Showing papers on "Renewal theory published in 2004"

A fractional generalization of the Poisson processes

Abstract: It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.

Discrete-Time Semi-Markov Model for Reliability and Survival Analysis

Abstract: In this paper, we define a discrete-time semi-Markov model and propose a computation procedure for solving the corresponding Markov renewal equation, necessary for all our reliability measurements. Then, we compute the reliability and its related measures, and we apply the results to a three-state system.

Stochastic representation of a class of non-Markovian completely positive evolutions

Abstract: By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong nonexponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects [S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)] is clarified in this context.

Strategically Seeking Service: How Competition Can Generate Poisson Arrivals

Abstract: We consider a simple game in which strategic agents select arrival times to a service facility. Agents find congestion costly and, hence, try to arrive when the system is underutilized. Working in discrete time, we characterize pure-strategy Nash equilibria for the case of ample service capacity. In this case, agents try to spread themselves out as much as possible and their self-interested actions will lead to a socially optimal outcome if all agents have the same well-behaved delay cost function. For even modest sized problems, the set of possible pure-strategy Nash equilibria is quite large, making implementation potentially cumbersome. We consequently examine mixed-strategy Nash equilibria and show that there is a unique symmetric Nash equilibrium. Not only is this equilibrium independent of the number of agents and their individual delay cost functions, the arrival pattern it generates approaches a discrete-time Poisson process as the number of agents and arrival points gets large. Our results extend to the case of time varying preferences. With an appropriate initialization, the results also extend to a system with limited capacity. Our model lends support to the traditional literature on managing service systems. This work has generally ignored customers strategically choosing arrival times. Rather it is commonly assumed that customers seek service according to some well-behaved process (e.g., that interarrival times follow a renewal process). We show that assuming Poisson arrivals is an acceptable assumption even with strategic customers if the population is large and the horizon is long.

Periodic review (s, S) inventory model with permissible delay in payments

Abstract: This paper investigates the effect of permissible delay in payments on ordering policies in a periodic review (s, S) inventory model with stochastic demand. A new mathematical model is developed, which is an extension to that of Veinott and Wagner (Mngt Sci 1965; 11: 525) who applied renewal theory and stationary probabilistic analysis to determine the equivalent average cost per review period. The performance of the model is validated using a custom-built simulation programme. In addition, two distribution-free heuristic methods of reasonable accuracy develop approximate optimal policies for practical purposes based only on the mean and the standard deviation of the demand. Numerical examples are presented with results discussed.

The compound Poisson immigration process subject to binomial catastrophes

Abstract: Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p , independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.

Transient analysis of a repairable system, using phase-type distributions and geometric processes

Abstract: The transient behavior of a system with operational and repair times distributed following phase-type distributions is studied. These times are alternated in the evolution of the system, and they form 2 separate geometric processes. The stationary study of this system when the repair times form a renewal process has been made . This paper also considers that operational times are partitioned into two well-distinguished classes successively occupied: good, and preventive. An algorithmic approach is performed to determine the transition probabilities for the Markov process which governs the system, and other performance measures beyond those in are calculated in a well-structured form. The results are applied to a numerical example, and the transient quantities are compared with the ones obtained in the stationary case. The computational implementation of the mathematical expressions formulated are performed using the Matlab program.

Weak Regeneration in Modeling of Queueing Processes

Abstract: A survey of recent results on weak regeneration in queueing processes is given in which an embedded process of regeneration points is renewal, but unlike classical regeneration, a dependence between adjacent cycles is allowed. We develop a unified two-step approach to stability analysis based on a characterization of the limit behavior of the forward renewal time. This employs an extended construction (initially proposed by Foss and Kalashnikov [13]), which is widely used to transform an original one-dependent regenerative process into weakly regenerative one. It is shown that the approach simplifies stability analysis of many queuing processes. (The tightness of the queueing processes plays an important role in the proofs.) In particular, we consider both well-known classical queues and multi-server queues with regenerative input and non-identical servers. Included is a stability analysis of a feed-forward network with regenerative input and non-identical servers in the nodes.

