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Renewal theory
About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.
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TL;DR: The interspike intervals in steady-state neuron firing are assumed to be independently and identically distributed random variables and two models are demonstrated that may produce peaks in the power spectrum near 40 Hz.
Abstract: The interspike intervals in steady-state neuron firing are assumed to be independently and identically distributed random variables. In the simplest model discussed, each interval is assumed to be the sum of a random neuron refractory period and a statistically independent interval due to a stationary external process, whose statistics are assumed known. The power spectral density (hence the autocorrelation) of the composite neuron-firing renewal process is derived from the known spectrum of the external process and from the unknown spectrum of the neuron-refraction process. The results are applied to spike trains recorded in a previous study [2] of single neurons in the visual cortex of an awake monkey. Two models are demonstrated that may produce peaks in the power spectrum near 40 Hz.
38 citations
Journal Article•
TL;DR: In this article, the renewal process of Mittag-Leffler type and of Wright type are compared with each other, and the renewal processes with reward and numerically their long-time behaviour.
Abstract: After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and ofWright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour.
38 citations
TL;DR: Numerical results show that the distribution type of call time and/or cell residence time has influence on the performance measure and that the exponential case may underestimate the system performance.
Abstract: We present an analytic model for the performance evaluation of hierarchical cellular systems, which can provide multiple routes for calls through overflow from one cell layer to another. Our model allows the case where both the call time and the cell residence time are generally distributed. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of channel occupancy time distribution for each call type (new call, handoff call, and overflow call) is derived as a function of the Laplace transform of cell residence time. In particular, overflow calls are modeled by using a renewal process. Performance measures are derived based on the product form solution of a loss system with capacity limitation. Numerical results show that the distribution type of call time and/or cell residence time has influence on the performance measure and that the exponential case may underestimate the system performance.
38 citations
TL;DR: In this paper, the authors considered a triviariate stochastic process with random shocks at random intervals with random system states and derived the distribution of system lifetime, its moments and a related exponential limit theorem.
Abstract: A trivariate stochastic process is considered, describing a sequence of random shocks {Xn } at random intervals {Y n} with random system state {Jn }. The triviariate stochastic process satisfies a Markov renewal property in that the magnitude of shocks and the shock intervals are correlated pairwise and the corresponding joint distributions are affected by transitions of the system state which occur after each shock according to a Markov chain. Of interest is a system lifetime terminated whenever a shock magnitude exceeds a prespecified level z. The distribution of system lifetime, its moments and a related exponential limit theorem are derived explicitly. A similar transform analysis is conducted for a second type of system lifetime with system failures caused by the cumulative magnitude of shocks exceeding a fixed level z.
38 citations
TL;DR: In this article, the authors considered a special model in which a population evolves according to the compound Poisson process and catastrophes occur according to a renewal process and derived an algorithmic procedure for its approximate computation.
Abstract: Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p , independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.
38 citations