Random Lifetime Research Articles

Characterization Results Based on Generalized Dynamic Information Measure

In this paper, we consider a generalized past entropy of order [Formula: see text] and type [Formula: see text] and have characterized some distributions of random lifetimes in continuous and discrete situations. Further, we propose a weighted version of generalized past entropy for continuous and discrete cases and have proved the characterization results. Also, we study a generalized weighted residual entropy for the discrete case. At the end, the concept of generalized cumulative past entropy are also discussed.

Battery-Level-Triggered Transmit Power Control for Energy Harvesting Communications

We consider a continuous-time energy harvesting (EH) communication system consisting of an EH transmitter with random lifetime and a receiver. We assume that neither the fading channel state nor the EH state is available to the transmitter and the transmitter can only observe its battery level, based on which it adjusts its transmit power. Specifically, we quantize the battery capacity of the transmitter into a number of levels, and whenever the energy in the battery hits a certain level, the EH transmitter updates its transmit power. Our main objective is to find the optimal battery-level-triggered (BLT) control policy to maximize the expected total throughput of the transmitter in its lifetime. We model the system as an extended two-dimensional stochastic fluid model (2D-SFM), and derive the Laplace-Stieltjes Transform (LST) matrices of the imbedded process on the decision time sequence, based on which, we formulate the power control problem as a Markov decision process (MDP). We obtain the BLT control policy and the maximum expected throughput by solving the 2D-SFM induced MDP. We evaluate the performance of the BLT policy by comparing it with the conventional uniform-in-time control policies, and results show that the proposed BLT power control policy provides significantly higher expected throughput under the same total energy consumption and the same average control frequency.

Analysis of the Past Lifetime in a Replacement Model through Stochastic Comparisons and Differential Entropy

A suitable replacement model for random lifetimes is extended to the context of past lifetimes. At a fixed time u an item is planned to be replaced by another one having the same age but a different lifetime distribution. We investigate the past lifetime of this system, given that at a larger time t the system is found to be failed. Subsequently, we perform some stochastic comparisons between the random lifetimes of the single items and the doubly truncated random variable that describes the system lifetime. Moreover, we consider the relative ratio of improvement evaluated at x ∈ ( u , t ) , which is finalized to measure the goodness of the replacement procedure. The characterization and the properties of the differential entropy of the system lifetime are also discussed. Finally, an example of application to the firing activity of a stochastic neuronal model is provided.

Generalized Entropies, Variance and Applications.

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

Bounds for the hazard rate and the reversed hazard rate of the convolution of dependent random lifetimes

AbstractAn upper bound for the hazard rate function of a convolution of not necessarily independent random lifetimes is provided, which generalizes a recent result established for independent random lifetimes. Similar results are considered for the reversed hazard rate function. Applications to parametric and semiparametric models are also given.

Continuous review (s, Q) inventory system with random lifetime and two demand classes

This paper examines a continuous review inventory model for perishable items with two demand classes. Demands for both classes occur according to Poisson process. The items in inventory are perishable products and have exponential lifetimes. The time after placing an order is an exponential random variable. When the on-hand inventory drops to pre-specified level s, only the priority customer demands are met whereas the demands from ordinary customers are lost. And also, the demand occurring stock-out periods are lost. The inventory system is characterized by continuous-time Markov process and steady-state probabilities are derived. The expected cost function is formulated and a numerical study is provided for optimization.

QMCD approach for perishability models: The (S, s) control policy with lead time

We consider cost minimization for an (S, s) continuous-review perishable inventory system with random lead times and times to perishability, and a state-dependent Poisson demand. We derive the stationary distributions for the inventory level using the Queueing and Markov Chain Decomposition (QMCD) methodology. Applying QMCD, we develop an intuitive approach to characterizing the distribution of the residual time for the next event in different states of the system. We provide comprehensive analysis of two main models. The first model assumes a general random lifetime and an exponential distributed lead time. The second model assumes an exponential distributed lifetime and a general lead time. Each model is analyzed under both backordering and lost sales assumptions. We consider a fixed cost for each order, a purchase cost, a holding cost, a cost for perished items, and a penalty cost in the case of shortage. Numerical examples are provided and show that variability of lead time is more costly than that of perishability time. Therefore, after reducing lead time and increasing perishability time, managers should focus on reducing variability of lead time.

