Stationary availability of a semi‐Markov system with random maintenance

We consider a reparable system with a finite state space, evolving in time according to a semi‐Markov process. The system is stopped for it to be preventively maintained at random times for a random duration. Our aim is to find the preventive maintenance policy that optimizes the stationary availability, whenever it exists. The computation of the stationary availability is based on the fact that the above maintained system evolves according to a semi‐regenerative process. As for the optimization, we observe on numerical examples that it is possible to limit the study to the maintenance actions that begin at deterministic times. We demonstrate this result in a particular case and we study the deterministic maintenance policies in that case. In particular, we show that, if the initial system has an increasing failure rate, the maintenance actions improve the stationary availability if and only if they are not too long on the average, compared to the repairs ( a bound for the mean duration of the maintenance actions is provided). On the contrary, if the initial system has a decreasing failure rate, the maintenance policy lowers the stationary availability. A few other cases are studied. Copyright © 2000 John Wiley & Sons, Ltd.     

Analyzing continuous state space failure time processes: Two further results*

In a recent paper, Petersen (1988) considered a continuous state space failure time process. The central result provided in that paper was that the destination‐specific rate of transition of the process can be specified in two steps. First, one specifies the overall rate at which a change occurs. Then, one specifies the probability density function of the destination state, given that a transition occurred. This two‐step property was used in deriving the likelihood of the data and was exploited for purposes of estimation. The overall rate of transition can be estimated from the data on durations between changes in the dependent variable. The density for the new value of the dependent variable, given a change, can be estimated from the data on the values of the dependent variable after the change.

This paper extends these results in two ways. First, it is shown that one can derive the likelihood of the process directly from the destination‐specific rate of transition, without going through its decomposition into the overall rate times the density of the destination state, given a transition. Once the likelihood is derived, estimation is comparatively straightforward. Second, it is shown how one can derive, at each point in time, a more standard regression function for the continuous dependent variable, where its value is expressed in terms of its conditional mean plus an error term.     

A curious renewal process average

The lifetimes of a renewal process observed during a fixed interval (0, t] are smaller, on the average, than the process mean lifetime; it is shown that the mean observed lifetime has a particularly simple form.     

Future pricing through homogeneous semi‐Markov processes

An Erratum for this article has been published in Applied Stochastic Models in Business and Industry 2005; (in press) This paper presents a future pricing model based on the discrete time homogeneous semi‐Markov process (DTHSMP). The model is adapted to the real data of the Italian primary future stock index. After showing the pricing model, the DTHSMP solution is given. The solution of the semi‐Markov process gives, for each period of the considered horizon time, and for each starting state, the probability distribution of the future price. Copyright © 2005 John Wiley & Sons, Ltd.     

ON THE EXTINCTION OF POPULATION-SIZE-DEPENDENT BISEXUAL GALTON-WATSON PROCESSES

In this article, the population-size-dependent bisexual Galton-Watson processes are considered. Under some suitable conditions on the mating functions and the offspring distribution, existence of the limit of mean growth rate per mating unit is proved. And based on the limit, a criterion to identify whether the process admits ultimate extinct with probability one is obtained.     

Integer Valued Autoegressive Process with Katz Arrivals and Its Application in Predicting the Count of the Cases of Respiratory Disease

The traditional PAR process (Poisson autoregressive process) assumes that the arrival process is the equi-dispersed Poisson process, with its mean being equal to its variance. Whereas the arrival process in the real DGP (data generating process) could either be over-dispersed, with variance being greater than the mean, or under-dispersed, with variance being less than the mean. This paper proposes using the Katz family distributions to model the arrival process in the INAR process (integer valued autoregressive process with Katz arrivals) and deploying Monte Carlo simulations to examine the performance of maximum likelihood (ML) and method of moments (MM) estimators of INAR-Katz model. Finally, we used the INAR-Katz process to model count data of hospital emergency room visits for respiratory disease. The results show that the INAR-Katz model outperforms the Poisson model, PAR(1) model, and has great potential in empirical application.     

Central Limit Theorem for Nonlinear Semi-Markov Reward Processes

Abstract

In this work, we obtain a central limit theorem for reward processes defined on a finite state space semi-Markov process, when reward functions assumed to have general forms and are not of constant rates. Martingale theory is the main tool which have been used for establishing the convergence of scaled and shifted reward process to a zero mean Brownian motion. The striking point in this article is considering general forms for the reward functions which are realistic in applications. The conditions needed for these results are existence of variances for sojourn times in each state and second order integrability of reward functions with respect to sojourn times distributions.     

