Showing papers on "Renewal theory published in 2002"

Exact fill rates for (R, s, S) inventory control with gamma distributed demand

Abstract: For the familiar (R, s, S) inventory control system only approximate expressions exist for the fill rate, ie the fraction of demand that can be satisfied from stock. Best-known are the approximations derived from renewal theory, holding under specific conditions; in particular, S–s should be reasonably large. Here, an exact expression for the fill rate is derived, holding generally in the situation that demand has a gamma distribution with known integer-valued shape parameter, while lead time is constant. These exact results allow a check of the renewal theory based approximations. In addition, an extremely fast simulation program was written, obviously holding for general shape parameter values.

Modeling and analysis of hierarchical cellular networks with general distributions of call and cell residence times

Abstract: We present an analytic model for the performance evaluation of hierarchical cellular systems, which can provide multiple routes for calls through overflow from one cell layer to another. Our model allows the case where both the call time and the cell residence time are generally distributed. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of channel occupancy time distribution for each call type (new call, handoff call, and overflow call) is derived as a function of the Laplace transform of cell residence time. In particular, overflow calls are modeled by using a renewal process. Performance measures are derived based on the product form solution of a loss system with capacity limitation. Numerical results show that the distribution type of call time and/or cell residence time has influence on the performance measure and that the exponential case may underestimate the system performance.

Exact transient analysis of a planar random motion with three directions

Abstract: Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.

