Showing papers on "Renewal theory published in 1993"
TL;DR: This paper improves significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution and shows an interesting dependence of the system performance on the demand distribution.
Abstract: We analyze a class of stochastic and dynamic vehicle routing problems in which demands arrive randomly over time and the objective is minimizing waiting time. In our previous work ([6], [7]), we analyzed this problem for the case of uniformly distributed demand locations and Poisson arrivals. In this paper, using quite different techniques, we are able to extend our results to the more realistic case where demand locations have an arbitrary continuous distribution and arrivals follow only a general renewal process. Further, we improve significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution. We show that the leading behavior of the optimal system time has a particularly simple form that offers important structural insight into the behavior of the system. Moreover, by distinguishing two classes of policies our analysis shows an interesting dependence of the system performance on the demand distribution.
108 citations
TL;DR: In this article, a nonparametric estimator for the renewal function is proposed and its properties, including consistency, asymptotic normality, and asyptotic validity of bootstrap confidence regions are discussed.
Abstract: We introduce a nonparametric estimator for the renewal function and discuss its properties, including consistency, asymptotic normality and asymptotic validity of bootstrap confidence regions. The underlying theme is that stochastic models can be regarded as functionals or nonlinear operators. This view leads to nonparametric estimators in a natural way and statistical properties of the estimators can be related to the local behaviour of the functionals.
47 citations
TL;DR: In this article, the optimal stopping problem for semi-Markov processes is studied in a fairly general setting, where transitions are made from state to state in accordance with a Markov chain, but the amount of time spent in each state is random.
Abstract: In this paper, optimal stopping problems for semi-Markov processes are studied in a fairly general setting. In such a process transitions are made from state to state in accordance with a Markov chain, but the amount of time spent in each state is random. The times spent in each state follow a general renewal process. They may depend on the present state as well as on the state into which the next transition is made. Our goal is to maximize the expected net return, which is given as a function of the state at time t minus some cost function. Discounting may or may not be considered. The main theorems (Theorems 3.5 and 3.11) are expressions for the optimal stopping time in the undiscounted and discounted case. These theorems generalize results of Zuckerman [16] and Boshuizen and Gouweleeuw [3]. Applications are given in various special cases. The results developed in this paper can also be applied to semi-Markov shock models, as considered in Taylor [13], Feldman [6] and Zuckerman [15].
33 citations
TL;DR: Two relatively simple renewal processes whose power spectral densities vary as 1/f/sup D/ are constructed: 1) a standard renewal point process, with 0 > 1; and 2) a standards-based approach to this problem.
Abstract: Two relatively simple renewal processes whose power spectral densities vary as 1/f/sup D/ are constructed: 1) a standard renewal point process, with 0 >
28 citations
TL;DR: The Markov renewal viewpoint of single-part/multiple-state manufacturing systems under hedging control policies, subjected to a constant rate of demand for parts, is used to derive sufficient criteria to guarantee the ergodicity of the resulting parts surplus process.
Abstract: The Markov renewal viewpoint of single-part/multiple-state manufacturing systems under hedging control policies, subjected to a constant rate of demand for parts, is used to derive sufficient criteria to guarantee the ergodicity of the resulting parts surplus process. The criteria are simple and directly verifiable from an analysis of the Markov chain characterizing the dynamics of the discrete manufacturing system production state. A key intermediate result in reaching these criteria is a potentially very useful system of linear differential equations with boundary conditions which can be generalized to permit computation of moments of arbitrary order for sojourn times in the regions between successive hedging points in the parts surplus space. >
20 citations
TL;DR: In this article, the authors extended the existing theory to non-stationary problems which occur in transient loading situations and whenever structural resistances experience some deterioration in time, and proposed special computation schemes for the case when simple random variables are present including an importance sampling scheme.
19 citations
TL;DR: In this paper, the authors investigated the general repair models based on a virtual age concept and showed that the corresponding renewal process converges in some sense to a renewal process with identically distributed cycles.
19 citations
TL;DR: In this article, two renewal processes, known in reliability maintenance as minimal repair and replacement policy, are considered and their properties are studied in the case where the generating random sequence has a distribution with periodic failure rate.
15 citations
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TL;DR: 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.
Abstract: Optimal arrangements of cartridges and file partitioning schemes are examined in carousel type massstorage systems using Markov decision theory. It is shown that the Organ-Pipe Arrangement is optimalunder different storage configurations for both the anticipatory as well as the non-anticipatory versions ofthe problem. When requests arrive as per an arbitrary renewal process this arrangement is also shown tominimize the mean queueing delay and the time spent in the system by the requests.
15 citations
28 Mar 1993
TL;DR: A random packet selection policy for multicast switching is studied and it is shown that the old packets have a larger number of copies than the fresh packets, and the copy distribution is derived.
