Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.

Deducing queueing from transactional data: the queue inference engine, revisited

Abstract: R. Larson proposed a method to statistically infer the expected transient queue length during a busy period in O(n5) solely from the n starting and stopping times of each customer's service during the busy period and assuming the arrival distribution is Poisson. We develop a new O(n3) algorithm which uses these data to deduce transient queue lengths as well as the waiting times of each customer in the busy period. We also develop an O(n) on-line algorithm to dynamically update the current estimates for queue lengths after each departure. Moreover, we generalize our algorithms for the case of a time-varying Poisson process and also for the case of i.i.d. interarrival times with an arbitrary distribution. We report computational results that exhibit the speed and accuracy of our algorithms.

Asymptotic approximations for stationary distributions of many-server queues with abandonment

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 $N\rightarrow \infty$, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the $N\rightarrow \infty$ and $t\rightarrow \infty$ limits cannot always be interchanged.

Testing for a monotone trend in a modulated renewal process

Abstract: : In examining point processes which are overdispersed with respect to a Poisson process, there is a problem of discriminating between trends and the appearance in data of sequences of very long intervals. In this case the standard robust methods for trend analysis based on log transforms and regression techniques perform very poorly, and the standard exact test for a monotone trend derived for modulated Poisson processes is not robust with respect to its distribution theory when the underlying process is non-Poisson. However, experience with data and an examination of the departures from the Poisson distribution theory suggest a modification to the standard test for trend, both for modulated renewal and general point processes.

The economic design of control charts: a renewal theory approach

Abstract: Economic models for the design of control charts based on Duncan's approach1 have been well studied in the recent past We present an alternative approach to the development of a few of these models using renewal equations The main emphasis here is to study the role of the probability model associated with the process failure mechanism It is demonstrated that the expressions Tor the expected cycle length E( T) and the expected cost per cycle E( C) are easier to obtain by the proposed renewal equation approach than by adopting the traditional approach Furthermore, it is observed that certain non-Markovian shock models may be analyzed by adopting a renewal equation approach, whereas Duncan's approach has not been used with any non-Markovian model