Showing papers on "Renewal theory published in 1973"

Random difference equations and Renewal theory for products of random matrices

Abstract: where Mn and Qn are random d • d matrices respectively d-vectors and Yn also is a d-vector. Throughout we take the sequence of pairs (Mn, Q~), n >/1, independently and identically distributed. The equation (1.1) arises in various contexts. We first met a special case in a paper by Solomon, [20] sect. 4, which studies random walks in random environments. Closely related is the fact tha t if Yn(i) is the expected number of particles of type i in the nth generation of a d-type branching process in a random environment with immigration, then Yn = (Yn(1) ..... Yn(d)) satisfies (1.1) (Qn represents the immigrants in the nth generation). (1.1) has been used for the amount of radioactive material in a compar tment ([17]) and in control theory [9 a]. Moreover, it is the principal feacture in a model for evolution and cultural inheritance by Cavalli-Sforza and Feldman [2]. Notice also tha t the dth order linear difference equation

Shock Models and Wear Processes

Abstract: The life distribution $H(t)$ of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities $P_k$ of not surviving the first $k$ shocks Various properties of the discrete failure distribution $P_k$ are shown to be reflected in corresponding properties of the continuous life distribution $H(t)$ As an example, if $P_k$ has discrete increasing hazard rate, then $H(t)$ has continuous increasing hazard rate Properties of $P_k$ are obtained from various physically motivated models, including that in which damage resulting from shocks accumulates until exceedance of a threshold results in failure We extend our results to continuous wear processes Applications of interest in renewal theory are obtained Total positivity theory is used in deriving many of the results

The interrupted poisson process as an overflow process

Abstract: Traffic overflowing a first-choice trunk group can be approximated accurately by a simple renewal process called an interrupted Poisson process–a Poisson process which is alternately turned on for an exponentially distributed time and then turned off for another (independent) exponentially distributed time. The approximation is obtained by matching either the first two or three moments of an interrupted Poisson process to those of an overflow process. Numerical investigation of errors in the approximation and subsequent experience has shown that this method of generating overflow traffic is accurate and very useful in both simulations and analyses of traffic systems.

The heavy traffic approximation for single server queues in series

Abstract: A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

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.

A Characterization of M/G/1 Queues with Renewal Departure Processes

Abstract: Burke [Burke, P. J. 1956. The output of a queueing system. Oper. Res.4 699--704.] showed that the departure process from an M/M/1 queue with infinite capacity was in fact a Poisson process. Using methods from semi-Markov process theory, this paper extends this result by determining that the departure process from an M/G/1 queue is a renewal process if and only if the queue is in steady state and one of the following four conditions holds: 1 the queue is the null queue---the service times are all 0; 2 the queue has capacity excluding the server 0; 3 the queue has capacity 1 and the service times are constant deterministic; or 4 the queue has infinite capacity and the service times are negatively exponentially distributed M/M/1/∞ queue.

Linear Bounds on the Renewal Function

Abstract: Lower and upper linear bounds are obtained on the renewal function of an ordinary renewal process. We show that any linear function, with slope equal to the reciprocal of the mean of the time between renewals, when iterated in the renewal equation will converge on the renewal function. However, convergence may not be monotonic for all t. We find the “best” linear bounds, which we define to be the sharpest bounds which, when iterated, converge monotonically to $M( t )$ for all t. We also show that the failure rate function of an equilibrium distribution has less variation than the failure rate of the underlying distribution. This fact is used to show that the bounds in this paper are an improvement over previously published linear bounds.

The Transient Reliability Behavior of Series Systems or Superimposed Renewal Processes

Abstract: In studying the waiting times between failures of a series system whose components fail independently and are replaced upon failure, it is well known that for large systems in time equilibrium, the distribution of waiting times is approximately exponential. Correction terms are available to account for the finite number of components. Here, correction terms are developed for the finite age of the system as well. Also numerical studies are presented indicating the magnitude of the error in reliability predictions based on the assumption of time equilibrium when system age is really finite.

A theory of traffic-measurement errors for loss systems with renewal input

Abstract: A theory of traffic-measurement errors for loss systems with renewal input is developed. The results provide an accurate approximation for the variance of any differentiable function of one or more of the following basic traffic measurements taken during a given time interval: (i) The total number of attempts (peg count) (ii) The number of unsuccessful attempts (overflow count) (iii) The usage based on discrete samples (TUR measurement) or on continuous scan. The approximation is given in terms of the individual variances and covariance functions of the three measurements. Asymptotic approximations for these moments are obtained using the concept of a generalized renewal process, and are shown to be sufficiently accurate for telephone traffic-engineering purposes. As an application of the theory, we examine the variances of the standard estimates of the load and peakedness (variance-to-mean ratio) of an input traffic stream for a time interval of one hour. Other possible applications to Bell System trunking problems are discussed.

A Functional Central Limit Theorem Connected with Extended Renewal Theory

Abstract: If p = 0 and if ~l, ~2 . . . . , are assumed to be positive random variables, the theorem is contained in Billingsley [2], Theorem 17.3:, p. 148. If p = 0, the theorem has been proved by Basu [1] and Vervaat [11]. As is pointed out in [2], n may tend to infinity in a continuous manner. Since the projections from C to R k are continuous mappings, (see [2], 20), it follows that the finite-dimensional distributions converge to multidimensional normal distributions (cp. [2], 30). In particular, the following corollary holds.

Bounds on the availability function

Abstract: This paper gives bounds on the availability function for an alternating renewal process with exponential failure and general repair times. A bound on the error is also given. Several of the bounds with greatest practical consequence are worked out and illustrated. Repair distributions for which a lower bound on availability is easily computed are gamma (integer shape parameter), log normal, and Weibull. Finally, some simulation results for log normal repair versus gamma repair are given.

A Characterization of the Exponential Law

Abstract: Under a slight regularity condition we prove that if the spent and residual waiting times at $t$ in a renewal process are independent random variables for one value of $t = t_0$, then the process is Poisson.

The asymptotic frequencies of the types

Abstract: The mean square and almost sure convergence of W(t) = e - t Z(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration. MULTITYPE AGE-DEPENDENT BRANCHING PROCESS; IMMIGRATION; ASYMPTOTIC FREQUENCIES OF TYPES; MEAN SQUARE AND ALMOST SURE CONVERGENCE