Showing papers on "Renewal theory published in 1994"

The transient BMAP/G/l queue

Abstract: We derive the two-dimensional transforms of the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes. Numerical results are obtained via two-dimensional transform inversion algorithms based on the Fourier-series method. From the numerical examples we see that predictions of system performance based on transient and stationary performance measures can be quite different

Kitting process in a stochastic assembly system

Abstract: In small-lot, multi-product, multi-level assembly systems, kitting (or accumulating) components required for assembly plays a crucial role in determining system performance, especially when the system operates in a stochastic environment. This paper analyzes the kitting process of a stochastic assembly system, treating it as an assembly-like queue. If components arrive according to Poisson processes, we show that the output stream departing the kitting operation is a Markov renewal process. The distribution of time between kit completions is also derived. Under the special condition of identical component arrival streams having the same Poisson parameter, we show that the output stream of kits approximates a Poisson process with parameter equal to that of the input stream. This approximately decouples assembly from kitting, allowing the assembly operation to be analyzed separately.

Availability for repairable components and series systems

Abstract: The failure pattern of repairable components is often modelled by an alternating renewal process which implies that a failed component is perfectly repaired. In practice, repair is often imperfect. This paper proposes a generalized availability model for repairable components and series systems. The lifetime of a repaired component has a general distribution which can be different from that of a new component. Availability and some asymptotic quantities in these models are derived. An example illustrates the application of these models. >

Mr/gi/1 queues with positively correlated arrival stream

Abstract: The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times

Characterization of distributions by relationships between failure rate and mean residual life

Abstract: The characterizations described use relations between the failure rate function and conditional expectation. The theorems proved here extend the results of some authors and can be used in the context of the renewal process. The utility of these results is demonstrated by using them to characterize some common distributions. >

Classes of life distributions and renewal counting process

Abstract: We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Weibull renewal processes

Abstract: We study the class of renewal processes with Weibull lifetime distribution from the point of view of the general theory of point processes. We investigate whether a Weibull renewal process can be expressed as a Cox process. It is shown that a Weibull renewal process is a Cox process if and only if 0<α≤1, where α denotes the shape parameter of the Weibull distribution. The Cox character of the process is analyzed. It is shown that the directing measure of the process is continuous and singular.

Application of perturbation analysis to a class of periodic review (s, S) inventory systems

Abstract: In this article we apply perturbation analysis (PA), combined with conditional Monte Carlo, to obtain derivative estimators of the expected cost per period with respect to s and S, for a class of periodic review (s, S) inventory systems with full backlogging, linear holding and shortage costs, and where the arrivals of demands follow a renewal process. We first develop the general form of four different estimators of the gradient for the finite-horizon case, and prove that they are unbiased. We next consider the problem of implementing our estimators, and develop efficient methodologies for the infinite-horizon case. For the case of exponentially distributed demand interarrival times, we implement our estimators using a single sample path. Generally distributed interarrival times are modeled as phase-type distributions, and the implementation of this more general case requires a number of additional off-line simulations. The resulting estimators are still efficient and practical, provided that the number of phases is not too large. We conclude by reporting the results of simulation experiments. The results provide further validity of our methodology and also indicate that our estimators have very low variance. © 1994 John Wiley & Sons, Inc.

Identification of alternating renewal electric load models from energy measurements

Abstract: In statistical load modeling methodologies, aggregate electric load behavior is derived by propagating the ensemble statistics of an individual load process which is representative of the loads in the aggregate. Such a modeling philosophy tends to yield models whereby if physical meaning is present at the elemental level, it is preserved at the aggregate level. This property is essential for applications involving direct control of power system loads. The potential applicability of statistical load models is a strong function of one's ability to limit the volume of unusual data required to build those. An identification algorithm for a previously proposed stochastic hybrid-state Markov model of individual heating-cooling loads is presented. It relies only on data routinely gathered in power systems (device energy consumption over constant time intervals). It exploits an alternating renewal viewpoint of the load dynamics. After deriving some general results on the occupation statistics of time homogeneous alternating renewal processes, the analysis is focused on the specific model. In the process, however, some intriguing features likely to be shared by a wide class of alternating renewal processes are revealed. >

Non-parametric Confidence Intervals for the Renewal Function and the Point Availability

