A combinatorial approach to jumping particles: The parallel TASEP

In this article, we continue the combinatorial study of models of particles jumping on a row of cells which we initiated with the standard totally asymmetric simple exclusion process or TASEP (Duchi and Schaeffer, Journal of Combinatorial Theory, Series A, 110(2005), 1–29). We consider here the parallel TASEP, in which particles can jump simultaneously. On the one hand, the interest in this process comes from highway traffic modeling: it is the only solvable special case of the Nagel‐Schreckenberg automaton, the most popular model in that context. On the other hand, the parallel TASEP is of some theoretical interest because the derivation of its stationary distribution, as appearing in the physics literature, is harder than that of the standard TASEP. We offer here an elementary derivation that extends the combinatorial approach we developed for the standard TASEP. In particular, we show that this stationary distribution can be expressed in terms of refinements of Catalan numbers. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Trajectories of lattice particles using the new approach to no integrability

We study two systems which lead to a lattice when an integration path is specified in “aesthetic field theory”. One of these cases involves nonsoliton type particles (magnitudes of maxima and minima oscillate in time). The other system is made up of soliton type particles. The two systems are intrinsically three-dimensional. We speak of the third dimension as “time”. In one of our solutions, the particles move on straight line trajectories, insofar as our numerical work indicates. In the other solution, the soliton type particles undergo what appears to be simple harmonic motion in both the x- and y-directions (loop motion). We then study these two systems using the new approach to integrability which involves a superposition principle and is characterized by a unique change function at each point. We still find multi maxima and minima. The systems are not as symmetric as the lattice. The soliton characteristic is preserved by the new method. We investigated the motion of lattice particles. We found evidence of maxima (minima) regions coalescing so that the location of the maxima (minima) became difficult to follow. The concept of location of particles may not even have a well-defined meaning here. We find examples of soliton particles appearing and disappearing. We conclude that the manner of integration in a no integrability theory can transform a system with well-defined trajectories into a system where particles can no longer be followed in time.     

A complex tanh-function method applied to nonlinear equations of Schrödinger type

A complex tanh-function method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases and solutions. The scheme is implemented for obtaining multiple soliton solutions to the nonlinear cubic Schrödinger equation and a generalized Schrödinger-like equation. In additon. an ansätze is proposed to obtain stationary soliton solutions of the cubic Schrödinger equation.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

System of sticking diffusion particles of variable mass

We construct a mathematical model of an infinite system of diffusion particles with interaction whose masses affect the diffusion coefficient. The particles begin to move from a certain stationary distribution of masses. Their motion is independent up to their meeting. Then the particles become stuck and their masses are added. As a result, the diffusion coefficient varies as a function inversely proportional to the square root of the mass. It is shown that the mass transported by particles is also characterized by a stationary distribution.     

The effect of suspended particles on Rayleigh-Bénard convection II. A nonlinear stability analysis of a thermal disequilibrium model

The weakly nonlinear stability of the pure conduction solution for an appropriate aerosol one-layer Rayleigh-Bénard model of a Boussinesq particle-gas system retaining both the particle and collision pressures and considering particle to particle radiative effects while relaxing the usual assumption of thermal equilibrium between those particles and the gas is investigated. Then an analysis of the criteria governing the occurrence of supercritically re-equilibrated stationary rolls yields a minimum Rayleigh number and a critical wavelength which are completely compatible in their layer-depth behavior with normal convective and columnar instabilities observed in mixtures of smoke with air or carbon dioxide.     

Method of Immersion and Its Parametric Realization for the Korteweg–De Vries Equation

By using the method of immersion (imbedding) proposed in the author's previous works, we describe the space S of initial conditions of the Cauchy problem for the general differential Korteweg–de Vries equation. The space S is called a stationary soliton Korteweg–de Vries manifold because "stationary projections" of solitons fall into the space S. In addition, we introduce the notion of a space of Sturm–Liouville operators over a soliton Korteweg–de Vries manifold. For real functions and parameters, we formulate the spectral theorem for a commutative Lax pair over a real stationary soliton Korteweg–de Vries manifold.     

Asymptotic Stability of Some Nonlinear Boltzmann-Type Equations

A generalized version of the Tjon-Wu equation is considered. It describes the evolution of the energy distribution in a model of gas in which simultaneous collisions of many particles are permitted. Using the technique of Zolotarev metrics we show that the stationary solution is exponentially stable.     

Reflection and Refraction of Solitons by the KdV–Burgers Equation in Nonhomogeneous Dissipative Media

We study the behavior of the soliton that encounters a barrier with dissipation while moving in a nondissipative medium. We use the Korteweg–de Vries–Burgers equation to model this situation. The modeling includes the case of a finite dissipative layer similar to a wave passing through air–glass–air and also a wave passing from a nondissipative layer into a dissipative layer (similar to light passing from air to water). The dissipation predictably reduces the soliton amplitude/velocity. Other effects also occur in the case of a finite barrier in the soliton path: after the wave leaves the dissipative barrier, it retains the soliton form, but a reflection wave arises as small and quasiharmonic oscillations (a breather). The breather propagates faster than the soliton passing through the barrier.     

