. We utilize a typical sample path of the workload process as a physical guide, and simple level crossing theorems, to derive model equations for the steady-state pdf. A potential application is to the processing of electronic signals with error free components and components that require later confirmation before joining the queue. The confirmation is the arrival of the next signal.

We consider a variant of M/M/1 where customers arrive singly or in pairs. Each single and one member of each pair is called primary; the other member of each pair is called secondary. Each primary joins the queue upon arrival. Each secondary is delayed in a separate area, and joins the queue when “pushed” by the next arriving primary. Thus each secondary joins the queue followed immediately by the next primary. This arrival/delay mechanism appears to be new in queueing theory. Our goal is to obtain the steady-state probability density function (pdf) of the workload, and related quantities of interest. We utilize a typical sample path of the workload process as a physical guide, and simple level crossing theorems, to derive model equations for the steady-state pdf. A potential application is to the processing of electronic signals with error free components and components that require later confirmation before joining the queue. The confirmation is the arrival of the next signal.

Cite this paper

P. Brill and M. Hlynka, "Server Workload in an M/M/1 Queue with Bulk Arrivals and Special Delays,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2174-2177. doi: 10.4236/am.2012.312A298.

P. Brill and M. Hlynka, "Server Workload in an M/M/1 Queue with Bulk Arrivals and Special Delays,"

References

[1] M. Hlynka, “An M/M/1 Queue with Bulk Arrivals and Delays,” Canadian Operational Research Society Conference Presentation, Niagara Falls, June 2012.

[2] P. H. Brill, “Level Crossing Methods in Stochastic Models,” Springer, New York, 2008. doi:10.1007/978-0-387-09421-2

[3] J. W. Cohen, “On Regenerative Processes in Queueing Theory,” Lecture Notes in Economics and Mathematical Systems, Spring-Verlag, New York, 1976.

[1] M. Hlynka, “An M/M/1 Queue with Bulk Arrivals and Delays,” Canadian Operational Research Society Conference Presentation, Niagara Falls, June 2012.

[2] P. H. Brill, “Level Crossing Methods in Stochastic Models,” Springer, New York, 2008. doi:10.1007/978-0-387-09421-2

[3] J. W. Cohen, “On Regenerative Processes in Queueing Theory,” Lecture Notes in Economics and Mathematical Systems, Spring-Verlag, New York, 1976.