Total Travel Cost in Stochastic Assignment Models

Deterministic assignment models are sometimes used to approximate properties of more complex stochastic models. One property that is of particular interest from a system optimization viewpoint is total travel cost. This paper looks at the approximation of mean total travel cost. It is shown that deterministic models will underestimate this quantity in many common situations. Furthermore, discrepancies between total travel cost under the different modelling frameworks can lead to situations in which network modifications which are detrimental according to a stochastic model appear beneficial when using the natural deterministic approximation. We conclude that estimation of mean travel cost in stochastic assignment is often best done using simulation. Some suggestions are made regarding the implementation of traffic assignment simulation.     

Numerical study of a stochastic particle method for homogeneous gas-phase reactions

In this paper, we study a stochastic particle system that describes homogeneous gas-phase reactions of a number of chemical species. First, we introduce the system as a Markov jump process and discuss how relevant physical quantities are represented in terms of appropriate random variables. Then, we show how various deterministic equations, used in the literature, are derived from the stochastic system in the limit when the number of particles goes to infinity. Finally, we apply the corresponding stochastic algorithm to a toy problem, a simple formal reaction mechanism, and a real combustion problem. This problem is given by the isothermal combustion of a homogeneous mixture of heptane and air modelled by a detailed reaction mechanism with 107 chemical species and 808 reversible reactions. Heptane as described in this chemical mechanism serves as model-fuel for different types of internal combustion engines. In particular, we study the order of convergence with respect to the number of simulation particles, and illustrate the limitations of the method.     

Deterministic and stochastic model based run-to-run control for batch processes with measurement delays of uncertain duration

A novel run-to-run control algorithm integrating deterministic and stochastic model based control is developed for batch processes with measurement delays of uncertain duration. This control algorithm is referred to as deterministic and stochastic model based control (DSMBC). The deterministic component responds quickly to deterministic changes while the stochastic component minimizes the effects arising from measurement delays of uncertain duration. The deterministic component uses a linear process model with parameters that are updated online. The stochastic component uses an error probability density function (PDF) to characterize the effects due to measurement delays and this error PDF is determined from deviations between the set-point and the available process output. To integrate the two control algorithms, the control input is determined by minimizing the weighted sum of the predicted error from the deterministic model and the information entropy of the error probability density distribution. Using a simulated setting where the rate of chemical vapor deposition is controlled, the performance of the proposed DSMBC is shown to be superior to that of EWMA.     

机票定价与舱位控制两阶段决策方法

针对航空收益管理中定价与舱位控制联合决策问题,提出一种两阶段决策方法.以最大化总收益为目标,建立和分析相应的联合决策模型,包括非嵌套模型(确定性模型和随机模型)和嵌套模型.通过对模型的求解和仿真得到:在价格方面,随机模型定价最高,其次是嵌套模型,确定性模型定价最低;在对低票价等级的订座限制方面,随机模型限制最严,其次是确定性模型,嵌套模型限制最宽松;最终总收益方面,嵌套模型的总收益最高,而随机模型与确定性模型总收益的高低视情况而定.为应对求解大规模嵌套模型算例时的复杂性,分别将非嵌套模型计算所得的定价结果作为嵌套模型的输入价格,求得对应的座位分配结果.对所得到的两阶段策略进行仿真得到,随机模型与嵌套模型相结合所得到的两阶段策略表现更好,能够使总收益接近最优水平.     

Controller assessment for a class of non-linear systems

The use of autoregressive moving average (ARMA) models to assess the control loop performance for processes that are adequately described by the superposition of a linear dynamic model and linear stochastic or deterministic disturbance model is well known. In this paper, classes of non-linear dynamic/stochastic systems for which a similar result can be obtained are established for single-input single-output discrete system. For these systems, lower mean-square error bounds on performance, can be estimated from the closed-loop routine operating data by using non-linear autoregressive moving average with exogenous inputs (NARMAX) models. It is necessary to know the process time delay. The fitting of these models is greatly facilitated by using efficient algorithms, such as Orthogonal Least Squares or other fast orthogonal search algorithms. These models can also be used to assess the predictive importance of non-linearities over multiple-time horizons.     

