A stable recursion for the steady state vector in markov chains of m/g/1 type

Ramaswami, V.

doi:10.1080/15326348808807077

Cited by 263 publications

(102 citation statements)

References 6 publications

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“…The stationary probability vectors π 0,i can be calculated recursively using the Ramaswami formula [10]. Tailoring it to this particular system gives:…”

Section: Analysis Of the Zero Levelmentioning

confidence: 99%

Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

Horváth¹

2012

Performance Evaluation

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The analysis of priority queues where both the arrival and the service processes are correlated does not have a long literature. Only a few results are known, that attack the problem with the matrix geometric machinery. Unfortunately, these results have some restrictions that limit their usability significantly, for example they require the calculation of infinite series of matrices and infinite summations, that can be implemented only by truncation.The method presented in this paper calculates only the queue length moments, but without relying on infinite series of matrices and provides procedures to calculate the arising infinite sums accurately in an efficient way by means of linear equations, matrix-quadratic equations and a coupled matrix-quadratic equation.The numerical examples demonstrate that the presented method is several orders of magnitudes faster than the existing ones. From the large number of queue length moments it is possible to obtain lower and upper bounds for the queue length distribution by using existing moment based distribution estimation results.

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“…The stationary probability vectors π 0,i can be calculated recursively using the Ramaswami formula [10]. Tailoring it to this particular system gives:…”

Section: Analysis Of the Zero Levelmentioning

confidence: 99%

Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

Horváth¹

2012

Performance Evaluation

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“…The key to the matrix-analytical approach is the solution to a nonlinear matrix equation which can be solved by quadratical convergence rate algorithms like the logarithmic reduction algorithm for [14] and the iterative scheme of [15] for QBD (Quasi Birth and Death) type Markov chains, and the cyclic reduction algorithm of [16], the invariant subspace approach of [17], and the technique proposed in [18] for M/G/1 type Markov chains. Once a solution for this matrix equation is obtained, one can then find the queue length probabilities recursively [19]. Given the steady-state queue length probabilities, the waiting time distribution and its moments can be obtained, although not in a very compact form [2].…”

Section: Introductionmentioning

confidence: 99%

System-theoretical algorithmic solution to waiting times in semi-Markov queues

Akar¹,

Sohraby²

2009

Performance Evaluation

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a b s t r a c tMarkov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples.

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“…In this paper we present an exact method, based on matrix-analytic techniques [22,23] to determine the equilibrium joint queue length distribution. In particular, it appears to be possible to avoid the use of infinite series and truncation of the state space.…”

Section: Introductionmentioning

confidence: 99%

Joint queue length distribution of multi-class, single-server queues with preemptive priorities

et al. 2015

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In this paper we analyze an M/M/1 queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments, we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an M/M/1 queueing system with non-preemptive priorities between customer classes.

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A stable recursion for the steady state vector in markov chains of m/g/1 type

Cited by 263 publications

References 6 publications

Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

System-theoretical algorithmic solution to waiting times in semi-Markov queues

Joint queue length distribution of multi-class, single-server queues with preemptive priorities

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