Abstract: A random packet selection policy for multicast switching is studied. An input packet generates a fixed number of primary copies plus a random number of secondary copies. Assuming a constant number of contending packets during a slot, the system is modeled as a discrete-time birth process. A difference equation describing the dynamics of this process is derived, the solution of which gives the distribution of the number of packets chosen. This result is extended to the steady-state distribution through an embedded Markov chain analysis. It is shown that the old packets have a larger number of copies than the fresh packets, and the copy distribution is derived. The packet and copy throughputs taking into account the old packets have been determined. The asymptotic distribution of the number of packets chosen is obtained for large switch sizes under saturation by applying results from renewal theory. >
14 citations
TL;DR: In this article, a self-consistent mathematical model of the surface-renewal theory of interphase mass transfer by resorting to the theories and methodologies of stochastic processes based on the Markovian assumption is presented.
TL;DR: In this article, it was shown that for a fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process and the common distribution function of the interarrival times is geometric when F is discrete.
Abstract: Let γ t and δ t denote the residual life at t and current life at t , respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G , under mild conditions, as long as holds for a single positive integer n , then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t , we find that for some fixed positive integer n , if is independent of t , then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t .
29 Nov 1993
TL;DR: Presents 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), and sees that predictions of system performance based on transient and stationary performance measures can be quite different.
Abstract: Presents 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 the authors see that predictions of system performance based on transient and stationary performance measures can be quite different. >
TL;DR: In this article, the authors considered supercritical linear birth-and-death processes under the influence of disasters that arrive as a renewal process independently of the population size and the novelty of this paper lies in assuming that the killing probability in a disaster is a function of the time that has elapsed since the last disaster.
TL;DR: A Markovian stochastic model for a system subject to random shocks is introduced in this paper, where it is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount.
Abstract: A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold a. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if X(t) I> a. The stationary case is also discussed.
TL;DR: In this article, the authors studied the superposition of finitely many Markov renewal processes with countable state spaces and defined the S-Markov renewal equations associated with the superposed process.
Abstract: In this paper, we study the superposition of finitely many Markov renewal processes with countable state spaces. We define the S-Markov renewal equations associated with the superposed process. The solutions of the S-Markov renewal equations are derived and the asymptotic behaviors of these solutions are studied. These results are applied to calculate various characteristics of queueing systems with superposition semi-Markovian arrivals, queueing networks with bulk service, system availability, and continuous superposition remaining and current life processes.
TL;DR: This work presents a non-stationary generalization of renewal processes, using the ideas of renewal theory, and shows that a renewal-type integral equation that arises in this generalization has a unique solution.
TL;DR: In this article, the authors consider a dam model with a compound Poisson input having positive jumps and derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps.
Abstract: We consider a dam model with a compound Poisson input having positive jumps in which the release rule can be dynamically controlled between two release functions, r1 and r2. The dam process has two switchover levels, a and b, 0 < a < b < ∞, which are prescribed to switch the release rule from r2 to r1 only when the dam downcrosses levels and switch from r1 to r2 only when the dam upcrosses level b. We derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps. We also compute an expression for the average number of switches from r1 to r2, and vice versa; consider an approximation to the general case (with b − a being sufficiently large with respect to r1 and r2 and the jump sizes), which allows the use of asymptotic results from renewal theory; and study the wet period analysis for the case of exponential jumps.
TL;DR: In this article, the authors derived closed form expressions for the first order light traffic limit,, in the cases where the total arrival process is Poisson, a phase type renewal process, and a superposition of independent phase-type renewal processes.
Abstract: Let be the m thmoment of the sojourn time distribution for some particular customer class in an open queueing system, where λ is the overall arrival rate. We derive closed form expressions, in terms of the basic system data, for the first order light traffic limit, , in the cases where the total arrival process is Poisson, a phase-type renewal process, and a superposition of independent phase-type renewal processes. For certain phase-type renewal processes, the k th order light traffic limit is zero for n≷k. In these cases we derive the kth order limit . The expressions are numerically tractable. The most difficult operation is a matrix inversion
TL;DR: In this article, the authors study the matched queueing system, MoPH/G/1, where the type-I input is a Poisson process, the Type-II input is PH renewal process, and the service times are i.i.d. random variables.
Abstract: In this paper, we study the matched queueing system, MoPH/G/1, where the type-I input is a Poisson process, the type-II input is a PH renewal process, and the service times are i.i.d. random variables. A necessary and sufficient condition for the stationariness of the system is given. The expectations of the length of the non-idle period and the number of customers served in a non-idle period are obtained.
TL;DR: Two mutually independent traffic streams whose arrivals are modeled as renewal processes are considered and the class of nonpreemptive scheduling policies which satisfy the delay constraint while maintaining the system stability is found.
