Finite Observation Period Research Articles

Optimal inspection for randomly triggered hidden deterioration processes

AbstractThis paper deals with degradation processes whose onset is triggered at a random time and which stay hidden until they are discovered through inspection or when they begin to show symptoms. This is applicable in many healthcare and industrial scenarios, for example, in the modeling of breast cancer or termite infestation. In our model, we assume that symptoms appear after hitting a random critical threshold and that inspections may have a sensitivity less than one as well as a nonzero false positive rate. The expected cost of repair is derived, and the inspection rate is optimized for a cycle (which lasts from degradation‐free to repaired state). This gives results for three cases: the first is for a finite observation period with no degradation recurrence, the second for infinite time horizon allowing recurrence. In the third case, we derive an upper bound for the expected cost in a given constant time period. Finally, the model is applied to determine the optimal strategy for breast cancer screening with regard to the effects of different parametrizations.

UNCOVERING CIRCUMBINARY PLANETARY ARCHITECTURAL PROPERTIES FROM SELECTION BIASES

ABSTRACT Studying newly discovered circumbinary planetary systems improves our understanding of planetary system formation. Learning the architectural properties of these systems is essential for constraining the different formation mechanisms. We first revisit the stability limit of circumbinary planets. Next, we focus on eclipsing stellar binaries and obtain an analytical expression for the transit probability in a realistic setting, where a finite observation period and planetary orbital precession are included. Our understanding of the architectural properties of the currently observed transiting systems is then refined, based on Bayesian analysis and a series of tested hypotheses. We find that (1) it is not a selection bias that the innermost planets reside near the stability limit for eight of the nine observed systems, and this pile-up is consistent with a log uniform distribution of the planetary semimajor axis; (2) it is not a selection bias that the planetary and stellar orbits are nearly coplanar (≲3°), and this—along with previous studies—may imply an occurrence rate of circumbinary planets similar to that of single star systems; (3) the dominance of observed circumbinary systems with only one transiting planet may be caused by selection effects; (4) formation mechanisms involving Lidov–Kozai oscillations, which may produce misalignment and large separation between planets and stellar binaries, are consistent with the lack of transiting circumbinary planets around short-period stellar binaries, in agreement with previous studies. As a consequence of (4), eclipse timing variations may better suit the detection of planets in such configurations.

Statistical Inference for Unified Garch–Itô Models with High‐Frequency Financial Data

The existing estimation methods for the model parameters of the unified GARCH–Itô model (Kim and Wang, ) require long period observations to obtain the consistency. However, in practice, it is hard to believe that the structure of a stock price is stable during such a long period. In this article, we introduce an estimation method for the model parameters based on the high‐frequency financial data with a finite observation period. In particular, we establish a quasi‐likelihood function for daily integrated volatilities, and realized volatility estimators are adopted to estimate the integrated volatilities. The model parameters are estimated by maximizing the quasi‐likelihood function. We establish asymptotic theories for the proposed estimator. A simulation study is conducted to check the finite sample performance of the proposed estimator. We apply the proposed estimation approach to the Bank of America stock price data.

Estimating interevent time distributions from finite observation periods in communication networks.

A diverse variety of processes-including recurrent disease episodes, neuron firing, and communication patterns among humans-can be described using interevent time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this paper, we illustrate how this length bias is born and how it can be corrected without assuming any particular shape for the IET distribution. To do this, we model event sequences using stationary renewal processes, and we formulate simple heuristics for determining the severity of the bias. To illustrate our results, we focus on the example of empirical communication networks, which are temporal networks that are constructed from communication events. The IET distributions of such systems guide efforts to build models of human behavior, and the variance of IETs is very important for estimating the spreading rate of information in networks of temporal interactions. We analyze several well-known data sets from the literature, and we find that the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions.

Statistical analysis and modeling of Skype VoIP flows

The traffic characterization of adaptive real-time applications provides a challenging engineering task of the current Internet. Here the arrival and packet length processes arising from a unidirectional VoIP flow of Skype packets are investigated by statistical means. The mean traffic load offered by the packet flow to a virtual link of limited capacity in a finite observation period is calculated and the mean and upper bounds for the time without loss are derived. A possible characterization of the packet arrival process by a properly chosen non-linear GARCH (1, 1) model is presented.

The effect of transient exposures on the risk of an acute illness with low hazard rate.

There are many situations where intermittent short-term exposures of a certain kind are thought to temporarily enhance the risk of onset of an adverse health event (illness). When the hazard rate of the illness is small it is desirable to investigate this possible association using only data on cases occurring in a finite observation period. Here we extend a method for such an analysis by allowing the baseline hazard for the illness to depend on the increasing age over the observation period and using age at the times of exposure, a time dependent variable, as a covariate in the effect of the transient exposure. The method is illustrated with a study of the possible association of long-haul air travel and hospitalization for venous thromboembolism over an observation period of 19 years. It is demonstrated that allowing for aging over the observation period can avoid bias in the estimated effect size when the baseline hazard for the illness increases with age and exposures occur irregularly over time.

