Exact analysis of asymmetric random polling systems with single buffers and correlated input process

We introduce a simple approach for modeling and analyzing asymmetric random polling systems with single buffers and correlated input process. We consider two variations of single buffers system: the conventional system and the buffer relaxation system. In the conventional system, at most one customer may be resided in any queue at any time. In the buffer relaxation system, a buffer becomes available to new customers as soon as the current customer is being served. Previous studies concentrate on conventional single buffer system with independent Poisson process input process. It has been shown that the asymmetric system requires the solution ofm 2 ^m –1) linear equations; and the symmetric system requires the solution of 2 ^m–1–1 linear equations, wherem is the number of stations in the system. For both the conventional system and the buffer relaxation system, we give the exact solution to the more general case and show that our analysis requires the solution of 2 ^m –1 linear equations. For the symmetric case, we obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, loss probability, throughput, and the expected delay observed by a customer.     

Effect of viscoelastic relaxation modes on stability of extrusion film casting process modeled using multi-mode Phan-Thien-Tanner constitutive equation

Extrusion film casting (EFC) is a commercially important process that is used to produce a significant quantity of polymer films, sheets and coatings for both industrial and household applications. Recently, we have demonstrated the influence of polymer chain architecture on the extent of necking under isothermal as well as non-isothermal film casting operation for commercially relevant polyolefin based materials [1], [2], [3], [4]. In the present research, we focus on another instability that frequently occurs in high-speed EFC process called as draw resonance. Draw resonance manifests itself as an instability that causes periodic fluctuations in both the width as well as thickness of the extruded molten film above a critical draw ratio (DR). In this work, we have carried out a linear stability analysis of the isothermal EFC process using a multi-mode Phan-Thien-Tanner (PTT) constitutive equation to determine the onset of draw resonance. We show that as the number of relaxation modes is increased there is a dramatic change in the stability regions. In particular, there is a marked variation in the stability regions obtained by simulating the multi-mode model and those obtained by taking averaged relaxation time of the modes. Additionally, as the number of faster-relaxing modes in a multi-mode spectrum is progressively increased, the process becomes increasingly stable as the level of elasticity in the melt decreases. Finally, the addition of a long relaxation mode in a multi-mode spectrum is akin to adding a long chain branch to a linear polymer that leads to a reduction in film necking and in many cases to enhanced process stability.     

The output of a buffered data communication system

The output process in a buffer-multiplexer data communication system with N identical input lines and one output line is considered. It is assumed that the rate of transmission on the input lines and on the output line is 1, that idle periods on the input lines follow an exponential distribution, and that active periods are allowed to have an arbitrary distribution function. The main result is that active periods on the output line follow the same distribution function as the busy period in a certain M/G/1 queue. First passage times from any state of the system to complete idleness are also considered and their Laplace transforms are derived.     

Noise-induced effects in nonlinear relaxation of condensed matter systems

Noise-induced phenomena characterise the nonlinear relaxation of nonequilibrium physical systems towards equilibrium states. Often, this relaxation process proceeds through metastable states and the noise can give rise to resonant phenomena with an enhancement of lifetime of these states or some coherent state of the condensed matter system considered. In this paper three noise induced phenomena, namely the noise enhanced stability, the stochastic resonant activation and the noise-induced coherence of electron spin, are reviewed in the nonlinear relaxation dynamics of three different systems of condensed matter: (i) a long-overlap Josephson junction (JJ) subject to thermal fluctuations and non-Gaussian, Lévy distributed, noise sources; (ii) a graphene-based Josephson junction subject to thermal fluctuations; (iii) electrons in a n-type GaAs crystal driven by a fluctuating electric field. In the first system, we focus on the switching events from the superconducting metastable state to the resistive state, by solving the perturbed stochastic sine-Gordon equation. Nonmonotonic behaviours of the mean switching time versus the noise intensity, frequency of the external driving, and length of the junction are obtained. Moreover, the influence of the noise induced solitons on the mean switching time behaviour is shown. In the second system, noise induced phenomena are observed, such as noise enhanced stability (NES) and stochastic resonant activation (SRA). In the third system, the spin polarised transport in GaAs is explored in two different scenarios, i.e. in the presence of Gaussian correlated fluctuations or symmetric dichotomous noise. Numerical results indicate an increase of the electron spin lifetime by rising the strength of the random fluctuating component. Furthermore, our findings for the electron spin depolarization time as a function of the noise correlation time point out (i) a non-monotonic behaviour with a maximum in the case of Gaussian correlated fluctuations, (ii) an increase up to a plateau in the case of dichotomous noise. The noise enhances the coherence of the spin relaxation process.     

Global fluctuations in general Dyson’s Brownian motion

We consider a system of diffusing particles on the real line in a quadratic external potential and with a logarithmic interaction potential. The empirical measure process is known to converge weakly to a deterministic measure-valued process as the number of particles tends to infinity. Provided the initial fluctuations are small, the rescaled linear statistics of the empirical measure process converge in distribution to a Gaussian limit for sufficiently smooth test functions. For a large class of analytic test functions, we derive explicit general formulae for the mean and covariance in this central limit theorem by analyzing a partial differential equation characterizing the limiting fluctuations.     

