On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

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Study on the Queue-Length Distribution inGeo/G(MWV)/1/NQueue with Working Vacations

Discrete Dynamics in Nature and Society ◽

10.1155/2015/160697 ◽

2015 ◽

Vol 2015 ◽

pp. 1-15

Author(s):

Chuanyi Luo ◽

Jiangyi Li ◽

Yinghui Tang

Keyword(s):

Cost Function ◽

Queue Length ◽

Operating Cost ◽

Control Variable ◽

Length Distribution ◽

Buffer Size ◽

Service Rate ◽

Finite Buffer ◽

Queue Length Distribution ◽

Working Vacations

This paper analyzes a finite buffer size discrete-timeGeo/G/1/Nqueue with multiple working vacations and different input rate. Using supplementary variable technique and embedded Markov chain method, the queue-length distribution solution in the form of formula at arbitrary epoch is obtained. Some performance measures associated with operating cost are also discussed based on the obtained queue-length distribution. Then, several numerical experiments follow to demonstrate the effectiveness of the obtained formulae. Finally, a state-dependent operating cost function is constructed to model an express logistics service center. Regarding the service rate during working vacation as a control variable, the optimization analysis on the cost function is carried out by using parabolic method.

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The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000012479 ◽

1998 ◽

Vol 40 (2) ◽

pp. 207-221 ◽

Cited By ~ 7

Author(s):

Yang Woo Shin ◽

Chareles E. M. Pearce

Keyword(s):

Queue Length ◽

Length Distribution ◽

Arrival Process ◽

Service Discipline ◽

Single Server ◽

Server Vacation ◽

Batch Markovian Arrival Process ◽

Queue Length Distribution ◽

Special Cases ◽

Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

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An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

RAIRO - Operations Research ◽

10.1051/ro/2017027 ◽

2019 ◽

Vol 53 (2) ◽

pp. 367-387

Author(s):

Shaojun Lan ◽

Yinghui Tang

Keyword(s):

Discrete Time ◽

Queue Length ◽

Length Distribution ◽

Probability Generating Function ◽

Renewal Theory ◽

Function Technique ◽

Single Server ◽

Bernoulli Feedback ◽

Queue Length Distribution ◽

Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

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Infinite- and finite-buffer Markov fluid queues: a unified analysis

Journal of Applied Probability ◽

10.1017/s0021900200014509 ◽

2004 ◽

Vol 41 (02) ◽

pp. 557-569 ◽

Cited By ~ 5

Author(s):

Nail Akar ◽

Khosrow Sohraby

Keyword(s):

Steady State ◽

Queue Length ◽

Length Distribution ◽

Matrix Exponential ◽

Divide And Conquer ◽

Buffer System ◽

Finite Buffer ◽

Fluid Queues ◽

Queue Length Distribution ◽

Markov Fluid Queues

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

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New results for the Jackson network

Advances in Applied Probability ◽

10.1017/s0001867800021844 ◽

1984 ◽

Vol 16 (1) ◽

pp. 7-7

Author(s):

William A. Massey

Keyword(s):

Queue Length ◽

Length Distribution ◽

Decomposition Theorem ◽

Jackson Network ◽

Jackson Networks ◽

Queue Length Distribution

Using operator methods, we prove a general decomposition theorem for Jackson networks. For its transient joint queue-length distribution, we can stochastically bound it above by a network that decouples into smaller independent Jackson networks.

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On the Queue Length Distribution in BMAP Systems

Journal of Communications Software and Systems ◽

10.24138/jcomss.v2i4.272 ◽

2017 ◽

Vol 2 (4) ◽

pp. 275 ◽

Cited By ~ 1

Author(s):

Andrzej Chydzinski

Keyword(s):

Poisson Process ◽

Queue Length ◽

Length Distribution ◽

Compound Poisson Process ◽

Arrival Process ◽

Finite Buffer ◽

Batch Markovian Arrival Process ◽

Queue Length Distribution ◽

Markovian Processes ◽

The Laplace Transform

Batch Markovian Arrival Process – BMAP – is a teletraffic model which combines high ability to imitate complexstatistical behaviour of network traces with relative simplicity in analysis and simulation. It is also a generalization of a wide class of Markovian processes, a class which in particular include the Poisson process, the compound Poisson process, the Markovmodulated Poisson process, the phase-type renewal process and others. In this paper we study the main queueing performance characteristic of a finite-buffer queue fed by the BMAP, namely the queue length distribution. In particular, we show a formula for the Laplace transform of the queue length distribution. The main benefit of this formula is that it may be used to obtain both transient and stationary characteristics. To demonstrate this, several numerical results are presented.

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Analysis of General Input State Dependent Working Vacation Queue with Changeover Time

ISRN Computational Mathematics ◽

10.1155/2014/248704 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Vijaya Laxmi Pikkala ◽

Suchitra Vepada

Keyword(s):

Queue Length ◽

Recursive Algorithm ◽

Length Distribution ◽

Input State ◽

Finite Buffer ◽

Queue Length Distribution ◽

State Dependent ◽

Stationary Queue Length Distribution ◽

The Moment ◽

Vacation Period

We consider a finite buffer GI/M(n)/1 queue with multiple working vacations and changeover time, where the server can keep on working but at a slower speed during the vacation period. Moreover, the amount of service demanded by a customer is conditioned by the queue length at the moment service is begun for that customer. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, some numerical results of the model are presented to show the parameter effect on various performance measures.

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Infinite- and finite-buffer Markov fluid queues: a unified analysis

Journal of Applied Probability ◽

10.1239/jap/1082999086 ◽

2004 ◽

Vol 41 (2) ◽

pp. 557-569 ◽

Cited By ~ 21

Author(s):

Nail Akar ◽

Khosrow Sohraby

Keyword(s):

Steady State ◽

Queue Length ◽

Length Distribution ◽

Matrix Exponential ◽

Divide And Conquer ◽

Buffer System ◽

Finite Buffer ◽

Fluid Queues ◽

Queue Length Distribution ◽

Markov Fluid Queues

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

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