Environmental Public Health Tracking/Surveillance in Canada:A Commentary

This paper studies an infinite-server queue in a random environment, meaning that the arrival rate, the service requirements, and the server work rate are modulated by a general càdlàg stochastic background process. To prove a large deviations principle, the concept of attainable parameters is introduced. Scaling both the arrival rates and the background process, a large deviations principle for the number of jobs in the system is derived using attainable parameters. Finally, some known results about Markov-modulated infinite-server queues are generalized and new results for several background processes and scalings are established in examples

Large deviations of an infinite-server system with a linearly scaled background process

This paper studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. We focus on the probability that the number of jobs in the system attains an unusually high value. Scaling the arrival rates ¿i¿i by a factor NN and the transition rates ¿ij¿ij of the background process as well, a large-deviations based approach is used to examine such tail probabilities (where NN tends to 88). The paper also presents qualitative properties of the system’s behavior conditional on the rare event under consideration happening. Keywords: Queues; Infinite-server systems; Markov modulation; Large deviation

Rare event analysis of Markov-modulated infinite-server queues: a Poisson limit

This article studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates (i) by a factor N and the rates (ij) of the background process by N1+E (for some E>0), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that N tends to . In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean

Functional Large Deviations for Cox Processes and Cox/G/∞Cox/G/\infty Queues, with a Biological Application

We consider an infinite-server queue into which customers arrive according to a Cox process and have independent service times with a general distribution. We prove a functional large deviations principle for the equilibrium queue length process. The model is motivated by a linear feed-forward gene regulatory network, in which the rate of protein synthesis is modulated by the number of RNA molecules present in a cell. The system can be modelled as a tandem of infinite-server queues, in which the number of customers present in a queue modulates the arrival rate into the next queue in the tandem. We establish large deviation principles for this queueing system in the asymptotic regime in which the arrival process is sped up, while the service process is not scaled.Comment: 36 pages, 2 figures, to appear in Annals of Applied Probabilit

Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation

We consider a linear stochastic fluid network under Markov modulation, with a focus on the probability that the joint storage level attains a value in a rare set at a given point in time. The main objective is to develop efficient importance sampling algorithms with provable performance guarantees. For linear stochastic fluid networks without modulation, we prove that the number of runs needed (so as to obtain an estimate with a given precision) increases polynomially (whereas the probability under consideration decays essentially exponentially); for networks operating in the slow modulation regime, our algorithm is asymptotically efficient. Our techniques are in the tradition of the rare-event simulation procedures that were developed for the sample-mean of i.i.d. one-dimensional light-tailed random variables, and intensively use the idea of exponential twisting. In passing, we also point out how to set up a recursion to evaluate the (transient and stationary) moments of the joint storage level in Markov-modulated linear stochastic fluid networks

Tail asymptotics of a Markov-modulated infinite-server queue

This paper analyzes large deviation probabilities related to the number of customers in a Markov modulated infinite-server queue, with state-dependent arrival and service rates. Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by $N$ (for large $N$), whereas in the second in addition the Markovian background process is sped up by a factor $N^{1+\epsilon}$, for some $\epsilon>0$. In both regimes, (transient and stationary) tail probabilities decay essentially exponentially, where the associated decay rate corresponds to that of the probability that the sample mean of i.i.d.\ Poisson random variables attains an atypical value

Rare event analysis of Markov-modulated infinite-service queues: A Poisson limit

This paper studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates $\lambda_i$ by a factor $N$ and the rates $\nu_{ij}$ of the background process by N^{1+\vareps} (for some \vareps > 0), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that $N$ tends to $\infty$. In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean