Quasi-birth-and-death processes and multivariate orthogonal polynomials

Fernández, Lidia; Iglesia, Manuel D. de la

doi:10.1016/j.jmaa.2021.125029

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…The existence of such a formula is essentially due to the tridiagonal structure of the Q-matrix. For the same reason, it is delicate to extend such an approach to generalized BDPs: pure birth processes [14], one-sided skip-free CTMCs [13,15,16], and recently (higher-dimensional) quasi-birth-death processes (QBDPs) with tridiagonal block structure of the Q-matrix [22]. There are no restrictions on the types of transition rates for the results in [28] to hold, except the BDP assumption.…”

Section: Comparison With Results In the Literaturementioning

confidence: 99%

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

Hansen

Wiuf

2022

Adv. Appl. Probab.

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This paper provides a full classification of the dynamics for continuous-time Markov chains (CTMCs) on the nonnegative integers with polynomial transition rate functions and without arbitrary large backward jumps. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence versus transience, positive recurrence versus null recurrence, certain absorption, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and nonexistence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded forward jumps. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth–death processes. The approach is based on a mixture of Lyapunov–Foster-type results, the classical semimartingale approach, and estimates of stationary measures.

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Section: Comparison With Results In the Literaturementioning

confidence: 99%

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

Hansen

Wiuf

2022

Adv. Appl. Probab.

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“…Here, we are interested in the stationary distribution function at the slot edges, and then we have a birth-death model with β(0) � β. Terefore, (16) gives the stationary distribution function.…”

Section: Queues With Eas Rulementioning

confidence: 99%

“…Ozawa [14], and Ozawa and Kobayashi [15] , consider discrete-time two-dimensional quasi birthdeath processes. Fernández and de la Iglesia [16] study quasi-birth and death multivariate processes. Sasaki [17] gives examples of exactly solvable birth-death processes.…”

Section: Introductionmentioning

confidence: 99%

A Review of Birth-Death and Other Markovian Discrete-Time Queues

El-Taha

2023

Advances in Operations Research

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In this review article, we consider discrete-time birth-death processes and their applications to discrete-time queues. To make the analysis simpler to follow, we focus on transform-free methods and consider instances of non-birth-death Markovian discrete-time systems. We present a number of results within one discrete-time framework that parallels the treatment of continuous time models. This approach has two advantages; first, it unifies the treatment of several discrete-time models in one framework, and second, it parallels to the extent possible the treatment of continuous time models. This allows us to draw parallels and contrasts between the discrete and continuous time queues. Specifically, we focus on birth-death applications to the single server discrete-time model with Bernoulli arrivals and geometric service times and provide the reader with a simple rigorous detailed analysis that covers all five scheduling rules considered in the literature, with attention to stationary distributions at slot edges, slot centers, and prearrival epochs. We also cover the waiting time distributions. Moreover, we cover three Markovian models that fit the global balance equations. Our approach provides interesting insights into the behavior of discrete-time queues. The article is intended for those who are familiar with queueing theory basics and would like a simple, yet rigorous introductory treatment to discrete-time queues.

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“…This is illustrated by the following sampling. Karlin and McGregor themselves [13] and others [14,15,16,17] used multivariate orthogonal polynomials to analyze multidimensional or composition BDP. For applications of matrix orthogonal polynomials to quasi-birth and death processes one may consult [18] or [19,20] in the discrete time case and again for discrete time, random walks that multiple polynomials entail have been looked at recently in [21].…”

Section: Introductionmentioning

confidence: 99%

Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials

Miki,

Tsujimoto,

Vinet

2022

Preprint

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Quasi-birth-and-death processes and multivariate orthogonal polynomials

Cited by 7 publications

References 29 publications

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

A Review of Birth-Death and Other Markovian Discrete-Time Queues

Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials

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