A Short Note on the System-Length Distribution in a Finite-Buffer $$GI^X/C$$-MSP/1/N Queue Using Roots

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.

Download Full-text

Finite Buffer GI/M(n)/1 Queue with Bernoulli-Schedule Vacation Interruption under N-Policy

International Scholarly Research Notices ◽

10.1155/2014/392317 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

P. Vijaya Laxmi ◽

V. Suchitra

Keyword(s):

Performance Measures ◽

Finite Buffer ◽

Supplementary Variable ◽

Working Vacation ◽

Variable Technique ◽

Special Cases ◽

Vacation Interruption ◽

System Length ◽

Vacation Period ◽

Bernoulli Schedule

We study a finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption. The server works with a slower rate during vacation period. At a service completion epoch during working vacation, if there are at least N customers present in the queue, the server interrupts vacation and otherwise continues the vacation. Using the supplementary variable technique and recursive method, we obtain the steady state system length distributions at prearrival and arbitrary epochs. Some special cases of the model, various performance measures, and cost analysis are discussed. Finally, parameter effect on the performance measures of the model is presented through numerical computations.

Download Full-text

ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000349 ◽

2012 ◽

Vol 26 (2) ◽

pp. 221-244 ◽

Cited By ~ 13

Author(s):

M. L. Chaudhry ◽

S. K. Samanta ◽

A. Pacheco

Keyword(s):

Length Distribution ◽

Renewal Theory ◽

Computationally Efficient ◽

Form Analysis ◽

Matrix Geometric Method ◽

Alternative Approach ◽

The Matrix ◽

Sojourn Time Distribution ◽

System Length ◽

Markov Renewal Theory

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

Download Full-text

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.

Download Full-text

DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595914500031 ◽

2014 ◽

Vol 31 (01) ◽

pp. 1450003 ◽

Cited By ~ 3

Author(s):

SHAN GAO ◽

ZAIMING LIU ◽

QIWEN DU

Keyword(s):

Discrete Time ◽

Finite Buffer ◽

Waiting Time Distribution ◽

Service Period ◽

Imbedded Markov Chain ◽

Working Vacations ◽

Vacation Interruption ◽

Service Times ◽

System Length ◽

Vacation Period

In this paper, we study a discrete-time finite buffer batch arrival queue with multiple geometric working vacations and vacation interruption: the server serves the customers at the lower rate rather than completely stopping during the vacation period and can come back to the normal working level once there are customers after a service completion during the vacation period, i.e., a vacation interruption happens. The service times during a service period, service times during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

Download Full-text

An LCFS finite buffer model with finite source batch input

Journal of Applied Probability ◽

10.1017/s0021900200027352 ◽

1989 ◽

Vol 26 (02) ◽

pp. 372-380

Author(s):

Nico M. Van Dijk

Keyword(s):

Queueing Systems ◽

Length Distribution ◽

Source Distribution ◽

Closed Form Expression ◽

Finite Buffer ◽

Form Expression ◽

Batch Arrivals ◽

Queue Length Distribution ◽

Recursive Computation ◽

Batch Input

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

Download Full-text

Optimization of Balking and Reneging Queue with Vacation Interruption underN-Policy

Journal of Optimization ◽

10.1155/2013/683708 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

P. Vijaya Laxmi ◽

V. Goswami ◽

K. Jyothsna

Keyword(s):

Performance Measures ◽

Search Method ◽

Model Parameters ◽

Finite Buffer ◽

Working Vacation ◽

Swarm Optimization ◽

Multiple Working Vacations ◽

Working Vacations ◽

Vacation Interruption ◽

System Length

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging, and vacation interruption underN-policy. In the working vacation, a customer is served at a lower rate and at the instants of a service completion; if there are at leastNcustomers in the queue, the vacation is interrupted and the server switches to regular busy period otherwise continues the vacation. Using Markov process and recursive technique, we derive the stationary system length distributions at arbitrary epoch. Various performance measures and some special models of the system are presented. Cost analysis is carried out using particle swarm optimization and quadratic fit search method. Finally, some numerical results showing the effect of model parameters on key performance measures of the system are presented.

Download Full-text

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

Journal of Applied Probability ◽

10.1239/jap/1110381381 ◽

2005 ◽

Vol 42 (1) ◽

pp. 199-222 ◽

Cited By ~ 7

Author(s):

Yutaka Sakuma ◽

Masakiyo Miyazawa

Keyword(s):

Stability Condition ◽

Queue Length ◽

Length Distribution ◽

Buffer Size ◽

Finite Buffer ◽

Jackson Network ◽

Numerical Examples ◽

Queue Length Distribution ◽

Special Cases ◽

The Stability

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.

Download Full-text