Asymptotic fluctuations of a particle system with singular interaction

A particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force is considered. The empirical measure process is known to converge weakly in a space of continuous-measure-valued functions. In this work we show that the fluctuations around the limiting process, appropriately scaled, converge weakly to a Gaussian-distribution-valued process.     

Modifying the BFGS update by a new column scaling technique

LetB be a positive definite symmetric approximation to the second derivative matrix of the objective function of a minimization calculation. We study modifications of the BFGS method that apply a scaling technique to the columns of a conjugate direction matrixZ satisfyingZ ^T BZ = I. For a simple scaling technique similar to the ones considered by Powell (1987) and (1989) we show that, due to a two iteration cycle, linear convergence can occur when the method is applied to a quadratic function and Wolfe's line search is employed, although for exact line searches quadratic termination can be proved. We then suggest a different scaling technique that prevents certain columns thought to contain important curvature information from being scaled. For this algorithm we prove global and superlinear convergence and demonstrate the superiority of our method over the BFGS formula on a range of illconditioned optimization problems. Moreover, we present an implementation of our algorithm that requires only 3n ² +O(n) multiplications per iteration.     

Limit process of stationary TASEP near the characteristic line

The totally asymmetric simple exclusion process (TASEP) on\input amssym ${\Bbb Z}$ with the Bernoulli‐ρ measure as an initial condition, 0 < ρ < 1, is stationary. It is known that along the characteristic line, the current fluctuates at an order of t^1/3. The limiting distribution has also been obtained explicitly. In this paper we determine the limiting multipoint distribution of the current fluctuations moving away from the characteristics by the order t^2/3. The main tool is the analysis of a related directed last percolation model. We also discuss the process limit in tandem queues in equilibrium. © 2010 Wiley Periodicals, Inc.     

Extensions to online delay management on a single train line: new bounds for delay minimization and profit maximization

We present extensions to the Online Delay Management Problem on a Single Train Line. While a train travels along the line, it learns at each station how many of the passengers wanting to board the train have a delay of δ. If the train does not wait for them, they get delayed even more since they have to wait for the next train. Otherwise, the train waits and those passengers who were on time are delayed by δ. The problem consists in deciding when to wait in order to minimize the total delay of all passengers on the train line. We provide an improved lower bound on the competitive ratio of any deterministic online algorithm solving the problem using game tree evaluation. For the extension of the original model to two possible passenger delays δ ₁ and δ ₂, we present a 3-competitive deterministic online algorithm. Moreover, we study an objective function modeling the refund system of the German national railway company, which pays passengers with a delay of at least Δ a part of their ticket price back. In this setting, the aim is to maximize the profit. We show that there cannot be a deterministic competitive online algorithm for this problem and present a 2-competitive randomized algorithm.     

Stochastic Modelling of Wind Speed Power Production Correlations

Based on measurements we investigate the velocity-power characteristic of a 2 MW wind turbine. We apply a stochastic analysis where we describe the evolution of the power output with a Langevin equation, with special respect to short-time fluctuations in wind speed. Standard procedures, such as the IEC 61400-12 standard, are limited due to the fact that only mean values over several minutes of wind speed and power output are considered. According to this, short-time dynamics of wind and power fluctuations are usually not taken into account. We introduce an improved method which enables us to extract these dynamics of the power characteristic from the measured data. In particular, we get the response dynamics of the power L (u (t)) via the estimation of Kramers-Moyal coefficients, describing its evolution in time (t) with a Langevin equation where we separate the power output into a relaxation and a noise part. A fixed-point analysis provides the required power characteristic. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enhancing RLT relaxations via a new class of semidefinite cuts

In this paper, we propose a mechanism to tighten Reformulation-Linearization Technique (RLT) based relaxations for solving nonconvex programming problems by importing concepts from semidefinite programming (SDP), leading to a new class of semidefinite cutting planes. Given an RLT relaxation, the usual nonnegativity restrictions on the matrix of RLT product variables is replaced by a suitable positive semidefinite constraint. Instead of relying on specific SDP solvers, the positive semidefinite stipulation is re-written to develop a semi-infinite linear programming representation of the problem, and an approach is developed that can be implemented using traditional optimization software. Specifically, the infinite set of constraints is relaxed, and members of this set are generated as needed via a separation routine in polynomial time. In essence, this process yields an RLT relaxation that is augmented with valid inequalities, which are themselves classes of RLT constraints that we call semidefinite cuts. These semidefinite cuts comprise a relaxation of the underlying semidefinite constraint. We illustrate this strategy by applying it to the case of optimizing a nonconvex quadratic objective function over a simplex. The algorithm has been implemented in C++, using CPLEX callable routines, and two types of semidefinite restrictions are explored along with several implementation strategies. Several of the most promising lower bounding strategies have been implemented within a branch-and-bound framework. Computational results indicate that the cutting plane algorithm provides a significant tightening of the lower bound obtained by using RLT alone. Moreover, when used within a branch-and-bound framework, the proposed lower bound significantly reduces the effort required to obtain globally optimal solutions.     

