Automatic construction of Markov transition matrices under state and functional dependences

In a previous article, a new method allowing the treatment of large Markovian problems was presented. Based on a graph describing the influences between the components of the system, it performs successive approximate aggregations on the exact Markovian system to reduce its size. The main drawback of this method, as of any approximate method, is to assess its validity. That is why we develop a new presentation of the method here and we define, from this presentation, error bounds for the approximate results. They are then tested for two applications, one being very large with more than 10³⁰ states for the exact Markovian system. We also extend the method, initially defined for availability problems, to reliability problems.     

Control charts for traffic intensity monitoring of Markovian multiserver queues

A number of recent research studies have applied queueing theory as an approximate modeling tool to mathematically describe industrial systems, which include manufacturing, distribution, and service, for instance. Among the main observable characteristics in queues, the number of users in the system can be controlled to keep waiting times as minimal as possible. The design of efficient control charts is an attempt to monitor and control such systems. Control charts are proposed to monitor infinite queues with Markovian arrivals, exponential service times, and s identical parallel servers. The proposed charts monitor traffic intensities, which are the ratio between the arrival rate and the service rate, estimated through the number of users in the queueing system at random epochs. The effectiveness and efficiency of the proposed approaches in terms of the average run lengths are established by a comprehensive set of Monte Carlo simulations.     

Time integrators based on approximate discontinuous Hamiltonians

We introduce a class of time integration algorithms for finite dimensional mechanical systems whose Hamiltonians are separable. By partitioning the system's configuration space to construct an approximate potential energy, we define an approximate discontinuous Hamiltonian (ADH) whose resulting equations of motion can be solved exactly. The resulting integrators are symplectic and precisely conserve the approximate energy, which by design is always close to the exact one. We then propose two ADH algorithms for finite element discretizations of nonlinear elastic bodies. These result in two classes of explicit asynchronous time integrators that are scalable and, because they conserve the approximate Hamiltonian, could be considered to be unconditionally stable in some circumstances. In addition, these integrators can naturally incorporate frictionless contact conditions. We discuss the momentum conservation properties of the resulting methods and demonstrate their performance with several problems, such as rotating bodies and multiple collisions of bodies with rigid boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Semi-Markovian reliability models for systems with testable components and general test/outage times

Semi-Markov models for systems undergoing periodic test and maintenance are developed. In particular, systems undergoing specific changes of state at predetermined instances of time and transiting to states with generally distributed sojourn times are considered. Problems addressed by the models are those concerned with optimum assessment of test intervals, and allowable outage times. Equivalent Markovian models allowing for the decomposition of a system of dimensionality N+M into two smaller problems of dimensionality N and M, respectively are developed. The general model is also specialized to systems with instantaneously testable components, unmonitored components undergoing tests (repair, maintenance) of fixed duration, and systems containing components characterized by limited allowable outage time (under test, or repair). Approximate equivalent Markov models are derived in these cases. Simple numerical examples are also presented.     

Method of Estimating the State of Complex Technical Objects in Order to Synthesize High-Speed Predicting Systems

An original two-stage method for forming an operational estimate of the state of complex Markovian objects is given which can be used as a basis for synthesizing high-speed predicting systems. The method makes it possible to specify a priori the accuracy of estimating the stochastic characteristics which describe the state of a technical object.     

Decoherence rates in large-scale quantum computers and macroscopic quantum systems

Markovian regime decoherence effects in quantum computers are studied in terms of the fidelity for the situation where the number of qubits N becomes large. A general expression giving the decoherence time scale in terms of Markovian relaxation elements and expectation values of products of system fluctuation operators is obtained, which could also be applied to study decoherence in other macroscopic systems such as Bose condensates and superconductors. A standard circuit model quantum computer involving three-state lambda system ionic qubits is considered, with qubits localized around well-separated positions via trapping potentials. The centre of mass vibrations of the qubits act as a reservoir. Coherent one and two qubit gating processes are controlled by time-dependent localized classical electromagnetic fields that address specific qubits, the two qubit gating processes being facilitated by a cavity mode ancilla, which permits state interchange between qubits. With a suitable choice of parameters, it is found that the decoherence time can be made essentially independent of N.     

Reduced‐order modelling of linear discrete‐time systems using cauer third continued‐fraction method

Abstract

In this paper a generalization of the Cauer third continued‐fraction expansion (CFE) is introduced for the reduced‐order modelling of linear time‐invariant discrete‐time systems. First, a new algorithm is presented for obtaining the Cauer third continued‐fraction expansion about z=1 and z=8 of the z‐transfer function of a discrete‐time system from its time‐moments and Markov parameters. Next, a realization of the Cauer third continued‐fraction expansion about z=1 and z=8, and the corresponding CFE canonical state‐space model are derived. Finally, an approximate aggregation matrix is constructed for relating the state‐vector of the reduced‐order CFE canonical state‐space model to that of the original system. An example is included to illustrate the use of the method.     

