Showing papers on "Renewal theory published in 2018"
TL;DR: The almost surely globally asymptotical stability and almost surely exponential stability are investigated for switched systems with semi-Markovian switching, Markovians switching, and renewal process switching signals, respectively by using the probability analysis method.
Abstract: This paper is concerned with the stability problem of randomly switched systems. By using the probability analysis method, the almost surely globally asymptotical stability and almost surely exponential stability are investigated for switched systems with semi-Markovian switching, Markovian switching, and renewal process switching signals, respectively. Two examples are presented to demonstrate the effectiveness of the proposed results, in which an example of consensus of multiagent systems with nonlinear dynamics is taken into account.
139 citations
01 Oct 2018
TL;DR: An energy harvesting sensor that is sending status updates to a destination through an erasure channel is considered, in which transmissions are prone to being erased with some probability $q$, independently from other transmissions.
Abstract: An energy harvesting sensor that is sending status updates to a destination through an erasure channel is considered, in which transmissions are prone to being erased with some probability $q$, independently from other transmissions. The sensor, however, is unaware of erasure events due to lack of feedback from the destination. Energy expenditure is normalized in the sense that one transmission consumes one unit of energy. The sensor is equipped with a unit-sized battery to save its incoming energy, which arrives according to a Poisson process of unit rate. The setting is online, in which energy arrival times are only revealed causally after being harvested, and the goal is to design transmission times such that the long term average age of information $(AoI)$, defined as the time elapsed since the latest update has reached the destination successfully, is minimized. The optimal status update policy is first shown to have a renewal structure, in which the time instants at which the destination receives an update successfully constitute a renewal process. Then, for $ q\leq \frac {1}{2}$, the optimal renewal policy is shown to have a threshold structure, in which a new status update is transmitted only if the AoI grows above a certain threshold, that is shown to be a decreasing function of $q$. While for $q \gt \frac {1}{2}$, the optimal renewal policy is shown to be greedy, in which a new status update is transmitted whenever energy is available.
73 citations
TL;DR: In this article, it was shown that the underlying structure of all open linear autonomous compartmental systems is the phase-type distribution, which generalizes the exponential distribution from its application to one-compartment systems to multiple compartments.
Abstract: Linear compartmental models are commonly used in different areas of science, particularly in modeling the cycles of carbon and other biogeochemical elements. The representation of these models as linear autonomous compartmental systems allows different model structures and parameterizations to be compared. In particular, measures such as system age and transit time are useful model diagnostics. However, compact mathematical expressions describing their probability distributions remain to be derived. This paper transfers the theory of open linear autonomous compartmental systems to the theory of absorbing continuous-time Markov chains and concludes that the underlying structure of all open linear autonomous compartmental systems is the phase-type distribution. This probability distribution generalizes the exponential distribution from its application to one-compartment systems to multiple-compartment systems. Furthermore, this paper shows that important system diagnostics have natural probabilistic counterparts. For example, in steady state the system’s transit time coincides with the absorption time of a related Markov chain, whereas the system age and compartment ages correspond with backward recurrence times of an appropriate renewal process. These relations yield simple explicit formulas for the system diagnostics that are applied to one linear and one nonlinear carbon-cycle model in steady state. Earlier results for transit-time and system-age densities of simple systems are found to be special cases of probability density functions of phase-type. The new explicit formulas make costly long-term simulations to obtain and analyze the age structure of open linear autonomous compartmental systems in steady state unnecessary.
41 citations
TL;DR: Numerical experiments show the superiority of Quantity-and-Quality-based policy over Quality- based policy in most operational environment and the cost improvement under Quantity- and-Quality -based policy than other existing shipment consolidation policies.
Abstract: Based on Quality-based policy and Quantity-and-Quality-based policy proposed in this paper, the corresponding perishable products shipment consolidation stochastic models by integrating freshness-keeping effort decision are formulated. For each policy, analytical results give conditions and expressions of adopting these policies and freshness-keeping effort. Numerical experiments show: (i) the positive effect of fresh investment on cost, (ii) the changing tends of policy parameters, freshness-keeping efforts and cost based on all parameters, (iii) the superiority of Quantity-and-Quality-based policy over Quality-based policy in most operational environment and (iv) the cost improvement under Quantity-and-Quality-based policy than other existing shipment consolidation policies.
