Showing papers on "Renewal theory published in 1984"

Approximating a Point Process by a Renewal Process, II: Superposition Arrival Processes to Queues

Abstract: We develop an approximation for a queue having an arrival process that is the superposition of independent renewal processes, i.e., ∑GI1/G/1. This model is useful, for example, in analyzing networks of queues where the arrival process to an individual queue is the superposition of departure processes from other queues. If component arrival processes are approximated by renewal processes, the ∑GI1/G/1 model applies. The approximation proposed is a hybrid that combines two basic methods described by Whitt. All these methods approximate the complex superposition process by a renewal process and yield a GI/G/1 queue that can be solved analytically or approximately. In the hybrid method, the moments of the intervals in the approximating renewal process are a convex combination of the moments determined by the basic methods. The weight in the convex combination is identified using the asymptotic properties of the basic methods together with simulation. When compared to simulation estimates, the error in hybrid ...

A property of longtailed distributions

Abstract: We investigate sufficient conditions so that F,(x)= foF(y)dy is subexponential. Here F is a distribution function on [0,4 [, with finite mean. Some applications to risk theory and rates of convergence in renewal theory are given.

Inventory systems for perishable commodities with renewal input and poisson output

Abstract: We consider an inventory system to which arrival of items stored is a renewal process, and the demand is a Poisson process. Items stored have finite and fixed lifetimes. The blood-bank model inspired this study. Three models are studied. In the first one, we assume that each demand is for one unit and unsatisfied demands leave the system immediately. Using results on this model one is able to study a model in which arrival of items is Poisson but demands are for several units, and a model in which demands are willing to wait. We compute ergodic limits for the lost demands and the lost items processes and the limiting distribution of the number of items stored. The main tool in this analysis is an analogy to M/G/1 queueing systems with impatient customers.

Limit theorems for first-passage times in linear and non-linear renewal theory

Abstract: A local limit theorem for P{Ta = n, S7 - a 5 x} is obtained, where Ta is the first time a random walk S, with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

Solution of the volterra equation of renewal theory with the galerkin technique using cubic splines

Abstract: A numerical method for solving the renewal equation is proposed. The method which generates a cubic spline approximation of the renewal function by the Galerkin technique is tested on Gamma lifetime densities of various shapes. Results are compared against known analytical solutions and earlier approximation.

Bounds for Classical Ruin Probabilities

Abstract: We derive upper and lower bounds for the ruin probability over infinite time in the classical actuarial risk model (usual independence and equidistribution assumptions; the claim-number process is Poisson). Our starting point is the renewal equation for the ruin probability, but no renewal theory is used, except for the elementary facts proved in the note. Some bounds allow a very simple new proof of an asymptotic result akin to heavy-tailed claim-size distributions.

Strong Approximation of Extended Renewal Processes

Abstract: We develop a strong approximation approach to extended multidimensional renewal theory. The consequences of this approximation are a Bahadur-Kiefer type representation of the renewal process in terms of partial sums, Strassen and Chung type laws of the iterated logarithm. We also give a characterization of the renewal process by four classes of deterministic curves in the sense of Revesz (1982). We generalize our results to the case of non-independent and/or nonidentically distributed random vectors.

Model for chromatographic separations based on renewal theory

Abstract: A simple but reasonably realistic model is formulated for describing the behavior of chromatographic peaks. Our approach is based on statistical concepts and completely avoids the physically nonexistent theoretical plates of classical theory. This work complements the rate theory of chromatography in that we provide a more detailed look at the resistance to mass-transfer process (what we call the interphase process). The model is a stochastic one; because molecular level processes are random in nature, we feel that this is a natural approach. Although a variety of stochastic models have been proposed previously, they have been damaged by the necessity of assuming a particular mechanism. The present theory is largely immune from this criticism. The paper makes use of results from the theory of renewal processes, but the results should be comprehensible to anyone with only a modest acquaintance with statistical notions.

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.