Relaxed Simultaneous Congruences

 In certain digital signal processing applications, residues of an input integer signal x are measured with respect to integer moduli . The signal x is then recovered by solving the system of simultaneous linear congruences . Since the residues r _i are measured quantities, they are subject to noise contamination. To provide noise protection, the moduli m _i may be chosen to possess common factors. Accurate approximations for x may then be determined by solving approximate or ``relaxed" simultaneous congruences. This paper presents a coherent mathematical theory for the approximate solution of simultaneous congruences with inaccurate residues when no exact solution exists. After precisely formulating the notion of relaxed congruences, it is found that, under nonrestrictive technical assumptions, unique solutions to these congruences always exist. A variety of examples illustrating characteristics of solutions of relaxed congruences are provided, and a fast, efficient algorithm for solving them numerically is presented. The problem of finding an optimal approximate solution is then discussed. Several optimality criteria are proposed and procedures for finding optimal approximate solutions are outlined. Error bounds are derived which specify the maximum amount that an approximate solution based upon inaccurate residues may differ from the corresponding true solution based upon the exact residues. Received: January 12, 2001; revised version: December 23, 2002 Keywords: Residue number systems, Error correction, Error detection, Measurement noise, Linear simultaneous congruences.     

Estimation of failure intensity for the Weibull process

A Weibull process/non-homogeneous Poisson process is commonly used to analyze the failure behavior of repairable systems. The object of the present study is to obtain exact estimates of the failure intensity of this model at the time of n failures. The resulting MLE estimate is biased and the corrected version for biasedness along with some approximate estimates is given. An analytical and numerical comparison of the relative efficiencies of the MLE of the exact biased, approximated biased, exact unbiased and approximated unbiased of the intensity function is presented. It will be shown that for small n (n < 30) there is quite a large relative difference between the mean squared errors of the exact and approximate estimates of the intensity function. Real failure data are used to illustrate the difference between the exact and approximate estimates of the intensity function.     

On the efficiency of synchronization of electronic clocks with reference time and frequency signals

A Markovian mathematical model is proposed for comparative analysis of noise suppression efficiency of systems for synchronization of electronic clocks with time codes transmitted as a part of standard time and frequency signals. Analytic expressions were derived for noise suppression efficiency as a function of probability of error in the received time code signal.This work was financially supported by the Russian Foundation for Basic Research (Grant No. 93-02-15424).Translated from Izmeritel'naya Tekhnika, No. 10, pp. 28–32, October, 1994.     

含分数阶导数项的随机Duffing振子的稳态响应分析

本文对一个含有分数阶导数项阻尼的、Gaussian白噪声激励下的Duffing振子进行了稳态响应分析。首先,基于能量平衡理论,运用等效线性化方法,计算等效系统的线性阻尼及自然频率,建立统计意义下的等效线性化系统。然后,利用平均法建立随机Ito方程,得到随机响应的Markovian近似;给出描述振子振幅概率密度函数演化的Fokker-Planck方程,并得到它的稳态解。进一步,对于含有响应振幅的等效线性系统,借助由Laplace变换得到的转换函数,得到原系统的条件功率谱密度,结合振幅的稳态概率密度作为权重函数,给出原系统功率谱密度的估计,以及响应的统计量的估计。数值模拟的结果说明所提出的功率谱密度的近似解析表达式是可靠的,它甚至适用于Duffing振子具有强非线性回复力的情形,因为它可以较好的表现出功率谱密度共振频谱加宽及多峰现象的出现。     

Evaluation of circuits for synchronization of electronic clocks with standard time and frequency signals

A Markovian mathematical model is proposed for a comparative analysis of noise suppression and locking time efficiency of different electronic clock synchronization systems with time codes transmitted by the State Time and Frequency Service. The synchronization circuit of the ChK7-50 commercial clock was found the most efficient.The work was financially supported by the Russian Foundation for Basic Research (Grant No. 92-02-15424).Translated from Izmeritel'naya Tekhnika, No. 12, pp. 24–26, December, 1995.     

Some Comments on Spectral Analysis of Time Series

The Average Run Length of a Cusum chart for controlling a normal mean is calculated by solving the systems of linear equations which approximate the integral equations for the required quantities. The accuracy of approximation by this method is numerically evaluated and the results are compared with those obtained by other approximate methods. The construction and use of a new nomogram based on the contours of Average Run Lengths L_a . and L_r drawn in the h√n/σ—|μ – k|√n/σ plane is discussed. Numerical examples are given to illustrate the flexibility and convenience provided by this nomogram in the design of Cusum charts.     

