Showing papers on "Renewal theory published in 2013"

On the Integral Equation of Renewal Theory

Abstract: Feller’s paper is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation.

Introduction to Markov Chains

Abstract: The topic of Markov processes is huge. A number of volumes can be, and in fact were, written on this topic. We have no intentions to be complete in this area. What is given in this chapter is the minimum required in order to follow what is presented afterwards. In particular, we will refer at times to this chapter when results we present here are called for. For more comprehensive coverage of the topic of Markov chains and stochastic matrices, see [9, 19, 41] or [42].

Analysis of Distance-Based Location Management in Wireless Communication Networks

Abstract: The performance of dynamic distance-based location management schemes (DBLMS) in wireless communication networks is analyzed. A Markov chain is developed as a mobility model to describe the movement of a mobile terminal in 2D cellular structures. The paging area residence time is characterized for arbitrary cell residence time by using the Markov chain. The expected number of paging area boundary crossings and the cost of the distance-based location update method are analyzed by using the classical renewal theory for two different call handling models. For the call plus location update model, two cases are considered. In the first case, the intercall time has an arbitrary distribution and the cell residence time has an exponential distribution. In the second case, the intercall time has a hyper-Erlang distribution and the cell residence time has an arbitrary distribution. For the call without location update model, both intercall time and cell residence time can have arbitrary distributions. Our analysis makes it possible to find the optimal distance threshold that minimizes the total cost of location management in a DBLMS.

Delayed renewal process with uncertain interarrival times

Abstract: Delayed renewal process is a special type of renewal process in which the first interarrival time is quite different from the others. This paper first proposes an uncertain delayed renewal process whose interarrival times are regarded as uncertain variables. Then it gives an uncertainty distribution of delayed renewal process and an elementary delayed renewal theorem.

Generalized telegraph process with random jumps

Abstract: We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

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.

Mixing for some non-uniformly hyperbolic systems

Abstract: In this work we obtain mixing (and in some cases sharp mixing rates) for a reasonable large class of invertible systems preserving an infinite measure. The examples considered here are the invertible analogue of both Markov and non Markov unit interval maps. Moreover, we obtain results on the decay of correlation in the finite case of invertible non Markov maps, which, to our knowledge, were not previously addressed. The present method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig [39], with the framework of function spaces of distributions developed in the recent years along the lines of Blank, Keller and Liverani [9].

Markovian loop clusters on graphs

Abstract: We study the loop clusters induced by Poissonian ensembles of Markov loops on a finite or countable graph (Markov loops can be viewed as excursions of Markov chains with a random starting point, up to re-rooting). Poissonian ensembles are seen as a Poisson point process of loops indexed by 'time'. The evolution in time of the loop clusters defines a coalescent process on the vertices of the graph. After a description of some general properties of the coalescent process, we address several aspects of the loop clusters defined by a simple random walk killed at a constant rate on three different graphs: the integer number line $\mathbb{Z}$, the integer lattice $\mathbb{Z}^d$ with $d\geq 2$ and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.

Catalytic Branching Processes via Spine Techniques and Renewal Theory

Abstract: In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.

Optimal Rate Scheduling via Utility-Maximization for J-User MIMO Markov Fading Wireless Channels with Cooperation

Abstract: We design a dynamic rate scheduling policy of Markov type by using the solution (a social optimal Nash equilibrium point) to a utility-maximization problem over a randomly evolving capacity set for a stochastic system of generalized processor-sharing queues in a random environment whose job arrivals to each queue follow a doubly stochastic renewal process (DSRP). Both the random environment and the random arrival rate of each DSRP are driven by a finite state continuous time Markov chain. The scheduling policy optimizes in a greedy fashion with respect to each queue and environmental state. Since the closed-form solution for the performance of such a queuing system under the policy is difficult to obtain, we establish a reflecting diffusion with regime-switching model for its measures of performance. Furthermore, we justify its asymptotic optimality by deriving the stochastic fluid and diffusion limits for the corresponding system under heavy traffic. In addition, we identify a cost function related to th...

