A Simple and Complete Computational Analysis of MAP/R/1 Queue Using Roots

Chaudhry, M. L.; Singh, Gagandeep; Gupta, Umesh C.

doi:10.1007/s11009-011-9266-3

Cited by 32 publications

(21 citation statements)

References 18 publications

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“…Proof A similar proof was presented by Chaudhry et al (2013) for C-MAP arrival and service time distribution with rational Laplace-Stieltjes transform and is applicable in our case with some modification. We present it as follows.…”

Section: Appendixmentioning

confidence: 75%

“…Since various distributions can be either represented or approximated by PH-distribution, we take inter-batch arrival time distribution to be of PH-type having the representation (α, T), where α and T are of dimension ν. Then S(z) can be derived as follows using the procedure adopted by Chaudhry et al (2013).…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…In this connection, readers are referred to Chaudhry et al (1987), Chaudhry et al (1990), Tijms (2003) and Chaudhry et al (2012Chaudhry et al ( , 2013, who have used the roots method. In particular, regarding the details about the use of Rouché's theorem in the analysis of characteristic equation, the readers are referred to Cohen and Down (1996), Adan et al (2006) etc.…”

Section: Introductionmentioning

confidence: 99%

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A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

2015

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We consider a batch arrival infinite-buffer single-server queue with generally distributed inter-batch arrival times with arrivals occurring in batches of random sizes. The service process is correlated and its structure is governed by a Markovian service process in continuous time. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at a pre-arrival epoch. We also obtain the steady-state probability distribution at an arbitrary epoch using the classical argument based on Markov renewal theory. Some important performance measures such as the average number of customers in the system and the mean sojourn time have also been obtained. Later, we have established heavy-and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

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Section: Appendixmentioning

confidence: 75%

Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

2015

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“…basic knowledge of queueing, spectral theory, and computer programming to computationally implement it. Encountering the difficulty to numerically implement the present queueing model by Nishimura's method, we apply the recently developed CGG method (see [16]) to obtain the steady state vectors. For the sake of completeness, we present the CGG approach in stepwise manner in the following section.…”

Section: Remarkmentioning

confidence: 99%

“…It seems from the investigations that Nishimura's approach is tedious and difficult to employ for practical purposes. The objective of this paper is to analyze the same model with an alternative methodology proposed by Chaudhry et al (2013) (to be referred to as CGG method). The CGG method appears to be rather simple, mathematically tractable, and easy to implement as compared to Nishimura's approach.…”

mentioning

confidence: 99%

A Comparative Numerical Study of the Spectral Theory Approach of Nishimura and the Roots Method Based on the Analysis of BDMMAP/G/1 Queue

Maity

Gupta

2015

International Journal of Stochastic Analysis

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This paper considers an infinite-buffer queuing system with birth-death modulated Markovian arrival process (BDMMAP) with arbitrary service time distribution. BDMMAP is an excellent representation of the arrival process, where the fractal behavior such as burstiness, correlation, and self-similarity is observed, for example, in ethernet LAN traffic systems. This model was first apprised by Nishimura (2003), and to analyze it, he proposed a twofold spectral theory approach. It seems from the investigations that Nishimura's approach is tedious and difficult to employ for practical purposes. The objective of this paper is to analyze the same model with an alternative methodology proposed by Chaudhry et al. (2013) (to be referred to as CGG method). The CGG method appears to be rather simple, mathematically tractable, and easy to implement as compared to Nishimura's approach. The Achilles tendon of the CGG method is the roots of the characteristic equation associated with the probability generating function (pgf) of the queue length distribution, which absolves any eigenvalue algebra and iterative analysis. Both the methods are presented in stepwise manner for easy accessibility, followed by some illustrative examples in accordance with the context.

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Delay Analysis of a Queue with General Service Demands and Correlated Service Capacities

Muynck

Bruneel

Wittevrongel

2018

Queueing Theory and Network Applications

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A Simple and Complete Computational Analysis of MAP/R/1 Queue Using Roots

Cited by 32 publications

References 18 publications

A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

A Comparative Numerical Study of the Spectral Theory Approach of Nishimura and the Roots Method Based on the Analysis of BDMMAP/G/1 Queue

Delay Analysis of a Queue with General Service Demands and Correlated Service Capacities

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