Supercurrent versus temperature characteristics of cross-type Josephson junctions nearT c

The temperature dependence of the maximum Josephson supercurrent is studied in ranges nearT _c . The results are discussed in the framework of an approximate analysis for proximity systems based on the de Gennes theory and its application to several junctions.     

Performance of a Petrov–Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift‐diffusion equations

This study compares the performance of a relatively new Petrov–Galerkin smoothed aggregation (PGSA) multilevel preconditioner with a nonsmoothed aggregation (NSA) multilevel preconditioner to accelerate the convergence of Krylov solvers on systems arising from a drift‐diffusion model for semiconductor devices. PGSA is designed for nonsymmetric linear systems, Ax=b, and has two main differences with smoothed aggregation. Damping parameters for smoothing interpolation basis functions are now calculated locally and restriction is no longer the transpose of interpolation but instead corresponds to applying the interpolation algorithm to A^T and then transposing the result. The drift‐diffusion system consists of a Poisson equation for the electrostatic potential and two convection–diffusion‐reaction‐type equations for the electron and hole concentration. This system is discretized in space with a stabilized finite element method and the discrete solution is obtained by using a fully coupled preconditioned Newton–Krylov solver. The results demonstrate that the PGSA preconditioner scales significantly better than the NSA preconditioner, and can reduce the solution time by more than a factor of two for a problem with 110 million unknowns on 4000 processors. The solution of a 1B unknown problem on 24 000 processor cores of a Cray XT3/4 machine was obtained using the PGSA preconditioner. Copyright © 2010 John Wiley & Sons, Ltd.     

Structural dynamics analysis using an unsymmetric block Lanczos algorithm

This paper describes the way in which an unsymmetric block Lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and/or repeated (or closely spaced) eigenvalues. To reduce a square, unsymmetric matrix which may have repeated eigenvalues to block-tridiagonal form, an unsymmetric block Lanczos algorithm is developed. Right and left Lanczos vectors, generated by the reduction process, are employed to define the reduced-order model based on a discrete analytical model of general linear, time-invariant dynamic systems. A special type of starting vector is also introduced, which automatically includes the static displacement. This reduced-order model is employed to compute approximate eigenvalues/vectors and dynamic responses of the original system. To verify the proposed algorithm, examples of an 8-DOF beam-rotor system are provided for eigensolutions and dynamic responses due to step and random external forces. These examples showed promising results, indicating that the proposed algorithm is worthy of further study.     

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X ₁),...,f(X _n ). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension. The research was partially supported by grant #37 from the CONICYT and by grant 5-81089 from NSERC.     

Ultrafast sum-frequency polarization beats in twin Markovian stochastic correlation

An analytic closed form for the second-order or fourth-order Markovian stochastic correlation of attosecond sum-frequency polarization beat (ASPB) can be obtained in the extremely Doppler-broadened limit. The homodyne detected ASPB signal is shown to be particularly sensitive to the statistical properties of the Markovian stochastic light fields with arbitrary bandwidth. The physical explanation for this is that the Gaussian-amplitude field undergoes stronger intensity fluctuations than a chaotic field. On the other hand, the intensity (amplitude) fluctuations of the Gaussian-amplitude field or the chaotic field are always much larger than the pure phase fluctuations of the phase-diffusion field. The field correlation has weakly influence on the ASPB signal when the laser has narrow bandwidth. In contrast, when the laser has broadband linewidth, the ASPB signal shows resonant-nonresonant cross correlation, and the sensitivities of ASPB signal to three Markovian stochastic models increase as time delay is increased. A Doppler-free precision in the measurement of the energy-level sum can be achieved with an arbitrary bandwidth. The advantage of ASPB is that the ultrafast modulation period 900as can still be improved, because the energy-level interval between ground state and excited state can be widely separated.     

On extensions of subshifts by finite groups

λ-Graph systems are labelled Bratteli diagrams with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of λ-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems are G-strong shift equivalent if and only if their presented subshifts are G-conjugate.     

Estimating Component Failure Rates From Combined Component and Systems Data: Exponentially Distributed Component Lifetimes

The situation in which test results are available on systems consisting of some of k independent but nonidentical components in series is studied. The underlying distribution of the lifetime of the component is assumed to be an exponential. The method of maximum likelihood is used to obtain estimates, a chi squared approximation is used to approximate the mean and variance of the maximum likelihood estimate, and a method for reducing its bias is presented.