Renewal approximations of the doubly stochastic Poisson processes

In this paper, we analyze two approaches to approximate a doubly stochastic Poisson (DSP) process by a renewal process. A DSP process consists of Poisson processes whose rates alternate between two levels. Each rate remains for renewal times forming an alternating renewal process. Such processes can be used to model situations in reliability, inventory, queueing and production systems. We develop two expressions for the Laplace-Stieltjes transforms of the interrenewal time distributions of the renewal processes that approximate a DSP process. We then use these approximations to develop expressions for the stationary loss probability in a G/M/1/0 system. We evaluate the quality of these approximations by comparing them against exact results. Our approximations should be of significant use in several practical applications.     

New cumulative damage models for failure using stochastic processes as initial damage

Based on a generalized cumulative damage approach with a stochastic process describing initial damage for a material specimen, a broad class of statistical models for material strength is developed. Plausible choices of stochastic processes for the initial damage include Brownian motion, geometric Brownian motion, and the gamma process; and additive & multiplicative cumulative damage functions are considered. The resulting general statistical model gives an accelerated test form of the inverse Gaussian distribution, special cases of which include some existing models in addition to several new models. Model parameterizations & estimation by maximum likelihood from accelerated test data are discussed, and the applicability of the general model is illustrated for three sets of strength data. The proposed models are compared with the power-law Weibull model, and the inverse Gaussian generalized linear models.     

Pseudodynamic cost limit replacement model under minimal repair

This paper presents a mathematical model to evaluate pseudodynamic cost limit replacement policies for a system that follows a general time-to-failure distribution. When the failed system requires repair, it is first inspected and the repair cost is estimated. Minimal repair is only then undertaken if the estimated cost is less than the exponentially declining repair cost limit. A negative exponential distribution of estimated repair cost is assumed for analytic tractability. An example with a Weibull time-to-failure distribution is given to illustrate the computational results.     

Transient processes under stochastic synchronization of oscillators in colored-noise medium

Equations that describe mutual and induced chaotic synchronization of n (n = 2, 3, 4,...) autooscillation systems in colored-noise medium are derived. It is demonstrated that external colored noise acting upon a system of coupled oscillators and a chain of unidirectionally coupled oscillators causes a decrease in the duration of transient processes that lead to the chaotic synchronization (both mutual and induced) of nonidentical oscillators and a decrease in the time needed for stabilization of the symmetric state of the system when the parameters of oscillators are identical.     

Detection of stochastic processes

This paper reviews two streams of development, from the 1940's to the present, in signal detection theory: the structure of the likelihood ratio for detecting signals in noise and the role of dynamic optimization in detection problems involving either very large signal sets or the joint optimization of observation time and performance. This treatment deals exclusively with basic results developed for the situation in which the observations are modeled as continuous-time stochastic processes. The mathematics and intuition behind such developments as the matched filter, the RAKE receiver, the estimator-correlator, maximum-likelihood sequence detectors, multiuser detectors, sequential probability ratio tests, and cumulative-sum quickest detectors, are described     

Outage probability of the free-space optical channel with doubly stochastic scintillation

We study the asymptotic outage probability of multiple-input multiple-output free-space optical communication with pulse-position modulation. In particular, we consider doubly stochastic scintillation models, lognormal-Rice and I-K distributions. First we consider the case when channel state information is available at the receiver only. Then we consider the case when it is also available at the transmitter.     

Moment equations for stochastic distributed-parameter processes

Moment equations are derived for a general class of distributed-parameter dynamic processes corrupted by white in time distributed disturbances. An integrodifferential equation for the expected value of an arbitrary functional of the process is first derived simply by using the whiteness property of the stochastic disturbance. In this way there is no need to utilise the forward Kolmogorov's equation, which at present is not available for distributed-parameter processes. The moment equations are then derived as special cases of this functional integrodifferential equation.     

On computing the repair cost of a maintained reliability system with known repair cost function and probability of repair

In this paper, a simplified analytic cost model for maintained reliability system under opportunistic repair scheme is discussed. Life cycle cost curves under various operating life cycle times and linear repair cost function is derived.     

Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_n; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence{r(n)}, wherer(n) = E[x(l_{m+n}) overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(t). Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if{t_n}is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (fort leq t_oort geq t_o) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known{ r(n) }.     

Information, error, and imaging in deadtime-perturbed doubly stochastic Poisson counting systems

The detection of a fluctuating signal in the presence of noise is considered for a doubly stochastic Poisson counting system that is subject to fixed nonparalyzable detector deadtime. Explicit expressions are obtained for the likelihood-ratio detection of a modulated source of arbitrary statistics in the presence of Poisson noise counts. Receiver operating characteristics (ROC curves) are presented for an unmodulated (amplitude-stabilized) source with detector dead-time as a parameter; increasing deadtime causes a decrease in the probability of detection for a fixed false-alarm rate. Probability of error curves are presented for an amplitude-stabilized source, both in the absence of modulation and in the presence of triangular modulation, illustrating the deleterious effects of modulation, noise, and deadtime on receiver performance. Expressions for the average mutual information and channel capacity of the system are obtained and graphically presented for the simple counting receiver and for the maximum-likelihood counting receiver; the channel capacity decreases with decreasing signal level and with increasing deadtime and modulation depth. Representative examples of the appropriate counting distributions are provided. Finally, a maximum-likelihood estimate of the mean signal level is obtained for a simple image detection system with a deadtime-perturbed counting array. An expression for the statistical confidence level of the estimate is also obtained. The results are valid for an arbitrary deadtime-perturbed doubly stochastic Poisson counting system and as such are expected to find application in a broad variety of disciplines including photon counting and lightwave communications, operations research, nuclear particle counting, and neural counting and psychophysics.     

