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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: In this paper, Thorin has shown that the Wiener-hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.
Abstract: During the latest few years much attention has been given to the study of the ruin problem of a risk business when the epochs of claims form a renewal process. The study of this problem was initiated by E. S. Andersen (1957). Thorin has then in a series of papers (Thorin, 1970, 1971a, 1971b) shown that the Wiener-Hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.
43 citations
TL;DR: In this article, a modified block replacement policy was proposed in which a unit is replaced at failure during (0, T0) and at scheduled replacement time T. The model with two variables was transformed into one variable and the optimum policy was discussed.
Abstract: This paper considers a modified block replacement policy in which a unit is replaced at failure during (0, T0) and at scheduled replacement time T. If a failure occurs in an interval (T0, T), then the unit remains as it is until T. The mean cost rate is obtained, using the results of renewal theory. The model with two variables is transformed into one variable and the optimum policy is discussed. An example shows how to compute the optimum T0* and T* when the failure time of the unit has a gamma distribution.
43 citations
TL;DR: The complete statistical description of a first-order correlative tracking system with periodic nonlinearity is shown to be embedded in a renewal process and the time-dependent probability density function of the phase error is computed.
Abstract: The complete statistical description of a first-order correlative tracking system with periodic nonlinearity is shown to be embedded in a renewal process. The time-dependent probability density function of the phase error, as well as the distribution of the cycle slips, is computed. The use of the renewal process approach makes it possible for the first time to compute the distribution of the positive and negative number of cycle slips within a given time interval. This information is sufficient to determine the probability density function of the absolute phase error.
43 citations
TL;DR: In this paper, the authors consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process.
Abstract: We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.
43 citations
01 Feb 1975
TL;DR: It is shown that the approach based upon renewal-process theory yields more information about the system's behavior than does the PE solution, and the pdf obtained via the renewal- process approach can be reduced to agree with thePE solution for the first-order loop in the steady state only.
Abstract: A new method is presented which describes the behavior of an (N + 1) th-order tacking system in which the nonlinearity is either periodic [phase-locked loop (PLL) type] or a nonperiodic [delay-locked loop (DLL) type]. The cycle slipping of such systems is modeled by means of renewal Markov processes. A fundamental relation between the probability density function (pdf) of the single process and the renewal process is derived which holds in the transient as well as in the stationary state. Based on this relation it is shown that the stationary pdf, the mean time between two cycle slips, and the average number of cycles to the right (left) can be obtained by solving a single Fokker-Planck equation of the renewal process. The method is applied to the special case of a PLL and compared with the so-called periodic-extension (PE) approach. It is shown that the pdf obtained via the renewal-process approach can be reduced to agree with the PE solution for the first-order loop in the steady state only. The reasoning and its implications are discussed. In fact, it is shown that the approach based upon renewal-process theory yields more information about the system's behavior than does the PE solution.
43 citations