The Reverse H‐free Process for Strictly 2‐Balanced Graphs

Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2‐balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs.     

Inference in lognormal multidimensional diffusion processes with exogenous factors: Application to modelling in economics

The multidimensional lognormal diffusion process with exogenous factors is treated using the Kolmogorov equations, and the mean vector and covariance matrix are estimated using discrete sampling by the maximum-likelihood method. Also, this process is constructed as a solution of a multidimensional stochastic differential equation, and an estimation is made through the maximum-likelihood method to infer the parameters of the exogenous factors, this time using continuous sampling. Finally, a test for a hypothesis based on these parameters is constructed.     

The impact of order batching and picking area zoning on order picking system performance

This paper proposes an approximation model based on queuing network theory to analyze the impact of order batching and picking area zoning on the mean order throughput time in a pick-and-pass order picking system. The model includes the sorting process needed to sort the batch again by order. Service times at pick zones are assumed to follow general distributions. The first and second moments of service times at zones and the visiting probability of a batch of orders to a pick zone are derived. Based on this information, the mean throughput time of an arbitrary order in the order picking system is obtained. Results from a real application and simulation show that this approximation model provides acceptable accuracy for practical purposes. Furthermore, the proposed method is simple and fast and can be easily applied in the design and selection process of order picking systems.     

A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing

A mean‐reverting model is proposed for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non‐Gaussian Ornstein–Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. In a simulation example it is demonstrated that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. The pricing of European call and put options written on electricity forward contracts is also discussed.     

A TWO‐STAGE INFORMATION‐THEORETIC APPROACH TO MODELING LANDSCAPE‐LEVEL ATTRIBUTES AND MAXIMUM RECRUITMENT OF CHINOOK SALMON IN THE COLUMBIA RIVER BASIN

ABSTRACT. Many anadromous salmonid stocks in the Pacific Northwest are at their lowest recorded levels, which has raised questions regarding their long‐term persistence under current conditions. There are a number of factors, such as freshwater spawning and rearing habitat, that could potentially influence their numbers. Therefore, we used the latest advances in information‐theoretic methods in a two‐stage modeling process to investigate relationships between landscape‐level habitat attributes and maximum recruitment of 25 index stocks of chinook salmon (Onocorhynchus tshawy‐tscha) in the Columbia River basin. Our first‐stage model selection results indicated that the Ricker‐type, stock recruitment model with a constant Ricker a, i.e., recruits‐per‐spawner at low numbers of fish) across stocks was the only plausible one given these data, which contrasted with previous unpublished findings. Our second‐stage results revealed that maximum recruitment of chinook salmon had a strongly negative relationship with percentage of surrounding subwatersheds categorized as predominantly containing U.S. Forest Service and private moderate‐high impact managed forest. That is, our model predicted that average maximum recruitment of chinook salmon would decrease by at least 247 fish for every increase of 33% in surrounding subwatersheds categorized as predominantly containing U.S. Forest Service and privately managed forest. Conversely, mean annual air temperature had a positive relationship with salmon maximum recruitment, with an average increase of at least 179 fish for every increase in 2°C mean annual air temperature.     

The steady‐state probabilities for regenerative semi‐Markov processes with application to prevention and screening

There is a growing interest in planning and implementing broad‐scale clinical trials with a focus on prevention and screening. Often, the data‐generating mechanism for such experiments can be viewed as a semi‐Markov process. In this communication, we develop general expressions for the steady‐state probabilities for regenerative semi‐Markov processes. Hence, the probability of being in a certain state at the time of recruitment to a clinical trial can be calculated. An application to breast cancer prevention is demonstrated. Copyright © 1999 John Wiley & Sons, Ltd.     

The inspection paradox in renewal-reward processes

To explore the inspection paradox in the context of a renewal-reward process, we obtain asymptotic expressions for the mean and distribution function of the reward associated with the spread (total life) of the process. These results also yield a simplified demonstration of the elementary renewal-reward theorem.     

Sur les natures des processus solutions de certaines equations stochastiques aux differences

A study of the natures of the processes that are solutions of some finite difference stochastic equations, the right‐hand member of which is a stationary process, is given in the paper. Since the principal application of the present work concerns ARIMA models with or without seasonal variation, these processes are named G‐ARIMA. First, a criterion for a G‐ARIMA process to be stationary is established and some properties of a special class of stationary G‐ARIMA processes are studied. Then, we deduce some conditions for a finite difference stochastic equation to uniquely possess non‐stationary solutions. These have the particular property that their backward shift‐operator may be either a linear or a non‐linear operator, depending on the initial conditions of the solutions. So criteria are established for a non‐stationary G‐ARIMA process to have a bounded linear backward shift operator. Finally, some further properties of the G‐ARIMA processes are given, by comparing them with the broad class of V‐bounded processes. This comparison shows that a non‐stationary process cannot be at the same time G‐ARIMA and V‐bounded.     

