Showing papers on "Renewal theory published in 1978"

Renewal processes of phase type

Abstract: This paper discusses a class of analytically and numerically tractable renewal processes, which generalize the Poisson process. When used to describe interarrival or service times in queues, these renewal processes lead to computationally explicit solutions which involve only real arithmetic. Previous modifications of the Poisson process, based on the Erlang or the hyperexponential distributions, appear as particular cases.

Uniform limit theorems for non-singular renewal and Markov renewal processes

Abstract: We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Renewal Process Approach to the Theory of Genetic Linkage: Case of No Chromatid Interference

Abstract: Models are presented in which the distribution of crossovers at a four-four-strand stage of meiosis results from a renewal process. Probability distributions are obtained for the number of crossover events on a meiotic bivalent and for the number of exchange points in random meiotic products. These distributions are found to fit the observed distribution of these variables reasonably well. Using these distributions and assuming no chromatid interference, relations between map distance and the recombination fraction are obtained. These relations give either better or equivalent fit to data when compared to relations that are designed to account for both chromatid and chiasma interference.

Waiting Time in a Continuous Review (s, S) Inventory System with Constant Lead Times.

Abstract: : The waiting time distribution in an (s, S) continuous review inventory system with constant lead times is derived in this paper. The demand process is assumed to be a renewal process, and demand sizes are iid integer valued random variables. Some relationships are given between waiting time and some common inventory measures. (Author)

Locating bright spots in a point process

Abstract: As an approach to modelling the 'matching' of optical receptors in animals to the objects they are designed to see, we study the problem of locating regions of high intensity in a point process on the real line, using the counts of points in a movable interval of fixed length. We define performance measures analogous to statistical size and power for this procedure and, for points forming a renewal process, give conditions on the quantiles of the convolutions of the interpoint distribution which ensure that the optimal length for the 'detector' is close to that of the 'object' to be detected. We show that these conditions are satisfied for a Poisson process. Similar conditions ensure that the optimal length is close to zero, and we give a class of distributions satisfying these conditions. Finally we show that the results can be extended to simple two-dimensional models.

Regenerative multivariate point processes

Abstract: A class of stationary multivariate point processes is considered in which the events of one of the point processes act as regeneration points for the entire multivariate process. Some important properties of such processes are derived including the joint probability generating function for numbers of events in an interval of fixed length and the asymptotic behaviour of such processes. The general theory is then applied in three bivariate examples. Finally, some simple monotonicity results for stationary and renewal point processes (which are used in the second example) are proved in two appendices.

Reliability and Availability of Two General Multi-Unit Systems

Abstract: This paper considers k-unit systems with repair. Reliability and availability integral equations are set up using renewal theory. This paper treats a k-unit system with s-dependent failure rate and general repair time distribution, and a k-unit cold-standby system with Erlang failure time distribution and general repair time distribution.

Weak convergence of probability measures relative to incompatible topology and σ-field with applications to renewal theory

Abstract: Necessary and sufficient conditions are given in order that a sequence of probability measures, weakly convergent relative to a given topology τ 0 and associated σ-field σ(τ 0), are weakly convergent (and satisfy a continuity theorem) relative to the σ(τ 0)-measurable functions which are continuous in some finer topology τ 1, even if μ does not extend to σ(τ 0). These conditions are shown to be applicable to a sequence of translated renewal measures. Alternate conditions (tightness, uniformity of weak convergence) are investigated and shown to be inappropriate.

'Reliability Growth' as an Artifact of Renewal Testing

Abstract: : It is shown that observed 'reliability growth' may be an artifact of limited-horizon renewal testing. The 'growth' of several estimators of failure rate and MTBF with test time is examined for a stationary renewal process. (Author)

Renewal processes with random numbers of delays: application to a conception and birth model.

Abstract: Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.

The power of the exponential scores test for an ordinary renewal process against trend alternatives

Abstract: The power properties of a statistic based on the use of exponential scores which may be used for testing whether a series of events occurring in time form an ordinary renewal process against trend alternatives are examined. Small sample power comparisons under a Lehmann trend alternative are made with an alternative nonparametric test based on a rank trend statistic and with the parametric test when the intervals are exponentially distributed. Finally, some asymptotic efficiency results are developed for limiting trend alternatives.

On the Busy Period of GI/G/1 Queue with Two Priority Levels and Feedbacks

Abstract: In this paper we are concerned with several queue length processes' and several flow processes in M/G/1 queues with instantaneous feedback. Such queues occur often in application, with the round robin model of computer analysis being a prime example. It will be shown that such queues have some unexpected properties. In particular, it is shown that except for trivial cases (no feedback), flow in the network is not a Poisson process and is not even a renewal process. Such a result raises interesting questions concerning network decomposition and interpretation of the Jackson result. One attempt is made to reinterpret these Jackson results.