Showing papers on "Renewal theory published in 2017"

Optimal Status Update for Age of Information Minimization with an Energy Harvesting Source

Abstract: In this paper, we consider a scenario where an energy harvesting sensor continuously monitors a system and sends time-stamped status updates to a destination. The destination keeps track of the system status through the received updates. We use the metric Age of Information (AoI), the time that has elapsed since the last received update was generated, to measure the "freshness" of the status information available at the destination. We assume energy arrives randomly at the sensor according to a Poisson process, and each status update consumes one unit of energy. Our objective is to design optimal online status update policies to minimize the long-term average AoI, subject to the energy causality constraint at the sensor. We consider three scenarios, i.e., the battery size is infinite, finite, and one unit only, respectively. For the infinite battery scenario, we adopt a best-effort uniform status update policy and show that it minimizes the long-term average AoI. For the finite battery scenario, we adopt an energy-aware adaptive status update policy, and prove that it is asymptotically optimal when the battery size goes to infinity. For the last scenario where the battery size is one, we first show that within a broadly defined class of online policies, the optimal policy should have a renewal structure, i.e., the status update epochs form a renewal process, and the length of each renewal interval depends on the first energy arrival over that interval only. We then focus on a renewal interval, and prove that if the AoI in the system is below a threshold when the first energy arrives, the sensor should store the energy and hold status update until the AoI reaches the threshold, otherwise, it updates the status immediately. We analytically characterize the long-term average AoI under such a threshold-based policy, and explicitly identify the optimal threshold.

Informational and Causal Architecture of Continuous-time Renewal Processes

Abstract: We introduce the minimal maximally predictive models ( $$\epsilon \text{-machines }$$ ) of processes generated by certain hidden semi-Markov models Their causal states are either discrete, mixed, or continuous random variables and causal-state transitions are described by partial differential equations As an application, we present a complete analysis of the $$\epsilon \text{-machines }$$ of continuous-time renewal processes This leads to closed-form expressions for their entropy rate, statistical complexity, excess entropy, and differential information anatomy rates

Time‐between‐events control charts for an exponentiated class of distributions of the renewal process

Abstract: Time-between-events control charts are commonly used to monitor high-quality processes and have several advantages over the ordinary control charts. In this article, we present some new control charts based on the renewal process, where a class of absolutely continuous exponentiated distributions is assumed for the time between events. This class includes the generalized exponential, generalized Rayleigh, and exponentiated Pareto distributions. Although we discuss the design structure for all the mentioned distributions, our main focus will be on the generalized exponential distribution due to its practical relevance and popularity. Since the generalized exponential distribution is a generalization of the traditional exponential distribution, the new control chart is more flexible than the existing exponential time-between-events charts. The control chart performance is evaluated in terms of some useful measures, including the average run length (ARL), the expected quadratic loss, continuous ranked probability, and the relative ARL. The effect of parameter estimation using the maximum likelihood and Bayesian methods on the ARL is also discussed in this article. The study also presents an illustrative example and 4 case studies to highlight the practical relevance of the proposal.

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 Gouezel 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.

Multiple mechanisms generate a universal scaling with dissipation for the air-water gas transfer velocity

Abstract: A large corpus of field and laboratory experiments support the finding that the water side transfer velocity kL of sparingly soluble gases near air-water interfaces scales as kL∼(νe)1/4, where ν is the kinematic water viscosity and e is the mean turbulent kinetic energy dissipation rate. Originally predicted from surface renewal theory, this scaling appears to hold for marine and coastal systems and across many environmental conditions. It is shown that multiple approaches to representing the effects of turbulence on kL lead to this expression when the Kolmogorov microscale is assumed to be the most efficient transporting eddy near the interface. The approaches considered range from simplified surface renewal schemes with distinct models for renewal durations, scaling and dimensional considerations, and a new structure function approach derived using analogies between scalar and momentum transfer. The work offers a new perspective as to why the aforementioned 1/4 scaling is robust.

Reliability analysis in railway repairable systems

Abstract: Purpose The purpose of this paper is to present the results of the application of various models to estimate the reliability in railway repairable systems. Design/methodology/approach The methodology proposed by the International Electrotechnical Commission (IEC), using homogeneous Poisson process (HPP) and non-homogeneous Poisson process (NHPP) models, is adopted. Additionally, renewal process (RP) models, not covered by the IEC, are used, with a complementary analysis to characterize the failure intensity thereby obtained. Findings The findings show the impact of the recurrent failures in the times between failures (TBF) for rejection of the HPP and NHPP models. For systems not exhibiting a trend, RP models are presented, with TBF described by three-parameter lognormal or generalized logistic distributions, together with a methodology for generating clusters. Research limitations/implications For those systems that do not exhibit a trend, TBF is assumed to be independent and identically distributed (i.i.d.), and therefore, RP models of “perfect repair” have to be used. Practical implications Maintenance managers must refocus their efforts to study the reliability of individual repairable systems and their recurrent failures, instead of collections, in order to customize maintenance to the needs of each system. Originality/value The stochastic process models were applied for the first time to electric traction systems in 23 trains and to 40 escalators with ten years of operating data in a railway company. A practical application of the IEC models is presented for the first time.

