On design of quantized fault detection filters with randomly occurring nonlinearities and mixed time-delays

This paper is concerned with the fault detection problem for a class of discrete-time systems with randomly occurring nonlinearities, mixed stochastic time-delays as well as measurement quantizations. The nonlinearities are assumed to occur in a random way. The mixed time-delays comprise both the multiple discrete time-delays and the infinite distributed delays that occur in a random way as well. A sequence of stochastic variables is introduced to govern the random occurrences of the nonlinearities, discrete time-delays and distributed time-delays, where all the stochastic variables are mutually independent but obey the Bernoulli distribution. The main purpose of this paper is to design a fault detection filter such that, in the presence of measurement quantization, the overall fault detection dynamics is exponentially stable in the mean square and, at the same time, the error between the residual signal and the fault signal is made as small as possible. Sufficient conditions are first established via intensive stochastic analysis for the existence of the desired fault detection filters, and then the explicit expression of the desired filter gains is derived by means of the feasibility of certain matrix inequalities. Also, the optimal performance index for the addressed fault detection problem can be obtained by solving an auxiliary convex optimization problem. A practical example is provided to show the usefulness and effectiveness of the proposed design method.     

Entropy and expected acceptance counts for finite automata

If a sequence of independent unbiased random bits is fed into a finite automaton, it is straightforward to calculate the expected number of acceptances among the first n prefixes of the sequence. This paper deals with the situation in which the random bits are neither independent nor unbiased, but are nearly so. We show that, under suitable assumptions concerning the automaton, if the difference between the entropy of the first n bits and n converges to a constant exponentially fast, then the change in the expected number of acceptances also converges to a constant exponentially fast. We illustrate this result with a variety of examples in which numbers following the reciprocal distribution, which governs the significands of floating-point numbers, are recoded in the execution of various multiplication algorithms.     

Teletraffic analysis and mobility modeling of PCS networks

Channel holding time is of primary importance in teletraffic analysis of PCS networks. This quantity depends on user's mobility which can be characterized by the cell residence time. We show that when the cell residence time is not exponentially distributed, the channel holding time is not exponentially distributed either, a fact also confirmed by available field data. In order to capture the essence of PCS network behaviour, including the characterization of channel holding time, a correct mobility model is therefore necessary. The new model must be good enough to fit field data, while at the same time resulting in a tractable queueing system. We propose a new mobility model, called the hyper-Erlang distribution model, which is consistent with these requirements. Under the new realistic operational assumption of this model, in which the cell residence time is generally distributed, we derive analytical results for the channel holding time distribution, which are readily applicable to the hyper-Erlang distribution models. Using the derived analytical results we demonstrate how the distribution of the cell residence time affects the channel holding time distribution. The results presented in this paper can provide guidelines for field data processing in PCS network design and performance evaluation     

The NBUT class of life distributions

A new class of life distributions, namely new better than used in the total time on test transform ordering (NBUT), is introduced. The relationship of this class to other classes of life distributions, and closure properties under some reliability operations, are discussed. We provide a simple argument based on stochastic orders that the family of the NBUT distribution class is closed under the formation of series systems in case of independent identically distributed components. Behavior of this class is developed in terms of the monotonicity of the residual life of k-out-of-n systems given the time at which the (n-k)-th failure has occurred. Finally, we discuss testing exponentially against the NBUT aging property.     

Failure Time for a Redundant Repairable System of Two Dissimilar Elements

The distribution of time to failure for a system consisting of two dissimilar elements or subsystems operating redundantly and susceptible to repair is discussed. It is assumed that the times to failure for the two system elements are independent random variables from possibly different exponential distributions, and that the repair times peculiar to each element are independently distributed in an arbitrary fashion. For this basic model a derivation is given of the Laplace-Stieltjes transform of the distribution function of time to system failure, i.e, the time until both elements are simultaneously down for repair, measured from an instant at which both are operating. An explicit formula is given for the mean or expected time to system failure, a natural approximation to the latter is exhibited, and numerical comparisons indicate the quality of this approximation for various repair time distributions. In a second model the possibility of system failures due to overloading the remaining element after a single element failure is explicitly recognized. The assumptions made for the basic model are augmented by a stochastic process describing the random occurrence of overloads. Numerical examples are given. Finally, it is shown how the above models may be easily modified to account for delays in initiating repairs resulting from only occasional system surveillance, and to account for random catastrophic failures.     