Stochastic projective methods for simulating stiff chemical reacting systems

In this paper, stochastic projective methods are proposed to improve the stability and efficiency in simulating stiff chemical reacting systems. The efficiency of existing explicit tau-leaping methods can often severely be limited by the stiffness in the system, forcing the use of small time steps to maintain stability. The methods presented in this paper, namely stochastic projective (SP) and telescopic stochastic projective (TSP) method, can be considered as more general stochastic versions of the recently developed stable projective numerical integration methods for deterministic ordinary differential equations. SP and TSP method are developed by fully re-interpreting and extending the key projective integration steps in the deterministic regime under a stochastic context. These new stochastic methods not only automatically reduce to the original deterministic stable methods when applied to simulating ordinary differential equations, but also carry the enhanced stability property over to the stochastic regime. In some sense, the proposed methods are stochastic generalizations to their deterministic counterparts. As such, SP and TSP method can adopt a much larger effective time step than is allowed for explicit tau-leaping, leading to noticeable runtime speedup. The explicit nature of the proposed stochastic simulation methods relaxes the need for solving any coupled nonlinear systems of equations at each leaping step, making them more efficient than the implicit tau-leaping method with similar stability characteristics. The efficiency benefits of SP and TSP method over the implicit tau-leaping is expected to grow even more significantly for large complex stiff chemical systems involving hundreds of active species and beyond.     

Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems

Uncertainty quantification and propagation in physical systems appear as a critical path for the improvement of the prediction of their response. Galerkin-type spectral stochastic methods provide a general framework for the numerical simulation of physical models driven by stochastic partial differential equations. The response is searched in a tensor product space, which is the product of deterministic and stochastic approximation spaces. The computation of the approximate solution requires the solution of a very high dimensional problem, whose calculation costs are generally prohibitive. Recently, a model reduction technique, named Generalized Spectral Decomposition method, has been proposed in order to reduce these costs. This method belongs to the family of Proper Generalized Decomposition methods. It takes part of the tensor product structure of the solution function space and allows the a priori construction of a quasi optimal separated representation of the solution, which has quite the same convergence properties as a posteriori Hilbert Karhunen-Loève decompositions. The associated algorithms only require the solution of a few deterministic problems and a few stochastic problems on deterministic reduced basis (algebraic stochastic equations), these problems being uncoupled. However, this method does not circumvent the “curse of dimensionality” which is associated with the dramatic increase in the dimension of stochastic approximation spaces, when dealing with high stochastic dimension. In this paper, we propose a marriage between the Generalized Spectral Decomposition algorithms and a separated representation methodology, which exploits the tensor product structure of stochastic functions spaces. An efficient algorithm is proposed for the a priori construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.     

Distributed agent paradigm for soft and hard computation

This paper describes a multiagent-based simulation paradigm, for hybrid (soft and hard) simulation of complex dynamical systems. The computations are interpreted as the outcome arising out of deterministic, nondeterministic or stochastic interaction among the agents, which includes the environment. These interactions are like chemical reactions and the evolution of the multiset of agents can mimic the evolution of the complex system, e.g. genetic, nature inspired self-organized criticality and active walker (swarm and ant intelligence) models. Since the reaction rules are inherently parallel, any number of actions can be performed cooperatively or competitively among the subsets of the agents, so that the system evolve reaches an equilibrium (or a chaotic or an emergent) state. Practical realisation of this paradigm is achieved through a coordination programming language using Multiagent and transactions.     

Deterministic balancing and stochastic model reduction

A recent approach to the deterministic model reduction problem is based on the notion of balancing. However, the original development of deterministic balancing did not contain any statistical considerations with which to develop a stochastic model reduction algorithm. Nevertheless, it is shown in this note that there are two stochastic model reduction algorithms in the literature which result in a deterministically balanced model. Their equivalence with deterministic balancing provides a stochastic interpretation to the deterministic algorithm.     

