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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: This article discusses a one-to-one ordering perishable system, in which reorders are processed in the order of their arrival and the processing times are arbitrarily distributed, and as such, the leadtimes are not independent.
37 citations
TL;DR: In this article, the authors considered a many-server queueing system where customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables.
Abstract: A many-server queueing system is considered in which customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables. Customers enter service in the order of arrival and are assumed to abandon the queue if the waiting time in queue exceeds the patience time. The state of the system with N servers is represented by a four-component process that consists of the forward recurrence time of the arrival process, a pair of measure-valued processes, one that keeps track of the waiting times of customers in queue and the other that keeps track of the amounts of time customers present in the system have been in service and a real-valued process that represents the total number of customers in the system. Under general assumptions, it is shown that the state process is a Feller process, admits a stationary distribution and is ergodic. It is also shown that the associated sequence of scaled stationary distributions is tight, and that any subsequence converges to an invariant state for the fluid limit. In particular, this implies that when the associated fluid limit has a unique invariant state, then the sequence of stationary distributions converges, as N → ∞, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the N → ∞ and t → ∞ limits cannot always be interchanged.
37 citations
TL;DR: This paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed (iid) fuzzy random interarrival times.
Abstract: Fuzzy random variable is a measure function from a probability space to a collection of fuzzy variables. Based on the fuzzy random theory, this paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed (iid) fuzzy random interarrival times. The relationship between the expected value of the fuzzy random renewal variable and the distribution functions of the alpha-pessimistic values and alpha-optimistic values of the interarrival times is discussed. Furthermore, the fuzzy random style of renewal equation is provided. Finally, fuzzy random Blackwell's renewal theorem and Smith's key renewal theorem are also given
37 citations
TL;DR: In this article, the extremal properties of the shot noise process X(t) are investigated in a natural way through a discrete-time process which records the states of (X(t)} at the points of ti.
Abstract: Consider the shot noise process X(t):= .ih(t - rj), t---0, where h is a bounded positive non-increasing function supported on a finite interval, and the r;'s are the points of a renewal process tj on [0, oo). In this paper, the extremal properties of (X(t)} are studied. It is shown that these properties can be investigated in a natural way through a discrete-time process which records the states of (X(t)} at the points of ti. The important special case where 7t is Poisson is treated in detail, and a domain-of-attraction result for the compound Poisson distribution is obtained as a by-product. EXTREME VALUE; POINT PROCESS; RENEWAL PROCESS
37 citations
TL;DR: In this article, the authors studied the stochastic ordering of random measures and point processes generated by a partial order for all bounded Borel subsets B of the state space and derived a condition that the former can be realized as a thinning of the latter.
37 citations