The M/G/1 processor sharing queue as the almost sure limit of feedback queues

In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.

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The M/G/1 processor sharing queue as the almost sure limit of feedback queues

Journal of Applied Probability ◽

10.2307/3214834 ◽

1990 ◽

Vol 27 (4) ◽

pp. 913-918 ◽

Cited By ~ 10

Author(s):

J. A. C. Resing ◽

G. Hooghiemstra ◽

M. S. Keane

Keyword(s):

Almost Sure Convergence ◽

Processor Sharing ◽

Regenerative Processes ◽

Arrival Times ◽

Sojourn Times ◽

Feedback Model ◽

Poisson Arrival ◽

Arrival Processes ◽

Feedback Queue ◽

Feedback Queues

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Tollbooth tandem queues with infinite homogeneous servers

Journal of Applied Probability ◽

10.1239/jap/1450802745 ◽

2015 ◽

Vol 52 (4) ◽

pp. 941-961 ◽

Cited By ~ 3

Author(s):

Xiuli Chao ◽

Qi-Ming He ◽

Sheldon Ross

Keyword(s):

Performance Measures ◽

System Performance ◽

Stochastic Ordering ◽

Tandem Queues ◽

Poisson Arrival ◽

Arrival Processes ◽

Poisson Arrivals ◽

Number Of Customers ◽

Steady State Solutions ◽

Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

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Sojourn times in finite-capacity processor-sharing queues

Next Generation Internet Networks, 2005 ◽

10.1109/ngi.2005.1431647 ◽

2005 ◽

Cited By ~ 13

Author(s):

S. Borst ◽

O. Boxma ◽

N. Hegde

Keyword(s):

Processor Sharing ◽

Finite Capacity ◽

Sojourn Times ◽

Processor Sharing Queues

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On optimal periodic dividend and capital injection strategies for spectrally negative Lévy models

Journal of Applied Probability ◽

10.1017/jpr.2018.85 ◽

2018 ◽

Vol 55 (4) ◽

pp. 1272-1286 ◽

Cited By ~ 1

Author(s):

Kei Noba ◽

José-Luis Pérez ◽

Kazutoshi Yamazaki ◽

Kouji Yano

Keyword(s):

Arrival Time ◽

Arrival Times ◽

Classical Sense ◽

Capital Injection ◽

Dividend Payments ◽

Poisson Arrival ◽

Optimal Dividend ◽

Injection Strategies

Abstract De Finetti’s optimal dividend problem has recently been extended to the case when dividend payments can be made only at Poisson arrival times. In this paper we consider the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival time and also reflects from below at 0 in the classical sense.

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Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes

INFORMS Journal on Computing ◽

10.1287/ijoc.2018.0828 ◽

2019 ◽

Vol 31 (2) ◽

pp. 347-366

Author(s):

Ran Liu ◽

Michael E. Kuhl ◽

Yunan Liu ◽

James R. Wilson

Keyword(s):

Modeling And Simulation ◽

Poisson Arrival ◽

Arrival Processes

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Estimation of Poisson Arrival Processes Under Linear Models

IEEE Transactions on Information Theory ◽

10.1109/tit.2018.2889489 ◽

2019 ◽

Vol 65 (6) ◽

pp. 3555-3564

Author(s):

Michael G. Moore ◽

Mark A. Davenport

Keyword(s):

Linear Models ◽

Poisson Arrival ◽

Arrival Processes

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A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953399000295 ◽

1999 ◽

Vol 12 (4) ◽

pp. 311-338 ◽

Cited By ~ 1

Author(s):

Charles Knessl

Keyword(s):

Diffusion Approximation ◽

Queue Length ◽

Heavy Traffic ◽

Transient Behavior ◽

Processor Sharing ◽

Parallel Queues ◽

Poisson Arrival ◽

Heavy Traffic Limit ◽

Service Rates ◽

Dependent Probability

We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate λ and have service rates equal to μ. When both queues are non-empty, the two systems behave independently of each other. However, when one of the queues becomes empty, the corresponding server helps in the other queue. This is called head-of-the-line processor sharing. We study this model in the heavy traffic limit, where ρ=λ/μ→1. We formulate the heavy traffic diffusion approximation and explicitly compute the time-dependent probability of the diffusion approximation to the joint queue length process. We then evaluate the solution asymptotically for large values of space and/or time. This leads to simple expressions that show how the process achieves its stead state and other transient aspects.

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