Value of the stochastic efficiency in data envelopment analysis

This article examines the potential benefits of solving a stochastic DEA model over solving a deterministic DEA model. It demonstrates that wrong decisions could be made whenever a possible stochastic DEA problem is solved when the stochastic information is either unobserved or limited to a measure of central tendency. We propose two linear models: a semi-stochastic model where the inputs of the DMU of interest are treated as random while the inputs of the other DMUs are frozen at their expected values, and a stochastic model where the inputs of all of the DMUs are treated as random. These two models can be used with any empirical distribution in a Monte Carlo sampling approach. We also define the value of the stochastic efficiency (or semi-stochastic efficiency) and the expected value of the efficiency.     

Bayesian estimation of motion vector fields

A stochastic approach to the estimation of 2D motion vector fields from time-varying images is presented. The formulation involves the specification of a deterministic structural model along with stochastic observation and motion field models. Two motion models are proposed: a globally smooth model based on vector Markov random fields and a piecewise smooth model derived from coupled vector-binary Markov random fields. Two estimation criteria are studied. In the maximum a posteriori probability (MAP) estimation, the a posteriori probability of motion given data is maximized, whereas in the minimum expected cost (MEC) estimation, the expectation of a certain cost function is minimized. Both algorithms generate sample fields by means of stochastic relaxation implemented via the Gibbs sampler. Two versions are developed: one for a discrete state space and the other for a continuous state space. The MAP estimation is incorporated into a hierarchical environment to deal efficiently with large displacements     

Grid refinement tests of a least-squares finite element method for advective transport of reactive species

A time splitting least-squares finite element method (LSFEM) and the ‘stiff ODEs’ solver LSODE are used to simulate the advective transport of reactive species. Specifically, the rotating cone problem with chemical reactions serves as a model to test the algorithm. A non-linear filter is used to suppress spurious oscillations at each advective time step. All simulations are carried out by using linear and quadratic elements on two mesh systems. Results from the coarse mesh system suggest that the use of robust numerical methods alone will not be able to provide accurate results and point to the need of grid refinement. The grid refinement tests are evaluated by pollutant peak concentrations, pollutant concentration distributions, average relative errors, species mass conservation and distribution ratios and CPU times. Results from the fine mesh system are satisfactory and imply that accurate simulations of the transport of reactive species require adequate grid resolution and robust numerical methods for each individual advective and reactive step.     

Criteria for informative experiments with subspace identification

Informative experiments are identification experiments which contain sufficient information for an identification algorithm to discriminate between different models in an intended model set. In this paper, a particular set of identification algorithms, namely subspace based identification, is considered. Criteria for experiments to be informative with these methods in the deterministic setup and the combined deterministic-stochastic setup are presented. It is pointed out that if these criteria are not satisfied, interesting phenomena, in which perfect cancellations of the deterministic components and the stochastic components occur in a subspace projection, may occur. It is further shown that such cancellations can indeed be avoided under mild conditions.     

Algebraic structure of some stochastic discrete event systems,with applications

Generalized semi-Markov processes (GSMPs) and stochastic Petri nets (SPNs) are generally regarded as performance models (as opposed to logical models) of discrete event systems. Here we take the view that GSMPs and SPNS are essentially automata (generators) driven by input sequences that determine the timing of events. This view combines the deterministic, logical aspects and the stochastic, timed aspects of the two models. We focus on two conditions, (M) and (CX) (which we previously developed to study monotonicity and convexity properties of GSMPs), and the antimatroid and lattice structure they imply for the language generated by a GSMP or SPN. We illustrate applications of these structural properties in the areas of derivative estimation, simulation variance reduction, parallel simulation, and optimal control.Research supported in part by NSF grant ECS-89-96230.     

A location model with queueing constraints

Many models used for locating service facilities assume that customer demand is deterministic and that service facilities are always available. Stochastic demands for service may cause queueing if the server is not available. We consider a model which combines the spatial aspects of locating service faculties with the service delays caused by stochastic customer demands. The resulting 0–1 integer linear program may be solved by existing algorithms. An example is presented to illustrate the effect of the queueing considerations.     