Efficient simulation techniques for stochastic model checking

Abstract: In this thesis, we focus on methods for speeding-up computer simulations of stochastic models. We are motivated by real-world applications in which corporations formulate service requirements in terms of an upper bound on a probability of failure. If one wants to check whether a complex system model satisfies such a requirement, computer simulation is often the method of choice We aim to aid engineers during the design phase, so a question of both practical and mathematical relevance is how the runtime of the simulation can be minimised. We focus on two settings in which a speed-up can be achieved. First, when the probability of failure is low, as is typical in a highly reliable system, the time before the first failure is observed can be impractically large. Our research involves importance sampling; we simulate using a different probability measure under which failure is more likely, which typically decreases the variance of the estimator. In order to keep the estimator unbiased, we compensate for the resulting error using the Radon-Nikodym theorem. If done correctly, the gains can be huge. However, if the new probability measure is unsuited for the problem setting the negative consequences can be similarly profound (infinite variance is even possible). In our work, we have extended an importance sampling technique with good performance (i.e., proven to have bounded relative error) that was previously only applicable in restricted settings to a broad model class of stochastic (Markovian) Petri nets. We have also proposed methods to alleviate two well-known problems from the rare event simulation literature: the occurrence of so-called high-probability cycles and the applicability to large time horizons. For the first we use a method based on Dijkstra’s algorithm, for the second we use renewal theory. Second, it often occurs that the number of needed simulation runs is overestimated. As a solution, we use sequential hypothesis testing, which allows us to stop as soon as we can say whether the service requirement is satisfied. This area has seen a lot of recent interest from the model checking community, but some of the techniques used are not always perfectly understood. In our research we have compared the techniques implemented in the most popular model checking tools, identified several common pitfalls and proposed a method that we proved to not have this problem. In particular, we have proposed a new test for which we bounded the probability of an incorrect conclusion using martingale theory.

Characterization of the Pólya-Aeppli Process

Abstract: In this article, we study the Polya-Aeppli process (PAP). We define PAP from three different points of view: as a compound Poisson process, as a delayed renewal process, and as a pure birth process. We show that these definitions are equivalent. Also, using these definitions we identify several interesting characterizations of PAP.

The trend-renewal process: a useful model for medical recurrence data

Abstract: Time-to-event data analysis has a long tradition in applied statistics. Many models have been developed for data where each subject or observation unit experiences at most one event during its life. In contrast, in some applications, the subjects may experience more than one event. Recurrent events appear in science, medicine, economy, and technology. Often the events are followed by a repair action in reliability or a treatment in life science. A model to deal with recurrent event times for incomplete repair of technical systems is the trend-renewal process. It is composed of a trend and a renewal component. In the present paper, we use a Weibull process for both of these components. The model is extended to include a Cox type covariate term to account for observed heterogeneity. A further extension includes random effects to account for unobserved heterogeneity. We fit the suggested version of the trend-renewal process to a data set of hospital readmission times of colon cancer patients to illustrate the method for application to clinical data.

Delay analysis of the Max-Weight policy under heavy-tailed traffic via fluid approximations

Abstract: We consider a single-hop switched queueing network with a mix of heavy-tailed (i.e., arrival processes with infinite variance) and light-tailed traffic, and study the delay performance of the Max-Weight policy, known for its throughput optimality and asymptotic delay optimality properties. Classical results in queueing theory imply that heavy-tailed queues are delay unstable, i.e., they experience infinite expected delays in steady state. Thus, we focus on the impact of heavy-tailed traffic on the light-tailed queues, using delay stability as performance metric. Recent work has shown that this impact may come in the form of subtle rate-dependent phenomena, the stochastic analysis of which is quite cumbersome. Our goal is to show how fluid approximations can facilitate the delay analysis of the Max-Weight policy under heavy-tailed traffic. More specifically, we show how fluid approximations can be combined with renewal theory in order to prove delay instability results. Furthermore, we show how fluid approximations can be combined with stochastic Lyapunov theory in order to prove delay stability results. We illustrate the benefits of the proposed approach in two ways: (i) analytically, by providing a sharp characterization of the delay stability regions of networks with disjoint schedules, significantly generalizing previous results; (ii) computationally, through a Bottleneck Identification algorithm, which identifies (some) delay unstable queues by solving the fluid model of the network from certain initial conditions.

Renewal theory for random walks on surface groups

Abstract: We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enters a particular type of a cone and never leaves it again. As a consequence, the trajectory of the random walk can be expressed as an "aligned union" of i.i.d. trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

Feller's renewal theorem for systems of renewal equations

Abstract: In this paper, tlte renewal theorem of Feller is extended to the cause of system of renewal equations. Also of refinement of the renewal theorem is given and several open problems are listed.

Aging renewal theory and application to random walks

Abstract: The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. We here discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long time properties, this process exhibits aging: events counted initially in a time interval [0,t] statistically strongly differ from those observed at later times [t_a,t_a+t]. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous time random walk. By this we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking. Along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.

