Markovian Polling Systems: Functional Computation for Mean Waiting Times and its Computational Complexity

We study the expected delay in a cyclic polling model with mixtures of exhaustive and gated service in heavy traffic. We obtain closed-form expressions for the mean delay under standard heavy-traffic scalings, providing new insights into the behaviour of polling systems in heavy traffic. The results lead to excellent approximations of the expected waiting times in practical heavy-load scenarios and moreover, lead to new results for optimizing the system performance with respect to the service disciplines.

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Stochastic comparisons for queueing models via random sums and intervals

Advances in Applied Probability ◽

10.2307/1427721 ◽

1992 ◽

Vol 24 (4) ◽

pp. 960-985 ◽

Cited By ~ 20

Author(s):

Alain Jean-Marie ◽

Zhen Liu

Keyword(s):

Point Process ◽

Response Times ◽

Waiting Times ◽

Random Variables ◽

Stochastic Ordering ◽

Partial Sums ◽

Polling Systems ◽

Convex Ordering ◽

Random Intervals ◽

Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

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Stochastic comparisons for queueing models via random sums and intervals

Advances in Applied Probability ◽

10.1017/s0001867800025039 ◽

1992 ◽

Vol 24 (04) ◽

pp. 960-985 ◽

Cited By ~ 1

Author(s):

Alain Jean-Marie ◽

Zhen Liu

Keyword(s):

Point Process ◽

Response Times ◽

Waiting Times ◽

Random Variables ◽

Stochastic Ordering ◽

Partial Sums ◽

Polling Systems ◽

Convex Ordering ◽

Random Intervals ◽

Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered. These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

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Expected delay analysis of polling systems in heavy traffic

Advances in Applied Probability ◽

10.1239/aap/1035228085 ◽

1998 ◽

Vol 30 (2) ◽

pp. 586-602 ◽

Cited By ~ 11

Author(s):

R. D. van der Mei ◽

H. Levy

Keyword(s):

Heavy Traffic ◽

Waiting Times ◽

Delay Analysis ◽

Polling Systems ◽

Heavy Load ◽

Polling Model ◽

Mean Delay ◽

Service Disciplines ◽

The Mean ◽

Cyclic Polling

We study the expected delay in a cyclic polling model with mixtures of exhaustive and gated service in heavy traffic. We obtain closed-form expressions for the mean delay under standard heavy-traffic scalings, providing new insights into the behaviour of polling systems in heavy traffic. The results lead to excellent approximations of the expected waiting times in practical heavy-load scenarios and moreover, lead to new results for optimizing the system performance with respect to the service disciplines.

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SERVER WAITING TIMES IN INFINITE SUPPLY POLLING SYSTEMS WITH PREPARATION TIMES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964815000339 ◽

2015 ◽

Vol 30 (2) ◽

pp. 153-184 ◽

Cited By ~ 1

Author(s):

Jan-Pieter L. Dorsman ◽

Nir Perel ◽

Maria Vlasiou

Keyword(s):

Waiting Time ◽

Queueing Networks ◽

Waiting Times ◽

Fixed Number ◽

Polling Systems ◽

Single Server ◽

Dynamic Allocation ◽

Mean Waiting Time ◽

Preparation Phase ◽

Stationary Waiting Time

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.

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Continuous Polling on Graphs

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800002874 ◽

1993 ◽

Vol 7 (2) ◽

pp. 209-226 ◽

Cited By ~ 5

Author(s):

E. G. Coffman ◽

Aleksandr Stolyar

Keyword(s):

Waiting Times ◽

Past Research ◽

Continuous Model ◽

Arbitrary Distribution ◽

Polling Systems ◽

Fluid Limit ◽

Absolutely Continuous ◽

General Service Times ◽

General Service

Past research on polling systems has been quite restricted in the form of the paths followed by the server. This paper formulates a general, continuous model of such paths that includes closed walks on graphs. Customers arrive by a Poisson process and have general service times. The distribution of arrivals over the path is governed by an absolutely continuous, but otherwise arbitrary, distribution. The main results include a characterization of the stationary state distribution and explicit formulas for expected waiting times. The formulas reveal an interesting decomposition of the system into two components: a fluid limit and an M/G/1 queue.

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