Showing papers on "Renewal theory published in 1992"
TL;DR: In this paper, a simplified renewal model for stress relaxation in entangled, reversibly breakable polymers (e.g., worm-like micelles) is proposed, which replaces the exact reaction kinetics by a Poisson jump process that neglects temporal correlations in the chain length experienced by a particular monomer or tube segment.
Abstract: The problem of stress relaxation in entangled, reversibly breakable polymers (e.g., wormlike micelles) is considered. In the case where the dominant diffusive mode for the polymers is reptation, this problem has been treated in earlier numerical work by coupling the full reaction kinetics of scissions and recombinations to the dynamics of reptation (represented by a one‐dimensional stochastic process). Here we study a simplified renewal model, which replaces the exact reaction kinetics by a Poisson jump process that neglects temporal correlations in the chain length experienced by a particular monomer or tube segment. Between jumps in chain length, the stress relaxation is presumed to follow that of an equivalent unbreakable chain. We apply the solution to the case of reptating flexible polymers and compare the resulting complex modulus with the earlier numerical treatments. It is found that agreement is very good. The renewal model is then used to analyze in detail, for the first time, the crossover to a rapid‐scission regime in which chain diffusion between scission events is dominated by breathing modes. A third regime, in which the motion between scission events is Rouse‐like, remains unsuitable for study with this model, for reasons that we explain. Various implications of the renewal model for the interpretation of experimental results are discussed. We also provide explicit estimates for chain lengths in CTAC/NaSal/NaCl systems using experimental Cole–Cole plots.
410 citations
TL;DR: In this article, the statistical inference for geometric processes by nonparametric methods has been studied and two statistics and a graphical technique have been suggested for testing whether a process is a geometric process and estimating the parameters a, λ and σ2 of the geometric process by using linear regression 2 method.
Abstract: A stochastic process {Xn,n = 1,2,} is a geometric n process if there exists a > 0 so that is a n renewal process This is a stochastically monotone process, and can be used for modelling a point process with trend In this paper, we study the statistical inference for geometric processes by nonparametric methods Two statistics and a graphical technique are suggested for testing whether a process is a geometric process Further, we can estimate the parameters a, λ and σ2 of the geometric process by using linear regression 2 method, where λ and σ2 are the mean and variance of X1 respectively
67 citations
TL;DR: A new O(n3) algorithm is developed which uses data from the starting and stopping times of each customer's service during the busy period and assuming the arrival distribution is Poisson to deduce transient queue lengths.
Abstract: R. Larson proposed a method to statistically infer the expected transient queue length during a busy period in O(n5) solely from the n starting and stopping times of each customer's service during the busy period and assuming the arrival distribution is Poisson. We develop a new O(n3) algorithm which uses these data to deduce transient queue lengths as well as the waiting times of each customer in the busy period. We also develop an O(n) on-line algorithm to dynamically update the current estimates for queue lengths after each departure. Moreover, we generalize our algorithms for the case of a time-varying Poisson process and also for the case of i.i.d. interarrival times with an arbitrary distribution. We report computational results that exhibit the speed and accuracy of our algorithms.
42 citations
TL;DR: In this paper, the block replacement policies are compared with the age replacement policies and several new results which connect the properties of block replacement policy with properties of the corresponding renewal function and the excess lifetimes are obtained.
Abstract: Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.
33 citations
TL;DR: In this paper, steady-state Markov chain models for the Heine and Euler distributions were proposed for the discrete renewal process for oil exploration strategies, which were reinterpreted as current-age models for discrete renewal processes.
Abstract: The paper puts forward steady-state Markov chain models for the Heine and Euler distributions. The models for oil exploration strategies that were discussed by Benkherouf and Bather (1988) are reinterpreted as current-age models for discrete renewal processes. Steady-state success-runs processes with non-zero probabilities that a trial is abandoned, Foster processes, and equilibrium random walks corresponding to elective M / M /1 queues are also examined.
33 citations
TL;DR: In this article, the authors give a complete description of the asymptotic distribution of sums made from the top k, extreme values, for any sequence k, such that k, - co, k,/t -0 as t -- o.
Abstract: Let X() < X(2) E? . X(N()) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where (N(t); t - 0) is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top k, extreme values, for any sequence k, such that k, - co, k,/t -0 as t -- o. We discuss applications to reinsurance policies based on large claims.
