Matrix geometric approach for random walks: Stability condition and equilibrium distribution

Kapodistria, Stella; Palmowski, Zbigniew

doi:10.1080/15326349.2017.1359096

Cited by 11 publications

(7 citation statements)

References 34 publications

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“…Proof 9 1. Using the indexing of the compensation parameters (18), Since as i → ∞, γ i → 0, δ i−1 → 0, and δi−1 δi = δi−1 γi γi δi → w + 1 w − , the assertion 2 is now proved.…”

Section: Lemmamentioning

confidence: 85%

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Analysis of the Symmetric Join the Shortest Orbit Queue

Dimitriou¹

2020

Preprint

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This work introduces the join the shortest queue policy in the retrial setting. We consider a Markovian single server retrial system with two infinite capacity orbits. An arriving job finding the server busy, it is forwarded to the least loaded orbit. Otherwise, it is forwarded to an orbit randomly. Orbiting jobs of either type retry to access the server independently. We investigate the stability condition, the stationary tail decay rate, and obtain the equilibrium distribution by using the compensation method.

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Section: Lemmamentioning

confidence: 85%

“…The power series algorithm (PSA) was applied in JSQ systems in [7,8]; see also [12,17] (non-exhaustive list) for more complicated models. We also mention the matrix geometric method; see e.g., [15,23], for which connections with CM was recently reported in [18].…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Symmetric Join the Shortest Orbit Queue

Dimitriou¹

2020

Preprint

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“…Numerical/approximation methods were also applied: see the power series algorithm (PSA), e.g., [8], and the matrix geometric method; see e.g., [16,35], for which connections with CM was recently reported in [18]. PSA is numerically satisfactory for relatively lower dimensional models, although, the theoretical foundation of this method is still incomplete.…”

Section: Related Workmentioning

confidence: 99%

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

Dimitriou¹

2021

Preprint

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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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“…Additional results have been derived recently (see e.g., [16,14,18,9,2]). We also refer the reader to a recent paper by Kapodistria and Palmowski [10] and references therein, in which MG approach for solving certain random walk processes is discussed where both the phase and the level dimensions are unbounded.…”

Section: Contributionmentioning

confidence: 99%

Explicit Solutions for Continuous-Time QBD Processes by Using Relations Between Matrix Geometric Analysis and the Probability Generating Functions Method

Hanukov¹,

Yechiali²

2020

Prob. Eng. Inf. Sci.

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Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$ ; and (ii) the stability condition is readily extracted.

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Matrix geometric approach for random walks: Stability condition and equilibrium distribution

Cited by 11 publications

References 34 publications

Analysis of the Symmetric Join the Shortest Orbit Queue

Analysis of the Symmetric Join the Shortest Orbit Queue

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

Explicit Solutions for Continuous-Time QBD Processes by Using Relations Between Matrix Geometric Analysis and the Probability Generating Functions Method

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