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Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2017年第2期46卷 1035-1061页

作者： Banik, A. D. Indian Inst Technol Bhubaneswar Sch Basic Sci Permanent Campus Argul Khurja 752050 Odisha India

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).

关键词： Batch Markovian arrival process infinite-buffer queue Matrix-analytic procedure Multiple working vacations Pade approximation Phase-type (PH) Rational Laplace-Stieltjes transform Single server Weibull distribution

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Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

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COMPUTERS & INDUSTRIAL ENGINEERING 2013年第3期65卷 455-465页

作者： Banik, Abhijit Datta Indian Inst Technol Sch Basic Sci Bhubaneswar 757013 Orissa India

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： infinite-buffer queue Batch Markovian arrival process Multiple adaptive vacations Single server Cost control Matrix-analytic procedure

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The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

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APPLIED MATHEMATICAL MODELLING 2009年第7期33卷 3025-3039页

作者： Banik, A. D. Tata Inst Fundamental Res Sch Technol & Comp Sci Bombay 400005 Maharashtra India

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model. (C) 2008 Elsevier Inc. All rights reserved.

关键词： infinite-buffer queue Batch Markovian arrival process Variant of multiple vacation policy Single server

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Loss probability in a finite queue with service interruptions and queue length distribution in the corresponding infinite queue

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PERFORMANCE EVALUATION 2006年第7期63卷 682-699页

作者： Ishizaki, Fumio Korea Univ Dept Math Seoul 136701 South Korea Korea Univ Telecommun Math Res Ctr Seoul 136701 South Korea

In this paper, we consider a discrete-time finite-buffer queue with correlated arrivals and service interruptions and the corresponding infinite-buffer queue. Under some assumptions, we derive an exact relation holding between the loss probability in the finite-buffer queue and the queue length distribution in the corresponding infinite-buffer queue. The exact relation is considered as an integration/generalization of the exact relations which have been derived in previous papers. By applying the exact relation, we also develop formulas to estimate the loss probability. (C) 2005 Elsevier B.V. All rights reserved.

关键词： loss probability finite-buffer queue infinite-buffer queue proportional relation exact relation

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Loss probability in a finite discrete-time queue in terms of the steady state distribution of an infinite queue

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queueING SYSTEMS 1999年第3-4期31卷 317-326页

作者： Ishizaki, F Takine, T Univ Tokushima Fac Engn Dept Informat Sci & Intelligent Syst Tokushima 7708506 Japan Kyoto Univ Grad Sch Informat Dept Appl Math & Phys Kyoto 6068501 Japan

We consider a discrete-time single-server queue with arrivals governed by a stationary Markov chain, where no arrivals are assumed to occur only when the Markov chain is in a particular state. This assumption implies that off-periods in the arrival process are i.i.d. and geometrically distributed. For this queue, we establish the exact relationship between queue length distributions in a finite-buffer queue and the corresponding infinite-buffer queue. With the result, the exact loss probability is obtained in terms of the queue length distribution in the corresponding infinite-buffer queue. Note that this result enables us to compute the loss probability very efficiently, since the queue length distribution in the infinite-buffer queue can be efficiently computed when off-periods are geometrically distributed.

关键词： loss probability finite-buffer queue infinite-buffer queue

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