Maximum queue lengths during a fixed time interval in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si41.gif" overflow="scroll"><mml:mrow><mml:mi>M</mml:mi><mml:mo>/</mml:mo><mml:mi>M</mml:mi><mml:mo>/</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:math>retrial queue

Gómez‐Corral, Antonio; García, M.J.

doi:10.1016/j.amc.2014.02.074

Cited by 5 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The authors wish to thank an anonymous referee for some insightful comments and for pointing out references [2,3,10]. Michel Mandjes' research is partly funded by the NWO Gravitation Project NETWORKSgrant number 024.002.003.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…The exceptions that the authors are aware of have occurred in the retrial queue literature. There [3,10] provided a computational method to derive the distribution of the maximum queue length attained, either in a busy period or a fixed time, for an M/M/c retrial queue and [2] gave a similar analysis for a more general model of a call center queue, with a specific application to a retrial queue. The objective of the present paper is to determine the distribution of the running max-imumX(t) attained by a level-dependent QBD.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Running Maximum of a Level-Dependent Quasi-Birth-Death Process

Mandjes

Taylor

2015

Prob. Eng. Inf. Sci.

View full text Add to dashboard Cite

The objective of this note is to study the distribution of the running maximum of the level in a level-dependent quasi-birth-death process. By considering this running maximum at an exponentially distributed “killing epoch” T, we devise a technique to accomplish this, relying on elementary arguments only; importantly, it yields the distribution of the running maximum jointly with the level and phase at the killing epoch. We also point out how our procedure can be adapted to facilitate the computation of the distribution of the running maximum at a deterministic (rather than an exponential) epoch.

show abstract

Section: Acknowledgementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Running Maximum of a Level-Dependent Quasi-Birth-Death Process

Mandjes

Taylor

2015

Prob. Eng. Inf. Sci.

View full text Add to dashboard Cite

show abstract

“…QBDs exploit the tri-diagonal structure of transitions occurring across groups or levels of states, and summary statistics in these systems can be analysed by means of computationally efficient algorithms (Latouche and Ramaswami 1999). While this approach is standard in areas such as queuing theory (Bean et al 1997;Gómez-Corral and López-García 2014) and population dynamics (Gómez-Corral and López-García 2015), there is potential for it to be adopted more widely in applications of dynamics on networks in other areas of network science.…”

Section: Introductionmentioning

confidence: 99%

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

Ward

López‐García

2019

Appl Netw Sci

View full text Add to dashboard Cite

We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology.

show abstract

“…Specifically, the interest is in level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes (see, e.g., section 7.2 in the work of Artalejo and Gómez‐Corral), which are CTMCs in two dimensions, the level and the phase , such that the process only jumps across either adjacent levels or the same level in one transition. In analyzing LD‐QBD processes, matrix‐analytic methods are popular as modeling tools that allow us to construct and study, under a unified and algorithmic tractable framework, a variety of stochastic models, such as epidemic models, inventory problems, reliability systems, retrial queues, and two‐species competition processes, among others. The starting point in our analysis is the paper by Gaver et al, where the emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first‐passage times in finite LD‐QBD processes.…”

Section: Introductionmentioning

confidence: 99%

Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

Gómez‐Corral

López‐García

2018

Numerical Linear Algebra App

Self Cite

View full text Add to dashboard Cite

SummaryIn this paper, our interest is in the perturbation analysis of level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes, which constitute a wide class of structured Markov chains. An LD‐QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal‐by‐blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first‐passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD‐QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multitype versions of the susceptible‐infective (SI) and susceptible‐infective‐susceptible (SIS) epidemic models to the spread of antibiotic‐sensitive and antibiotic‐resistant bacterial strains in a hospital ward.

show abstract

Maximum queue lengths during a fixed time interval in theM/M/cretrial queue

Cited by 5 publications

References 21 publications

The Running Maximum of a Level-Dependent Quasi-Birth-Death Process

The Running Maximum of a Level-Dependent Quasi-Birth-Death Process

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

Contact Info

Product

Resources

About