Renewal theory

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

Non-parametric Confidence Intervals for the Renewal Function and the Point Availability

Abstract: A large sample non-parametric method for constructing confidence intervals for the renewal function and the point availability is investigated. The method is based on a linearization and on the fact that the empirical distribution function converges weakly to a Gaussian process as the sample size increases. The technique is illustrated by the analysis of some hitherto unpublished data. Two of the most important functions arising in renewal theory are the renewal function, the expected number of renewals in a given interval, and the point availability, the probability that a system modelled by an alternating renewal process (ARP) is in a particular state at a specified time. See, for example, Karlin & Taylor (1975, Ch. 5), Ross (1970, Ch. 3) and Cox (1962) for a discussion of applications of these functions. If the functional forms of the distribution functions of the random variables generating the processes are known, and observations of the random variables are available, point estimates of these functions are readily constructed. Further, approximate (large sample) confidence intervals may, in principle, be calculated by an application of the delta method, assuming that the parameter estimates are asymptotically normally distributed. If, however, as is sometimes the case, the functional forms of the underlying distribution functions are unknown, a non-parametric approach is required. Frees (1986a, b, 1988) discussed some non-parametric estimators of the renewal function and constructed a non-parametric confidence interval for this quantity. See Schneider et al. (1990) for a study of these estimators. In this paper, we propose an alternative non-parametric confidence interval for the renewal function which is easier to compute than that of Frees (1986a) and which is appreciably narrower. In addition, we derive an analogous non-parametric confidence interval for the point availability. To the best, of our knowledge, this is the first non-parametric interval estimator of the point availability to have ,been proposed. Our methodology is based on the analysis of Harel et al. (1994), who prove that the empirical renewal function converges weakly to a Gaussian process as the sample size increases. A numerical study shows that our proposed confidence intervals are easy to compute, requiring only a few seconds of CPU time on a Sun Sparc Station, and are fairly narrow for moderate sample sizes.

Redundancy of the Context-Tree Weighting Method on Renewal and Markov Renewal Processes

Abstract: The Context Tree Weighting method (CTW) is shown to be almost adaptive on the classes of renewal and Markov renewal processes. Up to logarithmic factor, ctw achieves the minimax pointwise redundancy described by I. Csiszaacuter and P. Shields in IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 2065-2072, Nov. 1996. This result not only complements previous results on the adaptivity of the Context-Tree Weighting method on the relatively small class of all finite context-tree sources (which encompasses the class of all finite order Markov sources), it shows that almost minimax redundancy can be achieved on massive classes of sources (classes that cannot be smoothly parameterized by subsets of finite-dimensional spaces). Moreover, it shows that (almost) adaptive compression can be achieved in a computationally efficient way on those massive classes. While previous adaptivity results for CTW could rely on the fact that any Markov source is a finite-context-tree source, this is no longer the case for renewal sources. In order to prove almost adaptivity of CTW over renewal sources, it is necessary to establish that CTW carefully balances estimation error and approximation error

The Embedding Capacity of Information Flows Under Renewal Traffic

Abstract: Given two independent point processes and a certain rule for matching points between them, what is the fraction of matched points over infinitely long streams? In many application contexts, e.g., secure networking, a meaningful matching rule is that of a maximum causal delay, and the problem is related to embedding a flow of packets in cover traffic such that no timing analysis can detect it. We study the best undetectable embedding policy and the corresponding maximum flow rate-that we call the embedding capacity-under the assumption that the cover traffic can be modeled as an arbitrary renewal process. We find that computing the embedding capacity requires the inversion of a very structured linear system that, for a broad range of renewal models encountered in practice, admits a fully analytical expression in terms of the renewal function of the processes. This result enables us to explore the properties of the embedding capacity, obtaining closed-form solutions for selected distribution families and a suite of sufficient conditions on the capacity ordering. We test our solution on real network traces, which shows a remarkable match for tight delay constraints. A gap between the predicted and the actual embedding capacities appears for looser constraints, and further investigation reveals that it is caused by inaccuracy of the renewal traffic model rather than of the solution itself.