Showing papers on "Renewal theory published in 1985"

An introduction to stochastic modeling

Abstract: Stochastic Modeling. Probability Review. The Major Discrete Distributions. @rtant Continuous Distributions. Some Elementary Exercises. Useful Functions, Integrals, and Sums. Conditional Probability and Conditional Expectation: The Discrete Case. The Dice Game Craps. Random Sums. Conditioning on a Continuous Random Variable. Markov Chains: Introduction: Definitions. Transition Probability Matrices of a Markov Chain. Some Markov Chain Models. First Step Analysis. Some Special Markov Chains. Functionals of Random Walks and Success Runs. Another Look at First Step Analysis. The Long Run Behavior of Markov Chains: Regular Transition Probability Matrices. Examples. The Classification of States. The Basic Limit Theorem of Markov Chains. Reducible Markov Chains. Sequential Decisions and Markov Chains. Poisson Processes: The Poisson Distribution and the Poisson Processes. The Law of Rare Events. Distributions Associated with the Poisson Process. The Uniform Distribution and Poisson Processes. Spatial Poisson Processes. Compound and Marked Poisson Processes. Continuous Time Markov Chains: Pure Birth Processes. Ptire Death Processes. Birth and Death Processes. The Limiting Behavior of Birth and Death Processes. Birth and Death Processes with Absorbing States. Finite State Continuous Time Markov Chains. Set Valued Processes. Renewal Phenomena: Definition of a Renewal Process and Related Concepts. Some Examples of Renewal Processes. The Poisson Process Viewed as a Renewal Process. The Asymptotic ]3ehavior as Renewal Process.

Sequential Analysis: Tests and Confidence Intervals

Abstract: I Introduction and Examples.- II The Sequential Probability Ratio Test.- III Brownian Approximations and Truncated Tests.- IV Tests with Curved Stopping Boundaries.- V Examples of Repeated Significance Tests.- VI Allocation of Treatments.- VII Interval Estimation of Prescribed Accuracy.- VIII Random Walk and Renewal Theory.- IX Nonlinear Renewal Theory.- X Corrected Brownian Approximations.- XI Miscellaneous Boundary Crossing Problems.- Appendix 1 Brownian Motion.- Appendix 2 Queueing and Insurance Risk Theory.- Appendix 3 Martingales and Stochastic Integrals.- Appendix 4 Renewal Theory.- Bibliographical Notes.- References.

Allocating Independent Subtasks on Parallel Processors

Abstract: When using MIMD (multiple instruction, multiple data) parallel computers, one is often confronted with solving a task composed of many independent subtasks where it is necessary to synchronize the processors after all the subtasks have been completed. This paper studies how the subtasks should be allocated to the processors in order to minimize the expected time it takes to finish all the subtasks (sometimes called the makespan). We assume that the running times of the subtasks are independent, identically distributed, increasing failure rate random variables, and that assigning one or more subtasks to a processor entails some overhead, or communication time, that is independent of the number of subtasks allocated. Our analyses, which use ideas from renewal theory, reliability theory, order statistics, and the theory of large deviations, are valid for a wide class of distributions. We show that allocating an equal number of subtasks to each processor all at once has good efficiency. This appears as a consequence of a rather general theorem which shows how some consequences of the central limit theorem hold even when we cannot prove that the central limit theorem applies.

A Time-dependent Stopping Problem with Application to Live Organ Transplants

Abstract: We consider a time-dependent stopping problem and its application to the decision-making process associated with transplanting a live organ. "Offers" e.g., kidneys for transplant become available from time to time. The values of the offers constitute a sequence of independent identically distributed positive random variables. When an offer arrives, a decision is made whether to accept it. If it is accepted, the process terminates. Otherwise, the offer is lost and the process continues until the next arrival, or until a moment when the process terminates by itself. Self-termination depends on an underlying lifetime distribution which in the application corresponds to that of the candidate for a transplant. When the underlying process has an increasing failure rate, and the arrivals form a renewal process, we show that the control-limit type policy that maximizes the expected reward is a nonincreasing function of time. For non-homogeneous Poisson arrivals, we derive a first-order differential equation for the control-limit function. This equation is explicitly solved for the case of discrete-valued offers, homogeneous Poisson arrivals, and Gamma distributed lifetime. We use the solution to analyze a detailed numerical example based on actual kidney transplant data.

Renewal theory for several patterns

Abstract: Discrete renewal theory is generalized to study the occurrence of a collection of patterns in random sequences, where a renewal is defined to be the occurrence of one of the patterns in the collection which does not overlap an earlier renewal. The action of restriction enzymes on DNA sequences provided motivation for this work. Related results of Guibas and Odlyzko are discussed.

