General Stochastic Problem Research Articles

Supply Base Design for the Procurement of Multiple Items

Sourcing new products in advance entails significant risks for a buyer. To reduce the possibility of bringing a mediocre item to the market, buyers can build a large supply base and pick the best available products after uncertainties have materialized. Designing such a supply base is a challenging task: the buyer needs to decide how many suppliers to recruit, and how to split them across potential groups, such as product types or geographies. In this study, we propose a framework to design optimal supply bases when multiple items must be sourced from potential suppliers that are subject to uncertainties on the profitability of their supplies. We start with the case of independent supplier profits: we characterize the optimal supply base size and find that it is optimal to recruit a much larger number of suppliers compared to the number of products needed, which increases in the riskiness of supplier designs so as to better exploit the potential upside in profitability deviations. We then study the case where different supplier groups are correlated. While the exact solution can be obtained in simple cases, the general stochastic problem can only be solved numerically. Hence, we propose three approximations that provide excellent solutions, especially when the number of products to source is large. Our framework thus provides guidelines for buyers that want to control their sourcing risks through supply base design and to improve their profits by exploiting the option value of diversification.

Delay-aware spectrum sharing solutions for mixed cellular and D2D links

In this paper, we dynamically determine the spectrum share of cellular and overlay device to device communication (D2D) links to improve queuing delay performance. Along changing the portion of resource blocks (RBs) dedicated to each cellular and D2D network, we assign the required power and RBs to each link. We consider a cross-layer optimization problem taking channel state information (CSI) and queue state information (QSI) into account. Two spectrum access mechanisms of overlay D2D mode, known as orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) are investigated. First, the dynamic spectrum sharing and power assignment problem is formulated as a general stochastic problem, then it is transformed into a deterministic one using Lyapunov optimization. As the cellular and D2D links should be scheduled prior to each frame transmission, low-complexity approaches are necessary to deal with the dynamic nature of the optimization problem. Thereby, we obtain closed-form-iterated solutions for OMA and NOMA with low design complexity because they do not need to know the probability functions of the channel state and traffic arrival process. Furthermore, for computational complexity reduction, a linear-complexity heuristic approach is proposed. Numerical results demonstrate that the proposed dynamic spectrum sharing approaches outperform static approaches in terms of average delay and average power. In addition, the performance of dynamic NOMA is superior to the dynamic OMA approach at the expense of higher computational complexity which can be compensated by the heuristic approach with a small power computation increase.

Response probability density of a system with cross-correlated parametric and additive input noises

Data processing and subsequent mining is a widely followed task. Employment of suitable evaluation and interpretation procedures can significantly improve the effective resolution of measuring facility using an identical hardware equipment. Recording of time variable processes is accompanied by various internal disturbing effects as a rule. They influence parameters of the measuring facility, transducer-device transmitting, etc. These parasitic processes are usually of the random character and, consequently, they exercise as parametric noises. Moreover, the input signal mostly consists of a useful signal, which can be taken for deterministic, and of a random additive part. Due to interaction of additive noises with the device itself, the cross-correlation of both additive and multiplicative noises cannot be neglected as a rule. Various combinations of noises are the origin of random and also systematic measuring errors which can have under certain circumstances a cumulative character. Their influence deteriorates the output signal quality and can lead finally to the stochastic stability loss. These effects can be theoretically described using differential systems with stochastic coefficients and a stochastic right hand side considering all input and output processes to be of the Markov type. A direct investigation of the relevant Fokker-Planck equation is employed as the main tool. Two first stochastic moments (mathematical mean value and variance) as evolutionary processes are investigated for a general deterministic useful signal and subsequently for two special cases of this one. Both types of input random noises are considered. Conditions of stochastic stability with respect to intensities of input random processes are formulated. The probability density function is deduced as well, in order to illustrate the probabilistic character of the system response as a whole. The stochastic asymmetry of the output signal is identified. Limitation procedures show a smooth transition from a general stochastic problem to deterministic noise free input signal and its processing.

Monte-Carlo simulation of a stochastic differential equation

For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient, Monte Carlo (MC) techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem. Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.

On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established and successful approaches for solving general stochastic problems. A recently developed method based on stochastic reduced order models (SROMs) can also be used. Herein we provide a comparison of the three methods for some numerical examples; our evaluation only holds for the examples considered in the paper. The purpose of the comparisons is not to criticize the SC or SG methods, which have proven very useful for a broad range of applications, nor is it to provide overall ratings of these methods as compared to the SROM method. Rather, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.

Stochastic diagrams for critical point spectra

A new technique for calculating the time-evolution, correlations and steady state spectra for nonlinear stochastic differential equations is presented. To illustrate the method, we consider examples involving cubic nonlinearities in an N-dimensional phase-space. These serve as a useful paradigm for describing critical point phase transitions in numerous equilibrium and non-equilibrium systems. The technique presented here is not perturbative. It consists in developing the stochastic variable as a power series in time, and using this to compute the short time expansion for the correlation functions. This, in turn, is extrapolated to large times and Fourier transformed to obtain the spectrum. A stochastic diagram technique is developed to facilitate computation of the coefficients of the relevant power series expansion. Two different types of long-time extrapolation technique, involving either simple exponentials or logarithmic rational approximations, are evaluated for third-order diagrams. The analytical results thus obtained are compared with numerical simulations, together with exact results available in special cases. The agreement is excellent in the neighborhood of the critical point. Finally, we emphasize that the technique is also applicable to more general stochastic problems involving spatial variation in addition to temporal variation.

Semigroup solutions to stochastic unsteady groundwater flow subject to random parameters

Two methods for the solution of partial differential equations (PDE) for the general case of random in time physical parameters are presented and their application to the solution of unsteady regional groundwater flow equations are illustrated. The first method is the semigroup approach which directly offers a solution without resorting to “closure approximations” (hierarchy techniques), perturbation techniques, or Montecarlo simulation techniques. The semigroup approach can also handle the general stochastic problem when randomness also appears as initial conditions, boundary conditions or forcing terms. The second method is an approximation scheme to obtain the semigroup solution in complex cases and permits the solution of equations with more than one random coefficient.

Foundations of stochastic electrodynamics.

Stochastic electrodynamics is classical electrodynamics with the hypothesis of a random background radiation in the whole space. The probability distributions of the Fourier components of this electromagnetic radiation are assumed Gaussian. Lorentz invariance fixes the spectrum of the radiation except for a constant, measuring its intensity, which is identified with Planck’s constant. A formalism is developed to deal with general stochastic problems, associating a Hilbert space with every set of random variables. Then, a mapping is defined of the algebra of continuous functions of the random variables onto the algebra of operators on the Hilbert space. A correspondence is found between probability distributions of the random variables and vectors in the Hilbert space. The case of Gaussian random variables is considered in detail. The formalism is applied to the study of the radiation field when some known amount of radiation is present besides the random background. This closely resembles the usual quantization of the free radiation field, although there are significant differences. In particular, not all vectors of the Hilbert space represent physical states in the present theory.