37 citations
TL;DR: In this article, the authors investigate the large deviations of the number of renewals, the forward and backward recurrence times, the occupation time, and the time interval straddling the observation time.
Abstract: Renewal processes with heavy-tailed power law distributed sojourn times are commonly encountered in physical modeling and so typical fluctuations of observables of interest have been investigated in detail. To describe rare events, the rate function approach from large deviation theory does not hold and new tools must be considered. Here, we investigate the large deviations of the number of renewals, the forward and backward recurrence times, the occupation time, and the time interval straddling the observation time. We show how non-normalized densities describe these rare fluctuations and how moments of certain observables are obtained from these limiting laws. Numerical simulations illustrate our results, showing the deviations from arcsine, Dynkin, Darling-Kac, L\'evy, and Lamperti laws.
31 citations
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TL;DR: In this paper, an erasure channel is considered, in which transmissions are prone to being erased with some probability $q, independently from other transmissions. But the erasure event is not considered in this paper.
Abstract: An energy harvesting sensor that is sending status updates to a destination through an erasure channel is considered, in which transmissions are prone to being erased with some probability $q$, independently from other transmissions. The sensor, however, is unaware of erasure events due to lack of feedback from the destination. Energy expenditure is normalized in the sense that one transmission consumes one unit of energy. The sensor is equipped with a unit-sized battery to save its incoming energy, which arrives according to a Poisson process of unit rate. The setting is online, in which energy arrival times are only revealed causally after being harvested, and the goal is to design transmission times such that the long term average age of information (AoI), defined as the time elapsed since the latest update has reached the destination successfully, is minimized. The optimal status update policy is first shown to have a renewal structure, in which the time instants at which the destination receives an update successfully constitute a renewal process. Then, for $q\leq\frac{1}{2}$, the optimal renewal policy is shown to have a threshold structure, in which a new status update is transmitted only if the AoI grows above a certain threshold, that is shown to be a decreasing function of $q$. While for $q>\frac{1}{2}$, the optimal renewal policy is shown to be greedy, in which a new status update is transmitted whenever energy is available.
31 citations
TL;DR: A new control chart is introduced based on the assumption of a renewal process with rewards, where the reward represents magnitude, and a magnitude-over-threshold condition represents the occurrence of an event.
19 citations
TL;DR: In this article, a physically motivated description of the history-dependent jammed state of disordered materials is proposed, which contrasts sharply with a continuous-time random walk (CTRW) with broadly distributed trapping times commonly used to fit aging data.
Abstract: Aging is a ubiquitous relaxation dynamic in disordered materials. It ensues after a rapid quench from an equilibrium "fluid" state into a nonequilibrium, history-dependent jammed state. We propose a physically motivated description that contrasts sharply with a continuous-time random walk (CTRW) with broadly distributed trapping times commonly used to fit aging data. A renewal process such as CTRW proves irreconcilable with the log-Poisson statistic exhibited, for example, by jammed colloids as well as by disordered magnets. A log-Poisson process is characteristic of the intermittent and decelerating dynamics of jammed matter usually activated by record-breaking fluctuations ("quakes"). We show that such a record dynamics provides a universal model for aging, physically grounded in generic features of free-energy landscapes of disordered systems.
17 citations
TL;DR: In this paper, a single-stage, continuous-time inventory model where unit-sized demands arrive according to a renewal process is studied and it is shown that an (s;S) policy is optimal under minimal assumptions on the ordering/procurement and holding/backorder cost functions.
Abstract: We study a single-stage, continuous-time inventory model where unit-sized demands arrive according to a renewal process and show that an (s;S) policy is optimal under minimal assumptions on the ordering/procurement and holding/backorder cost functions. To our knowledge, the derivation of almost all existing (s;S)-optimality results for stochastic inventory models assume that the ordering cost is composed of a fixed setup cost and a proportional variable cost; in contrast, our formulation allows virtually any reasonable ordering-cost structure. Thus, our paper demonstrates that (s;S)-optimality actually holds in an important, primitive stochastic setting for all other practically interesting ordering cost structures such as well-known quantity discount schemes (e.g., all-units, incremental and truckload), multiple setup costs, supplier-imposed size constraints (e.g., batch-ordering and minimum-order-quantity), arbitrary increasing and concave cost, as well as any variants of these. It is noteworthy that our proof only relies on elementary arguments.