21 citations
TL;DR: In this article, the authors investigated limit properties of the waiting time in k-server queues with renewal arrival process under light traffic conditions and derived formulas for the limits of the probability of waiting and waiting time moments for the two approaches of dilation and thinning of the arrival process.
Abstract: This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under light traffic conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussen's (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given
19 citations
TL;DR: In this article, the authors proved the equivalence of ten stability/ergodicity conditions on the transition law of the model, which imply the existence of average optimal stationary policies for an arbitrary continuous and bounded reward function.
Abstract: We are concerned with Markov decision processes with countable state space and discrete-time parameter. The main structural restriction on the model is the following: under the action of any stationary policy the state space is acommunicating class. In this context, we prove the equivalence of ten stability/ergodicity conditions on the transition law of the model, which imply the existence of average optimal stationary policies for an arbitrary continuous and bounded reward function; these conditions include the Lyapunov function condition (LFC) introduced by A. Hordijk. As a consequence of our results, the LFC is proved to be equivalent to the following: under the action of any stationary policy the corresponding Markov chain has a unique invariant distribution which depends continuously on the stationary policy being used. A weak form of the latter condition was used by one of the authors to establish the existence of optimal stationary policies using an approach based on renewal theory.
16 citations
TL;DR: In this paper, it was shown that if the inter-renewal time distribution is discrete DFR (decreasing failure rate), then both Ak and Zk are monotonically nondecreeasing in k in hazard rate ordering.
Abstract: In a discrete-time renewal process {Nk, k =0, 1, *}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k=0, 1,- } and {Zk, k=0, 1, --} are monotonically nondecreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+, - Nk, k =0, 1, -} to be nonincreasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.
15 citations
TL;DR: In this paper, the authors considered a renewal process whose interrenewal-time distribution is phase type with representation (α, T), where α is an appropriately modified initial probability vector.
Abstract: Consider a renewal process whose interrenewal-time distribution is phase type with representation (α, T). We show that the (time-dependent) excess-life distribution is phase type with representation (α′, T), where α′ is an appropriately modified initial probability vector. Using this result, we derive the (time-dependent) distributions for the current life and the total life of the phase-type renewal process. They in turn enable us to obtain the equilibrium distributions for the three random variables. These results simplify the computation of the respective distribution functions and consequently enhance the potential use of renewal theory in stochastic modeling—particularly in inventory, queueing, and reliability applications. © 1992 John Wiley & Sons, Inc.
11 citations
TL;DR: In this paper, the alternating Poisson process is used for work sampling studies and a spacing rule is presented for use in design of sampling studies, taking into account the mean length of time taken by the activity as well as the proportion of time it occupies.
Abstract: Statistics used in work sampling studies are normally evaluated only in terms of their capability to predict activity during the period of the study. In order to evaluate predictive ability beyond the period of the study it is necessary to have a model for the process. One possible model, the alternating Poisson process, is borrowed from renewal theory and explored for utility in systematic work sampling applications. A spacing rule is presented for use in design of sampling studies; the rule takes into account the mean length of time taken by the activity as well as the proportion of time it occupies. A method is given of modifying the rule in the presence of a cost constraint.
TL;DR: In this article, a new approach for calculating the long-term statistics of sea waves is proposed, which treats the sea-surface elevation as a random function of a fast time variable, and the time history of the spectral characteristics of the successive sea states, each of which possesses its own duration and intensity.
Abstract: A new approach for calculating the long-term statistics of sea waves is proposed. A rational long-term stochastic model is introduced which recognises that the wave climate at a given site in the ocean consists of a random succession of individual sea states, each sea state possessing its own duration and intensity. This model treats the sea-surface elevation as a random function of a "fast" time variable, and the time history of the spectral characteristics of the successive sea states as a random function of a "slow" time variable. By developing an appropriate conceptual framework, it becomes possible to express various probabilistic characteristics of the sea-surface elevation, which are sensible only in the fast-time scale, in terms of the statistic of sea-states duration and intensity, which is meaningful only in the slow-time scale. As an example, we study the random quantity Mu(T) = "number of maxima of the sea-surface elevation lying above the level u and occurring during a long-term time period [O,T]". Exploiting the proposed framework, it is shown that, under certain clearly defined assumptions, Mu(T) can be given the structure of a renewal-reward (cumulative) process, whose interarrival time correspond to the duration of successive sea states. Thus, using renewal theory, the complete characterisation of the probability structure of Mu(T) is obtained. As a consequence, the long-term probability distribution function of the individual wave height is rigorously defined and calculated. The relation of the present results with corresponding ones previously obtained is thoroughly discussed. The proposed model can be extended twofold: either by replacing some of the simplifying assumptions by more realistic ones, or by extending the model for treating the corresponding problems for ship and structures response.
