A generalization of the matrix M/G/l paradigm for Markov chains with a tree structure

Takine, Tetsuya; Sengupta, Bhaskar; Yeung, Raymond W.

doi:10.1080/15326349508807353

Cited by 51 publications

(36 citation statements)

References 4 publications

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“…Matrices {G(k), 1 ≤ k ≤ K} are the minimal nonnegative solutions to the matrix equations (Takine, Sengupta, and Yeung [16]):…”

Section: Fundamental Periods and Busy Periodsmentioning

confidence: 99%

Computational analysis ofMMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

Alfa

2000

Naval Research Logistics

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This paper studies a queueing system with a Markov arrival process with marked arrivals and P H-distribution service times for each type of customer. Customers (regardless of their types) are served on a mixed first-come-first-served (FCFS) and last-come-first-served (LCFS) nonpreemptive basis. That is, when the queue length is N (a positive integer) or less, customers are served on an FCFS basis; otherwise, customers are served on an LCFS basis. The focus is on the stationary distribution of queue strings, busy periods, and waiting times of individual types of customers. A computational approach is developed for computing the stationary distribution of queue strings, the mean of busy period, and the means and variances of waiting times. The relationship between these performance measures and the threshold number N is analyzed in depth numerically. It is found that the variance of the virtual (actual) waiting time of an arbitrary customer can be reduced by increasing N .

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“…Matrices {G(k), 1 ≤ k ≤ K} are the minimal nonnegative solutions to the matrix equations (Takine, Sengupta, and Yeung [16]):…”

Section: Fundamental Periods and Busy Periodsmentioning

confidence: 99%

Computational analysis ofMMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

Alfa

2000

Naval Research Logistics

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“…The key feature of our approach lies in reformulating the traditional three queue problem into a combined queue and stack problem. We show that the behavior of the reformulated problem can be captured by a Markov chain with a tree structured state space [16,20]. This Markov chain-that neither fits within the GI/M/1 or M/G/1 paradigm-is reduced to a binary treelike process [3,17], which is a special case of a tree structured QBD Markov chain [19].…”

Section: Introductionmentioning

confidence: 99%

Analyzing priority queues with 3 classes using tree-like processes

Houdt

Blondia

2006

Queueing Syst

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In this paper we demonstrate how tree-like processes can be used to analyze a general class of priority queues with three service classes, creating a new methodology to study priority queues. The key result is that the operation of a 3-class priority queue can be mimicked by means of an alternate system that is composed of a single stack and queue. The evolution of this alternate system is reduced to a tree-like Markov process, the solution of which is realized through matrix analytic methods. The main performance measures, i.e., the queue length distributions and loss rates, are obtained from the steady state of the tree-like process through a censoring argument. The strength of our approach is demonstrated via a series of numerical examples.

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“…Depending on the transitions allowed in such a tree-structured Markov chain, several classes have been identified: (a) M/G/1-type (Takine et al 1995), (b) GI/M/1-type (Yeung and Sengupta 1994), (c) QBD-type (Yeung and Alfa 1999) and (d) tree-like processes (Bini et al 2003), where the latter two classes were shown to be equivalent (Van Houdt and Blondia 2003). In each of these models the transitions fulfill a spacial homogeneity property, that is, the transition probability between two nodes depends only on the spacial relationship between the two nodes and not on their specific values.…”

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confidence: 99%

QBD Markov chains on binomial-like trees and its application to multilevel feedback queues

2007

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A matrix analytic paradigm, termed Quasi-Birth-Death Markov chains on binomial-like trees, is introduced and a quadratically converging algorithm to assess its steady state is presented. In a bivariate Markov chain {(X t , N t ), t ≥ 0}, the values of the variable X t are nodes of a binomial-like tree of order d, where the ith child has i children of its own. We demonstrate that it suffices to solve d quadratic matrix equations to yield the steady state vector, the form of which is matrix geometric. We apply this framework to analyze the multilevel feedback scheduling discipline, which forms an essential part in contemporary operating systems.Keywords QBD Markov chains · Tree-like processes · Matrix-analytic methods · Multilevel feedback queues Over the last few decades, broad classes of frequently encountered queueing models have been analyzed by matrix-analytic methods (Bini et al. 2005;Latouche and Ramaswami 1999;Neuts 1981Neuts , 1989. The embedded Markov chains in these models are two-dimensional generalizations of the classic M/G/1 and GI/M/1 queues, and quasi-birthdeath (QBD) processes. Matrix-analytic models include notions such as the Markovian arrival process (MAP) and the phase-type (PH) distribution, both in discrete and continuous time. Considerable efforts have been put into the development of efficient and numerically stable methods for their analysis (Bini et al. 2005). There is also an active search for accurate matching algorithms for both PH distributions and MAP arrivals when modeling communication systems (e.g., Asmussen et al. 1996;Bobbio et al. 2003;Breuer 2002;Horváth et al. 2005;Ryden 1996).One of the more recent paradigms within the field of matrix-analytic methods has been its generalization to discrete-time bivariate Markov chains {(X t , N t ), t ≥ 0}, in which the values of X t are the nodes of a d-ary tree (and N t takes integer values between 1 and h).

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A generalization of the matrix M/G/l paradigm for Markov chains with a tree structure

Cited by 51 publications

References 4 publications

Computational analysis ofMMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

Computational analysis ofMMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

Analyzing priority queues with 3 classes using tree-like processes

QBD Markov chains on binomial-like trees and its application to multilevel feedback queues

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