Sigma-delta modulation with i.i.d. Gaussian inputs

The response of a single-loop sigma-delta modulator to an independent identically distributed (i.i.d.) Gaussian input signal is analyzed. A continuous-time stochastic model is developed and the connection of the model to the system is described. A condition is given so that the difference between the behavior of the model and that of the true system can be made arbitrarily small. Theories from renewal and Wiener processes are applied to show the convergence and mixing properties of the output sequence. Also derived is the power spectrum of the quantization noise. Compared to the spectrum when the input is DC, the i.i.d. Gaussian random process smears the discrete spectrum into band structures     

Probabilistic Analysis of a 2-Unit Cold-Standby System with a Single Repair Facility

The reliability and the availability characteristics of a 2-unit cold standby system with a single repair facility are analyzed under the assumption that the failure and the repair times are both generally distributed. System breakdown occurs when the operating unit fails while the other unit is undergoing repair. The system is characterized by the probability of being up or down. Integral equations corresponding to different initial conditions are set up by identifying suitable regenerative stochastic processes. The probability of the first passage to the down-state starting from specified initial conditions is obtained by the same method. An explicit expression for a Laplace Transform of the probability density function (pdf) of the downtime during an arbitrary time interval is obtained when the repair time is exponentially distributed. A general method is suggested for the calculation of the moments of the downtime when the repair time is arbitrarily distributed.     

基于超几何分解的随机运算系统分析方法

基于Bernoulli分布的方差及期望传递方程一直是随机运算系统的数学基础,针对这种传统分析方法在实际应用中的不准确性和片面性,该文提出一种全新的数学方法:超几何分解(hypergeometic decomposition),用来解决在更复杂情况下期望与方差在随机运算系统中的传播规律.基于超几何分解,提出4组更加精确的期望及方差传递方程,在数学上证明了随机运算体系更加广泛的适用性,并且通过随机运算系统在图像处理中的应用,提出了基于方差的系统评价方法,相比于传统按位仿真方法,基于方差的系统分析方法具有耗时短、准确和全面的优点.新的方差传递方程首次将随机信号源的类型引入性能分析,证明了具有特定码流长度的随机序列可以使系统性能达到最优.     

Equivalence of Redundant Systems With Respect to Time to Failure

The time to failure of an m out of n component system, with component failure times exponentially and identically distributed, is treated. This failure time is shown to be identically distributed to the failure time of an n - m + 1 component standby system, with component failure times exponentially but not identically distributed. The proof of this result is based on a recursion formula for the distribution of time to system failure.     

A maintenance inspection model for a single machine with general failure distribution

In this paper we develop a mathematical model for determining a periodic inspection schedule in a preventive maintenance program for a single machine subject to random failure. We formulate the problem as a profit maximization model with general failure time distribution. We show that under certain conditions on the probability density function of failure, a unique optimal inspection interval can be obtained. When the failure times are exponentially distributed, we propose alternative optimal and heuristic procedures to find exact and approximate inspection intervals. Our heuristic solution method is shown numerically to be more efficient than an earlier published heuristic procedure. We also investigated the sensitivity of the optimal inspection interval and expected profit per unit of time with respect to the changes in the two parameters of the Weibull time to failure distribution.     

