Showing papers on "Renewal theory published in 1972"

Classes of distributions applicable in replacement, with renewal theory implications,

Abstract: : Age and block replacement policies are commonly used to diminish in-service failures. Unfortunately, for some items (say, those with decreasing failure rate), use of these policies may actually increase the number of in-service failures. In this paper the largest classes of life distributions are determined for which age and block replacement diminishes, either stochastically or in expected value, the number of failures in service. Bounds are obtained on survival probability, moment inequalities, and renewal quantity inequalities for distributions in these classes. It is shown that under certain reliability operations on components in a class of life distributions (such as formation of systems, addition of life lengths, and mixtures of distributions), life distributions are obtained which remain within the class. (Author)

Queues with mixed renewal and poisson inputs

Abstract: In a queueing system with two independent input streams, as exists, for example, when first-routed and overflow traffic streams are offered to a common sender-group, the state of the system encountered by the two different types of customers upon their arrival will generally be different. Consequently, in a system where delayed customers wait for service, the service rendered to the individual streams may also be different. The delay distribution in a single-server queue for each type of customer is derived under the assumption that one stream is Poissonian and the other is described by a renewal process. The difference in service received by the two streams is examined with the aid of numerical examples for two interarrival time distributions of the renewal stream. We show for two cases that a practical indicator of service received by the renewal customers is the coefficient of variation of their interarrival time distribution. If the coefficient is less than unity, then the renewal customers receive better service than the Poissonian customers. The converse is true when the coefficient exceeds unity. The stationary distribution of the number of busy servers in an infinite-server system as seen by the two types of customers is also derived.

Technical Note-Approximations to the Renewal Function mt

Abstract: Models of queuing, inventory, and reliability processes often have a useful renewal process imbedded in the fundamental stochastic process. The number of renewals is sufficient to determine performance measures such as the total cost, shortages, etc. The limit theorems of renewal theory are unsatisfactory in obtaining the expected values of these performance measures over a finite time horizon. This paper compares an accurate numerical technique for calculating the expected number of renewals with an approximation that uses the asymptotic expansion of the dominating residues of the Laplace transform MI¸. Furthermore, when a parameter of the renewal process is uncertain except for its Bayesian prior distribution, an approximation that uses a modified exponential renewal process appears better.

The convergence of a super-critical age-dependent branching process allowing immigration at the epochs of a renewal process

Abstract: Let N(t) be the number of individuals in a super-critical age-dependent branching process allowing immigration at the epochs of a renewal process. The mean square and almost sure convergence of the random variable W(t) = e-αt N(t) are proved, where a is the Malthusian parameter of the branching process.

The accuracy of call-congestion measurements for loss systems with renewal input

Abstract: The concept of a generalized renewal process is used to derive an asymptotic approximation for the variance of the observed proportion of unsuccessful attempts on a trunk group during a given time-interval. Calls are assumed to arrive according to a general renewal process, and those which are blocked leave the system and do not return (loss system). As an application of our result we examine the special case of an overflow input–an important example from telephone networks with alternate routing. Comparison of our results with values obtained from simulation indicates that the approximation is quite accurate for telephone traffic-engineering purposes.

A model for interaction of a Poisson and a renewal process and its relation with queuing theory

