Showing papers on "Renewal theory published in 2014"

Stochastic Processes: Theory for Applications

Abstract: This definitive textbook provides a solid introduction to discrete and continuous stochastic processes, tackling a complex field in a way that instils a deep understanding of the relevant mathematical principles, and develops an intuitive grasp of the way these principles can be applied to modelling real-world systems. It includes a careful review of elementary probability and detailed coverage of Poisson, Gaussian and Markov processes with richly varied queuing applications. The theory and applications of inference, hypothesis testing, estimation, random walks, large deviations, martingales and investments are developed. Written by one of the world's leading information theorists, evolving over twenty years of graduate classroom teaching and enriched by over 300 exercises, this is an exceptional resource for anyone looking to develop their understanding of stochastic processes.

Diffusion with resetting in arbitrary spatial dimension

Abstract: We consider diffusion in arbitrary spatial dimension d with the addition of a resetting process wherein the diffusive particle stochastically resets to a fixed position at a constant rate r. We compute the nonequilibrium stationary state which exhibits non-Gaussian behaviour. We then consider the presence of an absorbing target centred at the origin and compute the survival probability and mean time to absorption of the diffusive particle by the target. The mean absorption time is finite and has a minimum value at an optimal resetting rate r which depends on dimension. Finally we consider the problem of a finite density of diffusive particles, each resetting to its own initial position. While the typical survival probability of the target at the origin decays exponentially with time regardless of spatial dimension, the average survival probability decays asymptotically as exp ( − A(ln t)d) where A is a constant. We explain these findings using an interpretation as a renewal process and arguments invoking extreme value statistics.

Performance Analysis of Complex Networks and Systems

Abstract: 1. Introduction Part I. Probability Theory: 2. Random variables 3. Basic distributions 4. Correlation 5. Inequalities 6. Limit laws Part II. Stochastic Processes: 7. The Poisson process 8. Renewal theory 9. Discrete-time Markov chains 10. Continuous-time Markov chains 11. Applications of Markov chains 12. Branching processes 13. General queueing theory 14. Queueing models Part III. Network Science: 15. General characteristics of graphs 16. The shortest path problem 17. Epidemics in networks 18. The efficiency of multicast 19. The hopcount and weight to an anycast group Appendix A. A summary of matrix theory Appendix B. Solutions to problems.

Aging Renewal Theory and Application to Random Walks

Abstract: We 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 ½0;tstatistically strongly differ from those observed at later times ½ta;ta þ t� . 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. 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.

Semi-Markov approach to continuous time random walk limit processes

Abstract: Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in $\mathbb{R}^{d}$ with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method.

Fractional Poisson process: long-range dependence and applications in ruin theory

Abstract: We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.

Life behavior of \delta -shock models for uniformly distributed interarrival times

Abstract: In this paper we study the life behavior of $$\delta $$ -shock models when the shocks occur according to a renewal process whose interarrival distribution is uniform. In particular, we obtain the first two moments of the corresponding lifetime random variables for general interarrival distribution, and survival functions when the interarrival distribution is uniform.

Phase Diagram in Stored-Energy-Driven Lévy Flight

Abstract: Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven Levy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.

Diffusion with resetting in arbitrary spatial dimension

Abstract: We consider diffusion in arbitrary spatial dimension d with the addition of a resetting process wherein the diffusive particle stochastically resets to a fixed position at a constant rate $r$. We compute the non-equilibrium stationary state which exhibits non-Gaussian behaviour. We then consider the presence of an absorbing target centred at the origin and compute the survival probability and mean time to absorption of the diffusive particle by the target. The mean absorption time is finite and has a minimum value at an optimal resetting rate $r^*$ which depends on dimension. Finally we consider the problem of a finite density of diffusive particles, each resetting to its own initial position. While the typical survival probability of the target at the origin decays exponentially with time regardless of spatial dimension, the average survival probability decays asymptotically as $\exp -A (\log t)^d$ where $A$ is a constant. We explain these findings using an interpretation as a renewal process and arguments invoking extreme value statistics.

