Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process

Ozawa, Toshihisa

doi:10.1007/s11134-022-09860-w

Cited by 3 publications

(2 citation statements)

References 26 publications

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“…The explicit analytical representation for the stationary distribution of Md-QBDs is unknown, and most work has been devoted to deriving asymptotic formulas of the stationary distribution. Asymptotic properties of the stationary distribution of 2d-QBD processes were studied in [41][42][43][44]. The conditions ensuring a positive-recurrent or transient 2d-QBD process were analyzed in [45].…”

Section: Introductionmentioning

confidence: 99%

A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

Naumov

2024

Mathematics

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We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=ℤ+M×J, where ℤ+ is the set of non-negative integers and J={1,2,…,n}. The number of states in J may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states (k,i) and (n,j) may depend on i,j, and n−k but not on the specific values of k and n. It is shown that the stationary probability vector of the process is expressed through square matrices of order n, which are the minimal non-negative solutions to nonlinear matrix equations.

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Section: Introductionmentioning

confidence: 99%

A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

Naumov

2024

Mathematics

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“…Daduna [13] gives a detailed analysis of alternating birthdeath processes. 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.…”

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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