M/G/1-Type Markov Processes: A Tutorial

Riska, Alma; Smirni, Evgenia

doi:10.1007/3-540-45798-4_3

Cited by 19 publications

(10 citation statements)

References 23 publications

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“…Moreover, in the context of queueing networks, using phase-type distributions leads to models that can be solved analytically using the matrix-analytic method [570]. In these models, the "real" servers in the system being studied are represented by networks of exponential servers that together lead to the desired service-time distribution; the transitions between them are made by networks that create the desired interarrival distribution.…”

Section: Usesmentioning

confidence: 99%

Workload Modeling for Computer Systems Performance Evaluation

Feitelson

2015

197

131

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Reliable performance evaluations require the use of representative workloads. This is no easy task since modern computer systems and their workloads are complex, with many interrelated attributes and complicated structures. Experts often use sophisticated mathematics to analyze and describe workload models, making these models difficult for practitioners to grasp. This book aims to close this gap by emphasizing the intuition and the reasoning behind the definitions and derivations related to the workload models. It provides numerous examples from real production systems, with hundreds of graphs. Using this book, readers will be able to analyze collected workload data and clean it if necessary, derive statistical models that include skewed marginal distributions and correlations, and consider the need for generative models and feedback from the system. The descriptive statistics techniques covered are also useful for other domains.

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Section: Usesmentioning

confidence: 99%

Workload Modeling for Computer Systems Performance Evaluation

Feitelson

2015

197

131

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“…It is well known, [7,16], that for the chain to be recurrent, the following condition has to be satisfied:…”

Section: Numerical Stability Of Newetaqa the Probabilities πmentioning

confidence: 99%

“…Traditional matrix-analytic algorithms are based on stochastic complementation [16] and compute the stationary probability vector of the Markov process with a recursive function. The key in the matrix-analytic solution is the computation of an auxiliary matrix called G, on which the recursion for the computation of the stationary probability vectors is based.…”

mentioning

confidence: 99%

Bridging ETAQA and Ramaswami’s formula for the solution of M/G/1-type processes

Stathopoulos

Riska

Zhong

et al. 2005

Performance Evaluation

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Abstract.For some time, the method of Ramaswami has been the established way to analyze M/G/1-type processes. The ETAQA method, proposed previously in [15], has offered a more efficient alternative for the exact computation of a general class of metrics for M/G/1-type processes. However, the stability of ETAQA and its relation to Ramaswami's method were not well understood. In this paper, we derive a new formulation that improves the numerical stability and computational performance of ETAQA. As with ETAQA, the resulting methodology, newETAQA, solves a homogeneous system of equations to obtain the aggregate probability of a finite set of classes of states from the state space. In contrast to ETAQA, newETAQA constructs its matrix X in a way similar to the method of Ramaswami, decoupling the computation of the probabilities of the first two initial classes of states from the computation of the aggregate probability. Because direct methods are used to solve this system, the decoupling implies an often significant speedup over ETAQA. In addition, we show that the matrix X is an M-matrix, and under certain conditions, X is also diagonally dominant and thus can be factored stably. More importantly, we show that the newETAQA method is just an efficient way to implement Ramaswami's method. We also discuss alternative normalization conditions for Ramaswami's method. Our numerical experiments demonstrate the stability of our method for both stiff and well behaved processes, and for both low and high system utilizations.

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“…For more details on stochastic INFORMS Journal on Computing 19(2), pp. 215-228, © 2007 INFORMS complementation and its application for the development of matrix-analytic algorithms, see Riska and Smirni (2002b).…”

Section: Introductionmentioning

confidence: 99%

“…The traditional matrix-analytic algorithms were developed based on the concept of stochastic complementation, as explained in Riska and Smirni (2002b) and provide a recursive function for computation of the probability vector i that corresponds to j , for j ≥ 1. This recursive function is based on G (for the case of M/G/1-type processes) or R (for the case of GI/M/1-type processes).…”

Section: Introductionmentioning

confidence: 99%

ETAQA Solutions for Infinite Markov Processes with Repetitive Structure

Riska

Smirni

2007

INFORMS Journal on Computing

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W e describe the ETAQA (efficient technique for the solution of quasi birth-death processes) approach for the exact analysis of M/G/1 and GI/M/1-type processes, and their intersection, i.e., quasi birth-death processes. ETAQA exploits the repetitive structure of the infinite portion of the chain and derives a finite system of linear equations. In contrast to the classic techniques for solution of such systems, the solution of this finite linear system does not provide the entire probability distribution of the state space, but simply allows calculation of the aggregate probability of a finite set of classes of states from the state space, appropriately defined. Nonetheless, these aggregate probabilities allow for computation of a rich set of measures of interest such as the system queue length or any of its higher moments. The proposed solution approach is exact and, for the case of M/G/1-type processes, compares favorably to the classic methods as shown by detailed time and space complexity analysis. Detailed experimentation further corroborates that ETAQA provides significantly less expensive solutions when compared to the classic methods.

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M/G/1-Type Markov Processes: A Tutorial

Cited by 19 publications

References 23 publications

Workload Modeling for Computer Systems Performance Evaluation

Workload Modeling for Computer Systems Performance Evaluation

Bridging ETAQA and Ramaswami’s formula for the solution of M/G/1-type processes

ETAQA Solutions for Infinite Markov Processes with Repetitive Structure

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