Showing papers on "Renewal theory published in 1970"

Strong renewal theorems with infinite mean

Abstract: Let F be a nonarithmetic probability distribution on (0, oo) and suppose 1 —F(f) is regularly varying at oo with exponent a, 00 fixed. Next we derive asymptotic relations for the convolution U*z(t), t —>■ oo, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function m(t) = f0 [1 — F(x)] dx, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case a = l to derive some limit theorems for the distributions of certain waiting times associated with a renewal process. 1. Principal theorems. Let A be a probability measure concentrated on [0, oo)(2) and let U be the associated renewal measure defined for any measurable set / by (l.i) t/{/} = !>\"•{/} 0 where Fn' denotes the «-fold convolution of F with itself (P°* is the probability measure concentrated at the origin). The series (1.1) converges to a finite number for every bounded I. (For this and other elementary properties of U see [3, VI. 6] ; for a probabilistic interpretation of U see §9 in this paper.) We write U(x) for U{[0, x]} and we shall henceforth ignore the distinction between U the measure and U the function. (This convention applies to other measures as well.) The main results of this paper deal primarily with the differences U(t+h) — U(t) for h>0 fixed, and t -*■ oo. The principal assumption is that Phas the form (1.2) \\-F(t) = t~aL(t), t>0, Received by the editors October 4, 1969. A MS Subject Classifications. Primary 6070, 6020, 6030; Secondary 4042, 4252.

On Procedures for Reliability Evaluations of Transmission Systems

Abstract: A step-by-step procedure for reliability evaluations of transmission systems is described. Simple algebraic results for prediction of frequency and duration of transmission contingencies are obtained with the use of renewal process theory.

Renewal Theoretic Aspects of Two-Unit Redundant Systems

Abstract: This expository paper discusses four two-unit redundant systems: 1) parallel redundancy; 2) standby redundancy; 3) standby redundancy with priority; 4) standby redundancy with noninstantaneous switchover. In models 1), 2), and 3) the switchover time is instantaneous. The integral equation of renewal theory is applied by using the concept of a cycle. Applying the integral equation of renewal theory and the cycle, we obtain systematically for each model the Laplace-Stieltjes transform of the distribution of the time to first failure and its mean.

A Controlled Transportation Queueing Process

Abstract: A transportation queueing process in which taxis arrive in a Poisson process and customers arrive as a renewal process independent of taxi-arrival process is controlled by calling extra taxis whenever the total number of customers lost to the system reaches a certain predetermined number. Transient and steady state behavior of this process is studied using renewal theoretic arguments. The optimum value of the control variable is also obtained so as to minimize the total cost to the system due to the waiting taxis and lost customers.

Identifiability for random translations of Poisson processes

Abstract: A system in which each point of a stationary Poisson process is subjected to a random displacement, the displacements being independently and identically distributed, is considered. It is shown that the displacement distribution is identifiable if we are given a realization of the original process and the corresponding realization of the displaced process but not the linkage between the two.

Selective interaction of a stationary point process and a renewal process

Abstract: The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.

The finite dam ii

Abstract: A weir of capacity K is considered in which the water inflow is a process with stationary independent increments. Unless the weir is empty, there is a continuous release of water at unit rate; if K is finite the weir may become full in which case the excess water overflows instantaneously. A weir for which K is infinite will be referred to as infinite dam. For the latter the transient behaviour is well known if the input possesses a second moment (cf. e.g., Prabhu [7]) and serves as the starting point for the present paper. This result is first extended to yield the Laplace transform (L.T.) of the trivariate Laplace-Stieltjes transform (L.S.T.) of the content v ( t ) at time t , the input X ( t ) in (0, t ) and the total time d ( t ) in the interval (0, t ) during which the dam is dry. (Incidentally, the last two quantities, for relevant time intervals, will be carried throughout.) Then we use a relation between the latter and the L.S.T. of the expected number of downward level y crossings of the v ( t ) process established in Roes [9]. Since the dam processes considered are Markov processes, we have therewith the L.S.T. of the renewal function of the renewal process imbedded at level y. From this, one finds the L.S.T.'s of first entrance and taboo first entrance times (for their definition see introduction). Next we calculate the first skip times for the infinite dam from the first entrance times and the L.T. of the L.S.T. of v ( t ). It is then a routine matter to determine the taboo first skip times. From the (taboo) first entrance and skip times we derive the first entrance times for the finite dam, which in turn lead to the renewal functions of the renewal processes imbedded in the finite dam content process v* ( t ) and hence to the transient behaviour of the finite dam. The advantage of the present approach over the one given in Roes [8] is that it is entirely probabilistic and avoids involved analytic arguments. As a result, the question of uniqueness of the solution does not arise, while more insight is obtained in the structure. The L.S.T. of several first entrance times and first skip times have been derived by Cohen [2] for compound Poisson input.

