Probability-generating function

About: Probability-generating function is a research topic. Over the lifetime, 752 publications have been published within this topic receiving 9361 citations.

Steady state analysis of a single server bulk queue with general vacation times and restricted admissibility of arriving batches

Abstract: We analyze the steady state behavior of a single server vacation queue with variable batch size arrivals in a compound Poisson Process, exponential service in batch of fixed size b following a min(b,n) rule and general server vacations. However, an arriving batch may or may not be allowed to join the system at all the times. We obtain explicit steady state results for the probability generating functions for the number of customers in the system, the average number of customers and the average waiting time in the queue and the system. Some special cases of interest have been derived and finally a numerical example is provided.

Detection of Changes in INAR Models

Abstract: In the present paper we develop on-line procedures for detecting changes in the parameters of integer valued autoregressive models of order one. Tests statistics based on probability generating functions are constructed and studied. The asymptotic behavior of the tests under the null hypothesis as well as under certain alternatives is derived.

On the convergence rate of ordinal comparisons of random variables

Abstract: The asymptotic exponential convergence rate of ordinal comparisons follows from well-known results in large deviations theory, where the critical condition is the existence of a finite moment generating function. In this note, we show that this is both a necessary and sufficient condition, and also show how one can recover the exponential convergence rate in cases where the moment generating function is not finite. In particular, by working with appropriately truncated versions of the original random variables, the exponential convergence rate can be recovered.

Recursive formulas for discrete distributions

Abstract: We apply power series techniques for differential equations on probability generating functions to derive recursive formulas for discrete compound distributions. Such formulas are computationally effective and useful in risk theory.

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

Abstract: Many 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 large-deviations 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.