Indicator Variogram Models: Do We Have Much Choice?

Abstract: Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.

Some probabilistic properties of fractional point processes

Abstract: In this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernstein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form where are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process.

Coupling Poisson Processes by Self-Decomposability

Abstract: We analyze a method to produce pairs of non-independent Poisson processes M(t), N(t) from positively correlated, self-decomposable, exponential renewals. In particular, the present paper provides the family of copulas pairing the renewals, along with the closed form for the joint distribution $$p_{m,n}(s,t)$$ of the pair (M(s), N(t)), an outcome which turns out to be instrumental to produce explicit algorithms for applications in finance and queuing theory. We finally discuss the cross-correlation properties of the two processes and the relative timing of their jumps.

Implicit renewal theory in the arithmetic case

Abstract: We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution X of the fixed point equations X =D AX + B and X =D AX ∨ B is l(x) q(x) x -κ, where q is a logarithmically periodic function q(x e h ) = q(x), x > 0, with h being the span of the arithmetic distribution of log A, and l is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevicius (1975) and Goldie (1991).

Delay Analysis of the Max-Weight Policy Under Heavy-Tailed Traffic via Fluid Approximations

Abstract: We consider switched queueing networks with a mix of heavy-tailed (i.e., arrival processes with infinite variance) and exponential-type traffic and study the delay performance of the max-weight policy, known for its throughput optimality and asymptotic delay optimality properties. Our focus is on the impact of heavy-tailed traffic on exponential-type queues/flows, which may manifest itself in the form of subtle rate-dependent phenomena. We introduce a novel class of Lyapunov functions (piecewise linear and nonincreasing in the length of heavy-tailed queues), whose drift analysis provides exponentially decaying upper bounds to queue-length tail asymptotics despite the presence of heavy tails. To facilitate a drift analysis, we employ fluid approximations, proving that if a continuous and piecewise linear function is also a “Lyapunov function” for the fluid model, then the same function is a “Lyapunov function” for the original stochastic system. Furthermore, we use fluid approximations and renewal theory i...

Population density equations for stochastic processes with memory kernels.

Abstract: We present a method for solving population density equations (PDEs)–-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

Analysis of non-processive molecular motor transport using renewal reward theory

Abstract: We propose and analyze a mathematical model of cargo transport by non-processive molecular motors. In our model, the motors change states by random discrete events (corresponding to stepping and binding/unbinding), while the cargo position follows a stochastic differential equation (SDE) that depends on the discrete states of the motors. The resulting system for the cargo position is consequently an SDE that randomly switches according to a Markov jump process governing motor dynamics. To study this system we (1) cast the cargo position in a renewal theory framework and generalize the renewal reward theorem and (2) decompose the continuous and discrete sources of stochasticity and exploit a resulting pair of disparate timescales. With these mathematical tools, we obtain explicit formulas for experimentally measurable quantities, such as cargo velocity and run length. Analyzing these formulas then yields some predictions regarding so-called non-processive clustering, the phenomenon that a single motor cannot transport cargo, but that two or more motors can. We find that having motor stepping, binding, and unbinding rates depend on the number of bound motors, due to geometric effects, is necessary and sufficient to explain recent experimental data on non-processive motors.

Monitoring the shape parameter of a Weibull renewal process

Abstract: This research arose from a challenge faced in real practice—monitoring changes to the Weibull shape parameter. From first-hand experience, we understand that a mechanism for such a purpose is very useful. This article is primarily focused on monitoring the shape parameter of a Weibull renewal process. We derive a novel statistic on the Weibull shape parameter making use of maximum likelihood theory, which is demonstrated to follow an approximately normal distribution. This desirable normality property makes the statistic well suited for use in monitoring the Weibull shape parameter. It also allows for a simple approach to constructing a Shewhart-type control chart, named the Beta chart. The parameter values required to design a Beta chart are provided. A self-starting procedure is also proposed for setting up the Phase I Beta chart. The Average Run Length (ARL) performance of the Beta chart is evaluated through Monte Carlo simulation. A comparison with a moving range exponentially weighted moving ...

