Use Of Diffusion Approximation Research Articles

Performance Analysis of Packet Aggregation Mechanisms and Their Applications in Access (e.g., IoT, 4G/5G), Core, and Data Centre Networks.

The transmission of massive amounts of small packets generated by access networks through high-speed Internet core networks to other access networks or cloud computing data centres has introduced several challenges such as poor throughput, underutilisation of network resources, and higher energy consumption. Therefore, it is essential to develop strategies to deal with these challenges. One of them is to aggregate smaller packets into a larger payload packet, and these groups of aggregated packets will share the same header, hence increasing throughput, improved resource utilisation, and reduction in energy consumption. This paper presents a review of packet aggregation applications in access networks (e.g., IoT and 4G/5G mobile networks), optical core networks, and cloud computing data centre networks. Then we propose new analytical models based on diffusion approximation for the evaluation of the performance of packet aggregation mechanisms. We demonstrate the use of measured traffic from real networks to evaluate the performance of packet aggregation mechanisms analytically. The use of diffusion approximation allows us to consider time-dependent queueing models with general interarrival and service time distributions. Therefore these models are more general than others presented till now.

Stochastic Diffusion Model for Analysis of Dynamics and Forecasting Events in News Feeds

One of the problems of forecasting events in news feeds, is the development of models which allow for work with semi structured information space of text documents. This article describes a model for forecasting events in news feeds, which is based on the use of stochastic dynamics of changes in the structure of non-stationary time series in news clusters (states of the information space) on the basis of use of diffusion approximation. Forecasting events in a news feed is based on their text description, vectorization, and finding the cosine value of the angle between the given vector and the centroids of various information space semantic clusters. Changes over time in the cosine value of such angles between the above vector and centroids can be represented as a point wandering on the [0, 1] segment. This segment contains a trap at the event occurrence threshold point, which the wandering point may eventually fall into. When creating the model, we have considered probability patterns of transitions between states in the information space. On the basis of this approach, we have derived a nonlinear second-order differential equation; formulated and solved the boundary value problem of forecasting news events, which allowed obtaining theoretical time dependence on the probability density function of the parameter distribution of non-stationary time series, which describe the information space evolution. The results of simulating the events instance probability dependence on time (with sets of parameter values of the developed model, which have been experimentally determined for already occurred events) show that the model is consistent and adequate (all the news events which have been used for the model verification occur with high values of probability (within the order of 80%), or if these are fictitious events, they can only occur over the course of inadmissible long time).

Revisiting the time until fixation of a neutral mutant in a finite population – A coalescent theory approach

Evaluation of the time scale of the fixation of neutral mutations is crucial to the theoretical understanding of the role of neutral mutations in evolution. Diffusion approximations of the Wright–Fisher model are most often used to derive analytic formulations of genetic drift, as well as for the time scales of the fixation of neutral mutations. These approximations require a set of assumptions, most notably that genetic drift is a stochastic process in a continuous allele-frequency space, an assumption appropriate for large populations. Here equivalent approximations are derived using a coalescent theory approach which relies on a different set of assumptions than the diffusion approach, and adopts a discrete allele-frequency space. Solutions for the mean and variance of the time to fixation of a neutral mutation derived from the two approaches converge for large populations but slightly differ for small populations. A Markov chain analysis of the Wright–Fisher model for small populations is used to evaluate the solutions obtained, showing that both the mean and the variance are better approximated by the coalescent approach. The coalescence approximation represents a tighter upper-bound for the mean time to fixation than the diffusion approximation, while the diffusion approximation and coalescence approximation form an upper and lower bound, respectively, for the variance. The converging solutions and the small deviations of the two approaches strongly validate the use of diffusion approximations, but suggest that coalescent theory can provide more accurate approximations for small populations.

On parameter estimation in population models III: Time-inhomogeneous processes and observation error

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the population size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Continuing previous work in this series on the use of diffusion approximations for efficient calibration of continuous-time Markov chains, I present efficient techniques for time-inhomogeneous chains and accounting for observation error. Observation error (partial observability) is accounted for by joint estimation using a scaled unscented Kalman filter for state-space models. The methodology will be illustrated with respect to models of disease dynamics incorporating seasonal transmission rate and in the presence of observation error, including application to two influenza outbreaks and measles in London in the pre-vaccination era.

Nondiffusion Theory of Electric Probes

A mathematical model is suggested for the calculation of ion current to probes of different configurations in a locally collisional nonisothermal plasma. Compared with the diffusion models, the suggested model includes additionally the ion inertia, the ionization friction, and the effect of directional velocity of ions on the frequency of their collisions with atoms. The procedure for solving this nonlinear boundary-value problem enables one to determine the value of ion current, the distribution of ion concentration and velocity in the plasma, and the size of the positive space-charge layer. The basic dimensionless parameters of the problem are revealed, and their effect on the pattern of solutions is studied. It is demonstrated that the use of diffusion approximation for the calculation of the values of ion concentration and velocity on the plasma boundary leads, as a rule, to an error in the order of magnitude of the quantities being determined. The error in determining the ion current within diffusion approximation is likewise considerable; therefore, the use of the nondiffusion model helps increase the accuracy of determining the plasma parameters.