Stochastic models in reliability and maintenance

Abstract: 1 Renewal Processes and Their Computational Aspects- 11 Introduction- 12 Basic Renewal Theory- 121 Continuous renewal theory- 122 Discrete renewal theory- 13 Some Useful Properties of the Renewal Function- 131 Specific examples- 132 Asymptotic properties- 14 Analytical Approximation Methods- 141 Phase renewal processes- 142 Gamma approximations- 143 Methods based on equilibrium distribution- 15 Bounds- 16 Numerical Methods- 161 Laplace inversion technique- 162 Cubic spline algorithm- 163 Discritization algorithm- 164 Approximation by rational functions- 17 Concluding Remarks- 2 Stochastic Orders in Reliability Theory- 21 Introduction- 22 Definitions and Basic Properties- 221 Stochastic orders generated from univariate functions- 222 Conditional stochastic orders- 223 Bivariate characterization of stochastic orders- 23 Applications in Reliability Theory- 231 Notions of aging- 232 Useful stochastic inequalities in reliability theory- 233 Stochastic comparisons of system reliabilities- 234 Redundancy improvement- 235 Stochastic comparisons of maintenance policies- 2351 Replacements upon failures- 2352 Age replacement- 2353 Block replacement- 2354 Minimal repair- 2355 Minimal repair with block replacement- 2356 Stochastic comparison of different maintenance policies- 2A TP2 Functions- 3 Classical Maintenance Models- 31 Introduction- 32 Block Replacement- 33 Age Replacement- 34 Order Replacement- 35 Inspection Strategies- 36 Conclusions- 4 A Review of Delay Time Analysis for Modelling Plant Maintenance- 41 Introduction- 42 Maintenance Practice- 43 The Delay Time Concept- 44 Basic Delay Time Maintenance Model: Complex Plant- 45 Basic Maintenance Model: Component Tracking- 46 Relaxation of Assumptions- 47 Non-perfect Inspection- 48 Non-steady-state Condition- 49 Non-homogeneous Defect Arrival Rate ?- 410 Condition-dependent Cost and Downtime for Repair- 411 Case Experience Using Subjective Data: Case Experience- 412 Revision of Subjectively Estimated Delay Time Distribution- 413 Correction for Sampling Bias- 414 Subjective Estimation of the Delay Time Distribution Directly- 415 Objective Estimation of Delay Time Parameters- 416 Case Experience Using Objective Data: HPP of Defect Arrival- 417 Discussion of Further Developments in Delay Time Modelling- 418 Conclusions- 5 Imperfect Preventive Maintenance Models- 51 Introduction- 52 Sequential Imperfect Preventive Maintenance- 521 Introduction- 522 Model A - age- 523 Model B - failure rate- 524 Numerical examples- 53 Shock Model with Imperfect Preventive Maintenance- 531 Introduction- 532 Model and expected cost- 533 Optimal policies- 54 Conclusions- 6 Generalized Renewal Processes and General Repair Models- 61 Background and Motivation- 62 Generalized Renewal Processes- 63 g-Renewal Processes in Discrete Time- 64 Monotonicity and Asymptotic Properties of the g-Renewal Density- 65 On the g-Renewal Function- 66 A General Repair Model- 7 Two-Unit Redundant Models- 71 Introduction- 72 Two-Unit Standby System- 721 Model and assumptions- 722 First-passage time distributions- 723 Expected numbers of visits to state- 724 Transition probabilities- 73 Preventive Maintenance of Two-Unit Systems- 731 Model and analysis- 732 Optimum preventive maintenance policies- 733 Replacement of a two-unit parallel system- 74 Other Two-Unit Systems- 741 Two-unit parallel system- 742 Two-unit priority standby system- 743 Two-unit standby system with imperfect switchover- 744 Other models- 8 Optimal Maintenance Problems for Markovian Deteriorating Systems- 81 A Basic Optimal Replacement Problem for a Discrete Time Markovian Deteriorating System- 811 Some conditions on transition probabilities and cost structure- 812 Formulation by Markovian decision process (MDP)- 813 Optimality of control limit rule- 82 An Optimal Inspection and Replacement Problem- 821 Transition probability- 822 Formulation by semi-Markov decision process (SMDP)- 823 Structure of optimal inspection and replacement policy- 83 An Optimal Inspection and Replacement Policy with Incomplete Information- 831 Some notations and conditions- 832 Formulation by partially observable Markov decision process (POMDP)- 833 Some properties of TP2 order- 834 Some properties of optimal function- 835 Structure of optimal inspection and replacement policy- 84 A Continuous Time Markovian Deteriorating System- 841 A continuous time Markovian deteriorating system- 842 Transition probability- 843 Formulation by semi-Markov decision process- 844 Structure of optimal policy- 85 An Optimal Maintenance Problem for a Queueing System- 851 Model description- 852 Formulation by semi-Markov decision process- 853 Properties of value function- 854 Structure of optimal policy- 9 Transient Analysis of Semi-Markov Reliability Models - A Tutorial Review with Emphasis on Discrete-Parameter Approaches- 91 Introduction- 92 Modelling Framework- 93 Dependability Measures- 94 Methods of Analysis- 941 Continuous-parameter models- 942 Discrete-parameter models- 95 Equations for the Dependability Measures- 96 Numerical Solution Techniques- 961 Solving the integral equations- 962 Discrete-parameter approximations- 97 Recent Developments, Conclusions and Further Work- 10 Software Reliability Models- 101 Introduction- 102 Definitions and Software Reliability Model- 103 Software Reliability Growth Modeling- 104 Imperfect Debugging Modeling- 1041 Imperfect debugging model with perfect correction rate- 1042 Imperfect debugging model for introduced faults- 105 Software Availability Modeling- 1051 Model description- 1052 Software availability measures- 106 Application of Software Reliability Assessment- 1061 Optimal software release problem- 10611 Maintenance cost model- 10612 Maintenance cost model with reliability requirement- 1062 Statistical software testing-progress control- 1063 Optimal testing-effort allocation problem- 11 Reliability Models in Data Communication Systems- 111 Introduction- 112 SW ARQ Model with Intermittent Faults- 1121 Intermittent faults- 1122 ARQ policy- 1123 Optimal retransmission number- 1124 Numerical examples and remarks- 113 SR ARQ Model with Retransmission Number- 1131 Model and analysis- 1132 Optimal policy- 1133 Numerical examples and remarks- 114 Hybrid ARQ Models with Response Time- 1141 Type-I hybrid ARQ- 1142 Type-II hybrid ARQ- 1143 Comparison of type-I and type-II hybrid ARQs- 1144 Numerical examples and remarks- 12 Quick Monte Carlo Methods in Stochastic Systems and Reliability- 121 Introduction- 122 The Problem with Direct Simulation- 123 Importance Sampling- 124 The Optimal Change of Measure- 1241 Remarks- 1242 Preliminary definitions- 1243 The recursive approach- 1244 Exact calculation of ?(x)- 125 Cases of Application of the Recursive Approach- 126 System Model- 127 Regenerative Simulation- 128 Failure Biasing Methods- 1281 Simple failure biasing (SFB)- 1282 Balanced failure biasing (BFB)- 1283 Bias2 failure biasing- 1284 Failure distance biasing (FDB)- 1285 Balanced 1 failure biasing (B1FB)- 1286 Balanced 2 failure biasing (B2FB)- 1287 Bounded relative error and failure biasing- 129 Unreliability Estimation- 1291 One-component system- 1292 General case- 1293 Example- 1210 Analytical-Statistical Methods- 1211 Concluding Remarks

