The decay function of nonhomogeneous birth-death processes, with application to mean-field models

Granovsky, Boris L.; Zeifman, Alexander

doi:10.1016/s0304-4149(97)00085-9

Cited by 39 publications

(47 citation statements)

References 19 publications

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“…We refer to [2], [3], [4], [7], [8], [9], [12], [13], [15], [18], [29], [30], [31], [32], and the references there for further information. Information about λ n , the smallest eigenvalue of Q, and hence of C, may be obtained by observing that −λ n is the largest eigenvalue of −C.…”

Section: Birth-death Processesmentioning

confidence: 99%

“…So the speed of convergence of the mean-field model Φ to its stationary regime may be characterized by the rate of convergence of the corresponding birth-death process X , and hence the techniques of the previous section may be applied to study convergence of mean-field models. This approach was adopted in [12], [13] and [15].…”

Section: Mean-field Modelsmentioning

confidence: 99%

“…In the first place we want to survey the method for studying the rate of convergence of a birth-death process that one of us, inspired by [11], initiated in the early 1990's (see [29], [30] and [31]), and developed further in cooperation with Granovsky (see [12], [13] and [14]). Secondly, we want to present some new results obtained by the method, related in particular to the asymptotic behaviour of the rate of convergence for some specific birth-death processes as the number of states tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes

Doorn¹,

Zeifman²,

Panfilova³

2010

Theory Probab. Appl.

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Abstract. We survey a method initiated by one of us in the 1990's for finding bounds and representations for the rate of convergence of a birth-death process. We also present new results obtained by this method for some specific birth-death processes related to mean-field models and to the M/M/N/N +R service system. The new findings pertain to the asymptotic behaviour of the rate of convergence as the number of states tends to infinity.

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Section: Birth-death Processesmentioning

confidence: 99%

Section: Mean-field Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes

Doorn¹,

Zeifman²,

Panfilova³

2010

Theory Probab. Appl.

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“…If α 0 = α then the problem of determining α 0 can be reduced to that of finding the decay parameter in a pure birth-death process, for which many results are available (see [5], [6], [12], [18], [19], [23], [25], and [27]- [29]). Indeed, defineX := {X(t), t ≥ 0} to be the birth-death process on C with birth and death rates…”

Section: Theorem 2 If α > 0 and Eventual Extinction Is Certain Thenmentioning

confidence: 99%

Extinction Probability in A Birth-Death Process with Killing

Doorn

Zeifman²

2005

J. Appl. Probab.

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We study birth-death processes on the nonnegative integers, where {1, 2, . . . } is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

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“…Although first results were published more than 30 years ago by Gnedenko and Soloviev [12] and Gnedenko [11], a remarkable progress was achieved not a long time ago (see [13,14,16,23,[30][31][32][33][34][35][36]). In particular, Zeifman developed a number of effective tools, permitting investigate successfully ergodicity conditions for special classes of time-nonhomogeneous birth-and-death processes including M t /M t /1, M t /M t /S and M t /M t /S/0 queues (for further discussion, see [14]).…”

Section: Introductionmentioning

confidence: 99%

On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

Abramov

Liptser

2004

Queueing Systems

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Abstract. In this paper, sufficient conditions are given for the existence of limiting distribution of a nonhomogeneous countable Markov chain with time-dependent transition intensity matrix. The method of proof exploits the fact that if the distribution of random process Q = (Q t ) t 0 is absolutely continuous with respect to the distribution of ergodic random processwhere π is the invariant measure of Q • . We apply this result for asymptotic analysis, as t → ∞, of a nonhomogeneous countable Markov chain which shares limiting distribution with an ergodic birth-anddeath process.

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The decay function of nonhomogeneous birth-death processes, with application to mean-field models

Cited by 39 publications

References 19 publications

Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes

Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes

Extinction Probability in A Birth-Death Process with Killing

On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

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