Asymptotic solutions of neutron transport equation and the limits of correct use of diffusion approximation for rocks

The diffusion approximation solution for neutron transport has been used in well-logging geophysics for calculating tool responses in boreholes, sometimes with success. The problem of the dimension of different materials to which it can be applied with success is important for the borehole environment. The results obtained show that the diffusion approximation can be used for distances greater than a few millimetre in some rock types. For iron, barium, and other highly absorbing media the use of the diffusion approximation is inappropriate even for large distances.

Transition events fluctuations in jump markovian dynamics. A general approach to Šolc's problem

We introduce a new formalism for computing the moments of transition events for nonhomogeneous Markov jump processes. Our method is applied directly to the master equation and does not involve the use of diffusion approximation. The general theory is applied to produce exact expressions for means and dispersions. For time homogeneous Markov processes with a finite number of connected states we are able to prove that both means and dispersions asymptotically increase linearly in time.

A singular perturbation approach to first passage times for Markov jump processes

We introduce singular perturbation methods for constructing asymptotic approximations to the mean first passage time for Markov jump processes. Our methods are applied directly to the integrai equation for the mean first passage time and do not involve the use of diffusion approximations. An absorbing interval condition is used to properly account for the possible jumps of the process over the boundary which leads to a Wiener-Hopf problem in the neighborhood of the boundary. A model of unimolecular dissociation is considered to illustrate our methods.

Probabilistic models of computer systems

In this paper we examine certain problems related to the use of diffusion approximations for the approximate modelling of computer systems. In particular we develop a model which allows us to handle waiting times and batch arrivals: these results are a new approach to the use of diffusion approximations. We also examine the effect of the distribution of holding times at boundaries: this question had remained open in earlier studies. We show that the stationary distributions associated with these diffusion models depend only on the average residence time of the process on the boundaries and not on the complete distribution function. This result justifies the use of exponential holding times as had been done in an earlier study.

Probabilistic models of computer systems

In this paper we examine certain problems related to the use of diffusion approximations for the approximate modelling of computer systems. In particular we develop a model which allows us to handle waiting times and batch arrivals: these results are a new approach to the use of diffusion approximations. We also examine the effect of the distribution of holding times at boundaries: this question had remained open in earlier studies. We show that the stationary distributions associated with these diffusion models depend only on the average residence time of the process on the boundaries and not on the complete distribution function. This result justifies the use of exponential holding times as had been done in an earlier study.

A Mendelian Markov process with binomial transition probabilities

In any finite genetic population, the effect of chance involved in the sampling by which one generation replaces another causes gene frequency change as long as fixation or extinction is not achieved. In genetic terminology this fluctuation in gene frequency due to the finiteness of a population is called 'random genetic drift'. Fisher (1922), using an equation of the diffusion type, was the first to make an effort to describe quantitatively the phenomenon of random genetic drift. In the corrected version of this work (1930), he found the general result that the rate of decrease of the genetic variance in the population due to drift was 1/(2N) per generation, where N is the number of individuals of mating potential. This result was confirmed by various authors using different methods (e.g. Wright, 1931, 1937; Malecot, 1944; Feller, 1950). The particular form of the diffusion model which has had the widest use was introduced into genetical literature by Wright (1945). This equation which plays a fundamental role in the theory of gene frequencies is the Kolmogorov forward equation or-as it is commonly known among physicists-the Fokker-Planck equation. Earlier Kolmogorov (1935) had introduced the steady-state form of the forward equation in genetics. The complete solutions to the problem, however, were obtained much later and are due to Kimura (1955, 1957). The problem of random genetic drift as a problem in finite Markov chains was first considered by Malecot (1944). Based on the largest non-unit characteristic root of the transition probability matrix Malecot obtained the asymptotic rate of decrease of heterozygosity. This is essentially the same as the previously known results due to Fisher and Wright for steady decay. Feller (1951) succeeded in finding a general expression for all the characteristic roots of the transition probability matrix. The use of diffusion approximations requires the assumption of an infinite population: while in fact the state space is discrete, it is treated as though it were continuous. The tacit assumption in such an approach is that for large population size the approximation of the discrete time, discrete space process by the continuous diffusion process does not result in serious error. It was only recently that a rigorous justification of the diffusion approach was given by Watterson (1962) with respect to the processes considered by Wright (1945) and Kimura (1954, 1955). The validity of the diffusion approximation as providing adequate numerical solution to the problem concerning the probability that a particular gene is lost or fixed by a certain time was one of the questions considered by Ewens (1963). Since exact expressions were not available his method of getting exact results consisted in simply powering the transition matrix using a high-speed computer and collecting the appropriate terms.