Showing papers on "Renewal theory published in 1987"

Applied Probability And Queues

Abstract: Preface SIMPLE MARKOVIAN MODELS: Markov Chains Markov Jump Processes Queueing Theory at the Markovian Level BASIC MATHEMATICAL TOOLS: Basic Renewal Theory Regenerative Processes Further Topics in Renewal Theory and Regenerative Processes Random Walks SPECIAL MODELS AND METHODS: Steady-state Properties of GI/G/1 Explicit Examples in the Theory of Random Walks and Single Server Queues Multi-Dimensional Methods Many-server Queues Conjugate Processes Insurance Risk, Dam and Storage Models Selected Background and Notation.

Nonlinear Renewal Theory in Sequential Analysis

Abstract: Randomly Stopped Sequences Random Walks The Sequential Probability Ratio Test Nonlinear Renewal Theory Local Limit Theorems Open-Ended Tests Repeated Significance Tests Multiparameter Problems Estimation Following Sequential Testing Sequential Estimation.

Convergence rates in the strong law for bounded mixing sequences

Abstract: Speed of convergence is studied for a Marcinkiewicz-Zygmund strong law for partial sums of bounded dependent random variables under conditions on their mixing rate. Though α-mixing is also considered, the most interesting result concerns absolutely regular sequences. The results are applied to renewal theory to show that some of the estimates obtained by other authors on coupling are best possible. Another application sharpens a result for averaging a function along a random walk.

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

Markov renewal processes, counters and repeated sequences in Markov chains

Abstract: The theory of Markov renewal processes is applied to study the occurrence of specific sequences of states in a Markov chain. Cinlar&s (1969) results are used to study both the basic process, and that obtained when the overlap of sequences is not permitted, as in the theory of counters. These results are applied to the fragments formed when DNA is digested using one, or more, restriction enzymes.

Theory prob. appl.

Abstract: Let F be a probability measure on and the probability measure subordinate to F with subordinator {pn}IN. In this paper we discuss the precise asymptotic behaviour of when is finite. Applications to transient renewal theory and infinite divisibility are given.

On Subordinated Distributions and Generalized Renewal Measures

Abstract: where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1/n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail.

One-Dimensional Circuit-Switched Networks

Abstract: This paper is concerned with the stationary distribution of a one-dimensional circuit-switched network. We show that if arrival rates decay geometrically with distance, then under the stationary distribution the number of circuits busy on successive links of the network at a fixed point in time is a Markov chain. When each link of the network has unit capacity we show that translation invariant arrival rates lead to a stationary distribution which can be described in terms of an alternating renewal process.

Approximations in Renewal Theory

Abstract: A renewal process [N(t), t ≥ 0] with interarrival times Xi, for i ≥ 1 and renewal function m (t) is considered. Let Gn, λ denote the gamma distribution with parameters n and λ–that is, dGn, λ(x) = λe–λx(λx)n-1/(n – 1)In Section 1 we show how m(t) can be approximated by f m(s) dGn, n/t(s). In addition, we show that these approximations constitute an increasing sequence of lower bounds when the interarrival distribution has the decreasing failure rate property. In Section 2 we show how the integrated renewal function can be approximated in a similar fashion by a decreasing sequence of upper bounds. In Section 3 we consider the problem of approximating the residual life (also called excess life) and the renewal age distribution and their means, and in Section 4 we consider the distribution of N(t). Finally, in Section 5 we remark on the relationship between our approximations and the Feller technique for inverting a Laplace transform.

A note on repeated sequences in Markov chains

Abstract: If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

On asymptotic approximations for (s,S) policies

Abstract: This paper is concerned with optimal policy approximations based on asymptotic renewal theory, and their accuracy conditions, for a class of periodic-review inventory systems We compare the performances of two such approximations with those of the optimal policies using a wide range of demand distributions and parameter settings Accuracy conditions are derived from new bounds on the optimal policy and from empirical considerations related to the rate of approach of the renewal function to its linear asymptot. Algorithms used in computing asymptotic approximations and optimal policies are also based on new theoretical findings, including a sufficient optimality condition

Estimation of sojourn time distributions for cyclic semi-Markov processes in equilibrium

Abstract: SUMMARY The EM algorithm is used to develop procedures for estimating the interoccurrence distributions when n independent and identically distributed cyclic semi-Markov processes, each being ergodic, irreducible and in equilibrium, are observed over finite windows. This work is an extension of Vardi (1982a, b) who considers a similar problem for renewal processes and develops the RT algorithm for estimation of the common interoccurrence distribution. The results are illustrated using simulations for alternating renewal processes and compared with an extension of an approximate approach of Denby & Vardi (1985).