Abstract: This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found. RENEWAL PROCESS; POISSON PROCESS; INTERACTION; QUEUING THEORY; FINITE WAITING ROOM 1. The model and its connection with queuing theory Ten Hoopen and Reuver [2] have suggested the following kind of interaction between two random point processes as a mathematical model for neuron firing. Points in a renewal process, the excitatory process, are eliminated by points in a Poisson process, the inhibitory process, according to the following rule: whenever one or more points of the inhibitory process arrive, the next point of the excitatory process is eliminated. In this paper this model is generalized by assuming that one or more points of the inhibitory process eliminate a random number of points of the excitatory process. Ten Hoopen and Reuver's model has been generalized by Coleman and Gastwirth [1] in a different way: the points of the inhibitory process are effective during a time period of constant or random length, during which they eliminate all arriving points of the excitatory process. We assume in the following that the excitatory process is a renewal process and denote by F the c.d.f. of the time intervals between points in this process. Furthermore, we assume that the inhibitory process is a Poisson process with intensity A and that one or more points of the inhibitory process eliminate the next k points of the excitatory process with probability Pk, 0 Pk = 1. We also assume that points of the inhibitory process arriving during a time period when points are eliminated by previous points of the inhibitory process, do not affect the elimination process. The case studied by Ten Hoopen and Reuver is characterized by pl = 1. Received in revised form 28 July 1971. 451 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:41:49 UTC All use subject to http://about.jstor.org/terms 452 LENNART RADE Under these assumptions the random process of remaining points of the excitatory process, the response process, is a renewal process. We denote by F, the c.d.f. of the time intervals between points of this process. There is a close connection between the problem of determining the c.d.f. F, of the response process for a given c.d.f. F of the renewal excitatory process and a given elimination rule and queuing theory. As a matter of fact, the first problem can be interpreted as a problem of determining the busy period of a suitably chosen queuing situation. Consider for instance the case originally studied by Ten Hoopen and Reuver, when one or more points of the inhibitory process eliminate exactly one point of the excitatory process. In this case F, is the c.d.f. of the length of the busy period for the M/G/1 queuing system, where the arrival process is a Poisson process with intensity ), the service times have c.d.f. F and there is a finite waiting room with space for one waiting customer. For the case studied in [1] by Coleman and Gastwirth, when the inhibitors are effective during a random time period, one has to consider a similar queuing situation but with the waiting time of the customers limited by a random variable. For the case studied in this paper the appropriate queuing situation is an M/G/1 system with the following characteristics. The arrival process is a Poisson process with intensity A^ and there is room for only one waiting customer. The busy period is initiated by a virtual (imaginary) customer whose service time has c.d.f. F, while the subsequent customers have a service time with c.d.f. ,km=0 PkF*, where F[ is the convolution of k functions F; the latter service time can be considered as consisting of k phases with probability Pk. Arrivals are permitted only during the service time of the virtual customer, and during the last phase of the service of the actual customer, but at no other time. Put P,(x) = Probability that the length of the busy period is at most x, and the number of customers (including the virtual customer) served during the busy period is n. We now get PA(x) = fe-tdF(t) + Po (1 e-t) dF(t), P(x) = (1 e-'t')P,,1(xtl t2) dF(tl) d Pk_ l(t2) , n> 1. 1t=0 2=0 k=1 Taking Laplace-Stieltjes transforms yields (we denote by f the Laplace-Stieltjes transform off) P1(s) = P(s + o) + po(i(s)[(s + 2)), 00 P(s) = P~1(s)((s)P(s + )) pk p,(s)"k, ,1 > I k=l This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:41:49 UTC All use subject to http://about.jstor.org/terms A modelfor interaction of a Poisson and a renewalprocess 453 But Fi(x) = ~,P= 1P(x) and thus Pl(s) = ~~, 1P(s). Summation of the formulas above then gives P(s) = Poo(s) + (1 po)(s + ) PA(s) (+(s) f(s +)) pkF(S)k-1 k=

A Central Limit Theorem for Present Values of Discounted Cash Flows

Abstract: A renewal process is defined to represent the present value of a stream of money payments occurring at random times.The objective is to partially close a gap in the state of knowledge regarding the properties (asymptotic behavior) of certain cashflow models, scheduling models, and inventory models, all of which can be embedded in the context of replacement-reliability theory in general. The contribution of this paper is a characterization of the asymptotic distribution of a general capitalized random variable as the time-invariant rate of discount approaches zero. The main result is a central limit theorem demonstrating that the standardized distribution of the discounted random variable converges to the unit normal as the discount rate approaches zero.

Limit theorems for infinite systems of renewal type integral equations arising in age-dependent branching processes

Abstract: This article, a continuation of [6], is divided into three sections. In Section I some of the necessary results of the previous article are collected, and in Section 2 some generalizations of famous theorems from renewal theory, including those of Blackwell and Smith, are obtained. Finally, Section 3 is devoted to applications of the theory to age-dependent branching processes with arbitrary state space.

An asymptotic problem in renewal theory

Abstract: Summary A special problem in renewal theory is considered. The asymptotic behavior of the renewal function was studied by W. L. Smith. Here we show that his result with an exponentially small remainder term follows from a theorem of De Bruijn on Volterra integral equations.