Relaxation Time for the Discrete D/G/1 Queue

When queueing models are used for performance analysis of some stochastic system, it is usually assumed that the system is in steady-state. Whether or not this is a realistic assumption depends on the speed at which the system tends to its steady-state. A characterization of this speed is known in the queueing literature as relaxation time.The discrete D/G/1 queue has a wide range of applications. We derive relaxation time asymptotics for the discrete D/G/1 queue in a purely analytical way, mostly relying on the saddle point method. We present a simple and useful approximate upper bound which is sharp in case the load on the system is not very high. A sharpening of this upper bound, which involves the complementary error function, is then developed and this covers both the cases of low and high loads.For the discrete D/G/1 queue, the stationary waiting time distribution can be expressed in terms of infinite series that follow from Spitzer’s identity. These series involve convolutions of the probability distribution of a discrete random variable, which makes them suitable for computation. For practical purposes, though, the infinite series should be truncated. The relaxation time asymptotics can be applied to determine an appropriate truncation level based on a sharp estimate of the error caused by truncating.This revised version was published online in June 2005 with corrected coverdate     

Multiparticle space-time transitions in the totally asymmetric simple exclusion process

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinant formula for the generalized Green’s function describing transitions between particle positions at given instants. As an example, we calculate the current correlation function, i.e., the joint probability distribution of times required by each particle to travel a given distance. An asymptotic analysis shows that current fluctuations converge to the Airy ₂ process.     

Dynamical Spin Susceptibility in the <Emphasis Type="Italic">t-J</Emphasis> Model: The Memory Function Method

Based on the method of the equations of motion for the relaxation function in terms of Hubbard operators, we evaluate the dynamical spin susceptibility for the t-J model in the paramagnetic phase. Using a Mori-type projection technique, we express the relaxation function in terms of the second-order memory function, which is evaluated in the approximation of coupled modes for hole excitations and spin fluctuations in the fourth order in the hopping parameter t and the exchange interaction J. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 240–255, November, 2005.     

Transition between Airy1 and Airy2 processes and TASEP fluctuations

We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions, starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy₁ and Airy₂ processes, whose one‐point distributions are the GOE and GUE Tracy‐Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one‐point distribution is a new interpolation between GOE and GUE edge distributions. © 2007 Wiley Periodicals, Inc.     

Forced vibrations of an oscillator in the presence of dry friction

We study the vibrations of an oscillator with two degrees of freedom in the presence of dry friction. We compare the nature of the damping of the free oscillations in a straight line with the general case. For the forced vibrations we determine the way in which the critical values of friction at which there exist periodic motions depend on the parameters of external action in resonance mode.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 49–54.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study the small fluctuations between the original system around its average. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type Gaussian process, which is a form of the functional central limit theorem. Furthermore, sharp rates for the above convergences are also obtained, and these convergences are shown to not depend on the regularities of the coefficients with respect to the fast variable, which reflect the effects of noises on the multi-scale systems.

     

Emergent dynamics of the first‐order stochastic Cucker‐Smale model and application to finance

In this paper, we study stochastic aggregation properties of the financial model for the N‐asset price process whose dynamics is modeled by the weakly geometric Brownian motions with stochastic drifts. For the temporal evolution of stochastic components of drift coefficients, we employ a stochastic first‐order Cucker‐Smale model with additive noises. The asset price processes are weakly interacting via the stochastic components of drift coefficients. For the aggregation estimates, we use the macro‐micro decomposition of the fluctuations around the average process and show that the fluctuations around the average value satisfies a practical aggregation estimate over a time‐independent symmetric network topology so that we can control the differences of drift coefficients by tuning the coupling strength. We provide numerical examples and compare them with our analytical results. We also discuss some financial implications of our proposed model.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances

We deal with the stationary acoustic waves propagating in a cluster of small particles enjoying high contrasts. Such contrasts allow the appearance of (complex valued) resonances that are close to the real line as the size of the particles becomes small. For single (but not necessarily small) particles, we derive the characteristic equation that generates a class of these resonances (the ones for which the corresponding eigenfunctions are uniformly constant). For multiple and small particles, we provide sufficient conditions on the contrasts that generates quasi‐resonances for which the corresponding eigenfunctions are uniformly constant. Precisely, we show that, if we distribute the particles on a uniform line, then the existence of such quasi‐resonances is related to the eigenvalues of the Harary matrix. To show these results, we take, as the small contrasted particles, small obstacles with high surface impedances λ of the form λ: = βa^?1 ? αβa^{?1 + h} where a is the maximum radi of the particles, with a < <1, and β is a universal and positive constant depending only on the shape of the particles (but not on their size). In this case, if the relative constant α is an eigenvalue of the Harary matrix, then the used frequency is a quasi resonance of the cluster of the small particles where the error of approximation is of the order for h ∈ (0,1) as a < <1.