Power-hierarchy of dependability-model types

This paper formally establishes a hierarchy, among the most commonly used types of dependability models, according to their modeling power. Among the combinatorial (non-state-space) model types, we show that fault trees with repeated events are the most powerful in terms of kinds of dependencies among various system components that can be modeled. Reliability graphs are less powerful than fault trees with repeated events but more powerful than reliability block diagrams and fault trees without repeated events. By virtue of the constructive nature of our proofs, we provide algorithms for converting from one model type to another. Among the Markov (state-space) model types, we consider continuous-time Markov chains, generalized stochastic Petri nets, Markov reward models, and stochastic reward nets. These are more powerful than combinatorial-model types in that they can capture dependencies such as a shared repair facility between system components. However, they are analytically tractable only under certain distributional assumptions such as exponential failure- and repair-time distributions. They are also subject to an exponentially large state space. The equivalence among various Markov-model types is briefly discussed     

Analytical Model for Connectivity in Vehicular Ad Hoc Networks

We investigate connectivity in the ad hoc network formed between vehicles that move on a typical highway. We use a common model in vehicular traffic theory in which a fixed point on the highway sees cars passing it that are separated by times with an exponentially distributed duration. We obtain the distribution of the distances between the cars, which allows us to use techniques from queuing theory to study connectivity. We obtain the Laplace transform of the probability distribution of the connectivity distance, explicit expressions for the expected connectivity distance, and the probability distribution and expectation of the number of cars in a platoon. Then, we conduct extensive simulation studies to evaluate the obtained results. The analytical model that we present is able to describe the effects of various system parameters, including road traffic parameters (i.e., speed distribution and traffic flow) and the transmission range of vehicles, on the connectivity. To more precisely study the effect of speed on connectivity, we provide bounds obtained using stochastic ordering techniques. Our approach is based on the work of Miorandi and Altman, which transformed the problem of connectivity distance distribution into that of the distribution of the busy period of an equivalent infinite server queue. We use our analytical results, along with common road traffic statistical data, to understand connectivity in vehicular ad hoc networks.      

Optimal operation of a markovian queueing system with a removable and non-reliable server

This paper deals with the optimal operation of a single removable and non-reliable server in a Markovian queueing system under steady-state conditions. The server can be turned on at arrival epochs or off at departure epochs. We assume that the server may break down only if working and requires repair at a repair facility. Interarrival and service time distributions of the customers are assumed to be exponentially distributed. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. The following cost structure is incurred to the system: a holding cost for each customer in the system per unit time, costs per unit time for keeping the server on or off, a breakdown cost per unit time when a server fails, and fixed costs for turning the server on or off. The total expected cost function per unit time is developed to obtain the optimal operating policy at minimum cost.     

Stochastic analysis of systems with primary and secondary failures

This paper presents the stochastic analysis of repairable systems involving primary as well as secondary failures. To this end, two models are considered. The first model represents a system with two identical warm standbys. The failure rates of units and the system are constant and independent while the repair times are arbitrarily distributed. The second system modeled consists of three repairable regions. The system operates normally if all three regions are operating, otherwise it operates at a derated level unless all three regions fail. The failure rates and repair times of the regions are constant and independent. The first model is analyzed using the supplemental variable technique while the second model is analyzed using the regenerative point technique in the Markov renewal process. Various expressions including system availability, system reliability and mean time to system failure are obtained.     

Robust filtering for a class of stochastic uncertain nonlineartime-delay systems via exponential state estimation

We investigate the robust filter design problem for a class of nonlinear time-delay stochastic systems. The system under study involves stochastics, unknown state time-delay, parameter uncertainties, and unknown nonlinear disturbances, which are all often encountered in practice and the sources of instability. The aim of this problem is to design a linear, delayless, uncertainty-independent state estimator such that for all admissible uncertainties as well as nonlinear disturbances, the dynamics of the estimation error is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are proposed to guarantee the existence of desired robust exponential filters, which are derived in terms of the solutions to algebraic Riccati inequalities. The developed theory is illustrated by numerical simulation     

