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Transient workload distribution in the M/G/1 finite-buffer queue with single and multiple vacations

ANNALS OF OPERATIONS RESEARCH 2016年第2期239卷 381-400页

作者： Kempa, Wojciech M. Silesian Tech Univ Inst Math Ul Kaszubska 23 PL-44100 Gliwice Poland

A finite-buffer M/G/1-type queueing system with exhaustive service and server vacations is considered. The single and multiple vacation policies are investigated separately. In the former case a single vacation is being initialized every time when the system becomes empty. After the vacation the server waits on standby for packets to start the service process. In the multiple vacation policy the server begins successive single vacation periods until at least one packet is present in the system. For both vacation policies systems of integral equations for the transient virtual waiting time v(t) distributions, conditioned by the numbers of packets present in the system at the start epoch, are found. The representations for the twofold Laplace transforms of the conditional distributions of v(t) are obtained and written in compact forms, convenient in numerical treatment. From these formulae stationary distributions of v(t) and their means can be computed via usual operations on transforms. Some numerical illustrations of theoretical results are attached as well.

关键词： finite-buffer queue Multiple vacation Poisson arrivals Single vacation Transient distribution Virtual waiting time

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A Short Note on the System-Length Distribution in a finite-buffer GI^X/C-MSP/1/N queue Using Roots 17th

A Short Note on the System-Length Distribution in a Finite-B...

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26th International Conference, on Analytical and Stochastic Modelling Techniques and Applications (ASMTA)

作者： Banik, Abhijit Datta Chaudhry, Mohan L. Wittevrongel, Sabine Bruneel, Herwig Indian Inst Technol Bhubaneswar Sch Basic Sci Permanent Campus Argul Khurja 752050 Odisha India Royal Mil Coll Canada Dept Math & Comp Sci POB 17000 Kingston ON K7K 7B4 Canada Univ Ghent Dept Telecommun & Informat Proc UGent Sint Pietersnieuwstr 41 B-9000 Ghent Belgium

ISBN: (纸本)9783030918255;9783030918248

This paper deals with a renewal input finite-buffer single-server queue, where the arrivals occur in batches and the server serves the customers singly. It is assumed that the inter-batch arrival times are generally distributed and the successive service times are correlated. The correlated single-service process is exhibited by a continuous-time Markovian service process (C-MSP). As the buffer capacity N (including the one in service) is finite, the partial-batch rejection policy is considered here. Steady-state distributions at different epochs, namely pre-arrival and arbitrary epochs are obtained. These distributions are used to obtain some important performance measures, e.g. the blocking probability of the first, an arbitrary, and the last customer of a batch, the average number of customers in the system and the mean waiting time in the system. The proposed analysis is based on the roots of a characteristic equation which is derived from the balance equations of an embedded Markov chain at pre-arrival epochs of a batch. For this non-renewal service finite-buffer queueing model, we implement a novel as well as simple procedure for deriving the characteristic equation and then finding the stationary probability vectors in terms of the roots of the characteristic equation. Finally, some numerical results are presented in the form of tables for the case of a phase-type inter-batch arrival distribution.

关键词： Renewal input Batch arrival finite-buffer queue Continuous-time Markovian service process (C-MSP) Roots

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A finite-buffer queue WITH A SINGLE VACATION POLICY: AN ANALYTICAL STUDY WITH EVOLUTIONARY POSITIONING

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INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND COMPUTER SCIENCE 2014年第4期24卷 887-900页

作者： Wozniak, Marcin Kempa, Wojciech M. Gabryel, Marcin Nowicki, Robert K. Silesian Tech Univ Inst Math PL-44100 Gliwice Poland Cestochowa Univ Technol Inst Computat Intelligence PL-42200 Czestochowa Poland

In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.

关键词： finite-buffer queue evolutionary strategy object positioning queueing system busy period idle time single vacation

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A detailed note on the finite-buffer queueing system with correlated batch-arrivals and batch-size-/phase-dependent bulk-service

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4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH 2022年第2期20卷 241-272页

作者： Ghosh, Souvik Banik, A. D. Walraevens, Joris Bruneel, Herwig Tel Aviv Univ Dept Stat & Operat Res IL-69978 Tel Aviv Israel Indian Inst Technol Bhubaneswar Argul Campus Khurja 752050 Odisha India Univ Ghent UGent Dept Telecommun & Informat Proc Sint Pietersnieuwstr 41 B-9000 Ghent Belgium

