Networks of infinite-server queues with multiplicative transitions

Fiems, Dieter; Mandjes, Michel; Patch, Brendan

doi:10.1016/j.peva.2018.03.003

Cited by 8 publications

(5 citation statements)

References 25 publications

(21 reference statements)

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“…Examples of papers on random processes on (dynamic) random graphs are [3,9,10]. The systematic study of queueing process on such a randomly evolving graph has hardly been looked at, a notable exception being the recent study [13]. The model considered in [13] complements the one studied in the present work.…”

Section: Introductionmentioning

confidence: 90%

“…The systematic study of queueing process on such a randomly evolving graph has hardly been looked at, a notable exception being the recent study [13]. The model considered in [13] complements the one studied in the present work. Most notably, the framework of [13] in particular facilitates modelling the effect of nodes going down every now and then, where the present paper has a focus on links going down.…”

Section: Introductionmentioning

confidence: 92%

“…Most notably, the framework of [13] in particular facilitates modelling the effect of nodes going down every now and then, where the present paper has a focus on links going down. The immediate consequence of this difference in modelling, is that in the framework of [13] diffusion limits do not apply, due to the instantaneous downward jumps of the network population vector at epochs that a node fails. Our analysis will be based on casting our model as a network of infinite-server queues under Markov modulated arrival and service rates.…”

Section: Introductionmentioning

confidence: 99%

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Queues on a Dynamically Evolving Graph

2018

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This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in parallel. The links that connect the queues have the special feature that they are unreliable, in the sense that their status alternates between 'up' and 'down'. If a link between two nodes is down, with a fixed probability each of the clients attempting to use that link is lost; otherwise the client remains at the origin node and reattempts using the link (and jumps to the destination node when it finds the link restored). For these networks we present the following results: (a) a system of coupled partial differential equations that describes the joint probability generating function corresponding to the queues' time-dependent behavior (and a system of ordinary differential equations for its stationary counterpart), (b) an algorithm to evaluate the (time-dependent and stationary) moments, and procedures to compute user-perceived performance measures which facilitate the quantification of the impact of the links' outages, (c) a diffusion limit for the joint queue length process. We include explicit results for a series relevant special cases, such as tandem networks and symmetric fully connected networks.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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Queues on a Dynamically Evolving Graph

2018

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“…Remark 3. Observe that Fiems et al [25] obtain moments for a network that contains the present tandem system as a special case, except that binomial transitions do not fit in their model; they present equations from which transient moments can be obtained, but they do not give explicit expressions for stationary moments.…”

Section: Momentsmentioning

confidence: 99%

ASIP tandem queues with consumption

Yeger,

Boxma,

Resing

et al. 2024

SIGMETRICS Perform. Eval. Rev.

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The Asymmetric Inclusion Process (ASIP) tandem queue is a model of stations in series with a gate after each station. At a gate opening, all customers in that station instantaneously move to the next station unidirectionally. We enhance the ASIP model by introducing the capability for individual customers to independently move from one station to the next, and by allowing both individual customers and batches from any station to exit the system.

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“…Plenty of work on multidimensional level process has been devoted in deriving asymptotic formulas of the stationary distributions and studying the stability conditions [34,35,36,26,30,31] (not exhaustive list). We also refer to [17], that dealt with a network of infinite server queues in which the joint probability generating function (pgf) is characterized in terms of a system of partial differential equations; see also [23,4,3] for some recent studies on Markov modulated queueing systems.…”

Section: Related Workmentioning

confidence: 99%

Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

Dimitriou¹

2020

Preprint

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In this work, we focus on the stationary analysis of a specific class of continuous time Markov modulated reflected random walk in the quarter plane with applications in the modelling of twonode Markov-modulated queueing networks with coupled queues. The transition rates of the twodimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. Using a queueing network example, we numerically validated the theoretical findings.

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Networks of infinite-server queues with multiplicative transitions

Cited by 8 publications

References 25 publications

Queues on a Dynamically Evolving Graph

Queues on a Dynamically Evolving Graph

ASIP tandem queues with consumption

Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

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