TL;DR: In this paper, the convergence of Markovian stochastic processes in continuous time to their stationary distribution has been studied and a simple lemma on ϵ-coupling has been proposed.
Abstract: This paper studies coupling methods for proving convergence in distribution of (typically Markovian) stochastic processes in continuous time to their stationary distribution. The paper contains: (a) a simple lemma on $\varepsilon$-coupling; (b) conditions for Markov processes to couple in compact sets; (c) new variants of the coupling proof of the renewal theorem; (d) a convergence result for stochastically monotone Markov processes in an ordered Polish space; and (e) a case study of a queue with superposed renewal input. In a companion paper with Foss, similar discussion is given for many-server queues in continuous time.
TL;DR: In this paper, the existence of an optimal stationary policy under structural restrictions on the model is proved; both the lim inf and lim sup average criteria are considered, and the arguments are based on well-known facts from Renewal Theory.
Abstract: We consider discrete-timeaverage reward Markov decision processes with denumerable state space andbounded reward function. Under structural restrictions on the model the existence of an optimal stationary policy is proved; both the lim inf and lim sup average criteria are considered. In contrast to the usual approach our results donot rely on the average regard optimality equation. Rather, the arguments are based on well-known facts fromRenewal Theory.
TL;DR: In this article, the authors study the NBUE and NWUE properties of Markov renewal processes and present sufficient conditions such that the first passage times of these processes are new better than used in expectation.
Abstract: In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.
01 Jan 1992
TL;DR: Random sample sizes naturally come up in such topics as sequential analysis, branching processes, damage models or rarefactions of point processes, and records as maxima, while their introduction in an applied model permits the user to select samples of varying sizes on different occasions.
Abstract: Random sample sizes naturally come up in such topics as sequential analysis, branching processes, damage models or rarefactions of point processes, and records as maxima, while their introduction in an applied model permits the user to select samples of varying sizes on different occasions. In the first group of examples, the random sample size is generated by the problem itself, hence the mathematician has no control over the dependence between the sample size and the underlying random variables. On the other hand, if one introduces the random sample size as an extension of a model (mainly for statistical inference), one can usually assume that it is independent of the underlying variables. Therefore, one has to be aware of the limitations of using a result developed under the assumption of the sample size’s independence of the main variables.
TL;DR: A method for computing a continuous-time search model with n random offer streams forming independent renewal processes so that the searcher wants to select an offer such that the expected discounted reward is maximized.
Abstract: We consider a continuous-time search model with n random offer streams forming independent renewal processes. The distribution of the offer size may depend on the arrival time and the type of the offer, and there is a constant search cost per unit time. The searcher wants to select an offer such that the expected discounted reward is maximized. We present a method for computing such a strategy and apply it to some numerical examples. In the case of Poisson offer streams a more direct approach is also given.
TL;DR: In this article, the authors show how to delineate the limit function v for processes X associated with crudely regenerative phenomena, including refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EX t.
Abstract: Limit Statements obtainable by the key renewal theorem are of the form EX t = v ( t ) + o (1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EX t .
TL;DR: In this paper, the concept of surface renewal is applied to an impinging heated planar jet to predict the local heat transfer coefficient distribution near impingement, and the results show very good agreement with experimental data when the jet centerline conditions are used to scale the initial velocity and temperature renewal process.
TL;DR: Using a result in multiple integrals, the joint distribution function of the backward and forward recurrence times for the time dependent ordinary renewal process is derived in this article, where a similar methodology is used to find the distribution of their sums.
Abstract: Using a result in multiple integrals, the joint distribution function of the backward and forward recurrence times for the time dependent ordinary renewal process is derived. A similar methodology is used to find the distribution of their sums. The well known limiting behavior of these distributions is recovered
TL;DR: In this paper, the authors consider the difference process N of two independent renewal (counting) processes and show that second-order approximations to the distribution function of the level crossing time are considerably better than the known first-order ones.