This paper analyzes a finite-buffer queueing system, where customers arrive in batches and the accepted customers are served in batches by a single server. The service is assumed to be dependent on the batch-size and follows a general bulk service rule. The inter-arrival times of batches are assumed to be correlated and they are represented through the batch Markovian arrival process (BMAP). Computation procedure of the queue-length distributions at the post-batch-service completion, an arbitrary, and the pre-batch-arrival epochs are discussed. Various performance measures along with the consecutive customer loss probabilities are studied considering batch-size-dependent renewal service time distributions. Further, the above finite-buffer bulk-service queueing model is also investigated considering correlated batch-service times which are presented through the Markovian service process (MSP). The phase-dependent consecutive loss probabilities for the correlated batch-service times are determined. In the form of tables and graphs, a variety of numerical results for different batch-service time distributions are presented in this paper.

关键词： finite-buffer queue Batch Markovian arrival process (BMAP) Markovian service process (MSP) Batch-size-dependent bulk service Performance measures Consecutive customer loss (CCL)

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On the Heuristic Computational Procedures of the Virtual Waiting-Time Distribution in a Non-renewal Input finite-buffer Bulk-Service queues: MAP/R(a,b)/1/N

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JOURNAL OF THE INDIAN SOCIETY FOR PROBABILITY AND STATISTICS 2025年 1-46页

作者： Banik, Abhijit Datta Chaudhry, Mohan Barik, Sitaram Singh, Gagandeep Indian Inst Technol Bhubaneswar Sch Basic Sci Permanent Campus Argul Jatni 752050 Khurda India Royal Mil Coll Canada Dept Math & Comp Sci STN ForcesPOB 17000 Kingston ON K7K 7B4 Canada Panjab Univ Dept Math Chandigarh 160014 Punjab India

In this paper, we discuss a finite-buffer single-server queue wherein arrivals occur according to a Markovian arrival process. The server serves customers in batches according to (a, b) bulk-service rule, that is, the server serves in batches of maximum size "b" with a minimum threshold size "a". The service time of each batch follows the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The queue capacity is N excluding the batch with the server. An alternative approach for finding the queue-length probability vector at a post-departure epoch has been proposed using the roots of a characteristic equation. queue-length probability vectors at random and pre-arrival epochs have also been obtained using the embedded Markov chain, Markov renewal theory, and the semi-Markov processes. Approximation to the phase-wise virtual waiting-time distribution (in the queue as well as in the system) of a random customer is derived using the functional relation between the vector generating function (vgf) for the queue-length probability vectors at a random epoch and the vector containing the phase-wise Laplace-Stieltjes transform (LST) of the queueing-time distribution of a random customer. Using the LST, we discuss the derivation of an approximate phase-wise probability density function of the virtual waiting-time of a random customer as well as its numerical implementations.

关键词： finite-buffer queue Markovian arrival process (a, b) bulk-service rule Rational Laplace-Stieltjes transforms Roots Heavy-tailed distribution

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Genetic Cost Optimization of the GI/M/1/N finite-buffer queue with a Single Vacation Policy

Genetic Cost Optimization of the <i>GI/M/1/N</i> Finite-Buff...

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12th International Conference on Artificial Intelligence and Soft Computing, ICAISC 2013

作者： Gabryel, Marcin Nowicki, Robert K. Wozniak, Marcin Kempa, Wojciech M. Czestochowa Tech Univ Inst Computat Intelligence Al Armii Krajowej 36 PL-42200 Czestochowa Poland Silesian Tech Univ Inst Math PL-44100 Gliwice Poland

ISBN: (纸本)9783642386091

In the artice, problem of the cost optimization of the GI/M/1/N-type queue with finite buffer and a single vacation policy is analyzed. Basing on the explicit representation for the joint transform of the first busy period, first idle time and the number of packets transmitted during the first busy period and fixed values of unit costs of the server's functioning an optimal set of system parameters is found for exponentially distributed vacation period and 2-Erlang distribution of inter arrival times. The problem of optimization is solved using genetic algorithm. Different variants of the load of the system are considered as well.

关键词： Busy period finite-buffer queue genetic algorithm idle time optimization single vacation

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queueing Delay in a finite-buffer Model with Failures and Bernoulli Feedback 38th

Queueing Delay in a Finite-Buffer Model with Failures and Be...