Space-Time Models of Asynchronous CSMA Protocols for Local Area Networks

Carrier sense multiple-access (CSMA) protocols are widely used in local area networks (LAN's) to control access to a shared communications channel such as a coaxial cable or radio frequency band. CSMA protocols are designed to exploit the property that the signal propagation time across the LAN is much smaller than the packet transmission time. Consequently, their performance depends on the exact timing of a sequence of asynchronous events originating at different points in the network. Previous models of CSMA protocols have not captured this essential space-time characteristic. Instead, the system has been reduced to a 1-dimensional model (i.e., a sequence of events on a single time line) by assuming either that events are synchronous (i.e., "slotted" operation of the protocol) or that all stations are mutually equidistant (so it can be assumed that each station's time line is identical). The novelty in our work is to describe the system state in terms of an (N + 1)-dimensional "ribbon" of space-time, allowing us to faithfully model the exact timings of events for LAN's where the stations are distributed over a fully connected region inN-space. For example, an Ethernet-like "bus" network can be viewed as a 1-dimensional LAN, a terrestrial packet radio network can be viewed as a 2dimensional LAN, and so on. First we highlight some of the key properties exhibited by such a model in the general case, including the notion of an embedded Markovian sequence of "idle points," and the decomposition of the space-time ribbon into cycles (each of which can be further decomposed into a number of "regions"). Then we present extensive results for the interesting special case of a 1-dimensional LAN, i.e., an Ethernet-like "bus" network. Our results show that previous models of bus LAN's can significantly underestimate their performance.     

Computation of Mean Time to Failure for a System with Simple Redundancy

A system consisting of several independent components in series can be made more reliable by adding an "identical" component in parallel to some of the components. Two mathematical series are presented for computing the mean time to failure for the system when the failure times of the components are exponentially distributed. For large systems the evaluation of these series is not computationally feasible, but one of the series yields an accurate approximation if the series is truncated, and also yields an explicit solution if all components have identical failure rates. The explicit solution is then used to check the accuracy of solutions obtained using numerical integration. The method of numerical integration is highly accurate.     

Energy‐constrained mobile sensor with motion plans for monitoring stochastic events

With the development of robotics and embedded system, utilizing mobile sensors to capture stochastic events is emerging as a promising method to monitor a region of interest (RoI). In previous work, the quality of monitoring (QoM) is evaluated based on the stochastic events capture without taking the energy of motion into consideration. Since sensor nodes are normally constrained by limited energy capability, it is desirable to guide the mobile sensor in an energy‐efficient motion to capture events information. In this paper, by analyzing the different kinds of surveillance that may result in different required QoM, we obtain the expected Information captured Per unit of Energy consumption (IPE), which is a function with multiple parameters including the event type, the event dynamics, and the velocity of the mobile sensor. Our analysis is based on a realistic energy model of motion, and can achieve suboptimal motion plans by adopting existing typical approximation algorithm, and thus enable the sensor velocity to be optimized for capturing stochastic events information. We propose approximation algorithms to enable the tradeoff between the computation and efficiency, which make motion plans more practical in some realistic scenarios. The efficiency and effectiveness of proposed algorithm are validated by the extensive simulations. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiple-antenna capacity in correlated Rayleigh fading with channel covariance information

We analyze a mobile multiple input multiple output wireless link with M transmit and N receive antennas operating in a spatially correlated Rayleigh flat fading environment. Only the correlations between the channel coefficients are assumed to be known at the transmitter and the receiver. The channel coefficients are correlated in space and uncorrelated in time from one coherence interval to another. These coefficients remain constant for a coherence interval of T symbol periods after which they change to another independent realization according to the spatial correlation model. For this system we characterize the structure of the input signal that achieves capacity. The capacity achieving transmit signal is expressed as the product of an isotropically distributed unitary matrix, an independent nonnegative diagonal matrix and a unitary matrix whose columns are the eigenvectors of the transmit fade covariance matrix. For the case where the number of transmit antennas M is larger than the channel coherence interval T, we show that the channel capacity is independent of the smallest M-T eigenvalues of the transmit fade covariance matrix. In contrast to the previously reported results for the spatially white fading model where adding more transmit antennas beyond the coherence interval length (M>T) does not increase capacity, we find that additional transmit antennas always increase capacity as long as their channel fading coefficients are spatially correlated with the other antennas. We show that for fast hopping or fast fading systems (T=1) with only channel covariance information available to the transmitter and receiver, transmit fade correlations are beneficial. Mathematically, we prove this by showing that capacity is a Schur-convex function of the vector of eigenvalues of the transmit fade correlation matrix. We also show that the maximum possible capacity gain due to transmitter fade correlations is 10logM dB.