Markov Renewal Model for Maximum Bridge Loading

Abstract: The prediction of maximum vehicle loadings on a bridge is studied. The stationary distribution of the static response of highway bridges under random truck loading is obtained using a Markov Renewal Model. This model is a generalization of Markov chains and renewal processes and can be used to model the arrival of trucks on a multilane bridge. The model also accounts for random truck characteristics such as axle weights, axle spacings, speed, and the headway distribution between trucks. The stationary distribution of the response is obtained assuming that the bridge (represented by its influence line) acts as a filter to the truck arrival process. The maximum lifetime response is obtained from the stationary distribution using an approximation to S. O. Rice's upcrossing rate formula. The results are then compared to a simulation program and acceptable agreement is reported.

An approximate solution of the integral equation of renewal theory

Abstract: In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

An order statistic characterization of the poisson renewal process

Abstract: Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

An optimal sequential policy for controlling a Markov renewal process

Abstract: This paper discusses a renewal process whose time development between renewals is described by a Markov process. The process may be controlled by choosing the times at which renewal occurs, the objective of the control being to maximise the long-term average rate of reward. Let γ ∗ denote the maximum achievable rate. We consider a specific policy in which a sequence of estimates of γ ∗ is made. This sequence is defined inductively as follows. Initially an (a priori)estimate γo is chosen. On making the nth renewal one estimates γ ∗ in terms of γ o, the total rewards obtained in the first n renewal cycles and the total length of these cycles. γ n then determines the length of the (n + 1)th cycle. It is shown that γ n tends to γ ∗ as n tends to∞, and that this policy is optimal. The time at which the (n + 1)th renewal is made is determined by solving a stopping problem for the Markov process with continuation cost γ n per unit time and stopping reward equal to the renewal reward. Thus, in general, implementation of this policy requires a knowledge of the transition probabilities of the Markov process. An example is presented in which one needs to know essentially nothing about the details of this process or the fine details of the reward structure in order to implement the policy. The example is based on a problem in biology.

New Results on Mean Time-to-First-Slip for a First-Order Loop

Abstract: Approximate results are derived from a new expression for the mean time-to-first-slip for a first-order loop with arbitrary periodic equivalent phase detector characteristic in the presence of frequency detuning. They are accurate for loop signal-to-noise ratios greater than 3 dB or less than 0 dB. The approximations found serve to provide insight into the mean first slip time dependence on the shape of the equivalent phase detector characteristic and the frequency detuning, and dependence on loop parameters. The new expression itself is also of theoretical interest, since it allows one to explicitly express the stationary phase error probability density function and probability current in terms of the mean first slip time. This sheds further light on the interconnections between the conventional modulo-2π approach and the renewal theory approach to the Fokker-Planck method. Extensions to nonperidic equivalent phase detector characteristics are also considered.

Nonlinear Renewal Theory

Abstract: This chapter is concerned with first passages of random walks to nonlinear boundaries. Suitable generalizations of the renewal theory of Chapter VIII are developed in order to justify and generalize the approximations suggested in IV.3.

Analysis of an exhibiting inventory system

Abstract: This paper deals with a continuous review (s,S) inventory system having one exhibiting item subject to random failure. It is assumed that the demand epochs form a renewal process and the probability distribution of demand magnitudes depend only on the time elapsed since the previous demand. Replenishment of stock is instantaneous. For this model expression for the limiting distribution of position inventory is derived by applying the techniques of semi-regenerative process. Some special cases are discussed in detail

Failure Time Models for Thinned Crime Commission Data

Abstract: In studying the effects of correctional treatments and other variables on criminal behavior, researchers often use time to arrest, instead of time until a crime is committed, as the criterion variable. We examine conditions under which inferences can validly be made from the effects of predictors on the time to the first arrest to their effects on the time to the first crime committed. A nonhomogeneous Poisson process and a renewal process are considered as possible models for crime commission. An individual's crimes are assumed to result in arrest independently, with some fixed probability less than one. Multiplicative regression models for the time to the first crime are invariant with respect to thinning for either a Poisson or a renewal process. Proportional hazards models for the time to the first crime are invariant under thinning for a Poisson process but not for a general renewal process. Some limit results for thinned point processes yield an interpretation of proportional hazard effects as multi...

Opimzation of Full-Time and Time Shared Noncoherent Code Tracking Loops

Abstract: Four noncoherent correlative code tracking loops are presented and analyzed using the Renewal Theory framework. The results include the standard noncoherent delay lock loop, the time shared version of that loop known as the tau dither loop, and modified versions of each. From the stochastic differential equation, the generalized S-curve and statistics of the equivalent noise at the input of the loop filter can be derived for each loop. Computation of the noise statistics (autocorrelation and power spectral density) allow determination of the intensity coefficients used as inputs to the renewal equations. The solutions from these steps lead to the probability density of the tracking error. Optimization of the loop structures to minimize the tracking error variance and mean time to lose lock is indicated by the simulation results.

