A large deviations principle for infinite-server queues in a random environment

Jansen, H. M.; Mandjes, Michel; Turck, Koen De; Wittevrongel, Sabine

doi:10.1007/s11134-015-9470-x

Cited by 15 publications

(9 citation statements)

References 21 publications

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“…where R(t, τ ) = cov G [X(t), X(τ )] is given in (18). We set δ = δ * := 1 b ; also b will be always chosen so that b ≥ σ −1 λ for large n. Under these conditions we finally obtain var G [μ G ] ≤ c 5 M σ 2γ−1 λ be 2σ λ (b+1/4) (λ 0 n) −1 .…”

Section: Repeatedly Integrating By Parts the Integralmentioning

confidence: 99%

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Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Goldenshluger

Koops

2019

Stochastic Systems

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This paper provides a mathematical framework for estimation of the service time distribution and the expected service time of an infinite-server queueing system with a non-homogeneous Poisson arrival process, in the case of partial information, where only the number of busy servers are observed over time. The problem is reduced to a statistical deconvolution problem, which is solved by using Laplace transform techniques and kernels for regularization. Upper bounds on the mean squared error of the proposed estimators are derived. Some concrete simulation experiments are performed to illustrate how the method can be applied and to provide some insight in the practical performance.Infinite-server queue First we provide some background on the M/G/∞ queueing model, which is well studied and could be considered as a standard model. In such queues there are arrivals according to a homogeneous Poisson process, each customer is served independently of all other customers and customers do not have to queue for service, because there is an infinite number of servers. The model has a wide variety of applications in e.g. telecommunication, road traffic and hospital modeling. It can be used as an approximation to M/G/n systems, where n is relatively large with respect to the arrival rate, but the model is also interesting in its own right. For example, it can be interpreted outside queueing theory as a model for the size of a population. In this paper we will use queueing terminology (servers, customers, etc.), but these terms could be adjusted according to the application. For example, 'customers' could be cars travelling between two locations and their 'service time' could be the travel time. In many applications it is seen that the arrival rate is not constant, but it varies over time. We will provide some examples below. This observation motivates studying the M t /G/∞ queue, where the arrival rate is assumed to be a nonhomogeneous Poisson process. This model is still particularly tractable (cf. [12]) and amenable for statistical analysis, as shown in this paper. Statistical queueing problemsQueueing theory studies probabilistic properties of random processes in service systems on the basis of a complete specification of system parameters. In this paper we are interested in inverse problems when unknown characteristics of a system should be inferred from observations of the associated random processes. Typically such observations are incomplete in the sense that individual customers cannot be tracked as they go through the service system. The importance of such statistical inverse problems with incomplete observations was emphasized in [3].The service time distribution and its expected value are important performance metrics of the infinite-server queue (note that waiting times are identical to service times in infinite-server queues). Our goal is to estimate these characteristics of the M t /G/∞ system from observations of the queue-length process. Specifically, let {τ j , j ∈ Z} be arrival epochs constituting a realizati...

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Section: Repeatedly Integrating By Parts the Integralmentioning

confidence: 99%

“…• Biology: As noted in [18], the production of molecules may be 'bursty', in the sense that periods of high production activity can be followed by periods of low activity. In [9], this is modeled using an interrupted Poisson process, i.e.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Goldenshluger

Koops

2019

Stochastic Systems

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“…However, to derive the large deviations principle, we take an approach that strongly differs from the approach in [12]. This different approach is inspired by the approach in [14] and leads to an expression of the large deviations rate function that is particularly amenable to numerical evaluation. The resulting expressions of the decay rate of the probability of interest are in terms of variational problems, with an insightful decomposition into (i) the impact of the background process and (ii) that of the driving Brownian motion (conditional on the background process).…”

Section: Dm (T) = (α(J(t)) − γ(J(t))m (T)) Dt + σ(J(t)) Db(t)mentioning

confidence: 99%

“…Related results on large deviations for modulated infinite-server queues can be found in e.g. [4,14,17], whereas diffusion-type processes are considered in e.g. [12,15].…”

Section: Dm (T) = (α(J(t)) − γ(J(t))m (T)) Dt + σ(J(t)) Db(t)mentioning

confidence: 99%

Rare-event analysis of modulated Ornstein–Uhlenbeck processes

Jansen

Mandjes

Turck

et al. 2017

Performance Evaluation

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ABSTRACT. This paper studies Ornstein-Uhlenbeck (OU) processes in a random environment. The OU model has found widespread use in networking, as a Gaussian approximation of the user-level dynamics that allows explicit analysis; adding modulation to it allows incorporating phenomena in which the users' activity level is affected by exogenous factors. The focus lies on rare-event analysis: under a specific scaling of the parameters involved, we establish the large deviations asymptotics of the probability that the process reaches an extreme value. The decay rate of this probability is generally only implicitly available (as the solution to a variational problem), but specializing to the case of Markov modulation we succeed in devising efficient numerical procedures.

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“…By contrast, to interpret the behavior of many aspects in our lives, there has been an increasing tendency to study queueing systems in a random environment. Excellent surveys on the infinite server queue in a random environment have been reported [5,7,9,18].…”

Section: Introductionmentioning

confidence: 99%

Fluid M/M/1 catastrophic queue in a random environment

Ammar

2021

RAIRO-Oper. Res.

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Our main objective in this paper is to investigate the stationary behavior of a fluid disaster queue of $M/M/1$ in a random multi-phase environment. Occasionally, the system experiences a catastrophic failure causing the loss of all current jobs. The system then goes into a process of repair. As soon as the system is repaired, it moves with probability \textit{q${}_{i}$} $\mathrm{\ge}$ 0 to phase \textit{i}. The distribution of the buffer content is determined using the probability generating function. In addition, some numerical results are provided to illustrate the effect of various parameters on the distribution of buffer content.

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A large deviations principle for infinite-server queues in a random environment

Cited by 15 publications

References 21 publications

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Rare-event analysis of modulated Ornstein–Uhlenbeck processes

Fluid M/M/1 catastrophic queue in a random environment

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