Abstract: A large sample non-parametric method for constructing confidence intervals for the renewal function and the point availability is investigated. The method is based on a linearization and on the fact that the empirical distribution function converges weakly to a Gaussian process as the sample size increases. The technique is illustrated by the analysis of some hitherto unpublished data. Two of the most important functions arising in renewal theory are the renewal function, the expected number of renewals in a given interval, and the point availability, the probability that a system modelled by an alternating renewal process (ARP) is in a particular state at a specified time. See, for example, Karlin & Taylor (1975, Ch. 5), Ross (1970, Ch. 3) and Cox (1962) for a discussion of applications of these functions. If the functional forms of the distribution functions of the random variables generating the processes are known, and observations of the random variables are available, point estimates of these functions are readily constructed. Further, approximate (large sample) confidence intervals may, in principle, be calculated by an application of the delta method, assuming that the parameter estimates are asymptotically normally distributed. If, however, as is sometimes the case, the functional forms of the underlying distribution functions are unknown, a non-parametric approach is required. Frees (1986a, b, 1988) discussed some non-parametric estimators of the renewal function and constructed a non-parametric confidence interval for this quantity. See Schneider et al. (1990) for a study of these estimators. In this paper, we propose an alternative non-parametric confidence interval for the renewal function which is easier to compute than that of Frees (1986a) and which is appreciably narrower. In addition, we derive an analogous non-parametric confidence interval for the point availability. To the best, of our knowledge, this is the first non-parametric interval estimator of the point availability to have ,been proposed. Our methodology is based on the analysis of Harel et al. (1994), who prove that the empirical renewal function converges weakly to a Gaussian process as the sample size increases. A numerical study shows that our proposed confidence intervals are easy to compute, requiring only a few seconds of CPU time on a Sun Sparc Station, and are fairly narrow for moderate sample sizes.

A Central Limit Theorem for Cumulative Processes

Abstract: A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Optimal arrangements of cartridges in carousel type mass storage systems

Abstract: Optimal arrangements of cartridges and file partitioning schemes are examined in carousel type mass storage systems using Markov decision theory. It is shown that the Organ-Pipe Arrangement is optimal under different storage configurations for both the anticipatory as well as the non-anticipatory versions of the problem. When requests arrive as per an arbitrary renewal process this arrangement is also shown to minimize the mean queueing delay and the time spent in the system by the requests

On termination time processes

Abstract: In this paper we study the behavior of a delayed compound renewal process, S, about some fixed level, L. Normally, a jump process S increases at random times rl, r2,. . ., in random increments until it crosses L. S would then be terminated in a random number v of phases at time 7,. In many applications, a more general termination scenario assumes that S may evolve either through v or a random phases, whichever of the two is smaller (denoted by T). The number T of actual phases is called the termination index, and we evaluate a joint functional of T, the termination time rT and the termination level ST. We also seek information about the process S a step before its termination, and derive a joint functional for all relevant processes. Examples of these processes and their applications to various stochastic models are discussed.

Limit theorems for stopped random sequences. I: Rates of convergence and asymptotic expansions

Abstract: Stopped random sequences with record regeneration are considered and limit theorems refining the normal approximation are proved. Particular cases of such sequences are stopped random walks, recurrent Markov renewal processes, and certain procedures of sequential estimation.

Functional approximation theorems for controlled renewal processes

Abstract: We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.

Renewal Approximations of the Switched Poisson Processes and Their Applications to Queueing Systems

Abstract: The switched Poisson process (SPP), also known as the doubly stochastic Poisson process, has been widely used in the modelling of point processes whose rates vary subject to some random mechanism. The class of SPP includes a wide range of both renewal and non-renewal processes with squared coefficients of variation being larger than one. In this paper, we survey various approaches to approximate a non-renewal process by a renewal process. We derive the expressions for the first two moments of the inter-renewal time of a renewal process that approximates the SPP. We illustrate the quality of these approximations with numerical results in queueing applications. We believe that our approximations have potential applications in areas such as reliability, inventory control, telecommunications and maintenance.

Change Point and Jump Estimates in an AMOC Renewal Model

Abstract: A simple renewal model is studied which allows for atmost one change (AMOC) in the distribution of the underlying interoccurence times. We discuss tests and estimates concerning the change point as well as the magnitude of jump at this point. The results include limiting null distributions, consistency of tests and estimates, and asymptotic normality. Proofs make use of invariance principles for the renewal process under consideration.

Some renewal theory results with application to fleet warranties

Abstract: The fleet warranty guarantees the purchaser of a large population of like items that the mean life of the fleet will meet or exceed some negotiated mean μL. If the mean life is less than μL, compensation may be given in terms of a number of free replacement parts R. The expected number of replacements E[R] is studied based upon how the mean life of items in the field is determined and on whether the sampling window starts at time t = 0 (ordinary renewal process) or at some arbitrarily large time w (equilibrium renewal process). Properties of E[R] are compared and examples are given. © 1994 John Wiley & Sons, Inc.