A renewal theorem for random walks in multidimensional time

Abstract: Soit X, X 1 , X 2 ... une famille de variables aleatoires i.i.d. a valeurs entieres, moyenne positive et variance finie positive. Soit Sn=X 1 +X 2 +...+Xn. On etudie le comportement asymptotique de la somme ponderee R(k)=Σa n P(Sn=k), avec sommation sur n≥1, lorsque k→+∞

Hypothesis testing for a non-homogeneous poisson process

Abstract: A displaced Poisson process is observed only at the shifted time points T1, T2, ..., . The displacement intervals are assumed to be independent and identically distributed with distribution function F, and it is desired to test various hypotheses about F. It is shown that the problem may be reduced to that of testing for a constant versus an increasing (decreasing) intensity function of a Poisson process. Monte Carlo simulations are performed to compare the three standard tests as they apply to the above problem. Applications of the two-stage failure model introduced, are given

A frequentistic and bayesian analysis of zellner's economic regression model under an informative prior

Abstract: Vector-valued observations Y1,Y2,... arrive sequentially and satisfy the general linear model The Xi's are either random or known matrices and, given β and σ2, the are iid The parameter β is estimated by the Bayes estimator under the conjugate prior and subject to a loss structure that is the sum of the cost due to sampling and a predictive loss due to estimation error. We show that the myopic rule is asymptotically nondeficient in that the difference between its Bayes risk and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of a single observation, as c → 0. The myopic stopping rule is also examined frequentistically through the use of nonlinear renewal theory.

Some approximations of ultimate ruin probability for finite initial surplus

Abstract: Based on newer results of renewal theory formulas for the ruin probability are given for finite initial surplus.

Random Self-Avoiding Walks in some One-Dimensional Lattices*

Abstract: Asymptotic results are given for various properties of very long random self-avoiding walks in some “one-dimensional” lattices. The proofs are based on some results from renewal theory which are developed in an appendix.

Optimal control of a queueing system with an exponential and an Erlangian server and renewal input stream

Abstract: Customers arrive in a renewal process at a queue which is served by an exponential and a two-stage Erlangian server. We prove the optimal policy for assignment of customers to the servers which for any t maximizes the expected number of served customers in [0,t].

A class of ruin problems

Abstract: Von Bahr (1974) and others have investigated the ruin problem of insurance risk in the case where the total amount of claims received by the company is formulated as a compound renewal process. Such processes, under the name of renewal-reward processes, were introduced by Jewell (1967), who extended to them the fluctuation theory of random walks. In this paper we establish further results for compound renewal processes and apply them to Von Bahr’s ruin problem and also to a generalized model in which the gross risk premium is represented by a subordinator with a drift, while the claim process is a compound renewal process.

Weibull renewal equation solution

Abstract: The existing solution methods for the Weibull Renewal Equation suffer from a lack of sufficient accuracy due to the singularity at the origin for some parameter values of the weibull density. The proposed method of solution provides accuracy to any desired degree of precision for all parameter values particularly in the singular range. The method utilizes a cubic spline approximation of the unknown renewal function and applies the Galerkin technique of integral equation solution. Gaussian quadratures are used to evaluate integrals. The singular nature of the integrand is handled by the Gauss-Jacobi quadrature. Results are compared with those obtained by simulation.

Transient renewal processes in the subexponential case

Abstract: A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

Analysis for loss system with finite sources which require several channels simultaneously

Abstract: This paper analyzes a loss system with finite sources. The idle period of a source forms a renewal process and each customer requires several channels simultaneously. The service time of customers which require the same number of channels exhibit a general independent identical distribution. This is a general traffic model for a subscriber concentration stage in a high-speed wide-band communication network. The distributions of the numbers of customers in the system at arbitrary and customer arrival epochs are derived using the supplementary variable technique, where the backward recurrence-time of call idle period and holding time are chosen as supplementary variables. It is shown that these distributions are determined only by the mean service time and the mean idle period of a source, irrespective of their distributions (robustness). Finally, some traffic characteristics, such as loss probability and circuit occupancy, are evaluated numerically for two-traffic models and unbalanced input traffic models.

Applications in queueing theory

Abstract: We prove strong approximations for partial sums indexed by a renewal process. The obtained results are optimal. The established probability inequalities are also used to get bounds for the rate of convergence of some limit theorems in queueing theory.