rM/G/1 queue: remaining service time dependent arrivals and iid service times.

Date

2017-08-03

Author

Sezer, Ali Devin

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We study a generalization of theM/G/1 system (denotedrM/G/1) withindependent and identically distributed (iid) service times and with an arrival processwhose arrival rateλ0f(r) depends on the remaining service timerof the currentcustomer being served. We derive a natural stability condition and provide a stationaryanalysis under it both at service completion times (of the queue length process) and incontinuous time (of the queue length and the residual service time). In particular, weshow that the stationary measure of queue length at service completion times is equalto that of a correspondingM/G/1 system. Forf >0 we show that the continuous timestationary measure of therM/G/1 system is linked to theM/G/1 system via a timechange. As opposed to theM/G/1 queue, the stationary measure of queue length of therM/G/1 system at service completions differs from its marginal distribution under thecontinuous time stationary measure. Thus, in general, arrivals of therM/G/1 systemdo not see time averages. We derive formulas for the average queue length, probabilityof an empty system and average waiting time under the continuous time stationarymeasure. We provide examples showing the effect of changing the reshaping function onthe average waiting time.

URI

https://hdl.handle.net/11511/72213
https://viasm.edu.vn/Cms_Data/Contents/viasm/Media/file/AbstractsWorkshopNabil.pdf

Conference Name

WorkshopStochastic processes - Actuarial science and Finance (31 Temmuz - 03 Ağustos 2017)

Collections

Graduate School of Applied Mathematics, Conference / Seminar

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Citation Formats

A. D. Sezer, “rM/G/1 queue: remaining service time dependent arrivals and iid service times.,” Vietnam, 2017, p. 4, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/72213.