Topic
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: A fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables is studied, and some limit theorems are proved on the basis of the continuous Archimedean triangular norm based arithmetics.
Abstract: Using extension principle associated with a class of continuous Archimedean triangular norms, this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables. Some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the continuous Archimedean triangular norm based arithmetics. Furthermore, we discuss the fuzzy random renewal process based on the obtained limit theorems, and derive a fuzzy random elementary renewal theorem for the long-run expected renewal rate. The renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process. Finally, two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem.
43 citations
TL;DR: In this paper, the authors investigate a remote estimation problem in which a transmitter observes a Markov source and chooses the power level to transmit it over a time-varying packet-drop channel.
Abstract: We investigate a remote estimation problem in which a transmitter observes a Markov source and chooses the power level to transmit it over a time-varying packet-drop channel. The channel is modeled as a channel with Markovian state where the packet drop probability depends on the channel state and the transmit power. A receiver observes the channel output and the channel state and estimates the source realization. The receiver also feeds back the channel state and an acknowledgment for successful reception to the transmitter. We consider two models for the source—finite state Markov chains and first-order autoregressive processes. For the first model, using ideas from team theory, we establish the structure of optimal transmission and estimation strategies and identify a dynamic program to determine optimal strategies with that structure. For the second model, we assume that the noise process has unimodal and symmetric distribution. Using ideas from majorization theory, we show that the optimal transmission strategy is symmetric and monotonic and the optimal estimation strategy is like Kalman filter. Consequently, when there are a finite number of power levels, the optimal transmission strategy may be described using thresholds that depend on the channel state. Finally, we propose a simulation-based approach (renewal Monte Carlo) to compute the optimal thresholds and optimal performance and elucidate the algorithm with an example.
43 citations
TL;DR: A framework which leads to expressing the expected costs in terms of complex functions arising in renewal theory is reviewed and several approximations are discussed, finding that a straight line approximation performs well.
Abstract: In this paper, we survey a portion of the literature in the analysis of warranty costs. In order to compare the price of different warranty policies, we review a framework which leads to expressing the expected costs in terms of complex functions arising in renewal theory. These functions and several approximations are discussed. In particular, a straight line approximation performs well.
43 citations
TL;DR: A local limit theorem for P{Ta = n, S7 - a 5 x} is obtained in this article, where Ta is the first time a random walk S with positive drift exceeds a.
Abstract: A local limit theorem for P{Ta = n, S7 - a 5 x} is obtained, where Ta is the first time a random walk S, with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.
43 citations
TL;DR: Very simple renewal theory concepts are used to calculate transition probabilities, and their limits, for a Markov process subjected to an independent Poisson process of killing events which reset the process to the zero state as mentioned in this paper.
Abstract: Very simple renewal theory concepts are used to calculate transition probabilities, and their limits, for a Markov process subjected to an independent Poisson process of killing events which reset the process to the zero state. A particular case, treated heretofore by more restrictive methods, is where the basic process has an absorbing zero state from which it can be resurrected. Simple examples illustrate the scope of the basic formulation
42 citations