Renormalized treatment of the double heterogeneity with the method of characteristics

Abstract: A numerical solution for the method of characteristics in structured meshes is proposed to treat media comprising stochastic regions. As opposed to an earlier approach, the new method uses the stochastic solution predicted by renewal theory to compute region transmission and a normalizing coefficient is applied then to ensure conservation. Numerical comparisons are presented for set of Kritzs experiments for MOX fuel, and for a 2D HTGR reactor with prismatic fuel elements. Comparisons with a Monte Carlo calculation show that the renormalized MOC treatment predicts assembly powers within 1%.

A growth–collapse model: Lévy inflow, geometric crashes, and generalized Ornstein–Uhlenbeck dynamics

Abstract: We introduce and study a stochastic growth–collapse model. The growth process is a steady random inflow with stationary, independent, and non-negative increments. Crashes occur according to an arbitrary renewal process, they are geometric, and their magnitudes are random and are governed by an arbitrary distribution on the unit interval. If the system's pre-crash level is X>0, and the crash magnitude is 0

Moments of passage times for Lévy processes

Abstract: We give necessary and sufficient conditions, in terms of characteristics of the process, for finiteness of moments of passage times of general Levy processes above horizontal, linear or certain curved boundaries. They apply in particular to processes which drift almost surely to infinity, and lead to estimates of the rate of growth of certain expectations, constituting generalised kinds of renewal theorems. Further results concern the inverse local time at the maximum and the ladder height process, the amount of time spent below a given level, and the overall minimum of the Levy process.

A single product perishing inventory model with demand interaction

Abstract: The paper describes a single perishing product inventory model in which items deteriorate in two phases and then perish. An independent demand takes place at constant rates for items in both phases. A demand for an item in Phase I not satisfied may be satisfied by an item in Phase II, based on a probability measure. Demand for items in Phase II during stock-out is lost. The re-ordering policy is an adjustable (S, s) policy with the lead-time following an arbitrary distribution. Identifying the underlying stochastic process as a renewal process, the probability distribution of the inventory level at any arbitrary point in time is obtained. The expressions for the mean stationary rates of lost demand, substituted demand, perished units and scrapped units are also derived. A numerical example is considered to highlight the results obtained.

Simulation of risk processes

Abstract: The simulation of risk processes is a standard procedure for insurance companies. The generation of simulated (aggregated) claims is vital for the calculation of the amount of loss that may occur. Simulation of risk processes also appears naturally in rating triggered step-up bonds, where the interest rate is bound to random changes of the companies? ratings.

Diffusion in random environment and the renewal theorem

Abstract: According to a theorem of Schumacher and Brox, for a diffusion X in a Brownian environment, it holds that (X t - b logt )/log 2 t → 0 in probability, as t → ∞, where b is a stochastic process having an explicit description and depending only on the environment We compute the distribution of the number of sign changes for b on an interval [1, x] and study some of the consequences of the computation; in particular, we get the probability of b keeping the same sign on that interval These results have been announced in 1999 in a nonrigorous paper by Le Doussal, Monthus and Fisher [Phys Rev E 59 (1999) 4795-4840] and were treated with a Renormalization Group analysis We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory Finally, we comment on the information these results give about the behavior of the diffusion

A note on minimum-variance theory and beyond

Abstract: We revisit the minimum-variance theory proposed by Harris and Wolpert (1998 Nature 394 780–4), discuss the implications of the theory on modelling the firing patterns of single neurons and analytically find the optimal control signals, trajectories and velocities. Under the rate coding assumption, input control signals employed in the minimum-variance theory should be Fitts processes rather than Poisson processes. Only if information is coded by interspike intervals, Poisson processes are in agreement with the inputs employed in the minimum-variance theory. For the integrate-and-fire model with Fitts process inputs, interspike intervals of efferent spike trains are very irregular. We introduce diffusion approximations to approximate neural models with renewal process inputs and present theoretical results on calculating moments of interspike intervals of the integrate-and-fire model. Results in Feng, et al (2002 J. Phys. A: Math. Gen. 35 7287–304) are generalized. In conclusion, we present a complete picture on the minimum-variance theory ranging from input control signals, to model outputs, and to its implications on modelling firing patterns of single neurons.