Truncated Sequential Change‐point Detection based on Renewal Counting Processes

Abstract: The typical approach in change-point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. Based on the, perhaps, more realistic situation that the process can only be partially observed, we consider the counting process related to the original process observed at equidistant time points, after which action is taken or not depending on the number of observations between those time points. In order for the procedure to stop also when everything is in order, we introduce a fixed time horizon n at which we stop declaring “no change” if the observed data did not suggest any action until then. We propose some stopping rules and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes and extreme value asymptotics for Gaussian processes.

Properties of a Subclass of Avakumović Functions and Their Generalized Inverses

Abstract: We study properties of a subclass of ORV functions introduced by Avakumovic and provide their applications for the strong law of large numbers for renewal processes.

On the convexity of loss probabilities

Abstract: For the M/G/c loss system, it is well known that Erlang's loss probability is convex in the number of servers. We extend this result firstly to renewal arrivals and exponential service, then to regenerative arrivals and exponential service, and finally to an arbitrary arrival process with i.i.d. service times that are independent of the arrival process.

Slip‐size‐dependent renewal processes and Bayesian inferences for uncertainties

Abstract: [1] This paper is initially concerned with uncertainty of hazard estimates using renewal process models where the parameters are poorly constrained because of the scarcity of large earthquakes occurring along the same fault segment. A Bayesian inference is adopted, taking account of the whole likelihood function of the parameters. Also, new statistical models are considered which make use of knowledge on the slip associated with earthquake ruptures, may help to improve the hazard estimate in a positive way. Including these models, the predictive efficiency is compared by using the Akaike's Bayesian information criterion (ABIC). Three data sets are analyzed for the illustrations. The Brownian Passage Time (BPT) model is selected to fit the first data set, consisting of 10 historical great earthquakes from Nanaki trough in Japan. However, its predictive hazard function shows a large uncertainty (>100 years) of likely occurrence time around 2070. The second data set consists of the last three events of the first data set but associated with record of slip sizes, for which the ABIC selected the extended lognormal renewal process model where the time intervals between successive events are normalized by the corresponding slip sizes of the starting events of the intervals. The estimated predictive hazard function implies that the next event is likely to occur around 2040 ± 10. The last data set, consisting of 4 events with estimated occurrence times and slip sizes from a submarine fault. The ABIC selected the slip-size-dependent BPT model for this data. This model indicates that the likelihood of occurrence time of the next event is decreasing from now, and the period of its half decay is more than several hundred years. For the data sets of the last two examples, it was also shown that the slip size records are useful for better prediction of the next event.

Weighted renewal functions: a hierarchical approach

Abstract: We extend classical renewal theorems to the weighted case. A hierarchical chain of successive sharpenings of asymptotic statements on the weighted renewal functions is obtained by imposing stronger conditions on the weighting coefficients.

Laplace-transforms and fast-repair approximations for multiple availability and its generalizations

Abstract: Stochastic models, describing multiple availability, are analyzed for a system with periods of operation and repair that form an alternating renewal process with exponential times to failure and repair. For the simplest case multiple availability is defined as the probability that the system is available in the interval [0, t) at each moment of demand. Instants of demand form a homogeneous Poisson process. This setting is generalized to considering a possibility of one or more points of unavailability in [0, t) as well as time redundancy. The corresponding integral equations are derived and solved (wherever possible) via the Laplace transform. A fast repair approach is also applied to each case under consideration and simple approximate relations for multiple availability are obtained. The fast repair approximation makes it possible to derive approximate solutions for problems that cannot be solved by the first approach. The accuracies of the fast repair approximations are analyzed. Generalizations to arbitrary failure and repair distributions are also discussed.