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14 citations
TL;DR: In this paper, the authors construct a renewal structure for random walks on surface groups, defined as times when the random walks enter a particular type of a cone and never leave it again.
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.
TL;DR: 1/ln(λ_{1}+1) is possibly the largest epidemic threshold for a general non-Markovian SIS process with a Poisson curing process under the mean-field approximation of the Weibullian susceptible-infected-susceptible (SIS) process.
Abstract: Since a real epidemic process is not necessarily Markovian, the epidemic threshold obtained under the Markovian assumption may be not realistic. To understand general non-Markovian epidemic processes on networks, we study the Weibullian susceptible-infected-susceptible (SIS) process in which the infection process is a renewal process with a Weibull time distribution. We find that, if the infection rate exceeds 1/ln(λ1+1), where λ1 is the largest eigenvalue of the network's adjacency matrix, then the infection will persist on the network under the mean-field approximation. Thus, 1/ln(λ1+1) is possibly the largest epidemic threshold for a general non-Markovian SIS process with a Poisson curing process under the mean-field approximation. Furthermore, non-Markovian SIS processes may result in a multimodal prevalence. As a byproduct, we show that a limiting Weibullian SIS process has the potential to model bursts of a synchronized infection.
TL;DR: The fundamental limits of transmission of information over a Gaussian multiple access channel (MAC) with the use of variable-length feedback codes and under a non-vanishing error probability formalism are characterized.
Abstract: We characterize the fundamental limits of transmission of information over a Gaussian multiple access channel (MAC) with the use of variable-length feedback codes and under a non-vanishing error probability formalism. We develop new achievability and converse techniques to handle the continuous nature of the channel and the presence of expected power constraints. We establish the $\varepsilon $ -capacity regions and bounds on the second-order asymptotics of the Gaussian MAC with variable-length feedback with termination codes and stop-feedback codes. We show that the former outperforms the latter significantly. Due to the multi-terminal nature of the channel model, we leverage tools from renewal theory developed by Lai and Siegmund to bound the asymptotic behavior of the maximum of a finite number of stopping times.
01 Mar 2018
TL;DR: In this article, the authors considered real populations of items that were incepted into operation at different instants of time and obtained some useful inequalities between the population age and the remaining lifetime using reasoning similar to that employed in population studies.
Abstract: First, we consider items that are incepted into operation having already a random (initial) age and define the corresponding remaining lifetime. We show that these random variables are identically distributed when the age distribution is equal to the equilibrium distribution of the renewal theory. Then we consider real populations of items that were incepted into operation (were born) at different instants of time and obtain some useful inequalities between the population age and the remaining lifetime using reasoning similar to that employed in population studies. We also discuss the aging properties of populations using different stochastic orders.
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TL;DR: First order linear differential equations are derived for the temporal evolution of both the moments and the moment generating function (MGF) of the age vector components of a stochastic hybrid systems approach.
Abstract: A source provides status updates to monitors through a network with state defined by a continuous-time finite Markov chain. An age of information (AoI) metric is used to characterize timeliness by the vector of ages tracked by the monitors. Based on a stochastic hybrid systems (SHS) approach, first order linear differential equations are derived for the temporal evolution of both the moments and the moment generating function (MGF) of the age vector components. It is shown that the existence of a non-negative fixed point for the first moment is sufficient to guarantee convergence of all higher order moments as well as a region of convergence for the stationary MGF vector of the age. The stationary MGF vector is then found for the age on a line network of preemptive memoryless servers. From this MGF, it is found that the age at a node is identical in distribution to the sum of independent exponential service times. This observation is then generalized to linear status sampling networks in which each node receives samples of the update process at each preceding node according to a renewal point process. For each node in the line, the age is shown to be identical in distribution to a sum of independent renewal process age random variables.
TL;DR: In this paper, Stein and Chow this paper developed Stein type two-stage and Chow and Robbins type purely sequential strategies to estimate the unknown variance under a modified Linex loss function, and control the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number.
