Showing papers on "Renewal theory published in 1995"

Time structure of the activity in neural network models

Abstract: Several neural network models in continuous time are reconsidered in the framework of a general mean-field theory which is exact in the limit of a large and fully connected network. The theory assumes pointlike spikes which are generated by a renewal process. The effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak-coupling expansion. It is shown that the resulting ``spike response model'' is the most general renewal model with linear inputs. The standard integrate-and-fire model forms a special case. In a network structure with several pools of identical spiking neurons, the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools. We derive explicit stability criteria for stationary (incoherent) and oscillatory (coherent) solutions. It is shown that the stationary state of noiseless systems is ``almost always'' unstable. Noise suppresses fast oscillations and stabilizes the system. Furthermore, collective oscillations are stable only if the firing occurs while the synaptic potential is increasing. In particular, collective oscillations in a network with delayless excitatory interaction are at most semistable. Inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations. Our general results allow a straightforward application to different network models with spiking neurons. Furthermore, the theory allows an estimation of the errors introduced in firing rate or ``graded-response'' models.

Analysis of an M/G/1 Queue with Two Types of Impatient Units

Abstract: This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later. We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/1 type and in expressions of 'Takacs' equation' type. QUEUES WITH IMPATIENT UNITS AND REPEATED ATTEMPTS; TAKACS' EQUATION;

The effect of a refractory period on the power spectrum of neuronal discharge

Abstract: The interspike intervals in steady-state neuron firing are assumed to be independently and identically distributed random variables. In the simplest model discussed, each interval is assumed to be the sum of a random neuron refractory period and a statistically independent interval due to a stationary external process, whose statistics are assumed known. The power spectral density (hence the autocorrelation) of the composite neuron-firing renewal process is derived from the known spectrum of the external process and from the unknown spectrum of the neuron-refraction process. The results are applied to spike trains recorded in a previous study [2] of single neurons in the visual cortex of an awake monkey. Two models are demonstrated that may produce peaks in the power spectrum near 40 Hz.

Analysis of Markov renewal shock models

Abstract: A trivariate stochastic process is considered, describing a sequence of random shocks {Xn } at random intervals {Y n} with random system state {Jn }. The triviariate stochastic process satisfies a Markov renewal property in that the magnitude of shocks and the shock intervals are correlated pairwise and the corresponding joint distributions are affected by transitions of the system state which occur after each shock according to a Markov chain. Of interest is a system lifetime terminated whenever a shock magnitude exceeds a prespecified level z. The distribution of system lifetime, its moments and a related exponential limit theorem are derived explicitly. A similar transform analysis is conducted for a second type of system lifetime with system failures caused by the cumulative magnitude of shocks exceeding a fixed level z.

Explicit Limit Results for Minimal Sufficient Statistics and Maximum Likelihood Estimators in Some Markov Processes: Exponential Families Approach

Abstract: Finite-state Markov chains with either a discrete or continuous time parameter, Markov renewal processes and Markov-additive processes are considered. We prove that their likelihood functions, in the nonsequential as well as in various sequential cases, belong to special $(n + k, n)$-curved exponential families in general, for which limit results are easily established. Subsequently, asymptotic normality of the corresponding nonsequential and sequential maximum likelihood estimators is established. Also in the case of Markov renewal and Markov-additive processes, stopping times are determined which reduce the corresponding curved exponential families in general to noncurved ones. The latter, together with results of Stefanov, are combined with results of Serfozo to imply explicit solutions in functional limit theorems for the considered processes. In particular, we derive explicit solutions for the important variance parameter in the functional central limit theorems and functional laws of iterated logarithm for those processes. Indeed, our explicit solutions cover more general cases than the known ones, even in the case of finite-state Markov chains. Moreover, we supply explicit solutions, not previously available, in functional limit theorems for Markov renewal processes and Markov-additive processes.