The Convergence Result for Critical Reversible Nearest Particle Systems

Abstract: We consider critical reversible nearest particle systems. We assume that the associated renewal measure has large moments as well as some regularity conditions. It is shown that such processes, started from a nontrivial ergodic translation invariant distribution, converge in distribution to the upper invariant measure.

Renewal approximations for the departure processes of batch systems

Abstract: In this paper we consider three standard batch arrival/service systems and develop renewal process approximations for the inter-departure time squared coefficient of variations. The systems considered are: (i) batch arrival with individual service; (ii) batching for setup reduction for a single server system; and (iii) batch processing. The random version of the batch arrival with individual service case naturally results from random branching of individuals following a batch service process. This case is also developed with the intent of improving the modeling results for downstream workstations following a batch service process.

On the Ruin Probability Under a Class of Risk Processes

Abstract: In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount distribution is a mixed exponential. As its consequence, a well-known result about ultimate ruin probability in the classical risk model is obtained.

Analysis of the Discrete-Time G(G)/Geom/c Queueing Model

Abstract: This paper presents the steady-state analysis of a discrete-time infinite-capacity multiserver queue with c servers and independent geometrically distributed service times. The arrival process is a batch renewal process, characterized by general independent batch interarrival times and general independent batch sizes. The analysis has been carried out by means of an analytical technique based on generating functions, complex analysis and contour integration. Expressions for the generating functions of the system contents during an arrival slot as well as during an arbitrary slot have been obtained. Also, the delay in case of a first-come-first-served queueing discipline has been analyzed.

Order estimation for a special class of hidden Markov sources and binary renewal processes

Abstract: We consider the estimation of the order, i.e., the number of hidden states, of a special class of discrete-time finite-alphabet hidden Markov sources. This class can be characterized in terms of equivalent renewal processes. No a priori bound is assumed on the maximum. permissible order. An order estimator based on renewal types is constructed, and is shown to be strongly consistent by computing the precise asymptotics of the probability of estimation error. The probability of underestimation of the true order decays exponentially in the number of observations while the probability of overestimation goes to zero sufficiently fast. It is further shown that this estimator has the best possible error exponent in a large class of estimators. Our results are also valid for the general class of binary independent-renewal processes with finite mean renewal times.

Optimal stopping of a risk process: model with interest rates.

Abstract: The following problem in risk theory is considered. An insurance company, endowed with an initial capital a > 0, receives premiums and pays out claims that occur according to a renewal process {N(t),t > 0}. The times between consecutive claims are independent and identically distributed (i.i.d.). The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate 2 [0, 1], claims increase at rate 2 [0, 1]. The aim is to find the stopping time that maximizes the capital of the company. A dynamic programming method is used to find the optimal stopping

Teletraffic analysis of cellular communication systems with general mobility based on hyper-Erlang characterization

Abstract: This paper proposes a new analytical model for the performance evaluation of cellular communication systems. Our model allows more general cases of the distributions of the call time and the cell residence time. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of the channel occupancy time distribution is derived as a function of the Laplace transform of the cell residence time distribution. In particular, the essential quality of service measure such as the forced termination probability is exactly obtained by using a delayed renewal process. From numerical results, it is shown that the non-exponential model has an obvious difference in performance as compared to exponential models.

Stone’s decomposition of the renewal measure via Banach-algebraic techniques

Abstract: A Banach-algebraic approach to Stone's decomposition of the renewal measure is discussed. Estimates of the rate of convergence in a key renewal theorem are given.

Statistical Inference for an Arithmetic Process

Abstract: A stochastic process {, n = 1, 2, ...} is an arithmetic process (AP) if there exists some real number, d, so that { + (n-1)d, n =1, 2, ...} is a renewal process (RP). AP is a stochastically monotonic process and can be used for modeling a point process, i.e. point events occurring in a haphazard way in time (or space), especially with a trend. For example, the vents may be failures arising from a deteriorating machine; and such a series of failures id distributed haphazardly along a time continuum. In this paper, we discuss estimation procedures for an AP, similar to those for a geometric process (GP) proposed by Lam (1992). Two statistics are suggested for testing whether a given process is an AP. If this is so, we can estimate the parameters d, and of the AP based on the techniques of simple linear regression, where and are the mean and variance of the first random variable respectively. In this paper, the procedures are, for the most part, discussed in reliability terminology. Of course, the methods are valid in any area of application, in which case they should be interpreted accordingly.

