Stability condition of a two-dimensional QBD process and its application to estimation of efficiency for two-queue models

Ozawa, Toshihisa

doi:10.1016/j.peva.2018.11.004

Cited by 15 publications

(19 citation statements)

References 8 publications

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“…). By Corollary 3.1 of Ozawa [23], the stability condition of the 2d-QBD process {Y n } is given as follows:…”

Section: Preliminaries 21 Stability Conditionmentioning

confidence: 99%

“…Each mean drift is represented in terms of the stationary distribution of the corresponding induced Markov chain, i.e., the background process of the corresponding induced MA-process; for their expressions, see Subsection 3.1 of Ozawa [23] and its related parts. We assume the following condition throughout the paper.…”

Section: Preliminaries 21 Stability Conditionmentioning

confidence: 99%

“…Stochastic models arising from various Markovian two-queue models and two-node queueing networks such as two-queue polling models and generalized two-node Jackson networks with Markovian arrival processes and phase-type service processes can be represented as two-dimensional continuous-time QBD processes, and their stationary distributions can be analyzed through the corresponding two-dimensional discrete-time QBD processes obtained by the uniformization technique; See, for example, Refs. [14,20,23]. In that sense, 2d-QBD processes are more versatile than 2d-RRWs, which have no background processes.…”

Section: Introductionmentioning

confidence: 99%

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

Ozawa¹

2022

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We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) {(X n , J n )} on Z 2 + × S 0 , where X n = (X 1,n , X 2,n ) is the level state, J n the phase state (background state) and S 0 a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in any direction. This result is an extension of the corresponding result for a certain two-dimensional reflecting random work without background processes, obtained by using the large deviation techniques. We also present a condition ensuring the sequence of the stationary probabilities geometrically decays without power terms, asymptotically. Asymptotic properties of the stationary tail distribution in the coordinate directions in a 2d-QBD process have already been studied in the literature. The results of this paper are also important complements to those results.

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“…). By Corollary 3.1 of Ozawa [23], the stability condition of the 2d-QBD process {Y n } is given as follows:…”

Section: Preliminaries 21 Stability Conditionmentioning

confidence: 99%

Section: Preliminaries 21 Stability Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Ozawa¹

2022

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“…Note that our model is described by a non-homogeneous Markov modulated two-dimensional nearest-neighbour random walk, and its stability condition, to our best knowledge, is still an open problem. We mention here that the stability condition of a homogeneous Markov modulated two-dimensional nearest neighbour random walk was recently investigated in [32] by using the concept of induced Markov chains.…”

Section: On the Stability Conditionmentioning

confidence: 99%

The generalized join the shortest orbit queue system: Stability, exact tail asymptotics and stationary approximations

Dimitriou¹

2021

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We introduce the generalized join the shortest queue model with retrials and two infinite capacity orbit queues. Three independent Poisson streams of jobs, namely a smart, and two dedicated streams, flow into a single server system, which can hold at most one job. Arriving jobs that find the server occupied are routed to the orbits as follows: Blocked jobs from the smart stream are routed to the shortest orbit queue, and in case of a tie, they choose an orbit randomly. Blocked jobs from the dedicated streams are routed directly to their orbits. Orbiting jobs retry to connect with the server at different retrial rates, i.e., heterogeneous orbit queues. Applications of such a system are found in the modelling of wireless cooperative networks. We are interested in the asymptotic behaviour of the stationary distribution of this model, provided that the system is stable. More precisely, we investigate the conditions under which the tail asymptotic of the minimum orbit queue length is exactly geometric. Moreover, we apply a heuristic asymptotic approach to obtain approximations of the steady-state joint orbit queue-length distribution. Useful numerical examples are presented and shown that the results obtained through the asymptotic analysis and the heuristic approach agreed.

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“…A discretetime two-dimensional quasi-birth-and-death process [9] (2d-QBD process for short) is a 2d-MMRW with reflecting boundaries on the x 1 and x 2 -axes, where the process (X 1,n , X 2,n ) is the level and J n the phase. Stochastic models arising from various Markovian two-queue models and two-node queueing networks such as two-queue polling models and generalized two-node Jackson networks with Markovian arrival processes and phase-type service processes can be represented as continuoustime 2d-QBD processes (see, for example, [7] and [9,10,11]) and, by using the uniformization technique, they can be deduced to discrete-time 2d-QBD processes. In that sense, (discrete-time) 2d-QBD processes are more versatile than two-dimensional skip-free reflecting random walks (2d-RRWs for short), which are 2d-QBD processes without a phase process and called double QBD processes in [4].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk

Ozawa

2020

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We consider a discrete-time two-dimensional process {(X 1,n , X 2,n )} on Z 2 with a background process {J n } on a finite set S 0 , where individual processes {X 1,n } and {X 2,n } are both skip free. We assume that the joint process {Y n } = {(X 1,n , X 2,n , J n )} is Markovian and that the transition probabilities of the two-dimensional process {(X 1,n , X 2,n )} vary according to the state of the background process {J n }. This modulation is assumed to be space homogeneous. We refer to this process as a two-dimensional skip-free Markov modulate random walk. For y, y ∈ Z 2 + × S 0 , consider the process {Y n } n≥0 starting from the state y and let qy,y be the expected number of visits to the state y before the process leaves the nonnegative area Z 2 + × S 0 for the first time. For y = (x 1 , x 2 , j) ∈ Z 2 + × S 0 , the measure (q y,y ; y = (x 1 , x 2 , j ) ∈ Z 2 + × S 0 ) is called an occupation measure. Our main aim is to obtain asymptotic decay rate of the occupation measure as the values of x 1 and x 2 go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.

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Stability condition of a two-dimensional QBD process and its application to estimation of efficiency for two-queue models

Cited by 15 publications

References 8 publications

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

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

The generalized join the shortest orbit queue system: Stability, exact tail asymptotics and stationary approximations

Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk

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