Stationary Joint Research Articles

Martingale approach for tail asymptotic problems in the gener­alized Jackson network

MARTINGALE APPROACH FOR TAIL ASYMPTOTIC PROBLEMS IN THE GENERALIZED JACKSON NETWORKWe study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network GJN for short, assumingits stability. For the two-station case, this problem has recently been solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase-type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.

Pattern Storage, Bifurcations, and Groupwise Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model.

Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

Triboacoustic diagnostic fixed joints of machines

The experimental results of the tribo-acoustic control in real time the integrity of nominally stationary friction joints under conditions of fretting are presented. The disturbance in the integrity of the tribosuppression as a result of wear has been reliably detected both by the change in the shape of the acoustic histograms of the operating noise and by the change in the noise spectrum due to the appearance of strong mid-frequency modes of the contact gap. The appearance of two oppositely directed wave energy cascades in the regime of fretting has been observed: a direct cascade from mod the pump in the direction of high frequencies and the opposite – side to the low infrasonic frequencies. The former has been associated with microplastic deformations in the zones of actual contact, while the latter has been associated with the dynamic generation of slow angular motions in the friction system.

Stochastic model of deteriorating cargo transshipment at port’s terminal under irregular arrival of ships

In the article the stochastic model is analyzed for deteriorating cargo transshipment taking into account the irregularity of arrived ships movement and cargo removing from warehouse with the regular surface transport. It is assumed that unloading rate of ships is given and fixed, but hold capacities of ships are randomly varying. During storage the cargo is subjected to deterioration with the given intensity. The problem of stationary probabilistic joint distribution of number of ships in terminal and quantity of cargo in warehouse is investigated. For this purpose the theory of Markov drift processes and methods of queueing theory are used. The problem of optimal value of deterioration rate is formulated and solved.

Tail asymptotics for a batch service polling system with retrials and nonpersistent customers

Perel and Yechiali 2014 [16] considered a multi-queue single-server retrial polling system with batch service of an unlimited size, i.e., the so called “Israeli queue” with retrial, where the system consists of a main queue and an orbit queue, and only the customer at the head of the orbit queue is allowed to try to access the main queue. In this present paper, we aim to give a further study on this queueing model, in which each of the retrial customers in the orbit queue independently repeatedly tries for receiving service, and customers may abandon at the arrival instant from outside or at the departure epoch from the orbit queue. By analysing this model, we find that it is difficult to obtain an explicit closed form solution for the joint stationary probability distribution of the number of retrial customers in the orbit queue and the number of groups in the main queue. Therefore, by making use of the matrix analytic approach and the censoring technique, we derive the tail asymptotics result for the stationary joint probabilities.

Fast computation of stationary joint probability distribution of sparse Markov chains

In this paper, we study a fast algorithm for finding stationary joint probability distributions of sparse Markov chains or multilinear PageRank vectors which arise from data mining applications. In these applications, the main computational problem is to calculate and store solutions of many unknowns in joint probability distributions of sparse Markov chains. Our idea is to approximate large-scale solutions of such sparse Markov chains by two components: the sparsity component and the rank-one component. Here the non-zero locations in the sparsity component refer to important associations in the joint probability distribution and the rank-one component refers to a background value of the solution. We propose to determine solutions by formulating and solving sparse and rank-one optimization problems via closed form solutions. The convergence of the truncated power method is established. Numerical examples of multilinear PageRank vector calculation and second-order web-linkage analysis are presented to show the efficiency of the proposed method. It is shown that both computation and storage are significantly reduced by comparing with the traditional power method.

Iterative Approach to Calculating the Order Fill Rate Under Purchase Dependence

For the business environment under purchase dependence that characterizes customer purchase patterns observed in such areas as marketing, manufacturing systems, and distribution systems, this paper proposes a new approximate approach to calculate the order fill rate, probability of filling an entire customer order immediately from the shelf. The new approximate approach divides customer orders into item orders and calculates fill rates of all order types to approximate the order fill rate. In order to avoid the curse of dimensionality and prevent the solution from diverging for the large instances, we develop the greed iterative search algorithm (GISA) based on the Gauss-Seidel method. Through the computational analysis that compares the GISA with the simulation, we demonstrate that the GISA is a dependable algorithm to derive the stationary joint distribution of on-hand inventories in the type-K pure system. In addition, we present some managerial insights.

