Tail asymptotics of a Markov-modulated infinite-server queue

Blom, Joke; Turck, Koen De; Kella, Offer; Mandjes, Michel

doi:10.1007/s11134-014-9412-z

Cited by 10 publications

(26 citation statements)

References 13 publications

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“…Again, the value of α determines what type of tail behavior dominates: for α < 1 this is effect (i), for α > 1 effect (ii), and for α = 1 a combination of effects (i) and (ii). These findings complement similar results that have been established for an infinite-server system with Markov-modulated input, where it is noted that the slow regime (α ∈ (0, 1)) was not covered in that setting [3,6]. We conclude Section 4 by pointing out how the large deviations results can be extended to the multidimensional setting.…”

Section: Introductionsupporting

confidence: 85%

“…For instance, one could use Markov-modulated Poisson processes (MMPPs) in which the arrival rate Λ(t) = λ J(t) is a function of a continuous-time Markov chain J(·) on a finite state space S and non-negative rates λ i for i ∈ S (see e.g. [1,3]). Although the MMPP is versatile and has various attractive properties, it has considerable drawbacks as well.…”

Section: Introductionmentioning

confidence: 99%

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Scaling limits for infinite-server systems in a random environment

Heemskerk¹,

Leeuwaarden²,

Mandjes³

2017

Stoch. Syst.

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Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditionsThis article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval, unless otherwise noted. For more information, contact permissions@informs.org.The Publisher does not warrant or guarantee the article's accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. Copyright © 2017, The author(s)Please scroll down for article-it is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. In our setup, a random environment is modeled by drawing an arrival rate Λ from a given distribution every Δ time units, yielding an i.i.d. sequence of arrival rates Λ1, Λ2, . . .. Applying a martingale central limit theorem, we obtain a functional central limit theorem for the scaled queue length process. We proceed to large deviations and derive the logarithmic asymptotics of the queue length's tail probabilities. As it turns out, in a rapidly changing environment (i.e., Δ is small relative to Λ) the overdispersion of the arrival process hardly affects system behavior, whereas in a slowly changing random environment it is fundamentally different; this general finding applies to both the central limit and the large deviations regime. We extend our results to the setting where each arrival creates a job in multiple infinite-server queues.

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Section: Introductionsupporting

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Scaling limits for infinite-server systems in a random environment

Heemskerk¹,

Leeuwaarden²,

Mandjes³

2017

Stoch. Syst.

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“…In the proofs we have seen that overflow is most likely caused by φ t ( J ) attaining a value 'close to' its maximal value a (+ , i ) t , which only happens when the jump epochs are close to those of some maximizing path (that was explicitly determined in Blom et al (2014) and for Models i and ii , respectively). We saw that the probability of φ t ( J ) being an amount in the order of δ away from its max-…”

Section: Computational Issuesmentioning

confidence: 91%

“…The proof of this property consists of two steps, and relies on the property that M ( N ) ( t ) obeys a Poisson distribution with random parameter (as was observed in e.g. Blom et al, 2014;D'Auria, 2008 ).…”

Section: (2)mentioning

confidence: 99%

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Refined large deviations asymptotics for Markov-modulated infinite-server systems

Blom

Turck

Mandjes

2017

European Journal of Operational Research

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a b s t r a c tMany networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers' arrival rates and service rates are modulated by a Markovian background process; additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obtain all moments. As a consequence, recent research efforts have pursued an asymptotic analysis in various limiting regimes, notably the central-limit regime (describing fluctuations around the average behavior) and the largedeviations regime (focusing on rare events). Many of these results use the property that the number of customers in the system obeys a Poisson distribution with a random parameter. The objective of this paper is to develop techniques to accurately approximate tail probabilities in the large-deviations regime. We consider the scaling in which the arrival rates are inflated by a factor N , and we are interested in the probability that the number of customers exceeds a given level Na . Where earlier contributions focused on so-called logarithmic asymptotics of this exceedance probability (which are inherently imprecise), the present paper improves upon those results in that exact asymptotics are established. These are found in two steps: first the distribution of the random parameter of the Poisson distribution is characterized, and then this knowledge is used to identify the exact asymptotics. The paper is concluded by a set of numerical experiments, in which the accuracy of the asymptotic results is assessed.

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Queues with Variable Service Speeds: Exact Results and Scaling Limits

Yajima,

Phung-Duc

2022

The Palgrave Handbook of Operations Research

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Tail asymptotics of a Markov-modulated infinite-server queue

Cited by 10 publications

References 13 publications

Scaling limits for infinite-server systems in a random environment

Scaling limits for infinite-server systems in a random environment

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Queues with Variable Service Speeds: Exact Results and Scaling Limits

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