On some continuity and differentiability properties of paths of Gaussian processes

The following path properties of real separable Gaussian processes ξ with parameter set an arbitrary interval are established. At every fixed point the paths of ξ are continuous, or differentiable, with probability zero or one. If ξ is measurable, then with probability one its paths have essentially the same points of continuity and differentiability. If ξ is measurable and not mean square continuous or differentiable at every point, then with probability one its paths are almost nowhere continuous or differentiable, respectively. If ξ harmonizable or if it is mean square continuous with stationary increments, then its paths are absolutely continuous with probability one if and only if ξ is mean square differentiable; also mean square differentiability of ξ implies path differentiability with probability one at every fixed point. If ξ is mean square differentiable and stationary, then on every interval with probability one its paths are either differentiable everywhere or nondifferentiable on countable dense subsets. Also a class of harmonizable processes is determined for which of the following are true: (i) with probability one paths are either continuous or unbounded on every interval, and (ii) mean square differentiability implies that with probability one on every interval paths are either differentiable everywhere or nondifferentiable on countable dense subsets.     

Large Eddy Simulation of a Compressible Boundary Layer

A large eddy simulation of a compressible boundary layer is performed. To generate an appropriate inflow distribution the rescaling technique for compressible flows is discribed. In this method Morkovin's hypothesis in which the total temperature fluctuations are neglected compared with the static temperature fluctuations is applied to rescale and generate the temperature profile at inlet. This new technique is used for various large eddy simulations of subsonic and supersonic three‐dimensional boundary layers of a flat plate. Simulation results for the time‐averaged mean flow and Reynolds stresses are compared with numerical and analytical data to demonstrate the high quality of the method.     

Curved exponential families of stochastic processes and their envelope families

Exponential families of stochastic processes are usually curved. The full exponential families generated by the finite sample exponential families are called the envelope families to emphasize that their interpretation as stochastic process models is not straightforward. A general result on how to calculate the envelope families is given, and the interpretation of these families as stochastic process models is considered. For Markov processes rather explicit answers are given. Three examples are considered some in detail: Gaussian autoregressions, the pure birth process and the Ornstein-Uhlenbeck process. Finally, a goodness-of-fit test for censored data is discussed.     

Contagion processes on urban bus networks in Indian cities

Bus transportation is the most convenient and cheapest way of public transportation in Indian cities. Due to cost‐effectiveness and wide reachability, buses bring people to their destinations every day. Although the bus transportation has numerous advantages over other ways of public transportation, this mode of transportation also poses a serious threat of spreading contagious diseases throughout the city. It is extremely difficult to predict the extent and spread of such an epidemic. Earlier studies have focused on the contagion processes on scale‐free network topologies; whereas, real‐world networks such as bus networks exhibit a wide‐spectrum of network topology. Therefore, we aim in this study to understand this complex dynamical process of epidemic outbreak and information diffusion on the bus networks for six different Indian cities using SI and SIR models. We identify epidemic thresholds for these networks which help us in controlling outbreaks by developing node‐based immunization techniques. © 2016 Wiley Periodicals, Inc. Complexity 21: 451–458, 2016     

The evolution of subcritical Achlioptas processes

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. AlthouSgh the evolution of such ‘local’ modifications of the Erd?s–Rényi random graph process has received considerable attention during the last decade, so far only rather simple rules are well understood. Indeed, the main focus has been on ‘bounded‐size’ rules, where all component sizes larger than some constant B are treated the same way, and for more complex rules very few rigorous results are known. In this paper we study Achlioptas processes given by (unbounded) size rules such as the sum and product rules. Using a variant of the neighbourhood exploration process and branching process arguments, we show that certain key statistics are tightly concentrated at least until the susceptibility (the expected size of the component containing a randomly chosen vertex) diverges. Our convergence result is most likely best possible for certain generalized Achlioptas processes: in the later evolution the number of vertices in small components may not be concentrated. Furthermore, we believe that for a large class of rules the critical time where the susceptibility ‘blows up’ coincides with the percolation threshold. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 174–203, 2015