A TWO-CLASS RETRIAL SYSTEM WITH COUPLED ORBIT QUEUES

We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, typeicustomer,i=1, 2, is routed to a separate typeiorbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queueitries to re-dispatch a blocked customer of typeito the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μito$\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

On the analytical approximation of joint aggregate queue-length distributions for traffic networks: A stationary finite capacity Markovian network approach

This paper is motivated by recent results in the design of signal plans for Manhattan that highlight the importance of providing signal control algorithms with an analytical description of between-link dependencies. This is particularly important for congested networks prone to the occurrence of spillbacks. This paper formulates a probabilistic network model that proposes an aggregate description of the queue-length, and then approximates the joint aggregate queue-length distribution of subnetworks. The goal is to model between-queue dependencies beyond first-order moments, yet to do so in a tractable manner such that these techniques can be used for optimization purposes.This paper models an urban road network as a finite space capacity Markovian queueing network. Exact evaluation of the stationary joint queue-length distribution of such a network with arbitrary size and topology can be obtained numerically. Nonetheless, the main challenge to such an approach remains the dimensionality of the network state space, which is exponential in the number of queues. This paper proposes to address the dimensionality issue by: 1) describing the state of the network aggregately, and 2) decomposing the network into overlapping subnetworks. We propose an analytical approximation of the stationary aggregate joint queue-length distribution of a subnetwork. The model consists of a system of nonlinear equations with a dimension that is linear, instead of exponential, in the number of queues and that is independent of the space capacity of the individual queues. The method is derived for tandem Markovian finite capacity queueing networks. The proposed model is computationally tractable and scalable, it can be efficiently used for the higher-order distributional analysis of large-scale networks. The model is validated versus simulation estimates and versus other decomposition methods. We then use it to address an urban traffic control problem. We show the added value of accounting for higher-order spatial between-queue dependency information in the control of congested urban networks.

Multi-class M/PH/1 queues with deterministic impatience times

ABSTRACTThis article considers computational procedures for the waiting time and queue length distributions in stationary multi-class first-come, first-served single-server queues with deterministic impatience times. There are several classes of customers, which are distinguished by deterministic impatience times (i.e., maximum allowable waiting times). We assume that customers in each class arrive according to an independent Poisson process and a single server serves customers on a first-come, first-served basis. Service times of customers in each class are independent and identically distributed according to a phase-type distribution that may differ for different classes. We first consider the stationary distribution of the virtual waiting time and then derive numerically feasible formulas for the actual waiting time distribution and loss probability. We also analyze the joint queue length distribution and provide an algorithmic procedure for computing the probability mass function of the stationary joint queue length.

Human Arm Motion Tracking by Orientation-Based Fusion of Inertial Sensors and Kinect Using Unscented Kalman Filter.

Due to various applications of human motion capture techniques, developing low-cost methods that would be applicable in nonlaboratory environments is under consideration. MEMS inertial sensors and Kinect are two low-cost devices that can be utilized in home-based motion capture systems, e.g., home-based rehabilitation. In this work, an unscented Kalman filter approach was developed based on the complementary properties of Kinect and the inertial sensors to fuse the orientation data of these two devices for human arm motion tracking during both stationary shoulder joint position and human body movement. A new measurement model of the fusion algorithm was obtained that can compensate for the inertial sensors drift problem in high dynamic motions and also joints occlusion in Kinect. The efficiency of the proposed algorithm was evaluated by an optical motion tracker system. The errors were reduced by almost 50% compared to cases when either inertial sensor or Kinect measurements were utilized.

Random response of dielectric elastomer balloon to electrical or mechanical perturbation

Random response of dielectric elastomer balloon disturbed by electrical or mechanical fluctuation is analytically investigated in this article. The stochastic differential equation governing the oscillating behavior around the stable equilibrium position is first derived by introducing the translation transformation. The stationary joint probability density about the disturbing stretch ratio and its rate of change is analytically established by adopting the stochastic averaging of energy envelope; the statistics quantities, such as the mean value and the standard deviation of the disturbing stretch ratio, are then subsequently calculated. Two special cases, the first case with only voltage fluctuation and the second one with only pressure variation, are discussed in detail, and the random response properties are summarized. The accuracy of the analytical solution is verified by comparing with the Monte Carlo simulation for the perturbation with weak intensity, and the valid ranges for the mean voltage and pressure are illustrated. This work provides an effective technique to evaluate the detection precision of new type of sensors based on the dielectric elastomer.

GPU computing for accelerating the numerical Path Integration approach

The paper discusses a novel approach of accelerating the numerical Path Integration method, used for generating a stationary joint response probability density function of a dynamic system subjected to a random excitation, by the GPU computing. The paper proposes the parallelization of nested loops technique and demonstrates the advantages of GPU computing. Two, three and four dimensional in space problems are investigated as a part of the pilot project and the achieved maximum accelerations are reported. Three degree-of-freedom system (6D) is approached by the Path Integration technique for the first time. The application of the proposed GPU methodology for problems of stochastic dynamics and reliability are discussed.