Efficient inference of Gaussian-process-modulated renewal processes with application to medical event data

Abstract: The episodic, irregular and asynchronous nature of medical data render them difficult substrates for standard machine learning algorithms. We would like to abstract away this difficulty for the class of time-stamped categorical variables (or events) by modeling them as a renewal process and inferring a probability density over non-parametric longitudinal intensity functions that modulate the process. Several methods exist for inferring such a density over intensity functions, but either their constraints prevent their use with our potentially bursty event streams, or their time complexity renders their use intractable on our long-duration observations of high-resolution events, or both. In this paper we present a new efficient and flexible inference method that uses direct numeric integration and smooth interpolation over Gaussian processes. We demonstrate that our direct method is up to twice as accurate and two orders of magnitude more efficient than the best existing method (thinning). Importantly, our direct method can infer intensity functions over the full range of bursty to memoryless to regular events, which thinning and many other methods cannot do. Finally, we apply the method to clinical event data and demonstrate a simple example application facilitated by the abstraction.

Phase Diagram in Stored-Energy-Driven L\'evy Flight

Abstract: Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven Levy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.

Internet traffic modelling: from superposition to scaling

Abstract: Internet traffic at various tiers of service providers is essentially a superposition or active mixture of traffic from various sources. Statistical properties of this superposition and a resulting phenomenon of scaling are important for network performance (queuing), traffic engineering (routing) and network dimensioning (bandwidth provisioning). In this article, the authors study the process of superposition and scaling jointly in a non-asymptotic framework so as to better understand the point process nature of cumulative input traffic process arriving at telecommunication devices (e.g., switches, routers). The authors further assess the scaling dynamics of the structural components (packets, flows and sessions) of the cumulative input process and their relation with superposition of point processes. Classical and new results are discussed with their applicability in access and core networks. The authors propose that renewal theory-based approximate point process models, that is, Pareto renewal process superposition and Weibull renewal process superposition can model the similar second-order scaling, as observed in traffic data of access and backbone core networks, respectively.

Dynamic Activation Policies for Event Capture in Rechargeable Sensor Network

Abstract: We consider the problem of event capture by a rechargeable sensor network. We assume that the events of interest follow a renewal process whose event inter-arrival times are drawn from a general probability distribution, and that a stochastic recharge process is used to provide energy for the sensors’ operation. Dynamics of the event and recharge processes make the optimal sensor activation problem highly challenging. In this paper we first consider the single-sensor problem. Using dynamic control theory, we consider a full-information model in which, independent of its activation schedule, the sensor will know whether an event has occurred in the last time slot or not. In this case, a simple and optimal greedy policy for the solution is developed. We then further consider a partial-information model where the sensor knows about the occurrence of an event only when it is active. This problem falls into the class of partially observable Markov decision processes (POMDP). Since the POMDP’s optimal policy has exponential computational complexity and is intrinsically hard to solve, we propose an efficient heuristic clustering policy and evaluate its performance. Finally, our solutions are extended to handle a network setting in which multiple sensors collaborate to capture the events. We also provide extensive simulation results to evaluate the performance of our solutions.

On transient queue-size distribution in the batch-arrivals system with a single vacation policy

Abstract: A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.

Availability-based optimal maintenance policies for repairable systems in military aviation by identification of dominant failure modes

Abstract: Modeling of imperfect repair through perfect renewal process uses an “as good as new” repair assumption and nonhomogeneous Poisson process uses an “ABAO” repair assumption. In practice, repair actions do not result in such extreme situations but in a complex transitional one, that is, general repair. This article discusses generalized renewal process for an aero engine as repairable component. Maximum likelihood estimators for the reliability parameters are estimated using generalized renewal process, for field failure data of an aero engine. The current practice designates repairable components, as high failure rate components based on intuition, experience and the number of unscheduled failures at repair depots. A methodology is developed to designate high failure rate components based on availability by taking into consideration the dominant failure modes of the aero engine. Then, a comparison is made with a “Black Box” approach. The present maintenance policy is then reviewed by reducing the present t...

