Unimodality preservation in Markov chains

Keilson, Julian; Kester, Adri

doi:10.1016/0304-4149(78)90016-9

Cited by 24 publications

(8 citation statements)

References 3 publications

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“…Since the p.m.f. 's are unimodal in y for each 0 and t, see Keilson and Kester (1978), so is any limit as O~O, which implies that any limit function is integrable. As in Theorem 3.5, we can apply the Lebesgue dominated convergence theorem (with Lemma 10.1) to get convergence of the associated c.d.f.…”

Section: Connections To Rbmmentioning

confidence: 97%

Transient behavior of the M/M/1 queue via Laplace transforms

Abate

Whitt

1988

Advances in Applied Probability

103

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This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in the M/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as . All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtaining M/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

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Section: Connections To Rbmmentioning

confidence: 97%

Transient behavior of the M/M/1 queue via Laplace transforms

Abate

Whitt

1988

Advances in Applied Probability

103

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“…s enough to ensure the TP 2 (RR 2 ) property of F (see [8], [10]). We next introduce an important class of discrete distributions.…”

Section: Some Preliminariesmentioning

confidence: 99%

Further monotonicity properties of renewal processes

Kijima

1992

Advances in Applied Probability

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In a discrete-time renewal process {Nk, k = 0, 1, ·· ·}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k = 0, 1, ·· ·} and {Zk, k = 0, 1, ·· ·} are monotonically non-decreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m – Nk, k = 0, 1, ·· ·} to be non-increasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

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“…Lindvall (1992) and Kijima (1997) are other prominent references in which stochastic orderings of various types are used to assess stability and other chain properties. Keilson and Kester (1978), A monotonicity in reversible Markov chains 487 Brown (1980), Shaked and Shanthikumar (1987), (1994), Liggett (1989), Hansen and Frenk (1991), and Lund (2001), (2002) are other references linking renewal theory, Markov chains, and stochastic orderings.…”

Section: Introductionmentioning

confidence: 99%

A monotonicity in reversible Markov chains

Lund

Zhao

Kiessler

2006

Journal of Applied Probability

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In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times to every state in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.

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Unimodality preservation in Markov chains

Cited by 24 publications

References 3 publications

Transient behavior of the M/M/1 queue via Laplace transforms

Transient behavior of the M/M/1 queue via Laplace transforms

Further monotonicity properties of renewal processes

A monotonicity in reversible Markov chains

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