Showing papers on "Renewal theory published in 2000"

Multihypothesis sequential probability ratio tests. II. Accurate asymptotic expansions for the expected sample size

Abstract: For pt. I see ibid. vol.45, p.2448-61, 1999. We proved in pt.I that two specific constructions of multihypothesis sequential tests, which we refer to as multihypothesis sequential probability ratio tests (MSPRTs), are asymptotically optimal as the decision risks (or error probabilities) go to zero. The MSPRTs asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution, under very general statistical models for the observations. In this paper, based on nonlinear renewal theory we find accurate asymptotic approximations (up to a vanishing term) for the expected sample size that take into account the "overshoot" over the boundaries of decision statistics. The approximations are derived for the scenario where the hypotheses are simple, the observations are independent and identically distributed (i.i.d.) according to one of the underlying distributions, and the decision risks go to zero. Simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes. The asymptotic results given here complete the analysis initiated by Baum and Veeravalli (1994), where first-order asymptotics were obtained for the expected sample size under a specific restriction on the Kullback-Leibler distances between the hypotheses.

NWU property of a class of random sums

Abstract: In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

Sampling Bias in Population Studies—How to Use the Lexis Diagram

Abstract: Modified versions of the lifetime distribution are often used in survival analysis. The modifications depend on how we choose individuals for the study and on the assumptions on the behaviour of the population. A rigorous point process description of the Lexis diagram is used to make the sampling mechanisms and the preconditions transparent. The point process description gives a framework to handle all possible sampling patterns. The set-up is generalized so it can handle more complicated life descriptions than just lifetimes, and the diability model is used as an example. Two set-ups can be used. Conditional on the birthtimes, the lifetime distribution is left truncated and subject to either right censoring or right truncation. Assuming that the birthtimes can be described by a Poisson process the modifications are length bias and the recurrence time distribution known from renewal theory.

A general shock model for a reliability system

Abstract: We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

A probabilistic approach to some asymptotics in noiseless communication

Abstract: Renewal theory is a powerful tool in the analysis of source codes. We use renewal theory to obtain some asymptotic properties of finite-state noiseless channels. We discuss the relationship between these results and earlier uses of renewal theory to analyze the Lempel-Ziv (1977, 1978) codes and the Tunstall (1967) code. As a new application of our results, we provide the asymptotic performance of two of the Perl, Garey and Even (1975) prefix condition codes.

The density of the extinction probability of a time homogeneous linear birth and death process under the influence of randomly occurring disasters

Abstract: Under the influence of randomly occurring disasters, the eventual extinction probability, q, of a birth and death process, Z, is a random variable. In this paper, we obtain an integral expression for the probability density function g(x) of q under the assumption that the population process Z is a time homogeneous linear birth and death process and the disasters occur according to an arbitrary renewal process so that its interarrival times have a density. An example is provided to demonstrate how to evaluate the integral numerically.

A Weak Convergence Result Relevant in Recurrent and Renewal Models

Abstract: In this paper we prove a general weak convergence result useful for establishing the distributional properties of processes, estimators, or test statistics arising in recurrent event and renewal process models.

Implications of fluctuations in substitution rates: impact on the uncertainty of branch lengths and on relative-rate tests.

Abstract: Many tests of the lineage dependence of substitution rates, computations of the error of evolutionary distances, and simulations of molecular evolution assume that the rate of evolution is constant in time within each lineage descended from a common ancestor. However, estimates of the index of dispersion of numbers of mammalian substitutions suggest that the rate has time-dependent variations consistent with a fractal-Gaussian-rate Poisson process, which assumes common descent without assuming rate constancy. While this model does not affect certain relative-rate tests, it substantially increases the uncertainty of branch lengths. Thus, fluctuations in the rate of substitution cannot be neglected in calculations that rely on evolutionary distances, such as the confidence intervals of divergence times and certain phylogenetic reconstructions. The fractal-Gaussian-rate Poisson process is compared and contrasted with previous models of molecular evolution, including other Poisson processes, the fractal renewal process, a Levy-stable process, a fractional-difference process, and a log-Brownian process. The fractal models are more compatible with mammalian data than the nonfractal models considered, and they may also be better supported by Darwinian theory. Although the fractal-Gaussian-rate Poisson process has not been proven to have better agreement with data or theory than the other fractal models, its Gaussian nature simplifies the exploration of its impact on evolutionary distance errors and relative-rate tests.

Diffusion approximations for a single node accessed by congestion-controlled sources

Abstract: We consider simple models of congestion control in high-speed networks, and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ON-OFF type with exponential ON and OFF times, then, under a certain scaling, the steady-state distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steady-state distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service-time distributions that are phase type, and show the impact of these distributions on the steady-state distribution of the number of sources in the system. We also establish an insensitivity to service-time distribution when the arrival process is Poisson. We use these results to relate the capacity of a bottleneck node to performance measures of interest for best effort traffic.

