Sojourn times in vacation and polling systems with Bernoulli feedback

We study sojourn times in M/G/1 multiple vacation systems and multiqueue cyclic-service (polling) systems with instantaneous Bernoulli feedback. Three service disciplines, exhaustive, gated, and 1-limited, are considered for both M/G/1 vacation and polling systems. The Laplace-Stieltjes transforms of the sojourn time distributions in the three vacation systems are derived. For polling systems, we provide explicit expressions for the mean sojourn times in symmetric cases. Furthermore a pseudo-conservation law with respect to the mean sojourn times is derived for a polling system with a mixture of the three service disciplines.

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Expected delay analysis of polling systems in heavy traffic

Advances in Applied Probability ◽

10.1017/s0001867800047443 ◽

1998 ◽

Vol 30 (02) ◽

pp. 586-602 ◽

Cited By ~ 1

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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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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Sojourn times in polling systems with various service disciplines

Performance Evaluation ◽

10.1016/j.peva.2009.05.004 ◽

2009 ◽

Vol 66 (11) ◽

pp. 621-639 ◽

Cited By ~ 27

Author(s):

Onno Boxma ◽

Josine Bruin ◽

Brian Fralix

Keyword(s):

Polling Systems ◽

Sojourn Times ◽

Service Disciplines

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On Two-Level State-Dependent Routing Polling Systems with Mixed Service

Mathematical Problems in Engineering ◽

10.1155/2015/109325 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 5

Author(s):

Guan Zheng ◽

Yang Zhijun ◽

Qian Wenhua ◽

He Min

Keyword(s):

Length Distribution ◽

Probability Generating Function ◽

Polling Systems ◽

Single Server ◽

Polling System ◽

The Novel ◽

Queue Length Distribution ◽

State Dependent ◽

Mean Waiting Time ◽

The Mean

Based on priority differentiation and efficiency of the system, we consider anN+1queues’ single-server two-level polling system which consists of one key queue andNnormal queues. The novel contribution of the present paper is that we consider that the server just polls active queues with customers waiting in the queue. Furthermore, key queue is served with exhaustive service and normal queues are served with 1-limited service in a parallel scheduling. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we derive the explicit closed-form expressions for the mean waiting time. Numerical examples demonstrate that theoretical and simulation results are identical and the new system is efficient both at key queue and normal queues.

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Analysis of polling systems with mixed service disciplines

Communications in Statistics Stochastic Models ◽

10.1080/15326349908807168 ◽

1990 ◽

Vol 6 (4) ◽

pp. 667-689 ◽

Cited By ~ 6

Author(s):

Oliver C. Ibe

Keyword(s):

Polling Systems ◽

Service Disciplines

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A note on sojourn times in M/G/1 queues with instantaneous, bernoulli feedback

Naval Research Logistics Quarterly ◽

10.1002/nav.3800280415 ◽

1981 ◽

Vol 28 (4) ◽

pp. 679-684 ◽

Cited By ~ 24

Author(s):

Ralph L. Disney

Keyword(s):

Bernoulli Feedback ◽

Sojourn Times

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On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

Journal of Applied Probability ◽

10.2307/3212332 ◽

1972 ◽

Vol 9 (3) ◽

pp. 642-649 ◽

Cited By ~ 8

Author(s):

Jacqueline Loris-Teghem

Keyword(s):

Generating Function ◽

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Queueing System ◽

Waiting Time Distribution ◽

Stieltjes Transform ◽

Sojourn Times ◽

Stieltjes Transforms ◽

Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

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On sojourn times at particular gene frequencies

Genetics Research ◽

10.1017/s0016672300015494 ◽

1975 ◽

Vol 25 (2) ◽

pp. 89-94 ◽

Cited By ~ 8

Author(s):

Edward Pollak ◽

Barry C. Arnold

Keyword(s):

Markov Chains ◽

Gene Frequency ◽

Overlapping Generations ◽

Finite Population ◽

Diffusion Method ◽

Gene Frequencies ◽

Sojourn Times ◽

The Mean ◽

Number Of Visits ◽

Finite Markov Chains

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

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The Service Time Properties of an Unreliable Server Characterize the Exponential Distribution

Advances in Applied Probability ◽

10.1017/s0001867800026069 ◽

1994 ◽

Vol 26 (01) ◽

pp. 172-182 ◽

Cited By ~ 1

Author(s):

Z. Khalil ◽

B. Dimitrov

Keyword(s):

Exponential Distribution ◽

Service Time ◽

Initial Conditions ◽

Mean Values ◽

Unreliable Server ◽

Service Disciplines ◽

The Mean ◽

Server Reliability ◽

Preemptive Repeat Different

Consider the total service time of a job on an unreliable server under preemptive-repeat-different and preemptive-resume service disciplines. With identical initial conditions, for both cases, we notice that the distributions of the total service time under these two disciplines coincide, when the original service time (without interruptions due to server failures) is exponential and independent of the server reliability. We show that this fact under varying server reliability is a characterization of the exponential distribution. Further we show, under the same initial conditions, that the coincidence of the mean values also leads to the same characterization.

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