On the exact order of normal approximation in multivariate renewal theory

Abstract: Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

A Sequential Probability Ratio Test Using a Biased Coin Design

Abstract: Consider a sequential probability ratio test comparing two treatments, where each subject receives only one of the treatments. Each subject's treatment assignment is determined by the flip of a biased coin, where the bias serves to balance the number of patients assigned to each treatment. The asymptotic properties of this test are studied, as the sample size approaches infinity. A renewal theorem is given for the joint distribution of the sample size, the imbalance in treatment assignment at the end of the experiment, and the excess over the stopping boundary. This theorem is used to calculate asymptotic expressions for the test's error probabilities.

EOR project networks

Abstract: This paper considers project networks all of whose nodes have an exclusive-or entrance and whose arc weights consist of the execution probabilities and distribution functions of the durations of the respective project activities. EOR networks have some nice properties, for example, a set of Markov renewal processes can be assigned to such a network. The renewal functions of those processes correspond to special activation functions of the network and can (approximately) be computed very efficiently. At last a method for the complete evaluation of an EOR project network in the sense of time planning is presented.

Random Walk and Renewal Theory

Abstract: Wald’s approximations to the power function and expected sample size of a sequential probability ratio test are based on ignoring the discrepancy between the (log) likelihood ratio and the stopping boundary, thus replacing a random variable by a constant. In what follows we develop methods to approximate this discrepancy and hence to obtain more accurate results. Some of the approximations have already been stated and used in III.5 and IV.3. The present chapter is concerned with linear stopping boundaries; and the more difficult non-linear case is discussed in Chapter IX. An alternative method for linear problems is given in Chapter X.

An application of the renewal theoretic selection principle: The first divisible sum

Abstract: This paper deals with the distribution of the first partial sum of a sequence of independent and identically distributed random variables which is divisible by some integerk>1. We give a formula for its characteristic function and obtain limit results fork→∞. A new characterization of the geometric distribution is also given. The results are applicable to models such as periodically observed self-renewing aggregates.

On some properties of Poisson processes

Abstract: The Poisson process, i.e., the simple stream, is defined by Khintchine as a stationary, orderly and finite stream without after-effects. A necessary and sufficient condition for a stream to be a simple stream is that the interarrival times are independent random variables with identical exponential distributions. This paper gives a simple and rigorous proof of the necessary and sufficient condition, and discusses the other necessary and sufficient conditions for a renewal process to be a Poisson process.

Filtering of markov renewal queues, iv: flow processes in feedback queues

Abstract: This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of 'arrivals', 'departures' and 'feedbacks', is a Markov renewal process (MRP). The filtering procedure of Cinlar (1969) was used in [12] to show that the queue length process embedded separately at 'arrivals', 'departures', 'feedbacks', 'inputs' (arrivals and feedbacks), 'outputs' (departures and feedbacks) and 'external' transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

Thinning of point processes-covariance

Abstract: Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

Frequency Fluctuations in Noisy Oscillators

Abstract: We study frequency and period fluctuations in a nonlinear oscillator driven by Gaussian white noise. We define the random period as the random time between two consecutive zero crossings by the random phase plane trajectory, and the random frequency as the number of such zero crossings per unit of time. These quantities are shown to be related by renewal theory. We find asymptotic expressions for the means and variances of the random period and random frequency, for small damping and small noise. The formulas are particularly useful for oscillators with high frequency.

On the limit behavior of certain quantities in a subcritical storage model

Abstract: The limit behavior of the content of a subcritical storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point (io, 0) of the process. By showing that the expected return time to (io, 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (O,t) is then established by showing the asymptotic independence of these two.

An optimal sequential policy for controlling

Abstract: This paper discusses a renewal process whose time development between renewals is described by a Markov process. The process may be controlled by choosing the times at which renewal occurs, the objective of the control being to maximise the long-term average rate of reward. Let y* denote the maximum achievable rate. We consider a specific policy in which a sequence of estimates {Vn}n0 of y* is made. This sequence is defined inductively as follows. Initially an (a priori) estimate To yis chosen. On making the nth renewal one estimates y* in terms of Yo, the total rewards obtained in the first n renewal cycles and the total length of these cycles. y, then determines the length of the (n + 1)th cycle. It is shown that y, tends to y* as n tends to oo, and that this policy is optimal. The time at which the (n + 1)th renewal is made is determined by solving a stopping problem for the Markov process with continuation cost y, per unit time and stopping reward equal to the renewal reward. Thus, in general, implementation of this policy requires a knowledge of the transition probabilities of the Markov process. An example is presented in which one needs to know essentially nothing about the details of this process or the fine details of the reward structure in order to implement the policy. The example is based on a problem in biology.