Introduction and Summary

6 Dependence notions in reliability theory

At first there was the Markov property. The theory of stochastic processes, which can be considered as an exten- sion of probability theory, allows the modeling of the evolution of systems through the time. It cannot be properly understood just as pure mathemat- ics, separated from the body of experience and examples that have brought it to life. The theory of stochastic processes entered a period of intensive develop- ment, which is not finished yet, when the idea of the Markov property was brought in. Not even a serious study of the renewal processes is possible without using the strong tool of Markov processes. The modern theory of Markov processes has its origins in the studies by A. A: Markov (1856-1922) of sequences of experiments connected in a chain and in the attempts to describe mathematically the physical phenomenon known as Brownian mo- tion. Later, many generalizations (in fact all kinds of weakenings of the Markov property) of Markov type stochastic processes were proposed. Some of them have led to new classes of stochastic processes and useful applications. Let us mention some of them: systems with complete connections [90, 91, 45, 86]; K-dependent Markov processes [44]; semi-Markov processes, and so forth. The semi-Markov processes generalize the renewal processes as well as the Markov jump processes and have numerous applications, especially in relia- bility.

Stochastic processes are powerful tools for the investigation of reliability and availability of repairable equipment and systems. Because of the involved models and in order to be mathematically tractable, these processes are generally confined to the class of regenerative stochastic processes. This contribution uses these processes in solving some reliability problems encountered in pratical applications. Investigations deal with different kinds of reliabilities and, availabilities for one item, series, parallel, and series/parallel structures.

Sleep modes are important for user equipment to reduce power consumption or to reduce battery reloading. To keep the user equipment connected, it has to be activated for uplink or downlink data transfer which causes overhead and additional delay. Wireless or mobile equipment has to be monitored repeatedly for location updates. These classes of problems can be modeled by queuing systems with modified busy and idle periods. In this paper, we consider a generalized infinite-buffer, single-server queuing systems with activation overhead and under various types of sleep modes during inactive periods. For the special assumption of Markovian arrival processes of data units and generally distributed activation times and vacation (sleep) periods, the models can be analyzed exactly by a mean value analysis method which is based on regenerative stochastic processes, renewal theory and classical queuing theorems. The method allows also for dynamic sleep periods which are dependent on each other as in the case of the standard IEEE 802.16m for wireless equipment. Explicit results are derived for average values of data unit delays, resource utilization and power- saving margins dependent on given parameters for activation times and wakeup signaling frequencies. The models include the special cases of M/G/1 queuing systems with a modified service time of the first arrival initiating a busy period and M/G/1 queuing systems with vacation periods, respectively, which are well known from literature providing general solutions for the state and delay distributions. Finally, all models are extended to the exact analysis of Batch Poisson arrival processes.

The literature on Bayesian methods for the analysis of discrete-time semi-Markov processes is sparse. In this paper, we introduce the semi-Markov beta-Stacy process, a stochastic process useful for the Bayesian non-parametric analysis of semi-Markov processes. The semi-Markov beta-Stacy process is conjugate with respect to data generated by a semi-Markov process, a property which makes it easy to obtain probabilistic forecasts. Its predictive distributions are characterized by a reinforced random walk on a system of urns.

We define a notion of regularity ordering among stochastic processes called directionally convex (dcx) ordering and give examples of doubly stochastic Poisson and Markov renewal processes where such ordering is prevalent. Further-more, we show that the class of segmented processes introduced by Chang, Chao, and Pinedo [3] provides a rich set of stochastic processes where the dcx ordering can be commonly encountered. When the input processes to a large class of queueing systems (single stage as well as networks) are dcx ordered, so are the processes associated with these queueing systems. For example, if the input processes to two tandem /M/c1→/M/c2→…→/M/cm queueing systems are dcx ordered, so are the numbers of customers in the systems. The concept of directionally convex functions (Shaked and Shanthikumar [15]) and the notion of multivariate stochastic convexity (Chang, Chao, Pinedo, and Shanthikumar [4]) are employed in our analysis.

