Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.

Markov-renewal programming

Abstract: : A special structure in dynamic programming is the problem of programming over a Markov chain. This paper extends the solution algorithms to programming over a Markov - renewal process - in which times between transitions of the system from state i to state j are independent samples from an inter-transition distribution which may depend on both i and j. For these processes, a general reward structure and a decision mechanism are postulated; the problem is to make decisions at each transition to maximize the total expected reward at the end of the planning horizon. For finite-horizon problems, or infinite-horizon problems with discounting, there is no difficulty; the results are similar to previous work, expect for a new dependency upon the transition time distributions being generally present. In the cases where the horizon extends towards infinity, or when discounting vanishes, however, a fundamental dichotomy in the optimal solutions may occur. It then becomes important to specify whether the limiting experiment is: (i) undiscounted, with the number of transitions n approaches infinity , (ii) undiscounted, with a time horizon t approaches infinity , or (iii) infinite n or t , with discount factor a approaches 0 . In each case, a limiting form for the total expected reward is shown, and an algorithm developed to maximize the rate of return. The problem of finding the optimal or near-optimal policies In the case of ties in rate of return is still computationally unresolved. Extensions to non-ergodic processes are indicated, and special results for the two-state process are presented. Finally, an example of machine maintenance and repair is used to illustrate the generality of the approach and the special problems which may arise.

On the Objective Function Behavior in s, S Inventory Models

Abstract: The most common measure of effectiveness used in determining the optimal s, S inventory policies is the total cost function per unit time, Es, Δ, Δ = S-s. In stationary analysis, this function is constructed through the limiting distribution of on-hand inventory, and it involves some renewal-theoretic elements. For Δ â‰¥ 0 given, Es, Δ turns out to be convex in s, so that the corresponding optimal reorder point, s1Δ, can be characterized easily. However, Es1Δ, Δ is not in general unimodal on Δ â‰¥ 0. This requires the use of complicated search routines in computations, as there is no guarantee that a local minimum is global. Both for periodic and continuous review systems with constant lead times, full backlogging and linear holding and shortage costs, we prove in this paper that E's1Δ, Δ = 0, Δ â‰¥ 0, is both necessary and sufficient for a global minimum Es1Δ, Δ is pseudoconvex on Δ â‰¥ 0 if the underlying renewal function is concave. The optimal stationary policy can then be computed efficiently by a one-dimensional search routine. The renewal function in question is that of the renewal process of periodic demands in the periodic review model and of demand.sizes in the continuous review model.

Stochastic renewal model of low-flow streamflow sequences

Abstract: It is shown that runs of low-flow annual streamflow in a coastal semiarid basin of Central California can be adequately modelled by renewal theory. For example, runs of below-median annual streamflows are shown to follow a geometric distribution. The elapsed time between runs of below-median streamflow are geometrically distributed also. The sum of these two independently distributed geometric time variables defines the renewal time elapsing between the initiation of a low-flow run and the next one. The probability distribution of the renewal time is then derived from first principles, ultimately leading to the distribution of the number of low-flow runs in a specified time period, the expected number of low-flow runs, the risk of drought, and other important probabilistic indicators of low-flow. The authors argue that if one identifies drought threat with the occurrence of multiyear low-flow runs, as it is done by water supply managers in the study area, then our renewal model provides a number of interesting results concerning drought threat in areas historically subject to inclement, dry, climate. A 430-year long annual streamflow time series reconstructed by tree-ring analysis serves as the basis for testing our renewal model of low-flow sequences.

Nonparametric Renewal Function Estimation

Abstract: : The renewal function is a basic tool used in many probabilistic models and sequential analysis. Based on a random sample of size n, a nonparametric estimator of the renewal function is introduced. Asymptotic properties of the estimator such as the almost sure consistency and local asymptotic normality are developed. A discussion of an application of the estimator is also provided. Keywords: U-statistics, reverse martingales, warranty analysis.

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Abstract: Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramer light-tail assumption on the claim size distribution.