A study on a single-unit repairable system with working and repair time omission under an alternative renewal process:

Abstract: In this article, we study a single-unit repairable model with working and repair time omission under an alternative renewal process. As the working time is shorter than threshold τ1, we regard some...

Evaluation of average travel delay caused by moving bottlenecks on highways.

Abstract: This paper presents a modelling framework to evaluate travel delay of all vehicles influenced by moving bottlenecks on highways. During the derivation of analytical formulas, the arrival of slow vehicles was approximated by a Poisson process based on the assumption that they occupied a constant low proportion of the traffic stream. The mathematical analysis process was developed from moving bottlenecks with the same velocity to those with multiple different velocities, and the closed-form expression of expected average travel delay was obtained by utilizing kinematic-wave moving bottleneck theory, gap acceptance theory, probability theory and renewal theory. Model validation and parameters sensitive analysis were conducted by simulation relying on the open source database of US highway 10. The maximum passing rate and the macroscopic parameters of initial traffic state with maximum delay could be found by means of approximate formulas. The proposed modeling framework can be applied for evaluating impacts of slow vehicles on highway operation quantifiably, based on which traffic managements like truck prohibited period decision and speed or lane restriction could be made more scientifically.

Cycle Factors and Renewal Theory

Abstract: For which values of k does a uniformly chosen 3-regular graph G on n vertices typically contain n/k vertex-disjoint k-cycles (a k-cycle factor)? To date, this has been answered for k = n and for k ≪ log n; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability (w.h.p.) most vertices do not lie on k-cycles. A major role in our study of this problem is played by renewal processes without replacement, where one wishes to estimate the probability that in a uniform permutation of a given set of positive integers, the partial sums hit a designated target integer. Using sharp tail estimates for these renewal processes, which may be of independent interest, we settle the cycle factor problem completely: the “threshold” for a k-cycle factor in G as above is κ0 log2 n with . To be precise, G contains a k-cycle factor w.h.p. if and w.h.p. does not contain one if . Thus, for most values of n the threshold concentrates on the single integer K0(n). As a byproduct, we confirm the “comb conjecture,” an old problem concerning the embedding of certain spanning trees in the random graph (n,p).© 2015 Wiley Periodicals, Inc.

Precise large deviations of aggregate claims in a compound size-dependent renewal risk model

Abstract: In this paper, we introduce a compound size-dependent renewal risk model driven by two sequences of random sources The individual claim sizes and their inter-arrival times form a sequence of independent and identically distributed random pairs with each pair obeying a specific dependence structure The numbers of claims caused by individual events form another sequence of independent and identically distributed positive integer-valued random variables, independent of the random pairs above Precise large deviations of aggregate claims for the compound size-dependent renewal risk model are investigated in the case of dominatedly varying claim sizes

A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory

Abstract: We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.

Renewal theorems and mixing for non Markov flows with infinite measure.

Abstract: We obtain results on mixing for a large class of (not necessarily Markov) infinite measure semiflows and flows. Erickson proved, amongst other things, a strong renewal theorem in the corresponding i.i.d. setting. Using operator renewal theory, we extend Erickson's methods to the deterministic (i.e. non-i.i.d.) continuous time setting and obtain results on mixing as a consequence. Our results apply to intermittent semiflows and flows of Pomeau-Manneville type (both Markov and nonMarkov), and to semiflows and flows over Collet-Eckmann maps with nonintegrable roof function.

High order expansions for renewal functions and applications to ruin theory

Abstract: A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function $g$ on a neighborhood of $0$. This expansion relies on complex analysis and is expressed in terms of the residues of the function $1/(1-g)$. Under the assumption that $g$ can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability for the standard compound Poisson risk model. This precises the well- known Cramer-Lundberg approximation of the ruin probability when the initial reserve is large.

Stable Weak Approximation at Work in Index-Linked Catastrophe Bond Pricing

Abstract: We consider the subject of approximating tail probabilities in the general compound renewal process framework, where severity data are assumed to follow a heavy-tailed law (in that only the first moment is assumed to exist). By using weak convergence of compound renewal processes to Levy motion, we derive such weak approximations. Their applicability is then highlighted in the context of an existing, classical, index-linked catastrophe bond pricing model, and in doing so we specialise these approximations to the case of a compound time-inhomogeneous Poisson process. We emphasise a unique feature of our approximation, in that it only demands finiteness of the first moment of the aggregate loss processes. Finally, a numerical illustration is presented. The behaviour of our approximations is compared to both Monte Carlo simulations and first-order single risk loss process approximations, and compares favourably.