Showing papers on "Renewal theory published in 1981"

Applications of renewal theory in analysis of the free‐replacement warranty

Abstract: Under a free-replacement warranty of duration W, the customer is provided, for an initial cost of C, as many replacement items as needed to provide service for a period W. Payments of C are not made at fixed intervals of length W, but in random cycles of length Y = W + γ(W), where γ(W) is the (random) remaining life-time of the item in service W time units after the beginning of a cycle. The expected number of payments over the life cycle, L, of the item is given by MY(L), the renewal function for the random variable Y. We investigate this renewal function analytically and numerically and compare the latter with known asymptotic results. The distribution of Y, and hence the renewal function, depends on the underlying failure distribution of the items. Several choices for this distribution, including the exponential, uniform, gamma and Weibull, are considered.

Approximating a Point Process by a Renewal Process: The View Through a Queue, an Indirect Approach

Abstract: This paper investigates simple approximations for stochastic point processes. As in several previous studies, the approximating process is a renewal process characterized by the first two moments of the renewal interval. The approximating renewal-interval distribution itself is a convenient distribution with these two moments; it is constructed from exponential building blocks, e.g., the hyperexponential distribution. Here the moments of the renewal interval are chosen to produce the same level of congestion when the renewal process serves as an arrival process in a test queueing system as is produced when the general point process is the arrival process. The procedure can be applied to predict the behavior of a new service mechanism in a queueing system with a complicated arrival process; then we use the system with the old service mechanism as the test system. But the test system can also be an artificial device to approximate any point process. This indirect approximation procedure extends the equivalent random method and related techniques widely used in teletraffic engineering.

Further monotonicity properties for specialized renewal processes

Abstract: : Define Z(t) to be the forward recurrence time at t for a renewal process with interarrival time distribution, F, which is assumed to be IMRL (increasing mean residual life). It is shown that E phi (z(t)) is increasing in t or = 0 for all increasing convex phi. An example demonstrates that Z(t) is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for F DFR (decreasing failure rate). (Author)

On the Mean Residual Life Function in Survival Studies

Abstract: In reliability studies, the expected additional life time given that a component has survived until time t is called the mean residual life function (MRLF). This MRLF determines the distribution function uniquely. In this paper an interpretation of MRLF in renewal theory is presented and some characterizations of the exponential distribution are obtained. Finally, considering the general MRLF, a method is developed for obtaining the mixing distribution when the original distribution is exponential. Some examples are discussed, in one of which Morrison’s (1978) result is obtained as a special case.

Technical Note—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 times are assumed to constitute a renewal process, and demand sizes are independent identically distributed (i.i.d.) integer valued random variables. Some relationships are given between waiting time and some common inventory measures.

ALGORITHMS FOR A CONTINUOUS-REVIEW (s, S) INVENTORY SYSTEM

Abstract: The steady-state distribution of the inventory position for a continuousreview (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the 'versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc. COMPUTATIONAL PROBABILITY; CONTINUOUS-REVIEW INVENTORY; N-PROCESS; PHASE-TYPE DISTRIBUTIONS

Interval Reliability of An n-Unit System With Single Repair Facility

Abstract: This paper derives the interval reliability of an n-unit standby redundant system with warm standbys. The derivation uses renewal theory and regeneration-point techniques. The solution is obtained in the form of Laplace transforms; the transforms can be readily inverted for some special cases.

The exact order of normal approximation in bivariate renewal theory

Abstract: Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed. RATES OF CONVERGENCE; CENTRAL LIMIT THEOREM: FIRST-PASSAGE TIMES; MAXIMUM PARTIAL SUMS; RENEWAL VARIABLES

Transitional phenomena of renewal theory in asymptotic problems of the theory of random processes. i

Abstract: The limit behavior of distributions of additive functionals of regenerating processes with several types of regeneration points is studied on the basis of transitional phenomena of renewal theory Bibliography: 11 titles

On excess life in certain renewal processes

Abstract: Excess life distributions for discrete renewal processes may be computed by using elementary discrete Markov chain concepts involving absorption probabilities. Excess life distributions in general may then be obtained by approximating the renewal process under study by a suitably chosen sequence of discrete renewal processes. The technique is illustrated in the cases of renewal processes with interarrival distributions which are a linear combination of two exponentials and uniform [0, 1]. A related algorithm is described for computer generated approximations of excess life distributions corresponding to continuous interarrival time distributions with an increasing c.d.f. Conditions for convergence of this algorithm are examined.

A characterization of the negative exponential distribution with application to reliability theory

Abstract: The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution. CHARACTERIZATION THEOREM; NEGATIVE EXPONENTIAL DISTRIBUTION; RELIABILITY; REVEALED AND UNREVEALED FAULTS; ALTERNATING RENEWAL PROCESS; TWO-STATE SEMI-MARKOV PROCESS; SYSTEM AVAILABILITY

A two‐state system with partial availability in the failed state

Abstract: A generalization of the alternating renewal model of a repairable system to permit partial availability in the failed state is introduced. It is shown how, by making use of an embedded alternating renewal process, we can readily derive expressions for various measures of system availability. Expressions for the point availability of the generalized process are presented.

Solvable models of temporally correlated random walk on a lattice

Abstract: We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet Uncorrelated diffusion occurs whenW is a Poisson distribution The solutions corresponding to two different families of distributionsW are found and discussed The Poissonian is a limiting case in each of these families This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumpsW is constructed in terms ofψ using the continuous-time random walk approach The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated) Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc, are discussed

On general versions of Erdős-Rényi laws

Abstract: Rather general versions of the Erdős-Renyi [6] new law of large numbers have recently been given by S. Csorgő [5] for sequences of rv's which have stationary and independent increments and satisfy a first order large deviation theorem. It is shown that Csorgő's results can be extended to cover also situations of stochastic processes where stationarity and independence of increments are not generally available, but for randomly chosen subsequences of the process. Examples demonstrate that the main result can be applied, for instance, to waiting-times in G/G/1 queuing models or cumulative processes in renewal theory, where Erdős-Renyi type laws cannot be derived from Csorgő's theorems.

Moments in Terms of the Mean Residual Life Function

Abstract: In reliability studies, the mean additional life time, given that a component has survived until time t, is called the mean residual life function (MRLF). This MRLF determines the distribution function uniquely. In this paper a method of obtaining the moments in terms of the MRLF, by employing a result in the context of renewal theory, is developed. The method is illustrated by an example.

On a general storage problem and its

Abstract: A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(.) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn+1(y) = S (y, w)g,(w) dw with I(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram-Charlier or generalized GramCharlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss-Hermite-Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.