Abstract: In a normal distribution with its mean unknown, we have developed Stein type two-stage and Chow and Robbins type purely sequential strategies to estimate the unknown variance under a modified Linex loss function. We control the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, . Under both proposed estimation strategies, we have emphasized (i) exact calculations of the distributions and moments of the stopping times as well as the biases and risks associated with our terminal estimators of , along with (ii) selected asymptotic properties. In developing asymptotic second-order properties under the purely sequential estimation methodology, we have relied upon nonlinear renewal theory. We report extensive data analysis carried out via (i) exact calculations as well as (ii) simulations when requisite sample sizes range from small to moderate to large. Both estimation methodologies have been implemented and illustrated with the help of real data ...
TL;DR: The modeling and analysis of the temporal performance variation experienced by a mobile user in a wireless network and its impact on system-level design are analyzed and a simple stochastic geometry model is considered, i.e., constant velocity on a straight line, is considered.
Abstract: This paper focuses on the modeling and analysis of the temporal performance variation experienced by a mobile user in a wireless network and its impact on system-level design. We consider a simple stochastic geometry model: the infrastructure nodes are Poisson distributed while the user’s motion is the simplest possible, i.e., constant velocity on a straight line. We first characterize variations in the signal-to-noise ratio (SNR) process and associated downlink Shannon rate, resulting from variations in the infrastructure geometry seen by the mobile. Specifically, by making a connection between stochastic geometry and queuing theory, the level crossings of the SNR process are shown to form an alternating renewal process whose distribution is completely characterized. For large/small SNR levels, and associated rare events, we further derive simple distributional (exponential) models. We then characterize the second major contributor to such variations, namely, changes in the number of other users sharing the infrastructure. Combining these two phenomena, we study what are the dominant factors (infrastructure geometry or sharing number) when a mobile experiences a very high/low shared rate. These results are then used to evaluate and optimize the system-level quality of experience of the mobile users sharing such a wireless infrastructure, including mobile devices streaming video which proactively buffer content to prevent rebuffering and mobiles which are downloading large files. Finally, we use simulation to assess the fidelity of this model and its robustness to factors which are presently not taken into account.
TL;DR: In this article, the conditional, non-homogeneous and doubly stochastic compound Poisson process with discounted claims is studied and the moment generating functions of these risk processes are derived.
Abstract: In this paper, we study the conditional, non-homogeneous and doubly stochastic compound Poisson process with stochastic discounted claims. We derive the moment generating functions of these risk processes and find their inverses, numerically or analytically, by using their corresponding characteristic functions. We then compare their distributions and some risk measures as the VaR and TVaR, and we examine the case where there is a possible dependence between the occurrence time and the severity of the claim.
TL;DR: In this article, an Anscombe-type theorem for the large deviations principle for trajectories of a random process is proved, and the moderate deviations principle is obtained for the compound renewal processes.
Abstract: An Anscombe-type theorem for the large deviations principle for trajectories of a random process is proved. As a consequence, the moderate deviations principle for the compound renewal processes is obtained.
TL;DR: This work analyses such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process.
Abstract: The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.
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TL;DR: It is shown that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms are derived.
Abstract: Stochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear time-varying dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for the inclusion of state-dependent and independent noise terms. Our key contribution is derivation of necessary and sufficient conditions for the stability of statistical moments, along with exact analytical expressions for the steady-state moments. These results are illustrated on an example from cell biology, where deterministic synthesis and decay of a gene product (RNA or protein) is coupled to random timing of cell-division events. As experimentally observed, cell-division events occur based on an internal timer that measures the time elapsed since the start of cell cycle (i.e., last event). Upon division, the gene product level is halved, together with a state-dependent noise term that arises due to randomness in the partitioning of molecules between two daughter cells. We show that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and derive unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms. Systematic analysis of the formulas reveal counterintuitive insights, such as, if the partitioning noise is large then making the timing of cell division more random reduces noise in gene product levels.
TL;DR: In this article, a functional limit theorem for symmetric U-statistics was extended to asymmetric Ustatistics, and some renewal theory results for U-Statistics were shown.
Abstract: We extend a functional limit theorem for symmetric U-statistics [Miller and Sen, 1972] to asymmetric U-statistics, and use this to show some renewal theory results for asymmetric U-statistics. Some ...
TL;DR: A multivariate extension to the RHawkes process will be proposed, which allows different event types to interact with self- and cross-excitation effects, termed the multivariate renewal Hawkes (MRHawkes) process model.
