Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.

On some problems in the counting statistics of nuclear particles Investigation of the dead time problems

Abstract: The probabilistic aspects of the detection of nuclear particles are discussed with the application of the methods of renewal theory. It is assumed that the inter-event times in the investigated event series are independent, identically distributed random variables. The detector efficiency and various types of the dead time are accounted for. Exact analytical results are derived for the probability distribution functions, the expectations and the variances of the number of detected particles. Also, a precise analysis of the coincidence problem is given. The results should serve for the evaluation of the measurements in view of the dead time correction effects for the higher moments of the detector counts.

Stochastic Models of Spike Trains

Abstract: This chapter reviews the theory of stochastic point processes. For simple renewal processes, the relation between the stochastic intensity, the inter-spike interval (ISI) distribution, and the survival probability are derived. The moment and cumulant generating functions and the relation between the ISI distribution and the autocorrelation is investigated. We show how the Fano factor of the spike count distribution depends on the coefficient of variation of the ISI distribution. Next we investigate models of renewal processes with variable rates and CV2, which is often used to assess the variability of the spike train in this case and compare the latter to the CV. The second half of the chapter deals with stochastic point processes with correlations between the intervals. Several examples of such processes are shown, and the basic analytical techniques to deal with these processes are expounded. The effect of correlations in the ISIs on the Fano factor of the spike count and the CV2 are also explored.

Efficient simulation techniques for stochastic model checking

Abstract: In this thesis, we focus on methods for speeding-up computer simulations of stochastic models. We are motivated by real-world applications in which corporations formulate service requirements in terms of an upper bound on a probability of failure. If one wants to check whether a complex system model satisfies such a requirement, computer simulation is often the method of choice We aim to aid engineers during the design phase, so a question of both practical and mathematical relevance is how the runtime of the simulation can be minimised. We focus on two settings in which a speed-up can be achieved. First, when the probability of failure is low, as is typical in a highly reliable system, the time before the first failure is observed can be impractically large. Our research involves importance sampling; we simulate using a different probability measure under which failure is more likely, which typically decreases the variance of the estimator. In order to keep the estimator unbiased, we compensate for the resulting error using the Radon-Nikodym theorem. If done correctly, the gains can be huge. However, if the new probability measure is unsuited for the problem setting the negative consequences can be similarly profound (infinite variance is even possible). In our work, we have extended an importance sampling technique with good performance (i.e., proven to have bounded relative error) that was previously only applicable in restricted settings to a broad model class of stochastic (Markovian) Petri nets. We have also proposed methods to alleviate two well-known problems from the rare event simulation literature: the occurrence of so-called high-probability cycles and the applicability to large time horizons. For the first we use a method based on Dijkstra’s algorithm, for the second we use renewal theory. Second, it often occurs that the number of needed simulation runs is overestimated. As a solution, we use sequential hypothesis testing, which allows us to stop as soon as we can say whether the service requirement is satisfied. This area has seen a lot of recent interest from the model checking community, but some of the techniques used are not always perfectly understood. In our research we have compared the techniques implemented in the most popular model checking tools, identified several common pitfalls and proposed a method that we proved to not have this problem. In particular, we have proposed a new test for which we bounded the probability of an incorrect conclusion using martingale theory.

Characterization of the Pólya-Aeppli Process

Abstract: In this article, we study the Polya-Aeppli process (PAP). We define PAP from three different points of view: as a compound Poisson process, as a delayed renewal process, and as a pure birth process. We show that these definitions are equivalent. Also, using these definitions we identify several interesting characterizations of PAP.

The on---off network traffic model under intermediate scaling

Abstract: The result provided in this paper helps complete a unified picture of the scaling behavior in heavy-tailed stochastic models for transmission of packet traffic on high-speed communication links. Popular models include infinite source Poisson models, models based on aggregated renewal sequences, and models built from aggregated on---off sources. The versions of these models with finite variance transmission rate share the following pattern: if the sources connect at a fast rate over time the cumulative statistical fluctuations are fractional Brownian motion, if the connection rate is slow the traffic fluctuations are described by a stable Levy motion, while the limiting fluctuations for the intermediate scaling regime are given by fractional Poisson motion. In this paper, we prove an invariance principle for the normalized cumulative workload of a network with m on---off sources and time rescaled by a factor a. When both the number of sources m and the time scale a tend to infinity with a relative growth given by the so-called 'intermediate connection rate' condition, the limit process is the fractional Poisson motion. The proof is based on a coupling between the on---off model and the renewal type model.