Abstract: Two mutually independent traffic streams whose arrivals are modeled as renewal processes are considered The arrivals are stored in an infinite capacity buffer One of the processes requires a strict upper bound on the total delay per arrival, and it is assumed that the processing time per arrival for both processes is a constant Given the above model, the class of nonpreemptive scheduling policies which satisfy the delay constraint while maintaining the system stability is found The delays induced by the latter policy are analyzed via a methodology based on the regenerative theorem A numerical example is given in which the policy that minimizes the expected delays for both traffic streams is also considered >
TL;DR: In this article, the inspection paradox in an ordinary renewal process is investigated in the very general case of the time dependent problem and arbitrary distribution of inter-renewal times with finite mean and a probability density function.
Abstract: The inspection paradox in an ordinary renewal process is investigated in the very general case of the time dependent problem and arbitrary distribution of inter–renewal times with finite mean and a probability density function
TL;DR: In this article, a continuous time Markov-renewal model is presented that generalizes the classical Young and Almond model for manpower systems with given size, and a regenerative approach to the wastage process is outlined and two numerical examples from the literature on manpower planning illustrate the theory.
Abstract: A continuous time Markov-renewal model is presented that generalizes the classical Young and Almond model for manpower systems with given size. The construction is based on the associated Markov-renewal replacement process and exploits the properties of the embedded replacement chain. The joint cumulant generating function of the grade sizes is derived and an asymptotic analysis provides conditions for these to converge in distribution to a multinominal random vector exponentially fast independently of the initial distribution, both for aperiodic and periodic embedded replacement chains. A regenerative approach to the wastage process is outlined and two numerical examples from the literature on manpower planning illustrate the theory.
TL;DR: In this paper, a local limit theorem for P{Ta= n,ZT a-a ≤ x, YT a = y}, a → ∞ is obtained, where Zn is a perturbed Markov random walk related to a finite Markov chain Yn, Ta = min {n > O: Zn > a}.
Abstract: A local limit theorem for P{Ta= n,ZT a-a ≤ x, YT a = y}, a → ∞ is obtained, where Zn is a perturbed Markov random walk related to a finite Markov chain Yn, Ta = min {n > O: Zn > a}. An important class of processes satisfying the hypotheses of his theorem are presented.
TL;DR: In this article, the necessary and sufficient conditions so considered are in terms of the operating characteristics of perfect repair (renewal) and minimal repair strategies for equipments with NBU and NBUE aging life distributions.
TL;DR: In this paper, the joint distribution of inter-renewal times and the number of renewals is used to derive joint and marginal distributions of order statistics of waiting times of an ordinary renewal process.
Abstract: The joint distribution of inter–renewal times and the number of renewals is used to derive joint and marginal distributions of order statistics of waiting times of an ordinary renewal process. Also, expressions are obtained for the covariance function of the number of renewals and of the renewal increments in an ordinary renewal counting process in terms of the renewal function
TL;DR: In this paper, a non-stationary generalization of renewal reward processes is constructed using the ideas of renewal theory and moments and distributions of work life pension benefits are studied as sojourn time problems associated with these processes.
Abstract: Using the ideas of renewal theory, certain non-stationary generalizations of renewal reward processes are constructed. Moments and distributions of work life pension benefits are then studied as sojourn time problems associated with these processes. An application to the U.S. private pension system is also discussed, involving the measurement of the impact of vesting rules on the distribution of work life pension benefits.
TL;DR: In this article, it was shown that if Var(γ(t)) is constant, then F will be exponentially or geometrically distributed under the assumption F is continuous or discrete respectively.
Abstract: Let γ(t) be the residual life at time t of the renewal process {A(t), t > 0}, which has F as the common distribution function of the inter-arrival times. In this article we prove that if Var(γ(t)) is constant, then F will be exponentially or geometrically distributed under the assumption F is continuous or discrete respectively. An application and a related example also are given.
TL;DR: In this article, a binomial-like expression where the p and q do not sum to one is given for the superimposed renewal process, which is a compromise between the simple Poisson approximation and the more accurate binomial approximation.
Abstract: When n independent identical renewal processes are superimposed, the number of events in 0, s is approximately Poisson distributed for large n. For small to moderate n, this approximation is inaccurate. If no component process has more than one event in 0, s, the probability of exactly r events for the superimposed process is given by a binomial-like expression where the p and q do not sum to one. Several approximations have been derived from this observation and compared to the exact probability for small n, when the underlying distribution of time between events is gamma. One of these binomial approximations gives excellent results for small n. The following application is discussed. When k new series systems, each consisting of m identical components, are tested for time s and failed components are replaced upon failure, a superimposed renewal process results. To design an acceptance sampling scheme for new series systems using exact probability computations is cumbersome. The Poisson approximation is convenient, but may give misleading estimates of the O.C. curve for small n. The new binomial approximation is a compromise: It is easier to use for test design than the exact computation and more accurate than the simpler Poisson approximation.