Abstract: We consider the difference process N of two independent renewal (counting) processes Second-order approximations to the distribution function of the level crossing time are given Direct application of the second-order approximation is complicated by the occurrence of an (in general) unknown term E[Mtilde], which denotes the expected minimum of the stationary version of N However, this number is obtained for a wide class of processes N, using matrix-geometric techniques Numerical experiments have been carried out, in which the new approximations were compared to simulation, first-order and/or exact results These results confirm that the second-order approximations are considerably better than the (known) first-order ones We consider the difference process N of two independent renewal (counting) processes Second-order approximations to the distribution function of the level crossing time are given Direct application of the second-order approximation is complicated by the occurrence of an (in general) unknown term E[Mtilde], which denotes the expected minimum of the stationary version of N However, this number is obtained for a wide class of processes N, using matrix-geometric techniques Numerical experiments have been carried out, in which the new approximations were compared to simulation, first-order and/or exact results These results confirm that the second-order approximations are considerably better than the (known) first-order ones
TL;DR: In this article, the limit theorems for the logarithm of the likelihood ratio were proven and the rate of decrease in the probability of errors of the second kind in the Neumann-Pierson criterion was established.
Abstract: Limit theorems for the logarithm of the likelihood ratio are proven. With their help the rate of decrease in the probability of errors of the second kind in the Neumann-Pierson criterion is established.
TL;DR: A case study is presented which shows excellent agreement between the two approaches, confirming the accuracy of the burst level method for ATM simulation studies.
Abstract: Renewal arrival processes are shown to be equivalent to a special class of general modulated deterministic process. This equivalence allows a direct comparison of cell loss probability results obtained from renewal theory and from burst-level simulation. A case study is presented which shows excellent agreement between the two approaches, confirming the accuracy of the burst level method for ATM simulation studies
TL;DR: An Extended Stochastic Petri Net is presented that allows the firing times of its transitions to non-exponential distributions and is used to model and analyze a multi-robot system with parallel and cooperative motions in the context of a generalized Markov Renewal Process.
Abstract: Stochastic Petri Nets have been developed to model and analyze systems involving concurrent activities with the time associated is exponentialy distributed. In this paper, we present an Extended Stochastic Petri Net that allows the firing times of its transitions to non-exponential distributions. We use it to model and analyze a multi-robot system with parallel and cooperative motions in the context of a generalized Markov Renewal Process. The modeling flexibility of Petri Net and the analyzing power of Markov Renewal Process are fully exploited.
TL;DR: The cost analysis of one and two-dimensional free replacement policies involve renewal functions and the simulation approach to obtaining such renewal functions is dealt with.
22 Jun 1992
TL;DR: A methodology is presented for the approximate solution of large Stochastic Petri Nets that are structured into independent submodels that are aggregated to substitute nets of lower complexity to achieve a state space reduction.
Abstract: A methodology is presented for the approximate solution of large Stochastic Petri Nets that are structured into independent submodels. These subnets are aggregated to substitute nets of lower complexity to achieve a state space reduction. This is based on the estimated traffic process at a submodel's input in steady state and on a token's residence time distribution in the original submodel as equivalence criterion for matching the substitute network to the submodel. The Markov renewal process at the input of the submodel is approximated by a renewal process. Its moments and the arrival instant probabilities at the submodel are computed by means of a traffic set approach. The technique is applied to Generalized Stochastic Petri Nets and compared to Flow Equivalent Aggregation.
TL;DR: It is shown that the so-called circular assignment policy is optimal under mild conditions on the initial queue-length distributions and the holding cost.
Abstract: We consider a parallel queueing system with identical exponential servers. Customers arrive according to a renewal process and upon arrival are immediately assigned to those queues. The problem is to find an optimal assignment policy minimizing the longrun average expected cost, without information about the current queue lengths, but with the initial queue-length distributions and information about the past arrival process and assignment of customers. In this paper, it is shown that the so-called circular assignment policy is optimal under mild conditions on the initial queue-length distributions and the holding cost.
01 Jan 1992
TL;DR: This chapter discussed continuous-time Markov chains in which the process can move from one state to another, where each interarrival time is distributed exponentially.
Abstract: We have discussed Markov chains in the preceding two chapters. In Chapter 5 we discussed discrete-time Markov chains in which the process can move from one state to another (including to itself) in discrete time. In Chapter 6 we discussed continuous-time Markov chains in which the process can move from one state to another, where each interarrival time is distributed exponentially. Note that only the exponential distribution has the memoryless property which plays an important role in analyzing the process.