Numerical simulation of stochastic replicator models in catalyzed RNA-like polymers

A stochastic model for replicators in catalyzed RNA-like polymers is presented and numerically solved. The model consists of a system of reaction–diffusion equations describing the evolution of a population formed by RNA-like molecules with catalytic capabilities in a prebiotic process. The diffusion effects and the catalytic reactions are deterministic. A stochastic excitation with additive noise is introduced as a force term. To numerically solve the governing equations we apply the stochastic method of lines. A finite-difference reaction–diffusion system is constructed by discretizing the space and the associated stochastic differential system is numerically solved using a class of stochastic Runge–Kutta methods. Numerical experiments are carried out on a prototype of four catalyzed selfreplicator species along with an activated and an inactivated residues. Results are given in two space dimensions.     

A one-dimensional reactive multi-component landfill leachate transport model

A one-dimensional reactive multi-component landfill leachate transport model coupled to three modules (geochemical equilibrium, kinetic biodegradation, and kinetic precipitation–dissolution) is presented to simulate the migration of contaminants in soils under landfills. A two-step sequential operator splitting method is applied to solve the coupled transport equations and the biogeochemical reaction equations. The geochemical module is based on the equilibrium speciation model (MINTEQA2), which uses ion-association equilibrium–constant approach to represent the various geochemical reactions. The biodegradation module describes the sequential biological degradation of organic compounds by multiple functional bacterial populations. Analytical equations based on macroscopic approach are used to model changes in porosity and permeability caused by biomass accumulation and mineral precipitation in soils. The model has been evaluated by comparing the model results with the widely used one-dimensional mixing-cell model PHREEQM for acidic mine tailings discharge in a carbonate aquifer. The composite leachate transport model is applied to a hypothetical landfill to simulate the effect of biological degradation of organic matter on the local inorganic geochemistry and also to demonstrate the effect of microbial activity on the evolution of porosity reduction of soils under the landfill.     

On Stochastic Simulation of Forward-Looking Models

The solution of a nonlinear macroeconometric model with expectations of future-dated variables generally has to be approximated by numerical simulation. A brief review of deterministic, and stochastic dynamic simulations of a backward-looking model is followed by a conceptual presentation of methods for dynamic simulation of a forward-looking (rational expectations) model. I distinguish between uncertainty faced by rational agents and by the modeller, and suggest different ways of simulating random variables in the model. Simulations of simple linear and nonlinear univariate time-series models illustrate the methods.     

Finding Minimum Vertex Covering in Stochastic Graphs: A Learning Automata Approach

Structural and behavioral parameters of many real networks such as social networks are unpredictable, uncertain, and have time-varying parameters, and for these reasons, deterministic graphs for modeling such networks are too restrictive to solve most of the real-network problems. It seems that stochastic graphs, in which weights associated to the vertices are random variables, might be better graph models for real-world networks. Once we use a stochastic graph as the model for a network, every feature of the graph such as path, spanning tree, clique, dominating set, and cover set should be treated as a stochastic feature. For example, choosing a stochastic graph as a graph model of an online social network and defining community structure in terms of clique, the concept of a stochastic clique may be used to study community structures’ properties or define spreading of influence according to the coverage of influential users; the concept of stochastic vertex covering may be used to study spread of influence. In this article, minimum vertex covering in stochastic graphs is first defined, and then four learning, automata-based algorithms are proposed for solving a minimum vertex-covering problem in stochastic graphs where the probability distribution functions of the weights associated with the vertices of the graph are unknown. It is shown that through a proper choice of the parameters of the proposed algorithms, one can make the probability of finding minimum vertex cover in a stochastic graph as close to unity as possible. Experimental results on synthetic stochastic graphs reveal that at a certain confidence level the proposed algorithms significantly outperform the standard sampling method in terms of the number of samples needed to be taken from the vertices of the stochastic graph.