A theory of polyspectra for nonstationary stochastic processes

Harmonizable processes constitute an important class of nonstationary stochastic processes. We present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic quantities: the nth-order moment function, the nth-order time-frequency polyspectrum, the nth-order ambiguity function, and the nth-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous nth-order moment and the conventional nth-order stationary polyspectrum, respectively. All quantities except the nth-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order n, which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels.     

Wavelet representations of stochastic processes and multiresolutionstochastic models

Deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years. In the context of stochastic processes, the use of wavelet bases has not yet been fully investigated. We use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in L² defined in the time domain. We derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales. We study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by Basseville et al. (see IEEE Trans. Inform. Theory, vol.38, p.766-784, Mar. 1992). We propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process. These models are simple due to the strong decorrelation of the wavelet transform. Experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated     

Uncertainty in measurements on spectrally correlated stochastic processes

In this paper, the class of the spectrally correlated stochastic processes is introduced. Processes belonging to this class exhibit a Loe/spl grave/ve (1963) bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. Thus, such processes have spectral components that are correlated. The introduced class generalizes the almost-cyclostationary (ACS) processes that are obtained as a special case when the separation between correlated spectral components assumes values only in a countable set. In such a case, the support curves are lines with unit slope. For the spectrally correlated processes, the amount of spectral correlation existing between two separate spectral components is characterized by the bifrequency spectral correlation density function, which is the density of the Loe/spl grave/ve bifrequency spectrum on its support curves. It is shown that, in general, when the location of the support curves is unknown, the time-smoothed cross-periodogram can provide a reliable (low bias and variance) single sample-path-based estimate of the bifrequency spectral correlation density function in those points of the bifrequency plane where the slope of the support curves is not too far from unity. Moreover, there exists a tradeoff between the departure of the nonstationarity from the almost-cyclostationarity and the reliability of spectral correlation measurements obtainable by a single sample-path. Furthermore, in general, the estimate accuracy cannot be improved as wished by increasing the data-record length and the spectral resolution.     

Efficient entropy estimation based on doubly stochastic models for quantized wavelet image data.

Under a rate constraint, wavelet-based image coding involves strategic discarding of information such that the remaining data can be described with a given amount of rate. In a practical coding system, this task requires knowledge of the relationship between quantization step size and compressed rate for each group of wavelet coefficients, the R-Q curve. A common approach to this problem is to fit each subband with a scalar probability distribution and compute entropy estimates based on the model. This approach is not effective at rates below 1.0 bits-per-pixel because the distributions of quantized data do not reflect the dependencies in coefficient magnitudes. These dependencies can be addressed with doubly stochastic models, which have been previously proposed to characterize more localized behavior, though there are tradeoffs between storage, computation time, and accuracy. Using a doubly stochastic generalized Gaussian model, it is demonstrated that the relationship between step size and rate is accurately described by a low degree polynomial in the logarithm of the step size. Based on this observation, an entropy estimation scheme is presented which offers an excellent tradeoff between speed and accuracy; after a simple data-gathering step, estimates are computed instantaneously by evaluating a single polynomial for each group of wavelet coefficients quantized with the same step size. These estimates are on average within 3% of a desired target rate for several of state-of-the-art coders.     

An age replacement policy with minimal repair and general random repair cost

An age replacement policy is introduced which incorporates minimal repair, replacement, and general random repair costs. If an operating unit fails at age y<T, it is either replaced by a new unit with probability p(y) at a cost c₀, or it undergoes minimal repair with probability q(y) = 1−p(y). Otherwise, a unit is replaced when it fails for the first time after age T. The cost of the i-th minimal repair of an unit at age y depends on the random part C(y) and the deterministic part c_i(y). The aim of the paper is to find the optimal T which minimizes the long run expected cost per unit time of the policy. Various special cases are considered.     

Optimum repair limit policies with a cost constraint

In this paper, we discuss the optimum repair limit policies with a cost constraint for continuous and discrete distributions, respectively. We apply the expected total discounted cost with a discount rate as a criterion, and obtain the optimum policies which minimize it. We show that, under certain conditions, there exist finite and unique optimum policies for continuous and discrete distributions, respectively.     

An age replacement policy with increasing minimal repair cost

A replacement policy for a system in which minimal repair cost increases in system age is considered. If a system fails before age T, it is minimally repaired. Otherwise, the system is replaced when if fails for the first time after age T. The mean cost rate is used as a criterion for optimization. It is shown that the optimal T minimizing the mean cost rate is finite and unique.