A Successive Lumping Procedure for a Class of Markov Chains

Katehakis, Michael N.; Smit, Laurens C.

doi:10.1017/s0269964812000150

Cited by 30 publications

(46 citation statements)

References 19 publications

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“…For the proof of part (ii), using Proposition 1 of [21] (where π Xm 2 (m, i) = v L m (m, i)) we obtain that Eq. (18) below is valid for all m ≤ ν:…”

Section: State Space Truncationsmentioning

confidence: 96%

“…This theorem is a consequence of Theorem 2 of [21]; to avoid the introduction of the notation used in the latter paper, we provide a separate proof in the appendix. Theorem 1.…”

Section: Explicit Solutionsmentioning

confidence: 99%

“…This is true for all m ≥ M 1 . Since L m = L m−1 ∪ L m it follows from the definition that the chain is successively lumpable: in the notation of [21], "D 0 " corresponds to L M1 and for m > M 1 :…”

Section: The Class Of Des Processesmentioning

confidence: 99%

“…. , L M2 } of the state space X of the chain, for any fixed m. We note that the sets Δm of [21] within the present context, are given by: Δ k = {(k 0)} ∪ L k ; where (k 0) represents the "lumped state".…”

mentioning

confidence: 99%

“…Remark A.A.1). By Proposition 1 of [21] (with (k, i) ∈ L m in place of (0, i) ∈ Δ 0 ) we know that for all k ≤ m: Copyright of Probability in the Engineering & Informational Sciences is the property of Cambridge University Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.…”

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confidence: 99%

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Des and Res Processes and Their Explicit Solutions

Katehakis

Smit

Spieksma³

2014

Prob. Eng. Inf. Sci.

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This paper defines and studies the down entrance state (DES) and the restart entrance state (RES) classes of quasi-skip free (QSF) processes specified in terms of the nonzero structure of the elements of their transition rate matrix Q. A QSF process is a Markov chain with states that can be specified by tuples of the form (m, i), where m ∈ Z represents the "current" level of the state and i ∈ Z + the current phase of the state, and its transition probability matrix Q does not permit one-step transitions to states that are two or more levels away from the current state in one direction of the level variable m. A QSF process is a DES process if and only if one step "down" transitions from a level m can only reach a single state in level m − 1, for all m. A QSF process is a RES process if and only if one step "up" transitions from a level m can only reach a single set of states in the highest level M 2 , largest of all m.We derive explicit solutions and simple truncation bounds for the steady-state probabilities of both DES and RES processes, when in addition Q insures ergodicity. DES and RES processes have applications in many areas of applied probability comprising computer science, queueing theory, inventory theory, reliability, and the theory of branching processes. To motivate their applicability we present explicit solutions for the well-known open problem of the M/Er/n queue with batch arrivals, an inventory model, and a reliability model.

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“…For the proof of part (ii), using Proposition 1 of [21] (where π Xm 2 (m, i) = v L m (m, i)) we obtain that Eq. (18) below is valid for all m ≤ ν:…”

Section: State Space Truncationsmentioning

confidence: 96%

“…This theorem is a consequence of Theorem 2 of [21]; to avoid the introduction of the notation used in the latter paper, we provide a separate proof in the appendix. Theorem 1.…”

Section: Explicit Solutionsmentioning

confidence: 99%

Section: The Class Of Des Processesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Des and Res Processes and Their Explicit Solutions

Katehakis

Smit

Spieksma³

2014

Prob. Eng. Inf. Sci.

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Level product form QSF processes and an analysis of queues with Coxian interarrival distribution

Ertiningsih

Katehakis

Smit

et al. 2016

Naval Research Logistics

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Abstract:In this article, we study a class of Quasi-Skipfree (QSF) processes where the transition rate submatrices in the skipfree direction have a column times row structure. Under homogeneity and irreducibility assumptions we show that the stationary distributions of these processes have a product form as a function of the level. For an application, we will discuss the Cox(k)/M Y /1 -queue that can be modeled as a QSF process on a two-dimensional state space. In addition, we study the properties of the stationary distribution and derive monotonicity of the mean number of the customers in the queue, their mean sojourn time and the variance as a function of k for fixed mean arrival rate.

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Reasoning About Proportional Lumpability

Piazza

Rossi

2021

Quantitative Evaluation of Systems

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In this paper we reason about the notion of proportional lumpability, that generalizes the original definition of lumpability to cope with the state space explosion problem inherent to the computation of the performance indices of large stochastic models. Lumpability is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Proportional lumpability formalizes the idea that the transition rates of a Markov chain can be altered by some factors in such a way that the new resulting Markov chain is lumpable. It allows one to derive exact performance indices for the original process. We prove that the problem of computing the coarsest proportional lumpability which refines a given initial partition is well-defined, i.e., it has always a unique solution. Moreover, we introduce a polynomial time algorithm for solving the problem. This provides us further insights on both the notion of proportional lumpability and on generalizations of partition refinement techniques.

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A Successive Lumping Procedure for a Class of Markov Chains

Cited by 30 publications

References 19 publications

Des and Res Processes and Their Explicit Solutions

Des and Res Processes and Their Explicit Solutions

Level product form QSF processes and an analysis of queues with Coxian interarrival distribution

Reasoning About Proportional Lumpability

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