Server waiting times in infinite supply polling systems with preparation times

J.-P.L. Dorsman, N. Perel, M. Vlasiou

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

Abstract

We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.
Original languageEnglish
Pages (from-to)153-184
JournalProbability in the Engineering and Informational Sciences
Volume30
Issue number2
DOIs
Publication statusPublished - Apr 2016

Cite this

@article{d3791bb4c7914eb4a1d2f5a4e0fd6f0f,
title = "Server waiting times in infinite supply polling systems with preparation times",
abstract = "We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.",
author = "J.-P.L. Dorsman and N. Perel and M. Vlasiou",
year = "2016",
month = "4",
doi = "10.1017/S0269964815000339",
language = "English",
volume = "30",
pages = "153--184",
journal = "Probability in the Engineering and Informational Sciences",
issn = "0269-9648",
publisher = "Cambridge University Press",
number = "2",

}

Server waiting times in infinite supply polling systems with preparation times. / Dorsman, J.-P.L.; Perel, N.; Vlasiou, M.

In: Probability in the Engineering and Informational Sciences, Vol. 30, No. 2, 04.2016, p. 153-184.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Server waiting times in infinite supply polling systems with preparation times

AU - Dorsman, J.-P.L.

AU - Perel, N.

AU - Vlasiou, M.

PY - 2016/4

Y1 - 2016/4

N2 - We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

AB - We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

U2 - 10.1017/S0269964815000339

DO - 10.1017/S0269964815000339

M3 - Article

VL - 30

SP - 153

EP - 184

JO - Probability in the Engineering and Informational Sciences

JF - Probability in the Engineering and Informational Sciences

SN - 0269-9648

IS - 2

ER -