Stochastic processes for line shapes and intensities

. This article focuses on the optimal maintenance policy of a system under multi-state with the unit damage area and repair time based on Markov renewal process. In our model, the system performs planned maintenance for a time interval T without failure, and it undergoes detecting the degree of damage immediately when the fail occurs, if the unit damage area is completed within a thresholds for minor repair and update state S, the system is returned to the minor repair state and repaired failed unit by workers, otherwise it returned to the update state and replaced by new one. At first, the mathematical model with the optimal objective is proposed, which is maximizing the expected reward rate. Next, in order to solve this model, we derive some theorems of renewal functions and transition probabilities in a Markov renewal process with multi-state: operation, detection of damage area, planned maintenance, minor repair, and update. Finally, the numerical example shows that the analytical solution of the optimal planned maintenance time T * and critical value of the damage area S * are obtained when the lifetime distribution is given; moreover, the sensitivity analysis is illustrated to validate the effectiveness of the optimal maintenance policy.

We describe aggregate inflation as a stochastic process in which the rate of change of the price level can be positive or zero, where the times spent in each state are of random duration. This class of processes includes Two-State Markov Chains and Renewal Processes as special cases. It is shown that the optimal pricing policy of a monopolistic firm with non-convex costs of price adjustment is (S, s) in its real-price, i.e. its nominal price relative to the price level. A basic certainty-equivalence result is proved: i.e. the firm behaves as if it faces a certain and fixed rate of inflation, higher than the actual expected rate, the difference between the two rates being a risk premium which depends on the real interest rate and the parameters of the stochastic process. One can thus apply previous results from the case of certainty (Sheshinski and Weiss, Review of Economic Studies, 1977) to obtain comparative static results. In particular, one finds that an increase in the variance of expected inflation leads firms to choose a pricing policy with larger amplitude in real price. The paper also addresses the question of consistency in firms' expectations when the price level is determined by the firms' actions. In this paper we consider pricing policies of individual firms in an inflationary environment. Each firm expects the general price level to increase and must determine the rate of increase of its own price. It is assumed that the firm incurs an adjustment cost when it changes its nominal price. Consequently, firms choose to change prices occasionally rather than continuously.

A new class of non-Markovian models is introduced that results from the combination of stochastic automata networks and a very general class of stochastic processes, namely, rational arrival processes, which are derived from matrix exponential distributions. It is shown that the modeling formalism allows a compact representation of complex models with large state spaces. The resulting stochastic process is non-Markovian, but it can be analyzed with numerical techniques like a Markov chain, and the results at the level of the automata are stochastic distributions that can be used to compute standard performance and dependability results. The model class includes stochastic automata networks with phase-type distributed and correlated event times and also includes models that have a finite state space but cannot be represented by finite Markov chains. The paper introduces the model class, shows how the descriptor matrix can be represented in compact form, presents some example models, and outlines methods to analyze the new models.

Consider the alternating renewal process { T , D , T2, D, * * *} specified by the density functions f(t) and g(t) for {T,} and {DA} respectively, whose distribution functions are F(t) and G(t) respectively. The process ends with the first DN > 5 (a large gap), so that the total process is {T1 , DT , * * * TN-1, DN-1 , TN, 6} and its duration W = ]I,.1 (Ti + D,) + TN + a. The probability law for W is useful in reliability analyses, in which {Ti} represents the times to failure and {Di} represents the repair times; then W represents the time from the beginning of the process until the first repair time exceeding 5. It is also useful in traffic studies of the time to cross a busy road. There { T, } represents the times for platoons of vehicles to pass, { Di } represents the times between platoons, 6 is the time to cross the road safely, and W the time from arrival at the road until the crossing is completed (it being assumed that a platoon was passing upon arrival). The objective of this note is to derive the probability law for W. Theorem: Let v(t) = the density function of W V(t) = the distribution function of W X = the variable of the Laplace transform ?(f) of the function f(x) Do = the random variable whose density function is gi(x) = g(x)/G(6)(O < x < 6), g,(x) = 0 (6 < x). Then