Investigations on the bifurcation of a noisy Duffing–Van der Pol oscillator

This study carried out investigations on the bifurcation characteristics of a Duffing–Van der Pol (DVDP) oscillator subjected to white noise excitations. Dynamical (or D-) bifurcations are characterised by dramatic changes in the dynamical behaviour leading to topological changes in the phase portrait. An additional mode of bifurcation - phenomenological (or P-) bifurcation, is observed in stochastically excited systems when the stationary joint probability density function of the state variables undergo topological changes in the probability space. While D-bifurcation analysis is quantified in terms of the sign changes in the largest Lyapunov exponent, P-bifurcation analysis is usually qualitative and through visual inspection. In this study, a new quantitative measure for P-bifurcations based on the Shannon entropy is proposed. A comparison of the parameter regimes of the noisy DVDP oscillator identified by the three bifurcation methodologies have been presented.

Exact gap statistics for the random average process on a ring with a tracer

We study the statistics of the gaps in the random average process on a ring with particles hopping symmetrically, except one tracer particle which could be driven. These particles hop either to the left or to the right by a random fraction η of the space available till next particle in the respective directions. The random fraction is chosen from a distribution . For a non-driven tracer, when satisfies a necessary and sufficient condition, the stationary joint distribution of the gaps between successive particles takes a universal form that is factorized except for a global constraint. Some interesting explicit forms of are found which satisfy this condition. In the case of a driven tracer, the system reaches a current-carrying steady state where such factorization does not hold. Analytical progress has been made in the thermodynamic limit, where we computed the single site distribution inside the bulk. We have also computed the two point gap–gap correlation exactly in that limit. Numerical simulations support our analytical results.

On the three-queue priority polling system with threshold service policy

In this paper, a priority polling system consisting of three M / M / 1 queues, served by a single server is investigated. Queue 1 has the Head-of-Line (HoL) priority and Queue 2 has a higher priority over Queue 3 with threshold N. All the switches are instantaneous and preempting. Using the Kernel method we derive the probability of generating functions of the stationary joint queue-length distributions, which yields the mean queue lengths and the mean sojourn times. Furthermore, we consider the limit behaviors in the light-traffic and heavy-traffic scenarios. And an interpolation approximation for the sojourn times utilizing the light and heavy traffic limits are illustrated. To test the validity, we also undertake some simulation works.

Numerical simulation for Human Quiet Standing System

Human quiet standing system was investigated with a 2-joint (ankle and hip) sagittal model of body movement with perturbation noise. In the case of the noise perturbations, the influence on the stability of time delay is discussed numerically in this paper. Analytical analyses on the mean of the station response reveal that the presence of time delay leads to periodic vibration in parameter space, and the mean value increases as the reflexes delay increasing. But the mean square of the stationary response varies with not only the noise but also the delay. We find the stochastic factor has no affection on the stability of human, but it will make the amplify oscillations diffusion, which can be seen from the shape change of the figure on the stationary joint probability density and the stationary probability density. All these conclusions are shown by the numerical simulation. In conclusion, the influence of reflexes delay and perturbation on the stability and quality of stance has been studied by successfully combined with the biomechanical concepts and tools..

Analytical approach to an integrate-and-fire model with spike-triggered adaptation.

The calculation of the steady-state probability density for multidimensional stochastic systems that do not obey detailed balance is a difficult problem. Here we present the analytical derivation of the stationary joint and various marginal probability densities for a stochastic neuron model with adaptation current. Our approach assumes weak noise but is valid for arbitrary adaptation strength and time scale. The theory predicts several effects of adaptation on the statistics of the membrane potential of a tonically firing neuron: (i) a membrane potential distribution with a convex shape, (ii) a strongly increased probability of hyperpolarized membrane potentials induced by strong and fast adaptation, and (iii) a maximized variability associated with the adaptation current at a finite adaptation time scale.

Plant biomass and soil moisture dynamics: analytical results

In semi-arid ecosystems, successful use of the limited water resources is of central importance in determining the evolutionary trends of the vegetation. The competition between different species and individuals for this resource is driven by variations in physiology and metabolic regulation strategies, expressed by such parameters as rooting depth, wilting point or stomatal opening, among others. It is typically not practical to simulate the full evolutionary dynamics of every plant individual in the landscape because of the difficulties introduced by the spatial heterogeneity, as well as the many timescales involved, ranging from hourly up to intergenerational. Instead, the amount of biomass of a given species assimilated per unit area of the landscape may serve as a proxy for its competitiveness and evolutionary success. It is the behaviour of the biomass, which must be described probabilistically due to the stochasticity of the rainfall, which is the subject here. This paper develops a new analytical description of the stationary and transient joint behaviour of plant biomass and soil moisture. Additionally, the effects of climatic fluctuations are considered, including the important case of a bi-seasonal climate regime consisting of alternating wet and dry seasons, which is characteristic of many ecosystems of interest.

Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits

We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type i customer $$(i=1,2)$$ finds a busy server, it will join the type i orbit. After an exponential time with a constant (retrial) rate $$\mu _i$$ , a type i customer attempts to get service. This model has been recently studied by Avrachenkov et al. (Queueing Syst 77(1):1–31, 2014) by solving a Riemann–Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.