TL;DR: In this paper, the authors consider the infinite divisibility of distributions of some well-known inverse subordinators and show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible.
Abstract: We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.
10 Sep 2018
TL;DR: In this paper, the authors considered a GI/GI/\(\infty \) queuing system with n types of customers under the assumption that customers arrive at the queue according to a renewal process and occupy random resource amounts, which are independent of their service times.
Abstract: In the paper we consider a GI/GI/\(\infty \) queuing system with n types of customers under the assumptions that customers arrive at the queue according to a renewal process and occupy random resource amounts, which are independent of their service times. Since, in general, the analytical solution of the corresponding Kolmogorov differential equations is not available, we focus on the amount of resources occupied by each class of customers under the assumption of infinitely growing arrival rate, and derive its first and second-order asymptotic approximations. In more detail, we show that the n-dimensional probability distribution of the total resource amount is asymptotically n-dimensional Gaussian, and we verify the accuracy of the asymptotics (in terms of Kolmogorov distance) by means of discrete event simulation.
TL;DR: The proposed models and algorithms are applied to examples involving reliability-related data of complex systems and the obtained results suggest GRP plus q-distributions are promising techniques for the analyses of repairable systems.
Abstract: The Generalized Renewal Process (GRP) is a probabilistic model for repairable systems that can represent the usual states of a system after a repair: as new, as old, or in a condition between new and old. It is often coupled with the Weibull distribution, widely used in the reliability context. In this paper, we develop novel GRP models based on probability distributions that stem from the Tsallis’ non-extensive entropy, namely the q-Exponential and the q-Weibull distributions. The q-Exponential and Weibull distributions can model decreasing, constant or increasing failure intensity functions. However, the power law behavior of the q-Exponential probability density function for specific parameter values is an advantage over the Weibull distribution when adjusting data containing extreme values. The q-Weibull probability distribution, in turn, can also fit data with bathtub-shaped or unimodal failure intensities in addition to the behaviors already mentioned. Therefore, the q-Exponential-GRP is an alternative for the Weibull-GRP model and the q-Weibull-GRP generalizes both. The method of maximum likelihood is used for their parameters’ estimation by means of a particle swarm optimization algorithm, and Monte Carlo simulations are performed for the sake of validation. The proposed models and algorithms are applied to examples involving reliability-related data of complex systems and the obtained results suggest GRP plus q-distributions are promising techniques for the analyses of repairable systems.
TL;DR: A model of relaxation in a disordered medium with traps and obstacles is proposed and two-point correlation functions for certain subordinated stochastic processes, particularly for the generalized Ornstein-Uhlenbeck process, were calculated.
Abstract: In this paper, subordinated stochastic processes are considered, where the renewal process acting as the operational time. It is assumed that the observation of the process begins at a certain time after the start of the renewal process. A recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process. Two-point correlation functions for certain subordinated stochastic processes, particularly for the generalized Ornstein-Uhlenbeck process, were calculated. A model of relaxation in a disordered medium with traps and obstacles is proposed.
TL;DR: In this article, the renewal-based volatility estimators were proposed to estimate the spot variance of a continuous martingale in terms of the conditional intensity or conditional duration density of renewal sampling times.
Abstract: This paper develops the idea of renewal time sampling, a novel sampling scheme constructed from stopping times of semimartingales. Based on this new sampling scheme we propose a class of volatility estimators named renewal based volatility estimators. In this paper we show that: (1) The spot variance of a continuous martingale can be expressed in terms of the conditional intensity or conditional duration density of renewal sampling times; (2) In an infill asymptotics setting, renewal based volatility estimators are consistent and jump-robust estimators of the integrated variance of a general semimartingale; (3) Renewal time sampling and range-based sampling have a higher sampling efficiency than equidistant return-based sampling.
TL;DR: In this article, a new renewal theorem is proposed to obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and apply these results to obtain the tail of the supremum of a perturbed random walk.
Abstract: We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest. We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.
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TL;DR: In this paper, the renewal theory for a class of extremal Markov sequences connected with the Kendall convolution was studied and an analogue of the Fredholm theorem was proved for all regular generalized convolutions algebras.
Abstract: The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem for renewal processes defined by Kendall random walks.