Renewal theory and source coding

Abstract: Renewal theory provides a wavy to derive fundamental results about source coding and is useful in the analysis and design of many lossless data compression algorithms We consider two very different applications of renewal theory to source coding The first one results in a variable-length counterpart to the asymptotic equipartition property for unifilar Markov sources The second application leads to the first analysis of variable-to-fixed length codes with plurally parsable dictionaries

Decomposition of General Tandem Queueing Networks with MMPP Input

Abstract: For tandem queueing networks with generally distributed service times, decomposition often is the only feasible solution method besides simulation. The network is partitioned into individual nodes which are analysed in isolation. In existing decomposition algorithms for continuous-time networks, the output of a queue is usually approximated as a renewal process, which serves as the arrival process to the next queue. In this paper, the internal traffic processes are described as semi-Markov processes (SMPs) and Markov modulated Poisson processes (MMPPs). Thus, correlations in the traffic streams, which are known to have a considerable impact on performance, are taken into account to some extent. A two-state MMPP, which arises frequently in communications modeling, serves as input to the first queue of the tandem network. For tandem networks with infinite or finite buffers, stationary mean queue lengths at arbitrary time computed quasi-promptly by the decomposition component of the tool TimeNET are compared to simulation.

Statistics of the occupation time for a class of Gaussian Markov processes

Abstract: We revisit the work of Dhar and Majumdar [Phys. Rev. E 59, 6413 (1999)] on the limiting distribution of the temporal mean $M_{t}=t^{-1}\int_{0}^{t}du \sign y_{u}$, for a Gaussian Markovian process $y_{t}$ depending on a parameter $\alpha $, which can be interpreted as Brownian motion in the scale of time $t^{\prime}=t^{2\alpha}$. This quantity, for short the mean `magnetization', is simply related to the occupation time of the process, that is the length of time spent on one side of the origin up to time t. Using the fact that the intervals between sign changes of the process form a renewal process in the time scale t', we determine recursively the moments of the mean magnetization. We also find an integral equation for the distribution of $M_{t}$. This allows a local analysis of this distribution in the persistence region $(M_t\to\pm1)$, as well as its asymptotic analysis in the regime where $\alpha$ is large. We finally put the results thus found in perspective with those obtained by Dhar and Majumdar by another method, based on a formalism due to Kac.

Characterizing the departure process of a single server queue from the embedded Markov renewal process at departures

Abstract: In the literature, performance analyses of numerous single server queues are done by analyzing the embedded Markov renewal processes at departures. In this paper, we characterize the departure processes for a large class of such queueing systems. Results obtained include the Laplace–Stieltjes transform (LST) of the stationary distribution function of interdeparture times and recursive formula for {c}_{n} ≡ the covariance between interdeparture times of lag n. Departure processes of queues are difficult to characterize and for queues other than M/G/1 this is the first time that c_{ n} can be computed through an explicit recursive formula. With this formula, we can calculate c_{ n} very quickly, which provides deeper insight into the correlation structure of the departure process compared to the previous research. Numerical examples show that increasing server irregularity (i.e., the randomness of the service time distribution) destroys the short-range dependence of interdeparture times, while increasing system load strengthens both the short-range and the long-range dependence of interdeparture times. These findings show that the correlation structure of the departure process is greatly affected by server regularity and system load. Our results can also be applied to the performance analysis of a series of queues. We give an application to the performance analysis of a series of queues, and the results appear to be accurate.

A general framework for some asymptotic reliability formulas

Abstract: The authors prove that certain reliability formulas which link asymptotic availability, mean normal operation time, mean time between failures, mean number of failures over a period of time and asymptotic Vesely rate, and which are well known in the case of modelling using a Markov jump process or an alternating renewal process, are also true in the context of more general modelling.

Nonparametric estimation in renewal theory. II: Solutions of renewal-type equations

Abstract: Nonparametric estimators of solutions of renewal-type equations are proposed in terms of the empirical renewal function. Using the approach of Grubel and Pitts [Ann .Stat .21 1431 –1451 (1993)], who studied asymptotic properties of the empirical renewal function, a number of properties of our estimators are established, including strong consistency, asymptotic normality and efficiency, and asymptotic validity of bootstrap confidence bounds. The results are illustrated by some particular examples.

Optimal strategies in a risk selection investment model

Abstract: We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

Statistical inference for k independent, homogeneous arithmetic processes

Abstract: For k=1,…, K, a stochastic process {An,k, n =1, 2,…} is an arithmetic process (AP) if there exists some real number, d, so that {An,k +(n-1)d, n =1, 2,…} is a renewal process (RP). AP is a stochastically monotonic process and can be used to model a point process, i.e., point events occurring in a haphazard way in time (or space), especially with a trend. For example, the events may be failures arising from a deteriorating machine; and such a series of failures is distributed haphazardly along a time continuum. In this paper, we discuss estimation procedures for K independent, homogeneous APs. Two statistics are suggested for testing whether K given processes come from a common AP. If this is so, we can estimate the parameters d, and of the AP based on the techniques of simple linear regression, where and are the mean and variance of the first average random variable , respectively. In this paper, the procedures are, for the most part, discussed in reliability terminology. Of course, the methods are valid in any area of application, in which case they should be interpreted accordingly.

On stationary renewal reward processes where most rewards are zero

Abstract: We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable couplings.