Analysis of Markov-Modulated Infinite-Server Queues in the Central-Limit Regime

Springer Proceedings in Mathematics &Amp; Statistics

Spreij

2016

Self Cite

In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the 'state of the economy', to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies.Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be analytically computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor.

Section: Introductionsupporting

confidence: 86%

Explicit Computations for Some Markov Modulated Counting Processes

Springer Proceedings in Mathematics &Amp; Statistics

Spreij

2016

Self Cite

“…When inflating the arrival rates by a factor N , and speeding up the background process by a factor N α (for some α > 0), in e.g. ( Anderson, Blom, Mandjes, Thorsdottir, and De Turck, 2014;Blom, De Turck, and Mandjes, 2015;Blom, De Turck, and Mandjes, 2016 ) it has been proven that the (transient as well as stationary) number of jobs present in the system is, after centering and normalizing, asymptotically Normally distributed. An interesting dichotomy was identified, in that the regimes α < 1 and α > 1 lead to qualitatively different asymptotics.…”

mentioning

confidence: 99%

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

Blom

Turck

European Journal of Operational Research

2017

Self Cite

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.

“…3.3] that Z n α satisfies the LDP in M t with speed n α and rate function given by Eqn. (5). Since this rate function is finite everywhere (cf.…”

Section: The Markov-modulated Ornstein-uhlenbeck Processmentioning

confidence: 99%

“…We then consider the special case in which the background process is relatively slow, in which we find intuitively appealing results; notably, we prove that, along the most likely path, the background process jumps at most 2d − 2 times, with d the number of states of the background process; this is in stark contrast with earlier findings for the infiniteserver queue in [4], where the background process jumps at most d − 1 times. In previous work on modulated mean-reverting processes there was a strong focus on the central limit regime [2,5,13]; under various conditions convergence to an ordinary (i.e., non-modulated) OU process has been established. Related results on large deviations for modulated infinite-server queues can be found in e.g.…”

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

Turck

et al. 2017

Performance Evaluation

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.