Connectivity in Molecular Communication With Random Time Constraints

The spatial randomness of nanomachines and propagation time of molecules play an essential role for determining the quality of molecular communication between nanomachines. In this study, we introduce a connectivity model in which the connection between a transmit nanomachine (TN), which is randomly distributed in space, and a receive nanomachine (RN) is achieved when a molecule emitted from the TN arrives at the RN within a time constraint. In particular, this time constraint is modeled as a random lifetime to explain the dissipation phenomenon of molecules in a medium or the random arrival time of interfering molecules. Then, we characterize the local connectivity of the RN in terms of the in-degree by averaging over the spatial randomness of nanomachines and the random first passage time of molecules, which is governed by an anomalous diffusion law.

Stochastic Comparisons and Dynamic Information of Random Lifetimes in a Replacement Model

We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order to assess conditions leading to an improvement of the system. Attention is also given to the relative ratio of improvement, which is proposed as a suitable index finalized to measure the goodness of the replacement procedure. Finally, we provide various results on the dynamic differential entropy of the lifetime of the improved system.

Simultaneous Pricing, Routing, and Inventory Control for Perishable Goods in a Two-echelon Supply Chain

Due to the rapid development of technology in recent years, market competition and customer expectations have increased more than ever. In this situation, it is vital for businesses survival to determine the appropriate policy for inventory control, pricing, and routing, and decisions regarding each of them are often made separately. If the products have a perishable nature, it will be more important to determ ine the above policies. For integration of the decision-making concerning the three key components of the supply chain ,i.e. pricing, routing, and inventory control, two mathematical models were developed for a two-echelon supply chain of perishable items with direct shipment, where fixed lifetime is assumed in one model and random lifetime in another, so that profit is maximized. The proposed mathematical model was solved using the CPLEX solver package of the GAMS software for specification of the optimal policy of the supply chain. The results demonstrated that the CPU time needed for solving the mathematical model for perishable items with random lifetime was less than that for fixed lifetime, while the value of the objective function for products with fixed lifetime was greater than that for products with random lifetime.

Some Results and Applications of Geometric Counting Processes

Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.

A Reinforcement-Learning Approach to Proactive Caching in Wireless Networks

We consider a mobile user accessing contents in a dynamic environment, where new contents are generated over time (by the user’s contacts) and remain relevant to the user for random lifetimes. The user, equipped with a finite-capacity cache memory, randomly accesses the system and requests all the relevant contents at the time of access. The system incurs an energy cost associated with the number of contents downloaded and the channel quality at that time. Assuming causal knowledge of the channel quality, the content profile, and the user-access behavior, we model the proactive caching problem as a Markov decision process with the goal of minimizing the long-term average energy cost. We first prove the optimality of a threshold-based proactive caching scheme, which dynamically caches or removes appropriate contents from the memory, prior to being requested by the user, depending on the channel state. The optimal threshold values depend on the system state and hence are computationally intractable. Therefore, we propose parametric representations for the threshold values and use reinforcement-learning algorithms to find near-optimal parameterizations. We demonstrate through simulations that the proposed schemes significantly outperform classical reactive downloading and perform very close to a genie-aided lower bound.

Decomposition of Schramm–Loewner evolution along its curve

We show that, for κ∈(0,8), the integral of the laws of two-sided radial SLEκ curves through different interior points against a measure with SLEκ Green’s function density is the law of a chordal SLEκ curve, biased by the path’s natural length. We also show that, for κ>0, the integral of the laws of extended SLEκ(−8) curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLEκ curve, biased by the path’s capacity length restricted in that set. Another result is that, for κ∈(4,8), if one integrates the laws of two-sided chordal SLEκ curves through different force points on R against a measure with density on R, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLEκ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem.

Reliability and Random Lifetime Models of Planetary Gear Systems

Conventional reliability models of planetary gear systems are mainly static. In this paper, dynamic reliability models and random lifetime models of planetary gear systems are developed with dynamic working mechanism considered. The load parameters, the geometric parameters, and the material parameters are taken as the inputs of the reliability models and the random lifetime models. Moreover, failure dependence and dynamic random load redistributions are taken into account in the models. Monte Carlo simulations are carried out to validate the proposed models. The results show that the randomness of the load distribution is obvious in the system working process. Failure dependence has significant influences on system reliability. Moreover, the dispersion of external load has great impacts on the reliability, lifetime distribution, and redundancy of planetary gear systems.