Analysis of single server preemptive priority queues with a renewal process as high‐priority input stream

Abstract: This paper considers a preemptive priority queueing model with two different kinds of priority calls: the inter-arrival time distributions of high-priority call and low-priority call are a general distribution and an exponential distribution, respectively, where the Laplace-Stieltjes transforms of a general distribution is meromorphic. Both service time distributions are different exponential ones. the service of a high-priority call begins even if a low-priority call is under service. Under these assumptions, the mean time in the system of each call is obtained, using the theory of piecewise-Markov processes. First, the generating functions of steady-state probabilities of an imbedded Markov chain are derived. Next, by the rate conservation principle of piecewise-Markov processes, the relation between steady-state probabilities at a regeneration point and at any time is derived. the average numbers of calls in the system are computed by these probabilities. Further, applying Little's formula to the average number of calls in the system, the mean times in the system are also computed. to evaluate these discussions, it is determined whether these results agree with those of M1, M2/M1, M2/1 preemptive priority queueing models, when the inter-arrival time distribution of priority call is exponential. Finally, a numerical example is given.

Omni Transforms: Applications to Renewal Theory.

Abstract: : The theory of the renewal process has an important bearing on the single server queue, especially on the model M/G/1 (although some results are obtained for G/G/1 also). Since a renewal process can be viewed as a single-server saturated system (i.e., a system whose source keeps the server perpetually busy) this affinity is not surprising. Moreover, in deriving some results for the renewal process we gain insight into the use of the conservation method, based on the application of ergodic variables and of the so-called omni-transform, to the treatment of single-server queues. The renewal process adds insight into the theory of mixtures of random variables and of omni-forms (i.e., linear functions of omni-transforms). It allows to convert, when possible in principle, an omni-equation involving derivatives into an omni-equation without derivatives, an equation which is a statement about a mixture of distributions. (Author)

Models for some stochastic systems

Abstract: Renewal theory is seen to be an appropriate tool for studying transitions in certain graded social processes. Here the length of service (LOS) in an organization is treated as a renewal theory. By fitting some distributions to data, approximate values of the renewal equations were calculated. The values of h°(T)and the rate of withdrawal at time T,for the logarithmic transformations of data were also obtained.

Chapter 7 – Renewal Phenomena

Abstract: Publisher Summary This chapter presents the definition of a renewal process and discusses related concepts. The study of stochastic systems began the theory of renewal phenomena. The evolution of this process started interspersing with renewals or regeneration times and viewed as a the study of general functions of independent, identically distributed, nonnegative random variables representing the successive intervals between renewalsat present. This chapter discusses the examples of renewal processes, block replacement, the Poisson process viewed as a renewal process, the elementary renewal theorem, the renewal theorem for continuous lifetimes and the limiting distribution of age and excess life of the asymptotic behavior of renewal processes. The generalizations and variations on renewal processes that includes delayed renewal processes, stationary renewal processes, cumulative and related processes, stationary renewal processes, cumulative and related processes, and discrete renewal theory and its theorems are also presented in the chapter.

Topics in stochastic point processes, by applications in hydrology

Abstract: The strong impetus for the research in the theory of point processes comes from applications, real or potential, to a multitude of engineering, industrial and biological problems. Here we discuss some particular topics in this area and their applications in hydrology. Specifically, we consider stochastic models of a variety of geophysical phenomena, such as the rainfall, floods, sediment transport, dispersion in porous media and some others. The first part of this presentation is concerned with point processses on R+. Our discussion includes the stochastic intensity, some remarks on the martingale approach and a brief expose of the elements of renewal theory. In addition, we discuss in some detail the case when the counting random function represents a Markov process. In the second part we give an introduction to the theory of point processes on an abstract topological space. Elements of marked point processes, which are of particular interest in hydrological investigations, are also included. The rest of the paper is concerned with some hydrological applications.

A note on the characteristics of poisson processes

Abstract: In this note two characteristic theorems of Poisson processes are given. If {N(t);t≧0} is a renewal process,U t ,V t are, respectively, the time since the last renewal and the time to the next renewal att, Z t =U t +V t , then a Poisson process can be characterized by the limiting independene of the joint distribution of (U t ,V t ) whent→∞, orEZ t , or the distribution ofZ t .