One-sided interval trees

Abstract: We give an alternative treatment and extension of some results of Itoh and Mahmoud on one-sided interval trees. The proofs are based on renewal theory, including a case with mixed multiplica- tive and additive renewals.

The trend-renewal process for statistical analysis of: repairable systems

Abstract: The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes, corresponding to minimal repairs, and renewal processes, corresponding to perfect repairs. This article introduces and studies a more general model for recurrent events, the trend-renewal process (TRP). The TRP is a time-transformed renewal process having both the ordinary renewal process and the nonhomogeneous Poisson process as special cases. Parametric inference in the TRP model is studied, with emphasis on the case in which several systems are observed in the presence of a possible unobserved heterogeneity between systems.

How special is a 'special' interval: modeling departure from length-biased sampling in renewal processes.

Abstract: SUMMARY Length-biased sampling occurs in renewal processes when the probability that an interval is selected is proportional to the length of the interval. This can occur when intervals are selected because they contain an event that is independent of the renewal process and occurs with constant hazard. For example, if the times between donations for repeat blood donors are independent and identically distributed, and if the donor seroconverts to HIV (develops antibodies that indicate infection with human immunodeficiency virus), then the interval between the last HIV seronegative and first HIV seropositive test is expected to be longer than that donor’s previous time intervals between donations. We develop hypothesis tests to determine if the relationship between the typical and length-biased intervals is as expected, or if there is departure from length-biased sampling. We further develop a regression method to determine if there are covariates that explain the departure from length-biased sampling. Our approach is motivated by the question of whether there is evidence that repeat blood donors who develop antibodies to HIV or other viral infections change their donation pattern in some way because of seroconversion.

Efficient Estimation for Semiparametric Semi-Markov Processes

Abstract: We consider semiparametric models of semi-Markov processes with arbitrary state space. Assuming that the process is geometrically ergodic, we characterize efficient estimators, in the sense of Hajek and Le Cam, for arbitrary real-valued smooth functionals of the distribution of the embedded Markov renewal process. We construct efficient estimators of the parameter and of linear functionals of the distribution. In particular we treat the two cases in which we have a parametric model for the transition distribution of the embedded Markov chain and an arbitrary conditional distribution of the inter-jump times, and vice versa.

Approximating the distribution of a renewal process using generalized poisson distributions

Abstract: The exact distribution of a renewal counting process is not easy to compute and is rarely of closed form. In this article, we approximate the distribution of a renewal process using families of generalized Poisson distributions. We first compute approximations to the first several moments of the renewal process. In some cases, a closed form approximation is obtained. It is found that each family considered has its own strengths and weaknesses. Some new families of generalized Poisson distributions are recommended. Theorems are obtained determining when these variance to mean ratios are less than (or exceed) one without having to find the mean and variance. Some numerical comparisons are also made.

Simple bounds on cumulative intensity functions of renewal and G-renewal processes with increasing failure rate underlying distributions.

Abstract: The article considers point processes most commonly used in reliability and risk analysis. Short-term and long-term behavior for the point processes used as models for repairable systems 1 are introduced. As opposed to the long term, the term short term implies that a process is observed during an interval limited by a time close to the mean (or the median) of the respective underlying distribution, A new simple upper bound is proposed on the cumulative intensity function of the renewal process and G-renewal process with an increasing failure rate underlying distribution. The new bound is compared with some known bounds for the renewal process. Finally, a formal definition of "a boundary point" between the short-term repairable system behavior and long-term behavior is introduced. This point can also be used as a lower time limit beyond which the "long-term" Barlow and Proschan bound for the renewal process with NBUE underlying distribution could be effectively applied.

Overshoot in the Case of Normal Variables

Abstract: A new formula, in the form of an infinite series, is derived for the limiting expected excess over the boundary of a sum of independent and identically distributed normal variables as the positive mean goes to zero. The first order correction for small positive mean is also derived by the same method.