Expiration dates and order quantities for perishables

We address the problem of how to set expiration dates for perishable products in the context of a retailer that sells a random lifetime product under periodic review. Although the lifetime is random, the retailer, in addition to deciding when and how much to order, must also determine an expiration date that corresponds to the maximum number of periods that inventory may remain available for sale before it must be discarded. Unlike other models of perishable inventory, but similar to real-world processes, it is possible to unknowingly sell products that have reached the limit of their retail shelf life, or to discard products that are still good for sale. As a result, retailers must balance hazard costs and perishing costs by setting a remaining shelf life that provides customers with a reasonable time for storage at home before the product actually spoils.We formulate the problem as a Markov Decision Process and also introduce two heuristic policies for the computation of order quantities and expiration dates that are of more practical relevance. We conduct an extensive simulation study, focusing on fresh packaged tomatoes as an illustrative example. Through this study, we evaluate the sensitivity of the computed expiration dates relative to the problem parameters and provide managerial insight into the role of expiration date setting for inventory control.

Some properties of cumulative Tsallis entropy

The cumulative entropy is an information measure which is alternative to the differential entropy. Indeed, the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. In this paper we propose a new generalized cumulative entropy based on Tsallis entropy (CTE) and its dynamic version (DCTE). We study some properties and characterization results for this measure.

Testing monotonic equilibrium residual entropy of N-State Random Evolution

ABSTRACTIn this study, we propose an information measure of uncertainty associated with the random equilibrium residual lifetime of a system driven by N-State Random Evolution. A U-statistic test driven by a moment inequality is proposed for testing the hypothesis that the uncertainty of equilibrium remaining life of a system remains unchanged (when system is in the steady state) against the alternative situation when system’s equilibrium residual life has increasing uncertainty over time (i.e., the life distribution has Increasing Equilibrium Residual Entropy property). Some numerical results such as tabulated critical values and empirical power of the proposed test statistic are presented as well.

Doubly stochastic Poisson pulse model for fine-scale rainfall

Stochastic rainfall models are widely used in hydrological studies because they provide a framework not only for deriving information about the characteristics of rainfall but also for generating precipitation inputs to simulation models whenever data are not available. A stochastic point process model based on a class of doubly stochastic Poisson processes is proposed to analyse fine-scale point rainfall time series. In this model, rain cells arrive according to a doubly stochastic Poisson process whose arrival rate is determined by a finite-state Markov chain. Each rain cell has a random lifetime. During the lifetime of each rain cell, instantaneous random depths of rainfall bursts (pulses) occur according to a Poisson process. The covariance structure of the point process of pulse occurrences is studied. Moment properties of the time series of accumulated rainfall in discrete time intervals are derived to model 5-min rainfall data, over a period of 69 years, from Germany. Second-moment as well as third-moment properties of the rainfall are considered. The results show that the proposed model is capable of reproducing rainfall properties well at various sub-hourly resolutions. Incorporation of third-order moment properties in estimation showed a clear improvement in fitting. A good fit to the extremes is found at larger resolutions, both at 12-h and 24-h levels, despite underestimation at 5-min aggregation. The proportion of dry intervals is studied by comparing the proportion of time intervals, from the observed and simulated data, with rainfall depth below small thresholds. A good agreement was found at 5-min aggregation and for larger aggregation levels a closer fit was obtained when the threshold was increased. A simulation study is presented to assess the performance of the estimation method.

Order statistics of uncertain random variables with application to k-out-of-n system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.

A fundamental law relating stock and end-of-life flow in cyclic manufacturing

Cyclic manufacturing (CM) emerges as a sine-qua-non for sustainability. Consumer product stocks of today are the end-of-life flows (EoLF) of tomorrow in circular economy. Enacted legislation fosters reuse/recycle of EoL consumer products, of chemicals, raw materials and hazardous products and components (batteries, brake fluids, printed circuit boards, cellular phones, computers). But when is tomorrow and how much of the stock will appear as EoLF? Efficient CM operations depend on cognizance of EoLF and accumulating stock, the pool from which EoLF emanates. Consumer uncertainty, personal income, economic cycles, advent of technology, social and health reasons, stricter eco-standards and random early product losses during use, render EoLF and product stock uncertain and unobservable. Conventional identification methods based on regression, sequential least squares, or actuary science methods presuming specific residual life distributions, may not provide reliable estimates under uncertainty and non-stationarities. An appealingly simple constitutive law is revealed, relating the mean stock and EoLF in terms of stock mean-age and EoLF mean-age, which are scaled, readily and reliably monitored variables, even from relatively small, decentralized samples. Valid under random lifetime early losses, the law encompasses any EoL exit distribution, enabling stock and EoLF identification, data consolidation and assessment. Being a linear algebraic expression between stock and EoLF, the law reduces computational burden and evades presumption of any survival or exit probability distribution, or cognizance of early loss history, or ex-post fitting of stochastic parameters. The results may prove useful in planning/sizing EoL facilities and recycling operations and in environmental policy or compliance assessment.