Quasistochastic matrices and Markov renewal theory

Abstract: Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Rumor processes on $\N$ and discrete renewal processes

Abstract: We study two rumor processes on $\N$, the dynamics of which are related to an SI epidemic model with long range transmission Both models start with one spreader at site $0$ and ignorants at all the other sites of $\N$, but differ by the transmission mechanism In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders The key step of our proofs is to show that, in each model, the position of the spreaders on $\N$ can be related to a suitably chosen discrete renewal process

On queueing delay in WSN with energy saving mechanism based on queued wake up

Abstract: The time-dependent queueing delay in wireless sen-sor network (WSN) with node radios being activated via the queued wake up mechanism is considered. The mathematical model of the system is based on the transient M/G/1/K-type queue with finite buffer and the N-policy, in which the server becomes active after the idle period if the fixed number N of packets are accumulated in the buffer queue. The closed-form representation for the Laplace transform of the queueing delay distribution at fixed time epoch is found. The approach is based on the idea of embedded Markov chain, total probability law, renewal theory and linear algebra. A numerical example is attached as well.

Operator renewal theory for continuous time dynamical systems with finite and infinite measure

Abstract: We develop operator renewal theory for flows and apply this to obtain results on mixing and rates of mixing for a large class of finite and infinite measure semiflows. Examples of systems covered by our results include suspensions over parabolic rational maps of the complex plane, and nonuniformly expanding semiflows with indifferent periodic orbits. In the finite measure case, the emphasis is on obtaining sharp rates of decorrelations, extending results of Gou\"ezel and Sarig from the discrete time setting to continuous time. In the infinite measure case, the primary question is to prove results on mixing itself, extending our results in the discrete time setting. In some cases, we obtain also higher order asymptotics and rates of mixing.

Renewal Theory and Its Applications

Abstract: We have seen that a Poisson process is a counting process for which the times between successive events are independent and identically distributed exponential random variables One possible generalization is to consider a counting process for which the times between successive events are independent and identically distributed with an arbitrary distribution Such a counting process is called a renewal process

Repairable Systems Reliability

Abstract: Nonrepairable systems fail just once, so models for nonrepairable systems must account for random lifetimes. It is often reasonable to assume that different units have random lifetimes that are independent and follow the same distribution, leading to the usual i.i.d. (independent and identically distributed) assumption. By contrast, models for repairable systems must account for successive failures in time. For a given system, these times between failure are often not independent and not identically distributed. Various assumptions about the failure process lead to different models for repairable systems. In this article, we present the nonhomogeneous Poisson process (NHPP), the renewal process, the piecewise exponential model, and the modulated power law process (a compromise between renewal process and NHPP). Inference for a single system as well as for multiple copies of a system is discussed. We also discuss briefly the use of covariates or concomitant variables. Keywords: Poisson process; power law process; renewal process; piecewise exponential; multiple systems

A hidden renewal model for monitoring aquatic systems biosensors

Abstract: This article proposes a method to model signals of oysters' openings over time using a four-state renewal process. Two of them are of particular interest and correspond to instants when the animals are open or closed. An estimator of the cumulative jump rate of the renewal process is provided. It relies on observations of the jumps between the four states. Here these measures are not available but the observed signal is assumed to take ranges of real values according to this underlying process. A procedure to estimate a probability density function that summarizes the information of the signal is explained. This leads to estimation of the hidden renewal process and of its cumulative jump rate for each oyster. We propose to classify these estimated functions for a group of oysters in order to discriminate these animals according to their health status. Such a diagnosis is essential when using these animals as biosensors for water quality assessment.

A renewal version of the Sanov theorem

Abstract: Large deviations for the local time of a process X( t) are investigated, where X(t)=xi for t∈[Si-1,Si[ and (x_j) are i.i.d. random variables on a Polish space, S_j is the j-th arrival time of a renewal process depending on (x_j). No moment conditions are assumed on the arrival times of the renewal process.

Asymptotic Analysis of Queueing Systems with Finite Buffer Space

Abstract: In the paper, we consider a single-server loss system in which each customer has both service time and a random volume. The total volume of the customers present in the system is limited by a finite constant (the system’s capacity). For this system, we apply renewal theory and regenerative processes to establish a relation which connects the stationary idle probability P 0 with the limiting fraction of the lost volume, Q loss, provided the service time and the volume are proportional. Moreover, we use the inspection paradox to deduce an asymptotic relation between Q loss and the stationary loss probability P loss. An accuracy of this approximation is verified by simulation.