Discrete-time Renewal Research Articles

Transient behavior of a discrete-time renewal population growth model subject to geometric catastrophes

Transient behavior of a discrete-time renewal population growth model subject to geometric catastrophes

Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m=1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Squirrels Can Little Remember: A Random Walk with Jump Reversals Induced by a Discrete-Time Renewal Process

Squirrels Can Little Remember: A Random Walk with Jump Reversals Induced by a Discrete-Time Renewal Process

Synchronized abandonment in discrete-time renewal input queues with vacations

<p style='text-indent:20px;'>We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.</p>

Forecasting intermittent and sparse time series: A unified probabilistic framework via deep renewal processes.

Intermittency are a common and challenging problem in demand forecasting. We introduce a new, unified framework for building probabilistic forecasting models for intermittent demand time series, which incorporates and allows to generalize existing methods in several directions. Our framework is based on extensions of well-established model-based methods to discrete-time renewal processes, which can parsimoniously account for patterns such as aging, clustering and quasi-periodicity in demand arrivals. The connection to discrete-time renewal processes allows not only for a principled extension of Croston-type models, but additionally for a natural inclusion of neural network based models—by replacing exponential smoothing with a recurrent neural network. We also demonstrate that modeling continuous-time demand arrivals, i.e., with a temporal point process, is possible via a trivial extension of our framework. This leads to more flexible modeling in scenarios where data of individual purchase orders are directly available with granular timestamps. Complementing this theoretical advancement, we demonstrate the efficacy of our framework for forecasting practice via an extensive empirical study on standard intermittent demand data sets, in which we report predictive accuracy in a variety of scenarios.

Asymmetric random walks with bias generated by discrete-time counting processes

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the ‘generator process’ of the walk. We refer the so defined walk to as ‘asymmetric discrete-time random walk’ (ADTRW). We highlight connections of the waiting time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and analyze recurrent/transient features of simple ADTRWs (walks with unit jumps in both directions). We explore the connections of the recurrence/transience with the bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple ADTRWs are either unbiased in the large time limit or ‘strictly unbiased’ at all times with symmetric Bernoulli generator process. In this analysis we highlight the connections of bias and light-tailed/fat-tailed features of the waiting time density in the generator process. As a prototypical example with fat-tailed feature we consider the ADTRW with Sibuya distributed waiting times.We also introduce time-changed versions: We subordinate the ADTRW to a continuous-time renewal process which is independent from the generator process and the jumps to define the new class of ‘asymmetric continuous-time random walk’ (ACTRW). This new class – apart of some special cases – is not a Montroll–Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may open large interdisciplinary fields in anomalous transport, birth–death models and others.

Large deviations in discrete-time renewal theory

We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning.

Variable Length Memory Chains: Characterization of stationary probability measures

Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, are an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability theory. The question of existence of stationary probability measures leads us to introduce a key combinatorial structure for words produced by a VLMC: the Longest Internal Suffix. This notion allows us to state a necessary and sufficient condition for a VLMC to admit a unique invariant probability measure. This condition turns out to get a much simpler form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the general case, enjoys a renewal property. Namely, a stable VLMC induces a semi-Markov chain on an at most countable state space. Unfortunately, this discrete time renewal process does not contain the whole information of the VLMC, preventing the study of a stable VLMC to be reduced to the study of its induced semi-Markov chain. For a subclass of stable VLMC, the convergence in distribution of a VLMC towards its stationary probability measure is established. Finally, finite state space semi-Markov chains turn out to be very special stable VLMC, shedding some new light on their limit distributions.

Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP$^{\text {(a,b)}}/1/\infty $ Queueing System

This paper analyzes an infinite-buffer single-server bulk-service queueing system in which customers arrive according to a discrete-time renewal process. The customers are served under the discrete-time Markovian service process according to the general bulk-service rule. The matrix-geometric method is used to obtain the queue-length distribution at prearrival epoch. The queue-length distributions at other various time epochs are also derived based on prearrival epoch probabilities. A simple approach has been developed to compute the waiting-time distribution of an arriving customer. We also carried out closed-form analytical expression for the service batch size distribution of an arriving customer. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions and model parameters to understand the system behaviour.

Large deviations in renewal models of statistical mechanics

In Zamparo (2019 (arXiv:1903.03527)) the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland–Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of statistical mechanics, such as the Fisher–Felderhof model of fluids, the Wako–Saitô–Muñoz–Eaton model of protein folding, and the Tokar–Dreyssé model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of equilibrium statistical physics and statistical mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland–Scheraga model and for also the renewal models of statistical mechanics, is one of these observables.

Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization

<p style='text-indent:20px;'>This paper analyzes an infinite-buffer single-server queueing system wherein customers arrive in batches of random size according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process. Based on the censoring technique, the UL-type <inline-formula><tex-math id="M1">\begin{document}$ RG $\end{document}</tex-math></inline-formula>-factorization for the Toeplitz type block-structured Markov chain is used to obtain the prearrival epoch probabilities. The random epoch probabilities are obtained with the help of classical principle based on Markov renewal theory. The system-length distributions at outside observer's, intermediate and post-departure epochs are obtained by making relations among various time epochs. The analysis of waiting-time distribution measured in slots of an arbitrary customer in an arrival batch has also been investigated. In order to unify the results of both discrete-time and its continuous-time counterpart, we give a brief demonstration to get the continuous-time results from those of the discrete-time ones. A variety of numerical results are provided to illustrate the effect of model parameters on the performance measures.

