A comparative analysis of the successive lumping and the lattice path counting algorithms

Abstract

This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al. (2009) andWinands (2006). The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models considered in Van Leeuwaarden et al. (2009). The main findings are threefold. First, when both methods are applicable, the SL-based algorithms outperform the lattice path counting algorithm (LPCA). Second, there are important classes of problems (for example, models with (level) nonhomogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. Third, another main advantage of SL algorithms over lattice path counting is that the former includes a method to compute the steady state distribution using this rate matrix.

Matrix geometric approach for random walks: Stability condition and equilibrium distribution

“…Regarding QBDs with a simple structure the literature is mainly devoted to stochastic processes for which the rate matrix has a simple property, see, e.g., [11] and the references therein, or to stochastic process with a special structure in the allowed transitions, see, e.g., [19,20] and the references therein. In [11], the authors consider an infinite rate matrix R that can be written as the product of a vector column times a row vector.…”

“…Additionally, they apply their results in the case of the Cox(k)/M Y /1 queue. In [19,20] and the references therein, the authors investigate stochastic processes with a special structure regarding the allowed transitions satisfying the so called successive lumpability property. This class of Markov chains is specified through its property of calculating the stationary probabilities of the system by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains.…”

“…Such models are a special class of the GI/M/1 or the M/G/1 type of QBD processes. In [19,20], the authors discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Furthermore, in [20], the authors compare the successive lumping methodology developed with the lattice path counting approach [28] for the calculation of the rate matrix of a queueing model.…”

“…In [19,20], the authors discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Furthermore, in [20], the authors compare the successive lumping methodology developed with the lattice path counting approach [28] for the calculation of the rate matrix of a queueing model. These two methodologies are compared both in terms of applicability requirements and numerical complexity by analysing their performance for some classical queueing models.…”

Time-dependent analysis of an M / M / c preemptive priority system with two priority classes

“…In Sleptchenko et al [27] an exact, recursive procedure is given for computing the joint stationary distribution of an M/M/1 preemptive priority queue that serves an arbitrary finite number of customer classes. The M/M/1 2-class priority model is briefly discussed in Katehakis et al [20], where they explain how the successive lumping technique can be used to study M/M/1 2-class priority models when both customer classes experience the same service rate. Interesting asymptotic properties of the stationary distribution of the M/M/1 2-class preemptive priority model can be found in the work of Li and Zhao [23].…”