Semi-Markov-Modulated Infinite-Server Queues: Approximations by Time-Scaling

Hellings, Ton; Mandjes, Michel; Blom, Joke

doi:10.1080/15326349.2012.699759

Cited by 8 publications

(16 citation statements)

References 11 publications

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“…Whereas in [4] we proved that under the Poisson scaling the input process converges to a Poisson process with rate λ ∞ , we show in Section 3 that the steadystate number of customers in the system converges to a Poisson distribution with mean µ. The transient variant of this result is established in Section 4.…”

Section: Introductionmentioning

confidence: 99%

“…STEADY-STATE, POISSON REGIME Let, following [4], M i denote the steady-state number of customers in the system when the background process enters state i. We denote by γ i (·) the probability generating function of M i : γ i (z) := Ez M i .…”

Section: Model Descriptionmentioning

confidence: 99%

“…Other variants of the semi-Markov-modulated queues, such as the single-server counterpart, have been widely studied (see for example the discussion in the introduction of [4]); considerably less attention has been paid to the infinite-server case. Without aiming at giving a full account of the existing literature, we mention that results for the system's steady-state behavior (mostly in terms of factorial moments) have already been available for some time; see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Essentially two effects are combined. By scaling the transition times by N 1+ε , by virtue of the findings in [4], the input process converges to a Poisson process with rate λ ∞ . The effect of scaling the λ i s as well is that a central-limit type of regime kicks in, as described in e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Also Markov-modulated infinite server queues with Erlang or hyperexponentially distributed service times have been addressed [3] . A new line of research started in [4], where time-scalings are imposed so as to obtain explicit expressions for the resulting limiting distribution. The main result of that paper relates to speeding up the transition times by a factor N ; the resulting arrival process turns out to be a Poisson process (with a rate λ ∞ that can be given explicitly in terms of the λ i and the invariant associated with the generator of the background process).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Time-Scaling Limits for Markov-Modulated Infinite-Server Queues

2013

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ABSTRACT. In this paper we study semi-Markov modulated M/M/∞ queues, which are to be understood as infinite-server systems in which the Poisson input rate is modulated by a Markovian background process (where the times spent in each of its states are assumed deterministic), and the service times are exponential. Two specific scalings are considered, both in terms of transient and steady-state behavior. In the former the transition times of the background process are divided by N , and then N is sent to ∞; a Poisson limit is obtained. In the latter both the transition times and the Poissonian input rates are scaled, but the background process is sped up more than the arrival process; here a central-limit type regime applies. The accuracy and convergence rate of the limiting results are demonstrated with numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Section: Model Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time-Scaling Limits for Markov-Modulated Infinite-Server Queues

2013

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A Central Limit Theorem for Markov-Modulated Infinite-Server Queues

Blom

Turck

Mandjes

2013

Analytical and Stochastic Modeling Techniques and Applications

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Markov-modulated infinite-server queues with general service times

et al. 2013

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ABSTRACT. This paper studies an infinite-server queue in a Markov environment, that is, an infiniteserver queue with an arrival rate that equals λ i when an external Markov process is in state i. The service times have a general distribution that depends on the state of the background process upon arrival. We start by setting up explicit formulas for the mean and variance of the number of particles in the system at time t ≥ 0, given the system started empty. The special case of exponential service times is studied in detail, resulting in a recursive scheme to compute the moments of the number of customers at an exponentially distributed time, as well as the steady-state moments. Then we consider an asymptotic regime in which the arrival rates are sped up by a factor N , and the transition times by a factor N 1+ε (for some ε > 0). Under this scaling it turns out that the number of customers at time t ≥ 0 is asymptotically Normally distributed; in addition convergence of finite-dimensional distributions is proven.

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Semi-Markov-Modulated Infinite-Server Queues: Approximations by Time-Scaling

Cited by 8 publications

References 11 publications

Time-Scaling Limits for Markov-Modulated Infinite-Server Queues

Time-Scaling Limits for Markov-Modulated Infinite-Server Queues

A Central Limit Theorem for Markov-Modulated Infinite-Server Queues

Markov-modulated infinite-server queues with general service times

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