Moment Matching Approach Research Articles

Asymptotic Probability Extraction for Nonnormal Performance Distributions

While process variations are becoming more significant with each new IC technology generation, they are often modeled via linear regression models so that the resulting performance variations can be captured via normal distributions. Nonlinear response surface models (e.g., quadratic polynomials) can be utilized to capture larger scale process variations; however, such models result in nonnormal distributions for circuit performance. These performance distributions are difficult to capture efficiently since the distribution model is unknown. In this paper, an asymptotic-probability-extraction (APEX) method for estimating the unknown random distribution when using a nonlinear response surface modeling is proposed. The APEX begins by efficiently computing the high-order moments of the unknown distribution and then applies moment matching to approximate the characteristic function of the random distribution by an efficient rational function. It is proven that such a moment-matching approach is asymptotically convergent when applied to quadratic response surface models. In addition, a number of novel algorithms and methods, including binomial moment evaluation, PDF/CDF shifting, nonlinear companding and reverse evaluation, are proposed to improve the computation efficiency and/or approximation accuracy. Several circuit examples from both digital and analog applications demonstrate that APEX can provide better accuracy than a Monte Carlo simulation with 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> samples and achieve up to 10times more efficiency. The error, incurred by the popular normal modeling assumption for several circuit examples designed in standard IC technologies, is also shown

Partially Exact and Bounded Approximations for Arithmetic Asian Options

This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. Firstly we consider the partial differential equation approach to the pricing of Asian options. We show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Vecer [2001]. For the latter two formulations we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Rogers and Shi's lower bound for the general case of multiple underlyings. Thirdly, we sharpen Thompson's [1999a,b] upper bound for the value of an Asian option. This is important for the practically relevant case of options with long maturities. Numerical results show that when the strike price is not extremely high, the resulting upper bound is tighter than recently introduced upper bounds in studies by Nielsen and Sandmann [2003] and Vanmaele et al. [2005]. Finally, we consider analytical approximations for the value of an Asian option. A much heard criticism on moment-matching approaches is that the error in the approximation is not known beforehand. We combine the traditional moment-matching approaches (e.g. Levy [1992]) with the conditioning approaches (e.g. Curran [1994]) and introduce a class of analytical approximations, which can be proven to lie between a sharp lower and upper bound. In numerical examples the accuracy of these new approximations is demonstrated. The approximations are found to outperform all of the current state-of-the-art upper bounds and approximations.

Transient analysis of lossy interconnects by modified method of characteristics

In this paper, the modified method of characteristics (MMC) is presented to analyze lossy interconnects. The method applies lower order Taylor approximation to model the characteristic admittance, and further applies lower order Pade approximation to model propagation functions of a transmission line. However, different from the single-point or multipoint moment-matching approaches, this method does not generate a reduced-order model of a transmission line prior to constructing the system matrix. Instead, it takes a special expansion and directly incorporates the coefficients of the expansion into the modified nodal admittance (MNA) matrix. On the basis of the specific expansion, a set of time-domain recursive formulae is derived, which concerns only the quantities at the ends of transmission lines. The recursive formulae are similar to those of method of characteristics (MC), with some added modifications. Based on the formulae of MMC, an equivalent time-domain macromodel of a uniform lossy transmission line is obtained. It can be easily implemented within the framework of an existing circuit simulator such as SPICE. The examples indicate that the method gives accurate time-domain simulation of interconnects in high-speed IC systems.

An algorithm for simulating power/ground networks using Pade approximants and its symbolic implementation

This paper presents an event-driven algorithm and its symbolic implementation for the analysis of power and ground (P/G) bus networks. The algorithm uses frequency-domain techniques and moment matching approaches based on Pade approximants to estimate the transfer function at each node in the P/G network. Afterwards, the transient waveforms are extracted for each node. The process requires repetitive simulation of a linear and time-variant (from one time event to the next) circuit model for the P/G network which is the reason a symbolic implementation was produced. The P/G network is modeled by a hierarchical combination of mesh and tree structures that are composed of a collection of RC-/spl pi/-segments and pulldown (or pullup) switches. The switches are symbolically represented by Boolean variables and a compiled symbolic code is generated only once for each P/G network. The transient waveforms are then produced by repetitive evaluation of the symbolic output. The results show that the symbolic implementation is an order of magnitude faster, with reasonably good accuracy, than using a traditional analog circuit simulator like SPICE.

Performance analysis of a batch service queue arising out of manufacturing system modelling

In this paper, we present an exact analysis of a queueing system with Poisson arrivals and batch service. The system has a finite numberS of waiting places and a batch service capacityb. A service period is initialized when a service starting thresholda of waiting customers has been reached. The model is denoted accordingly byM/G [a,b] /1−S. The motivation for this model arises from manufacturing environments with batch service work stations, e.g. in machines for computer components and chip productions. The method of embedded Markov chain is used for the analysis, whereby a representation of the general service time is obtained via a moment matching approach. Numerical results are shown in order to illustrate the dependency of performance measures on special sets of system parameters. Furthermore, attention is devoted to the issues of starting rules, where performance objectives like short waiting time, small blocking probability and minimal amount of work in progress are taken into account.

Performance analysis of finite capacity polling systems with nonexhaustive service

In performance investigations of token passing local area networks, communication subsystems in switching systems with distributed control, etc., the class of polling models, i.e., multiqueue systems with cyclic service, is often employed. This paper presents an approximate analysis method for this class of models, whereby realistic modelling assumptions like the finiteness of queue capacities and nonsymmetrical load conditions are taken into account. The method of imbedded Markov chain is used for the analysis, whereby the special case of exponential as well as the case of general service time are successively considered. The latter case is analyzed in conjunction with a moment matching approach for the cycle time. The validation of the approximation is done by means of computer simulations. Numerical results are shown in order to illustrate the accuracy of the calculation method and its dependency on system parameters.