Mathematically, stochastic hybrid systems can be described by Markov processes with discrete and continuous behaviours. This chapter provides in an incremental way the necessary mathematical background for understanding such complex systems. The chapter starts with discrete and continuous-time Markov chains. In this book, a Markov chain is understood as a stochastic process with the Markov property (or memoryless property, i.e. its future evolution depends only on the current state) defined on a discrete (finite or countable) state space. The Markov property is illustrated in the Chapman-Kolmogorov equation satisfied by the transition probabilities. Usually, in the continuous-time Markov case, the evolution of these transition probabilities is described by forward/backward Kolmogorov equations. These equations are expressed in terms of the stochastic matrix (called also infinitesimal generator) associated to the Markov chain. This is the matrix of transition rates and the practical use of Markov chains resides in our ability of handling it.The second part of this chapter is dedicated to Markov processes defined on continuous state spaces that can be equipped with additional structures as sigma-algebras, metrics, topologies, norms or others like specific algebraic structures (lattice, vector space). Characterisations of such processes are usually given using functional analysis operators like: the operator semigroup/resolvent, the infinitesimal generator or other operators that can be associated to a Markov process. At an abstract level, these operators describe the evolution the evolution of transition probabilities for a Markov process. Chapman-Kolmogorov equation, Kolmogorov equations and other properties of these operators (e.g. Dynkin formula, martingale problem) represent the natural tools for studying such Markov processes.KeywordsMarkov ChainTopological SpaceMarkov ProcessMarkov PropertyPolish SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Naval Meteorology and Oceanography Command (NAVMETOCCOM) produces operational image products referred to as performance surfaces. A performance surface is the projection of numerical values that reflect the performance of a system or phenomenon into a Geographic Coordinate System layer portraying a physical representation of the earth's atmosphere, surface, or ocean. That is, a performance surface is a graphic that indicates optimal performance through visual cues. To date, performance surface algorithms exist for AntiSubmarine Warfare (ASW) and Piracy, and future performance surfaces are anticipated. A characteristic of performance surfaces is their computational complexity, which can be either deterministic or stochastic in nature. The use of stochastic processes drives a significant computational burden based on the sheer volume of individual simulations that must be run and then combined to form a probabilistic prediction. To mitigate the difficulties imposed by large volume stochastic processes, NAVMETOCCOM chartered an Integrated Product Team (IPT) to characterize a suitable computational architecture for existing and future performance surfaces. The IPT was also tasked to document an engineering process by which emergent performance surfaces could implement suitable architectures. This engineering process used tailored DoD acquisition best practices that were agile and easily repeatable for use by future NAVMETOCCOM IPT efforts.

Spurious measurements frequently occur in surface data from technical components. Excluding or ignoring these spurious points may lead to incorrect surface characterization if these points inherit features of the surface. Therefore, data imputation must be applied to ensure that the estimated data points at spurious measurements do not deviate strongly from the true surface and its characteristics. Traditional surface data imputation methods rely on simple assumptions and ignore existing knowledge of the surface, resulting in suboptimal estimates. In this paper, we propose the use of stochastic processes for data imputation. This approach, which originates from surface texture simulation, allows a straightforward integration of a priori knowledge. We employ Gaussian processes with both stationary and non-stationary covariance structures to address missing values in surface data. In addition, we apply the method to a real-world scenario in which a spurious turned profile is obtained from an actual measurement. Our results demonstrate that the proposed method fills the missing values by maintaining the surface characteristics, particularly when surface features are missing.

The paper presents reliability analysis of a rodding anode plant in aluminum industry with multiple unit failure and single repairman. Manufacturing process of raw aluminum blocks in this plant passes through eight stations. Failure of any of the stations brings the plant to a complete halt, except the butt & thimble removal press stations because of the parallel standby arrangement, and does not affect the system operation completely unless both the units fail. Six years of real maintenance data on component failures, repairs and associated costs are used in this analysis. Measures of system effectiveness is gauged through reliability indices such as mean time to plant failure, availability of the plant, busy period of repairman and expected number of repairs. Effect of repair rate, failure rate and repair cost on system performance w.r.t. revenue is shown graphically. Theory of Semi-Markov and regenerative stochastic processes is used in the analysis.