Adaptive scheduling in random flexible manufacturing systems subject to machine breakdowns

In a FMS dynamic scheduling environment, frequent rescheduling to react to disruptions such as machine breakdowns can make the behaviour of the system hard to predict, and hence reduce the effectiveness of dynamic scheduling. Another approach to handle the disruptions is to update the job ready time and completion time, and machine status on a rolling horizon basis, and consider the machine availability explicitly in generating schedules. When machine downtime has a small variation, the operation completion time is estimated by using limiting (steady-state) machine availability. However, steady-state analysis is sometimes unlikely to provide a complete picture of the system when there is a large variation in machine downtime and repair time, and frequent disruptions (e.g. tool failures) exist. Transient analysis of machine availability will be more meaningful in such a situation during a finite observation period. In this paper, an adaptive scheduling approach is proposed to make coupled decisions about part/machine scheduling and operation/tool assignments on a rolling horizon basis, while the operation completion time is estimated by a transient machine availability analysis based on a two-state continuous time Markov process. The expected tool waiting time is explicitly considered in the job machine scheduling decision. The effectiveness of the proposed method is compared with other approaches based on various dispatching heuristics such as Apparent Tardiness Cost, Cost OVER Time, and Bottleneck Dynamics, etc under different shop load and machine downtime levels.

Point and expected interval availability analysis with stationarity detection

Interval availability is a dependability measure defined by the fraction of time during which a system is in operation over a finite observation period. The system is assumed to be modeled by a finite Markov process. Because the computation of the distribution of this random variable is very expensive, it is common to compute only its expectation. In this note, we propose a new algorithm to compute the expected interval availability and we compare it with respect to the standard uniformization technique from an execution time point of view. This new method is particularly interesting if the Markov chain is stiff. Moreover, a new algorithm for the stationarity detection is proposed in order to avoid excessive computation. Scope and purpose The advent of fault-tolerant computing systems has led to increased interest in analytic techniques for prediction of dependability measures such as the availability over a given period. Interval availability is defined by the fraction of time during which a system is in operation over a finite observation period. By modeling the system behaviour by a Markov process, we transform the problem of evaluating dependability cumulative measures into the computation of the cumulative measures on a Markov process. In this note we are interested in the expected interval availability. Generally, we are faced with the problem of the execution time, especially when the Markov model is stiff, i.e., when we have a highly available system. This note proposes a new technique which deals efficiently with such a class of processes.

Availability analysis of repairable computer systems and stationarity detection

Point availability and expected interval availability are dependability measures respectively defined by the probability that a system is in operation at a given instant and by the mean percentage of time during which a system is in operation over a finite observation period. We consider a repairable computer system and we assume, as usual, that the system is modeled by a finite Markov process. We propose in this paper a new algorithm to compute these two availability measures. This algorithm is based on the classical uniformization technique in which a test to detect the stationary behavior of the system is used to stop the computation if the stationarity is reached. In that case, the algorithm gives not only the transient availability measures, but also the steady state availability, with significant computational savings, especially when the time at which measures are needed is large. In the case where the stationarity is not reached, the algorithm provides the transient availability measures and bounds for the steady state availability. It is also shown how the new algorithm can be extended to the computation of performability measures.

Calculating transient distributions of cumulative reward

Markov reward models have been employed to obtain performability measures of computer and communication systems. In these models, a continuous time Markov chain is used to represent changes in the system structure, usually caused by faults and repairs of its components, and reward rates are assigned to states of the model to indicate some measure of accomplishment at each structure. A procedure to calculate numerically the distribution of the reward accumulated over a finite observation period is presented. The development is based solely on probabilistic arguments, and the final recursion is quite simple. The algorithm has a low computational cost in terms of model parameters. In fact, the number of operations is linear in a parameter that is smaller than the number of rewards, while the storage required is independent of the number of rewards. We also consider the calculation of the distribution of cumulative reward for models in which impulse based rewards are associated with transitions.

Interval availability analysis using denumerable Markov processes: application to multiprocessor subject to breakdowns and repair

Interval availability is a dependability measure defined by the fraction of time during which a system is operational over a finite observation period. The computation of its distribution allows the user to ensure that the probability that its system will achieve a given availability level is high enough. The system is assumed to be modeled as a Markov process with countable state space. We propose a new algorithm to compute the interval availability distribution. One of its main advantages is that, in some cases, it applies even to infinite state spaces. This is useful, for instance, in case of models taking into account contention with unbounded buffers. This important feature is illustrated on models of multiprocessor systems, subject to breakdowns and repair. When the model is finite, we show through a numerical example that the new technique can perform very well.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Interval-availability distribution of 2-state systems with exponential failures and phase-type repairs