The Importance of Power-tail Distributions for Telecommunication Traffic Models

Abstract: While the warnings given to designers of telecommunications networks are no doubt correct, the statistical analyses have not revealed how either simulation or analytic techniques can be applied to study the performance of such systems. On the other hand, it has been shown that `self-similar' data can be generated by a renewal process where the interarrival times come from a single power-tail distribution with a finite mean (but infinite variance). The simplest model for this would be a GI/M/1 queue. Alternatively, the results in [LIKH95] indicate that a Poisson process with a `disbursed' batch of packets whose number is distributed by a power-tail, can also generate self-similar data. In its simplest version, this can be transformed into an M/G/1 queue, where `G' is a power-tail distribution. We describe in detail the properties of power-tail distributions, and then present an analytic class of well-behaved distributions (a sub-class of which are Phase Distributions which can be used in Markov Chain models) that have truncated power-tails, and in the limit become power-tail distributions. This class was first used in [LIPS86] to explain the long-tail behavior of measured CPU times at Bellcore in 1986 [LELA86]. It was also used to show what might happen in data-retrieval systems which have power-tail file sizes [GARG92], and even to explain the distribution of medical insurance claims [LOWR93]. We then use these distributions to study the behavior of steady-state GI/M/1 queues as a model for telecommunications networks and present the results of a parametric study of the affects of different a's on the geometric parameter s [LIPS92] as a function of the utilization parameter r. The variance of a power-tail distribution is infinite if a <= 2, but our calculations show that the steady-state performance of these queues becomes worse only gradually as a drops below 2 with r fixed. The performance only becomes disastrous as a approaches 1 from above (i.e., the mean still exists). We also present calculations for distributions with truncated power-tails, and show that they too can yield extraordinarily large mean queue lengths. Of course, all this is done assuming steady-state behavior. But this may require inordinately many arrivals before such large queue lengths could be seen in reality. Discrete event simulation models must necessarilly suffer from the same problem. We present an argument showing that the closer a is to 1, the more arrivals must occur before any system's steady state can be approached.

On Computations of the Mean and Variance of the Number of Renewals: a Unified Approach

Abstract: This paper presents a unified approach to finding both the mean (the so-called renewal function) and variance of the number of renewals in continuous time. Basically, it deals with three mutually exclusive situations depending on whether the inter-renewal time distribution has a closed-form (i) rational Laplace-Stieltjes transform (L–ST); (ii) irrational L–ST or (iii) it cannot be represented by a closed-form L–ST. Explicit closed-form expressions are obtained (in terms of roots) for both the mean and the variance. Asymptotic expressions in terms of roots are also obtained for large t. Numerical results are discussed for a variety of inter-renewal time distributions and their accuracies have been checked against the well-known asymptotic results and also against other numerical results available in the literature. The method discussed here can be employed for a variety of other problems occurring in areas such as queueing, inventory and reliability. The real strength of the method lies in the fact that it gives excellent results particularly in cases (i) and (ii) mentioned above.

Estimation of transition probabilities and bootstrap in a semiparametric markov renewal model

Abstract: We consider parameter estimation based an iid sequence of censored observations of a finite state modulated renewal process. Its intensities assume a similar form as in Cox regression except that the baseline intensities are functions of the backwards recurrence time of the process while the covariates may depend on both calendar and duration time clock. Under conditions slightly stronger than in Andersen and gill we show that if the covariates depend on the backwards recurrence time only then the parameter estimates have the same asymptotic distribution as in the classical Cox regression. The choice of time independent covariates corresponds to Markov renewal processes. In this special case we discuss estimation of the renewal and transition probability matrices and show weak consistency of the bootstrap.

Renewal models for maneuvering targets

Abstract: Most model-based tracking algorithms represent the temporal dynamics of a maneuver acceleration with a Markov process. The sample paths so generated may not be plausible. A renewal process model is shown to be more realistic, but a tracker based upon it is seen to be more complicated to implement. The response of the renewal-model tracker is compared with a simpler estimator based upon a Markov model both for the case in which the tracker utilizes an imager, and the case where it does not. It is shown in the latter case that performance improvement is not commensurate with algorithmic complexity. For an image-enhanced tracker, a sophisticated model of target dynamics promises significant performance improvement. >

A nonparametrie maximum likelihood estimator for incomplete renewal data

Abstract: SUMMARY The problem of estimating the lifetime distribution based on data from independently and identically distributed stationary renewal processes is addressed. The data are incomplete. A nonparametric maximum likelihood estimate of the lifetime distribution is derived using the EM algorithm. The missing information principle is used to estimate the standard error of the estimated distribution. The methodology is applied to a problem in the nursing profession where nurses withdraw from active service for a period of time before returning to take up post at a later date. It is important that nurse manpower planners accurately predict this pattern of return. The data analysed are from the Northern Ireland nursing profession.

Variable-to-fixed length codes and the conservation of entropy

Abstract: For a large class of parsing rules, we introduce a "conservation of entropy" theorem for the output of a unifilar Markov source. Using this theorem and renewal theory, we find a procedure to generate asymptotically optimal generalized variable-to-fixed length codes.