Conservative systems of integral convolution equations on the half-line and the entire line

Abstract: The following system of integral convolution equations is considered: where the -matrix-valued function satisfies the conditions of conservativeness Here is the class of non-negative indecomposable -matrices and is the spectral radius of the matrix . For the equation in question is a conservative system of Wiener-Hopf integral equations. For this is the multidimensional renewal equation on the entire line. Questions of the solubility of the inhomogeneous and the homogeneous equations, asymptotic and other properties of solutions are considered.The method of non-linear factorization equations is applied and developed in combination with new results in multidimensional renewal theory.

The use of lifetime distributions in bridge replacement modelling

Abstract: This paper proposes a new method to determine lifetime distributions for concrete bridges and to compute the expected cost of replacing a bridge stock. The uncer- tainty in the lifetime of a bridge can best be represented with a Weibull distribution. It is recommended to fit this Weibull distribution on the basis of aggregating the lifetimes of demolished bridges (complete observations) and the ages of current bridges (right-censored observations). Using renewal theory, the future expected cost of replacing the bridge stock can then be easily determined while taking account of the current bridge ages and the cor- responding uncertainties in the future replacement times. The proposed methodology is used to estimate the cost of replacing the Dutch stock of concrete bridges as a function of time.

Renewal Processes and Their Computational Aspects

Abstract: In this chapter, we review classical renewal theory and focus on the computational aspects for the renewal function. As well known, the renewal processes have an important role in understanding the discrete event systems arising in queueing theory, production and inventory control, design of communication systems, performance evaluation in computer science and product warranty estimation and also in reliability and maintenance modeling. On the other hand, from the practical perspective, since the computation of the renewal function is not so easy, the system analyst tends to treat the renewal function via the simplest method for him or her. In fact, a large number of authors have discussed the computation problems of the renewal function. Nevertheless, no articles reporting those results Fin a systematic way have appeared in the literature. In this chapter, we survey the computational aspects of the renewal function based on the most recent results obtained up to the present stage. A comprehensive bibliography in this research area is also provided.

A Sample-Path Condition for the Asymptotic Uniform Distribution of Clearing Processes

Abstract: In this note we show that the asymptotic time-average distribution of a functional of cumulative input process associated with an imbedded point process follows as asymptotic uniform distribution almost surely under a mild regulatory sample-path condition. Examples from stochastic clearing processes, inventory systems and renewal theory are provided.

Analysis of the Discrete-Time GG/Geom/c Queueing Model

Abstract: This paper presents the steady-state analysis of a discrete-time infinite-capacity multiserver queue with c servers and independent geometrically distributed service times The arrival process is a batch renewal process, characterized by general independent batch interarrival times and general independent batch sizes The analysis has been carried out by means of an analytical technique based on generating functions, complex analysis and contour integration Expressions for the generating functions of the system contents during an arrival slot as well as during an arbitrary slot have been obtained Also, the delay in case of a first-come-first-served queueing discipline has been analyzed

Renewal processes and the Hurst effect

Abstract: Consider a pure recurrent positive renewal process generated by some inter-arrival waiting time. If the waiting time has power-law fall-off exponent (i.e. tail index) in (1, 2), and if the jump's amplitude has non-zero but finite mean and finite variance, the cumulative amplitude process is long-range dependent with Hurst exponent in (1/2, 1). (Results in this direction have been obtained by Daley under the sole assumptions that the waiting time has moment index in (1, 2)). If the amplitude has zero mean, up to a Brownian trend, the cumulative amplitude process exhibits a negative-dependence property with Hurst exponent in (0, 1/2). The case of delayed stationary renewal processes is also investigated, together with two classes of limiting renewal processes: the compound exponential and the Levy classes. These are of some interest when the average time between consecutive jumps tends to zero jointly with the probability mass of the jumps' height concentrating about zero in some precise way. Under suitable hypothesis, the Hurst effect is maintained in the limit.