Interval availability is a dependability measure defined as the fraction of time during which a system is in operation over a finite observation period. Usually, for computing systems, the models used to evaluate interval availability distribution are Markov models. Numerous papers using these models have been published, and only complex numerical methods have been proposed as solutions to this problem even in simple cases such as the 2-state Markov model. This paper proposes a new way to compute this distribution when the model is a 2-state semi-Markov process in which the holding times have an exponential distribution for the operational state and a phase-type distribution for the nonoperational one. The main contribution of this paper is to define a new algorithm to compute the interval availability distribution for systems having only one operational state. The computational complexity depends weakly on the number of states of the system, and sometimes it can deal also with infinite state spaces. Moreover, simple closed expressions of this distribution are shown when repair periods are of the Erlang type with eventually absorbing states. >

Interval availability analysis using operational periods

Consider a repairable computer system alternating periods during which it delivers useful service ( operational periods) with periods in which it is being repared ( unoperational periods). Interval availability is a dependability measure of such a system defined by the fraction of time during which a system is in operation over a finite observation period. In general, to evaluate the interval availability distribution of repairable computer systems analysts use Markov models. In this paper, we first deal with semi-Markov processes and under some conditions we develop a method to compute this distribution. It is based on the analysis of the length of the successive operational and unoperational periods. Particular attention is then devoted to the Markov case leading to a specific algorithm. As a byproduct, we obtain the distribution of the cumulative operational time up to the nth operational period.

Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period

Markov process are widely used to model computer systems. De Souza e Silva and Gail [3] calculated numerically the distribution of the cumulative operational time of repairable computer systems modelled by Markovian processes, that is, the distribution of the total time during which the system was in operation over a finite observation period. An extension of their approach is presented here. A closed-form solution is obtained for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period, which is theoretically and numerically interesting. We also give an application of this result to a fault-tolerant system.

Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period

Markov process are widely used to model computer systems. De Souza e Silva and Gail [3] calculated numerically the distribution of the cumulative operational time of repairable computer systems modelled by Markovian processes, that is, the distribution of the total time during which the system was in operation over a finite observation period. An extension of their approach is presented here. A closed-form solution is obtained for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period, which is theoretically and numerically interesting. We also give an application of this result to a fault-tolerant system.

Calculating availability and performability measures of repairable computer systems using randomization

Repairable computer systems are considered, the availability behavior of which can be modeled as a homogeneous Markov process. The randomization method is used to calculate various measures over a finite observation period related to availability modeling of these systems. These measures include the distribution of the number of events of a certain type, the distribution of the length of time in a set of states, and the probability of a near-coincident fault. The method is then extended to calculate performability distributions. The method relies on coloring subintervals of the finite observation period based on the particular application, and then calculating the measure of interest using these colored intervals.

Effects of Noise, Time-Domain Damping, Zero-Filling and the FFT Algorithm on the “Exact” Interpolation of Fast Fourier Transform Spectra

A frequency-domain Lorentzian spectrum can be derived from the Fourier transform of a time-domain exponentially damped sinusoid of infinite duration. Remarkably, it has been shown that even when such a noiseless time-domain signal is truncated to zero amplitude after a finite observation period, one can determine the correct frequency of its corresponding magnitude-mode spectral peak maximum by fitting as few as three spectral data points to a magnitude-mode Lorentzian spectrum. In this paper, we show how the accuracy of such a procedure depends upon the ratio of time-domain acquisition period to exponential damping time constant, number of time-domain data points, computer word length, and number of time-domain zero-fillings. In particular, we show that extended zero-filling (e.g., a “zoom” transform) actually reduces the accuracy with which the spectral peak position can be determined. We also examine the effects of frequency-domain random noise and roundoff errors in the fast Fourier transformation (FFT) of time-domain data of limited discrete data word length (e.g., 20 bit/word at single and double precision). Our main conclusions are: (1) even in the presence of noise, a three-point fit of a magnitude-mode spectrum to a magnitude-mode Lorentzian line shape can offer an accurate estimate of peak position in Fourier transform spectroscopy; (2) the results can be more accurate (by a factor of up to 10) when the FFT processor operates with floating-point (preferably double-precision) rather than fixed-point arithmetic; and (3) FFT roundoff errors can be made negligible by use of sufficiently large (&gt; 16 K) data sets.

Estimating the sojourn time sensitivity in queueing networks using perturbation analysis

The sample path perturbation analysis technique developed earlier for the analysis of throughput sensitivities (Refs. 1–3) is extended to the performance measures involving mean sojourn times of customers. The major features of the sojourn time sensitivity problem are twofold. Firstly, it is a performance associated with servers, and not with customers. Secondly, the average sojourn time in any finite observation period can be a discontinuous function of mean service times when blocking is involved in a system. This discontinuity causes errors which must be accounted for in the estimation of sensitivities. Numerical experiments and analysis validate this method of computation of the sensitivities.

Calculating Cumulative Operational Time Distributions of Repairable Computer Systems

We consider repairable computer systems, those for which repair can be performed to put the system back in operation. The behavior of the system is assumed to be modeled as a homogeneous Markov process. We calculate numerically the distribution of cumulative operational time, which is the distribution of the total time during which the system was in operation over a finite observation period. The method is based on the randomization technique. The main advantages include the ability to specify error tolerances in advance, numerical stability, and simplicity of implementation. We also show that other quantities of interest can be calculated as a byproduct of the method without any significant extra computational effort.