Analysis of batch Bernoulli process subject to discrete-time renewal generated binomial catastrophes

With the evolution of internet, there has been increased interest in the use of computer-communication networks and digital communication systems. But unusual events, like virus attack, that often results in abrupt change in the state of the system poses a major threat to these systems. The occurrence of a virus may cause immediate removal of some elements (packets) from the system. Such type of situations can be well represented as discrete-time catastrophe model where interruptions due to various types of virus attacks are referred to as catastrophe. In this paper we study a discrete-time catastrophe model in which population grows according to batch Bernoulli process and catastrophes occur according to discrete-time renewal process. When a catastrophe occurs, an element (individual or packet) of the population survives with probability p or dies with probability $$1-p$$. The analysis of the model has been done using supplementary variable technique and steady-state probability generating function of the distribution of population size at post-catastrophe, arbitrary and pre-catastrophe epochs are obtained under late arrival system with delayed access. In order to make the model more useful for practitioners, a step-wise computing process for evaluation of distribution of population size at various epochs for commonly used inter-catastrophe time distributions viz. geometric, deterministic and arbitrary are given. We also present recursive formulae to compute factorial moments of the population size at various epochs. Further to study the effect of critical model parameters, numerical results and some graphs are included.

Performance analysis of a GI/D- $$MSP/1/\infty$$ M S P / 1 / ∞ queueing system under different service phase initiations

The main objective of this paper is to derive some analytic results concerning GI/D- $$MSP/1/\infty$$ queueing model in which customers arrive according to a discrete-time renewal process. We assume that the service process is correlated and its configuration is designed through discrete-time Markovian service process under different service phase initiations to start a busy period. The system-length distributions at various time epochs have been obtained by employing the matrix-geometric approach. An explicit closed form expression for the waiting-time distribution in the queue measured in slots has also been derived. Computational experiences with a variety of numerical results are discussed to show the effect of system parameters on the performance measures. This queueing model has potential applications in the domain of computer networks, telecommunication systems, and wireless local communication networks for evaluating the various system performances. An application of this type of queueing model in the domain of network coding in wireless communications is also discussed. This provides relevant information to the system analysts, researchers, and industry people who are interested to investigate congestion problems using queueing theory.

Generalizations of the arithmetic case of Blackwell’s renewal theorem

Using elementary techniques, for discrete-time renewal processes, we provide asymptotic results for the probability of renewal at time n using the binomial moments of the underlying discrete distribution. Using these results, we also provide an alternative derivation of the asymptotics for the first two moments of the number of renewals up to time n.

Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model

In this paper, we focus on analyzing the relationship between the discounted aggregate claim costs until ruin and ruin-related quantities including the time of ruin. To facilitate the evaluation of quantities of our interest as an approximation to the ones in the continuous case, discrete-time renewal risk model with certain dependent structure between interclaim times and claim amounts is considered. Furthermore, to provide explicit expressions for various moment-based joint probabilities, a fairly general class of distributions, namely the discrete Coxian distribution, is used for the interclaim times. Also, we assume a combination of geometrics claim size with arbitrary interlciam time distribution to derive a nice expression for the Gerber-Shiu type function involving the discounted aggregate claims until ruin. Consequently, the results are applied to evaluate some interesting quantities including the covariance between the discounted aggregate claim costs until ruin and the discounted claim causing ruin given that ruin occurs.

Informational and Causal Architecture of Discrete-Time Renewal Processes

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose the entropy of a single measurement into that shared with the past, future, or both. The causal state equivalence relation defines a new subclass of renewal processes with a finite number of causal states despite having an unbounded interevent count distribution. We use the resulting formulae to analyze the output of the parametrized Simple Nonunifilar Source, generated by a simple two-state hidden Markov model, but with an infinite-state machine presentation. All in all, the results lay the groundwork for analyzing more complex processes with infinite statistical complexity and infinite excess entropy.

Potential Well Spectrum and Hitting Time in Renewal Processes

The potential well of a state can be interpreted physically as the energy that a stationary process needs to leave the state. We prove that for discrete time renewal processes, the potential well is the right scaling for the hitting and return time distributions of the state. We further detail the potential well spectrum of these processes by giving a complete classification of the states according to their potential well.

METHODS FOR COMPUTING THEMARGINAL DISTRIBUTION OF RENEWAL PROCESSES

In this paper we discuss methods for computing the marginal distribution of renewal processes by using transforms. For discrete time renewal processes we use the double generating function whereas for continuous time renewal processes we use the Laplace transform. Both of the transforms can be inverted numerically. In this paper we also discuss how to compute the marginal distribution of non-parametric renewal processes.

Discrete-time renewal input queue with balking and multiple working vacations

This paper analyzes a discrete-time finite buffer GI/Geo/1 queue with multiple working vacations wherein the customers may balk due to impatience. The service times during working vacation and vacation times are assumed to be geometrically distributed. Embedded Markov chain and supplementary variable techniques have been adopted to obtain the steady-state system length distributions at pre-arrival and arbitrary epochs, respectively. Various performance measures of the model and waiting-time distribution are also presented. A variety of numerical results showing the effect of model parameters on key performance measures are demonstrated.