Complexity and stability of soliton management in periodically modulated and random systems

A brief introduction is given to the concept of the soliton management, i.e., stable motion of localized pulses in media with strong periodic (or, sometimes, random) inhomogeneity, or conditions for the survival of solitons in models with strong time‐periodic modulation of linear or nonlinear coefficients. It is demonstrated that a class of systems can be identified, in which solitons remain robust inherently coherent objects in seemingly “hostile” environments. Most physical models belonging to this class originate in nonlinear optics and Bose‐Einstein condensation, although other examples are known too (in particular, in hydrodynamics). In this paper, the complexity of the soliton‐management systems, and the robustness of solitons in them are illustrated using a recently explored fiber‐optic setting combining a periodic concatenation of nonlinear and dispersive segments (the split‐step model) for bimodal optical signals (i.e., ones with two polarizations of light), which includes the polarization mode dispersion, i.e., random linear mixing of the two polarization components at junctions between the fiber segment. © 2008 Wiley Periodicals, Inc. Complexity, 2008     

The first-order perturbed SBS equations

Summary Using standard multiscale techniques, a first-order perturbation theory for SBS is developed. In the presence of small damping, we find that there is a stationary solution for a soliton which is a fixed point. The velocity of this soliton is determined by the damping coefficients. In addition, there is also a constant shift in the pump intensity in the region between the front of the backward moving soliton and the forward light cone of the pump.     

Evolution Theory,Periodic Particles,and Solitons in Cellular Automata

A theory for soliton automata is developed and applied to the analysis and prediction of patterns in their behavior. A complete characterization and method of construction of 1-periodic particles is given. A general evolution theorem (GET) is obtained which provides significant information for a state in terms of preceding states. Application of this theorem yields several interesting results predicting periodicity and solitonic collisions. The GET explains and is based on a fundamental property of soliton automata, observed and analyzed in this paper, namely that pieces of information are lost on the left and reappear on the right.     

The integrability and parametric representation of a hierarchy of soliton equations

In this paper after having obtained the Lax pair of a hierarchy of soliton equations,we discuss the parametric representation for finite-band solutions of the stationary solitonequation, and prove it can be represented as a Hamiltonian system which is integrable inLiouville sense. The nonconfocal involutive integral representations {Fm} are obtained also.In the condition of finite-band solutions of the soliton equation, the time and space can bedevided inio two Hamiltonian systems, so the fi…     

Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching

This paper addresses a stochastic SIS epidemic model with vaccination under regime switching. The stochastic model in this paper includes white and color noises. By constructing stochastic Lyapunov functions with regime switching, we establish sufficient conditions for the existence of a unique ergodic stationary distribution.     

Asymptotic behavior of solutions: An application to stochastic NLP

In this article we study the consistency of optimal and stationary (KKT) points of a stochastic non-linear optimization problem involving expectation functionals, when the underlying probability distribution associated with the random variable is weakly approximated by a sequence of random probability measures. The optimization model includes constraints with expectation functionals those are not captured in direct application of the previous results on optimality conditions exist in the literature. We first study the consistency of stationary points of a general NLP problem with convex and locally Lipschitz data and then apply those results to the stochastic NLP problem and stochastic minimax problem. Moreover, we derive an exponential bound for such approximations using a large deviation principle.

     

Computing the Stationary Distribution of an SRBM in an Orthant with Applications to Queueing Networks

In [15], a BNAfm (Brownian network analyzer with finite element method) algorithm was developed for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. In this companion paper, that BNAfm algorithm is extended to computing the stationary distribution of an SRBM in an orthant, which is achieved by constructing a converging sequence of SRBMs in hypercubes. The SRBM in the orthant serves as an approximation model of queueing networks with infinite buffers. We show that the constructed sequence of SRBMs in the hypercubes converges weakly to the SRBM in the orthant as the hypercubes approach the orthant. Under the conjecture that the set of the stationary distributions of the SRBMs in the hypercubes is relatively compact, we prove that the sequence of the stationary distributions of the SRBMs in the hypercubes converges weakly to the stationary distribution of the SRBM in the orthant. A three-machine job shop example is presented to illustrate the effectiveness of the SRBM approximation model and our BNAfm algorithm. The BNAfm algorithm is shown to produce good estimates for stationary probabilities of queueing networks.     

Inference for Spatial Autoregressive Models with Infinite Variance Noises

A self-weighted quantile procedure is proposed to study the inference for a spatial unilateral autoregressive model with independent and identically distributed innovations belonging to the domain of attraction of a stable law with index of stability α, α ∈ (0, 2]. It is shown that when the model is stationary, the self-weighted quantile estimate of the parameter has a closed form and converges to a normal limiting distribution, which avoids the difficulty of Roknossadati and Zarepour (2010) in deriving their limiting distribution for an M-estimate. On the contrary, we show that when the model is not stationary, the proposed estimates have the same limiting distributions as those of Roknossadati and Zarepour. Furthermore, a Wald test statistic is proposed to consider the test for a linear restriction on the parameter, and it is shown that under a local alternative, the Wald statistic has a non-central chisquared distribution. Simulations and a real data example are also reported to assess the performance of the proposed method.     

Equivalence of Lyapunov stability criteria in a class of Markov decision processes

We are concerned with Markov decision processes with countable state space and discrete-time parameter. The main structural restriction on the model is the following: under the action of any stationary policy the state space is acommunicating class. In this context, we prove the equivalence of ten stability/ergodicity conditions on the transition law of the model, which imply the existence of average optimal stationary policies for an arbitrary continuous and bounded reward function; these conditions include the Lyapunov function condition (LFC) introduced by A. Hordijk. As a consequence of our results, the LFC is proved to be equivalent to the following: under the action of any stationary policy the corresponding Markov chain has a unique invariant distribution which depends continuously on the stationary policy being used. A weak form of the latter condition was used by one of the authors to establish the existence of optimal stationary policies using an approach based on renewal theory.This research was supported in part by the Third World Academy of Sciences (TWAS) under Grant TWAS RG MP 898-152.     

Stationary probability distributions for multiresponse linear learning models

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.