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38th International Conference on Information Systems Architecture and Technology (ISAT)

作者： Kempa, Wojciech M. Silesian Tech Univ Inst Math Fac Appl Math Ul Kaszubska 2A PL-44100 Gliwice Poland

ISBN: (纸本)9783319672298;9783319672281

In the paper a finite-capacity queueing model with server breakdowns is considered. Jobs arrive according to a Poisson process and are being served during exponentially distributed processing time under the FIFO service discipline. During the operation of the system successive exponential failure-free times are followed by generally distributed repair periods. After the processing a job may rejoin the queue (feedback) with probability q or leave the system with probability 1 - q. Applying the analytical approach based on the idea of Markov chain and the formula of total probability, a system of integral equations for the transient queueing delay distribution is built, conditioned by the initial buffer state. Algebraic method based on the idea of Korolyuk's potential is used to obtain the solution of the corresponding system written for Laplace transforms in a closed form. Numerical example is attached as well.

关键词： Bernoulli feedback finite-buffer queue Integral equations queueing delay Transient state

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On Loss Process in a queueing System Operating Under Single Vacation Policy 38th

On Loss Process in a Queueing System Operating Under Single ...

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38th International Conference on Information Systems Architecture and Technology (ISAT)

作者： Kempa, Wojciech M. Silesian Tech Univ Inst Math Fac Appl Math Ul Kaszubska 2A PL-44100 Gliwice Poland

ISBN: (纸本)9783319672298;9783319672281

A finite-buffer queueing model with Poisson arrivals and generally distributed processing times is analyzed. Every time when the service station becomes idle a single vacation period is being initialized, during which the processing is suspended. A system of integral equations for the probability distribution of the length of the first loss series is built, using the idea of embedded Markov chain and the total probability law. The solution of the is obtained in a compact form by using the linear algebraic approach. The corresponding result for next series of losses is hence derived. Numerical utility of analytical formulae is presented in a computational example.

关键词： Embedded Markov chain finite-buffer queue Integral equation Loss process Single vacation policy

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Efficient Computational Analysis of Stationary Probabilities for the queueing System BMAP/G/1/N With or Without Vacation(s)

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INFORMS JOURNAL ON COMPUTING 2017年第1期29卷 140-151页

作者： Banik, A. D. Chaudhry, M. L. Indian Inst Technol Sch Basic Sci Permanent Campus Argul Khurja 752050 Odisha India Royal Mil Coll Canada Dept Math & Comp Sci Kingston ON K7K 7B4 Canada

We consider a finite-buffer single-server queue with batch Markovian arrival process. In the case of finite-buffer batch arrival queue, there are different customer rejection/acceptance strategies such as partial batch rejection, total batch rejection, and total batch acceptance policy. We consider partial batch rejection strategy throughout our paper. We obtain queue length distributions at various epochs such as pre-arrival, arbitrary, and post-departure as well as some important performance measures, like probability of loss for the first, an arbitrary, and the last customer of a batch, mean queue lengths, and mean waiting times. The corresponding queueing model under single and multiple vacation policy has also been investigated. Some numerical results have been presented in the form of tables by considering phase-type and Pareto service time distributions. The proposed analysis is based on the successive substitutions in the Markov chain equations of the queue-length distribution at an embedded post-departure epoch of a customer. We also establish relationships among the queue-length distributions at post-departure, arbitrary, and pre-arrival epochs using the classical argument based on Markov renewal theory and semi-Markov processes. Such queueing systems find applications in the performance evaluation of teletraffic part in 4G A11-IP networks.

关键词： finite-buffer queue batch Markovian arrival process single server partial batch rejection policy single- and multiple-vacation policy Pareto service time distribution

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Departure Process in finite-buffer queue with Batch Arrivals

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18th International Conference on Analytical and Stochastic Modelling and Applications (ASMTA)

作者： Kempa, Wojciech M. Silesian Tech Univ Inst Math Ul Kaszubska 23 PL-44100 Gliwice Poland

ISBN: (纸本)9783642217135

A finite-buffer queueing system with batch Poisson arrivals is considered. A system of integral equations for the distribution function of the number of customers h(t) served before t, conditioned by the initial state of the system, is built. A compact formula for probability generating function of the Laplace transform of distribution of h(t) is found using the potential technique. From this representation the mean of h(t) can be effectively calculated numerically using one of the inverse Laplace transform approximation algorithms. Moreover a limit behavior of departure process as the buffer size tends to infinity is investigated. Numerical examples are attached as well.

关键词： Departure process finite-buffer queue Numerical inverse Laplace transform Poisson arrivals Potential method

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