Output Process in Batch-Arrival Queue with N-Policy and Multiple Vacations

Abstract: In the paper the M X /G/1-type queueing system with the N-policy and multiple vacations is considered. The output process, counting successive departures, is studied using the approach consisting of two main stages. Firstly, introducing an auxiliary model with the N-policy and multiple vacations, and applying the formula of total probability, the analysis is brought to the case of the corresponding system without restrictions in the service process, on its first busy cycle. Next, defining a delayed renewal process of successive vacation cycles, the general results are obtained. The explicit formula for the probability generating function of the Laplace transform of the distribution of the number of packets completely served before a fixed moment t is derived and written using transforms of “input” distributions of the system, and components of the Wiener-Hopf-type factorization identity connected with them. Moreover, illustrative numerical results are presented.

A compound renewal model for medical malpractice insurance

Abstract: A renewal model for the aggregate discounted payments and expenses assumed by the insurer is proposed for the “medical malpractice” insurance, where real interest rates could be stochastic and the dependencies between the expenses, the payments and the delays of payment are examined through the theory of copulas. As a first approach to this problem, we present formulas for the first two raw moments and the first joint moment of this aggregate risk process. Examples are given for Erlang claims interoccurence times and delays of payment, Pareto payments and expenses, and the influence of the dependency is illustrated by the Joe copula. Finally the distribution, VaR and TVaR of our risk process are also considered through simulations.

Computing the Distribution Function of the Number of Renewals

Abstract: The method of Laplace transforms is used to find the distribution function, mean, and variance of the number of renewals of a renewal process whose inter-arrival time distribution has a rational Laplace transform. Where the Laplace transform is not rational, we use the Pade approximation method. We apply our method to certain examples and the results are compared to those reported by other researchers.

On quenched and annealed critical curves of random pinning model with finite range correlations

Abstract: This paper focuses on directed polymers pinned at a disordered and correlated interface. We assume that the disorder sequence is a q-order moving average and show that the critical curve of the annealed model can be expressed in terms of the Perron-Frobenius eigenvalue of an explicit transfer matrix, which generalizes the annealed bound of the critical curve for i.i.d. disorder. We provide explicit values of the annealed critical curve for $q=1$,$2$ and a weak disorder asymptotic in the general case. Following the renewal theory approach of pinning, the processes arising in the study of the annealed model are particular Markov renewal processes. We consider the intersection of two replicas of this process to prove a result of disorder irrelevance (i.e. quenched and annealed critical curves as well as exponents coincide) via the method of second moment.

A Class of Modified Weighted Weibull Distribution and Its Properties

Abstract: The weibull distribution and other related distribution like exponential, Rayleigh and extreme value distributions are very use full in survival, reliability renewal theory and branching processes can be seen in recent papers among the others Olnyede (2006) who shows that the weighted distribution are used to adjust the probabilities of the events as observed and recorded Gupta and Kundu (2009) discussed a new class of weighted weibull distributions.

Applied Stochastic Processes

Abstract: Probability and Stochastic Processes Probability Random variables and their distributions Mathematical expectation Joint distribution and independence Convergence of random variables Laplace transform and generating functions Examples of discrete distributions Examples of continuous distributions Stochastic processes Stopping times Conditional expectation Poisson Processes Introduction to Poisson processes Arrival and inter-arrival times of Poisson processes Conditional distribution of arrival times Poisson processes with different types of events Compound Poisson processes Nonhomogeneous Poisson processes Renewal Processes An introduction to renewal processes Renewal reward processes Queuing systems Queue lengths, waiting times, and busy periods Renewal equation Key renewal theorem Regenerative processes Queue length distribution and PASTA Discrete Time Markov Chains Markov property and transition probabilities Examples of discrete time Markov chains Multi-step transition and reaching probabilities Classes, recurrence, and transience Periodicity, class property, and positive recurrence Expected hitting time and hitting probability Stationary distribution Limiting properties Continuous Time Markov Chain Markov property and transition probability Transition rates Stationary distribution and limiting properties Birth and death processes Exponential queuing systems Time reversibility Hitting time and phase-type distributions Queuing systems with time-varying rates Brownian Motion and Beyond Brownian motion Standard Brownian motion and its maximum Conditional expectation and martingales Brownian motion with drift Stochastic integrals Ito's formula and stochastic differential equations A single stock market model Bibliography Index

A New Time-varying Component Outage Model for Power System Reliability Analysis

Abstract: It is a fundamental work to develop accurate component outage model for reliability analysis in power system.The traditional outage model cannot reflect the impact of the time-varying factors in the operating conditions and the repair after failure.In this paper,a staircase function is used to approximate the deterioration failure rate curve and a renewal process based model is proposed to estimate time-varying failure probabilities.The proposed time-varying outage model can both reflect the effects on failure rate caused by both components’ deterioration and repair activities in long terms.An example of a real transformer shows that the proposed model can precisely predict life cumulative probability distribution and steady state of availability.Compared to traditional constant model,the model proposed in this paper is more accurate and practical.