A note on matrix renewal function.

Abstract: : The Laplace-Steiltjes Transform of the matrix renewal function M(t) of a Markov Renewal process is expanded in powers of the argument s, in this paper, by using a generalized inverse of the matrix I-P sub 0, where P sub 0 is the transition probability matrix of the imbedded Markov chain. This helps in obtaining the values of moments of any order of the number of renewals and also of the moments of the first passage times, for large values of t, the time. All the results of renewal theory are hidden under the Laplacian curtain and this expansion helps to lift this curtain at least for large values of t and is thus useful in applications of Markov Renewal processes to inventory control of repairable items, and to counter theory. (Author)

Characterization and Modeling of Real Communication Channels

Abstract: : The paper presents new descriptive and generative models for the error-cluster and error gap patterns which occur in the binary, discrete-time stochastic processes observed as outputs of digital communication channels having memory. The slope of the error-gap distribution is used to uncover relationships between various channel models. One characterizes the memory mu of a process of error density Pe by its relative deviation in average conditional entropy from the discrete memoryless channel (D.M.C.), which one proves has maximum entropy for the class of (finite and infinite memory length) processes of density Pe. One obtains an upper bound for mu for real channels, derive mu for the general discrete renewal process from the error gap probability mass function (EGPMF) and prove that it is a lower bound for any processes having the same EGPMF. One demonstrates some limitations of finite error-free state models by showing that their EGPMF is bounded from above by a geometric series. To estimate the counting distribution with flexibility we introduce conditional gap distributions and multigap statistics; one uses these in implementing a denumerable Markov Chain model which, free from finite state model limitations and more general than renewal processes, allows the derivation of all classical statistics including entropy. (Author)

Markov chains governed by complicated renewal processes

Abstract: In this work Markov chains governed by complicated processes are introduced and investigated (Section 1). In Section 2 an ergodic theorem for these processes is formulated, while in Section 3 the sojourn time of the process in a fixed region is studied; in Section 4 some examples are considered. The processes studied are of practical importance in the description of mass service systems and the theory of reliability for which the time intervals between successive demands cannot be assumed to be mutually independent random variables. It is shown that the dependence parameter r of these processes, if it is sufficiently large, allows us to formulate a relationship between the time intervals in question. Special cases of Markov chains governed by complicated renewal processes (i.e., Markov chains, semi-Markov processes, Markov chains with semiMarkovian interference, semi-Markov processes of rth order) and their application to various problems of mass service theory, as well as reliability theory, have already been studied by different authors ([2]-[10]). Problems close to those considered in this work have been treated in [11]-[13]. It is of interest to note that the processes, which we refer to as complicated renewal processes, recommended for study in Smith's [14] comprehensive work on renewal theory, are a special case of those investigated here. In Section 2 the stationary distribution of these processes is determined, and a simultaneous derivation of Blackwell's theorem given. In the examples considered in Section 4 some of Takacs' results [15] are generalized to the case where the flow of demands forms a uni-tied renewal process.

Estimating probabilities for sums of independent random vectors

Abstract: An estimate is made for the probability of occurrence of the value for a sum of independent, identically distributed, random s-dimensional vectors in different families of spheres.