A Sequential Assignment Match Process with General Renewal Arrival Times

Abstract: This work studies sequential assignment match processes, in which random offers arrive sequentially according to a renewal process, and when an offer arrives it must be assigned to one of given waiting candidates or rejected. Each candidate as well as each offer is characterized by an attribute. If the offer is assigned to a candidate that it matches, a reward R is received; if it is assigned to a candidate that it does not match, a reward r ≤ R is received; and if it is rejected, there is no reward. There is an arbitrary discount function, which corresponds to the process terminating after a random lifetime. Using continuoustime dynamic programming, we show that if this lifetime is decreasing in failure rate and candidates have distinct attributes, then the policy that maximizes total expected discounted reward is of a very simple form that is easily determined from the optimal single-candidate policy. If the lifetime is increasing in failure rate, the optimal policy can be recursively determined: a solution algorithm is presented that involves scalar rather than functional equations. The model originated in the study of optimal donor-recipient assignment in live-organ transplants. Some other applications are mentioned as well.

Comparison of replacement policies via point processes

Abstract: First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

A continuous-time job search model: general renewal processes

Abstract: In this paper we study a continuous time job search model introduced by Zuckerman in [11]: Job offers are received randomly over time according to a (general) renewal process. The offer wages are assumed to be independent and identically distributed random variables. The objective of the «job searcher» is to choose a stopping time which maximizes his expected net return. The main contribution of this paper is that we give a unified approach to solve the job searcher's problem for general interarrival time distributions (in a finite and an infinite horizon setting). In general, an optimal strategy does not necessarly stop at a time at which a job is offered. Further, we give various examples of what the optimal strategy looks like in specific submodels. The results obtained by Zuckerman in [11], [12] and [13] follow immediately from the results presented here

Batch renewal process: exact model of traffic correlation

Abstract: The batch renewal process can conform exactly to any given (bounded) indices of dispersion or (infinite) sets of correlation functions and is the least biased choice of all possible processes given only such measures of correlation. It contains no feature extraneous to correlation. In particular, it implies no assumptions about traffic burst structure.

Variance Estimation Based on Invariance Principles

Abstract: Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.

Nonparametric confidence intervals for the renewal function with censored data

Abstract: An asymptotic nonparametric method for constructing confidence intervals for the renewal function using censored data is presented. The method is based on the fact that the productlimit estimator of the renewal function converges weakly to a Gaussian process as the sample size increases

Renewal creep theory

Abstract: The mathematics of probability are used to construct a framework that describes some key features of primary and secondary creep. The underlying assumption is that dislocation slip and annihilation are probabilistic events. The resulting mathematical framework takes the form of renewal theory from probability theory. Renewal creep theory provides a mathematical frame-work for primary creep that accommodates previously developed empirical descriptions. Renewal creep theory also predicts the existence of secondary creep as an asymptotically constant strain-rate phenomenon. Creep modeling techniques are demonstrated for three titanium alloys.

Estimating expected sojourn times for a markov renewal process

Abstract: We consider the estimation of the expected sojourn time in a Markov renewal process under the data condition that only the counts of the exits from the states are available for fixed intervals of time. For analytical and illustrative purposes we concentrate on the two-state process case. We present least squares and method of moments estimators and compare their statistical properties both analytically and empirically. We also present modified estimators with improved properties based upon an overlapping interval sampling strategy. The major results indicate that the least squares estimator is biased in general with the bias depending on the size of the sampling interval and the first two moments of the sojourn time distribution function. The bias becomes negligible as the size of the sampling interval increases. Analytical and empirical results indicate that the method of moments estimator is less sensitive to the size of the sampling interval and has slightly better mean squared error properties than th...

Applied Stochastic Processes : A Biostatistical and Population Oriented Approach

Abstract: Introduction to Stochastic Processes: Basic Concepts. Random Walk and Markov Process. Non-Markov Process and Renewal Theory. Martingales. Counter Theory and Age-Replacement Policy. Palm Probability and Its Applications. Stochastic Epidemic Processes. Stochastic Processes of Clinical Drug Trials. Techniques of Stochastic Processes in Mortality Analysis--Applications on Life Table. Techniques of Stochastic Processes in Fertility Analysis. Techniques of Stochastic Process for Demographic Analysis--Population Growth Indices. Stochastic Processes on Survival and Competing Risk Theory. Stochastic Processes in Genetics. Appendix. Index.

An elementary renewal theorem for random compact convex sets

Abstract: A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as H(K)= P{S, c K}, n=O where S, = A1 D - A are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point,