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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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Journal ArticleDOI
TL;DR: A Markovian stochastic model for a system subject to random shocks is introduced in this paper, where it is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount.
Abstract: A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold a. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if X(t) I> a. The stationary case is also discussed.

9 citations

Journal ArticleDOI
TL;DR: This work develops hypothesis tests and develops a regression method to determine if there are covariates that explain the departure from length-biased sampling and whether there is evidence that repeat blood donors who develop antibodies to HIV or other viral infections change their donation pattern in some way.
Abstract: SUMMARY Length-biased sampling occurs in renewal processes when the probability that an interval is selected is proportional to the length of the interval. This can occur when intervals are selected because they contain an event that is independent of the renewal process and occurs with constant hazard. For example, if the times between donations for repeat blood donors are independent and identically distributed, and if the donor seroconverts to HIV (develops antibodies that indicate infection with human immunodeficiency virus), then the interval between the last HIV seronegative and first HIV seropositive test is expected to be longer than that donor’s previous time intervals between donations. We develop hypothesis tests to determine if the relationship between the typical and length-biased intervals is as expected, or if there is departure from length-biased sampling. We further develop a regression method to determine if there are covariates that explain the departure from length-biased sampling. Our approach is motivated by the question of whether there is evidence that repeat blood donors who develop antibodies to HIV or other viral infections change their donation pattern in some way because of seroconversion.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the authors define a shot noise process with a random response function (response process) in which shots occur at arbitrary random times and provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes.
Abstract: By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

9 citations

Journal ArticleDOI
TL;DR: Some Markov or renewal process models can be introduced to describe some of these structures and can be used to generate such random structures, according to some ideas of Van Cutsem and Ycart (1993).

9 citations

Journal ArticleDOI
TL;DR: An analytic method is given for calculating the reliability function, its Laplace transform and the mean time to failure (MTTF) if the Laplaces transform of the inter-arrival law of the renewal process is explicit.

9 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202327
202260
202173
202083
201973
201886