On markovian traffic with applications to tes processes

Abstract: Markov processes arc an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose inter-arrival times (separating the time points of discrete arrivals) form a real-valued Markov chain. As such this paper aims to cxtcnd the classical results of renewal traffic, where interarriva] times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The paper derives integral equations for these functions in the transform domain. These arc then specialized to a subclass, TES +, of a versatile class of random sequences, called TES (Transform-Expan&SampIe), consisting of marginally uniform autoregressivc schemes with modu]o-i reduction, followed by various transformations. TES models arc designed to simultaneously capture both first-order and second-order statistics of empirical records, and consequently can produce high-fidelity models. Two theoretical solutions for TES + traffic functions are rived: an operator-based solution and a matric solution, both in the transform domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to obtain instructive closed-form representations for two measures of traffic burstincss: peakedness and index of dispersion, elucidating the relationship between them.

On performance comparison of MR/GI(1 queues

Abstract: In this paper we are interested in the effect that dependencies in the arrival process to a queue have on queueing properties such as mean queue length and mean waiting time. We start with a review of the well known relations used to compare random variables and random vectors, e.g., stochastic orderings, stochastic increasing convexity, and strong stochastic increasing concavity. These relations and others are used to compare interarrival times in Markov renewal processes first in the case where the interarrival time distributions depend only on the current state in the underlying Markov chain and then in the general case where these interarrivai times depend on both the current state and the next state in that chain. These results are used to study a problem previously considered by Patuwo et al. [14]. Then, in order to keep the marginal distributions of the interarrivai times constant, we build a particular transition matrix for the underlying Markov chain depending on a single parameter,p. This Markov renewal process is used in the Patuwo et al. [14] problem so as to investigate the behavior of the mean queue length and mean waiting time on a correlation measure depending only onp. As constructed, the interarrival time distributions do not depend onp so that the effects we find depend only on correlation in the arrival process. As a result of this latter construction, we find that the mean queue length is always larger in the case where correlations are non-zero than they are in the more usual case of renewal arrivals (i.e., where the correlations are zero). The implications of our results are clear.

The matched queueing system GIoPh/Ph/1

Abstract: We study the matched queueing system GIoPH/PH/1, where the type-I input is a renewal process, the type-II input is a PH renewal process, and the service times are i.i.d. random variables with PH-distributions. First, a condition is given for the stationarity of the system. Then the distributions of the number of type-I customers at the arrival epoches of type-I customers and the number of type-I customers at an arbitrary epoch are derived. We also discuss the occupation time and the waiting time. Their L.S. transforms are derived. Finally, we discuss some problems in numerical computation.

On continuous-time two person full-information best choice problem with imperfect observation

Abstract: A zero-sum game version of the continuous-time full-information best choice problem is considered. Two players observe sequentially a stream of iid random variables from a known continuous distribution appearing according to some renewal process with the object of choosing the largest one. The horizon of observation is a positive random variable independent of observations. The observations of the random variables are imperfect and the players are informed only whether it is greater than or less than some levels specified by both of them. The normal form of the game is derived. Poisson horizon case is examined in detail.

Dynamic response of non-linear systems to renewal-driven random pulse trains

Abstract: The moment equations technique is developed for the non-linear dynamical systems under random pulse trains driven by a class of renewal processes. Since the increments of the considered point process are not statistically independent, the direct application of the generalized Ito's differential rule does not yield the explicit equations for moments. Hence, the approach is suitably modified. First, the excitation term is recast, for an ordinary renewal process with gamma-distributed, with k = 2, interarrivai times, as a transformation of a Poisson counting process, which allows to perform the averaging of the differential rule. Next, for the additional unknown expectations which consequently appear in the equations for moments, the differential equations in the form of the correlation splitting formulae are derived. The technique developed is applied to a linear oscillator and to a Duffing oscillator. In the latter case, suitable closure approximations are used in order to truncate the hierarchy of moment equations. The analytical results (transient response moments up to fourth order) are verified against the results of Monte Carlo simulations.

Equivalence of renewal sequences and isomorphism of random walks

Abstract: We show that centred aperiodic random walks on ℤ d whose jump random variables are inL 2√log+ L have equivalent renewal sequences. An isomorphism theorem is deduced.

The correlation of the backward and forward recurrence times in a renewal process

Abstract: The joint complementary distribution function is used to obtain the covariance function of the backward and forward recurrence times in an ordinary renewal process for both the time dependent and the steady state cases. Hence, a closed form expression for the steady state correlation of the backward and forward recurrence times is obtained and special cases are investigated

Generating Antithetic Random Variates in Simulation of a Replacement Process by Rejection Method

Abstract: When the times between renewals in a renewal process are not exponentially distributed, simulation can become a viable method of analysis. The renewal function is estimated through simulation for a renewal process simulation for a renewal process with gamma distributed renewal times and the shape parameter a > 1. Gamma random deviates will be generated by means of the so called Acceptance Rejection method. In order to reduce the variance of the point estimator, the idea of antithetic variates will be incorporated in the sampling process. It is shown that such sampling scheme is capable of reducing the variance of the point estimator. Finally, an algorithm is developed and along with the experimental results is verified.