Stiffness-tolerant methods for transient analysis of stiff Markov chains

Various candidate sources of error for the discrepancy between previous direct numerical simulation (DNS) results of Wu et al. and experimental data of Bookey et al. are investigated. Deficiencies of current numerical methods for DNS of shockwave/turbulent boundary layer interactions (STBLI) are found. The weighted-essentially-non-oscillatory (WENO) method is improved by adding limiters in the smoothness measurement. In turn, the numerical dissipation is significantly reduced. For the compression ramp case, using the limiter results in increased size of the separation bubble and improved mean wallpressure distribution. In addition, a fourth-order low-dissipation Runge-Kutta method is found to give less dissipation than the typically used third-order low-storage Runge-Kutta method. Combining the limiter with the low-dissipation Runge-Kutta method gives better results. Good agreement is found for the size of the separation bubble and the wall-pressure distribution when comparing against the experimental data at the same conditions.

A high-order multi-zone cut-stencil method for numerical simulations of high-speed flows over complex geometries

The production of active carbon from bituminous coal

We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number X of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process {Zn}, where Zn is the number of PCU’s in a queue at the beginning of the n-th red phase, n → ∞. We approximate the number Yn of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that {Zn} follow an infinite-state Markov chain. We approximate its stationary distribution using a finite-state Markov chain. We show numerically that there is a strong dependence of the mean queue length E[Zn] in equilibrium on the input distribution of Yn and, in particular, on the ”over dispersion” parameter γ= Var[Yn ]/E[Yn ]. For Poisson input, γ = 1. γ > 1 indicates presence of heavy-tailed input. In reality it means that a relatively large ”portion” of PCU’s, considerably exceeding the average, may arrive with high probability during one red-green cycle. Empirical formulas are presented for an accurate estimation of mean queue length as a function of load and g of the input flow. Using the Markov chain technique, we analyze the mean ”virtual” delay time for a car which always arrives at the beginning of the red phase.

Numerical analysis of rational processes beyond Markov chains

The study and development of high-order discretization techniques for the modeling and simulation of multiphase flows in petroleum reservoirs are still a challenge from the computational viewpoint due to the difficulty posed by some physical features such as heterogeneity and anisotropy of the medium, that are of paramount importance in this class of applications. In this paper, we present a preliminary investigation of the very high order Spectral Finite Volume Method (SFVM) for the discretization of the saturation equation that describes the 1-D Oil-Water displacement through porous media. In this equation, the hyperbolic term is discretized using a high order SFVM, the capillary term is approximated using a second order CVFD (Control Volume Finite Difference) method and the integration in time is carried out using a third-order Runge–Kutta method. On the other hand, to prevent numerical oscillations near shocks, that are typical from higher order schemes, a slope limiter procedure is used in the reconstruction stage. To evaluate the performance of the SFVM some simple model problems, including capillarity and gravity, are solved, yielding accurate and efficient results. The use of very high order formulations can be of utmost importance, particularly for multidimensional and real large-scale applications with complex wave structures.

In this article, performance of sliding frame (SF) R-ALOHA protocols for real-time distributed wireless networks is investigated analytically and by simulation. First, a discrete Markov chain and Monte Carlo modeling are constructed to evaluate dynamic behavior of the protocol in transient state including distribution of the system stabilization time (SST) and the average number of successful terminals. Furthermore, a hierarchical decomposition is conducted to simplify steady state analysis, thus generating two one-dimensional Markov chains for closed-form performance of SF R-ALOHA under local wireless environment. Terminals with Poisson message arrivals and Poisson message length are analyzed. Consequently, performance indices, such as throughput, the average message delay, and packet dropping probability, are derived from the proposed analytic models. Capture effects on the SF R-ALOHA system in the presence of multipath and shadowing are also obtained by the Markov chain analysis. The numerical results from the analytic models are compared with that from simulation and equilibrium point analysis, proving correctness, accuracy, and scalability of the decomposition. The results also reveal performance characteristics of the SF R-ALOHA system.

Properties of results obtained on Markov processes analysis in medical applications have been illustrated. Calculation was done for measurement of Thoracic Fluid Content by means of impedance cardiograph for chosen medical patients population. Main medical aspect of Markov chain calculation has been underlined to exhibit some constraints in decision making with such math models. Markov chain analysis was done as optimization task. Effective algorithm for irregular data in continuous time has been proposed.

Experimental Investigation with Steady-State Detection in a Micro-ORC Test Bench

This article extends the theory of classical finite-difference\nsummation-by-parts (FD-SBP) time-marching methods to the generalized\nsummation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods\nare shown to retain: A and L-stability, as well as superconvergence of integral\nfunctionals when integrated with the quadrature associated with the\ndiscretization. This also implies that the solution approximated at the end of\neach time step is superconvergent. In addition GSBP time-marching methods\nconstructed with a diagonal norm are BN-stable. This article also formalizes\nthe connection between FD-SBP/GSBP time-marching methods and implicit\nRunge-Kutta methods. Through this connection, the minimum accuracy of the\nsolution approximated at the end of a time step is extended for nonlinear\nproblems. It is also exploited to derive conditions under which nonlinearly\nstable GSBP time-marching methods can be constructed. The GSBP approach to time\nmarching can simplify the construction of high-order fully-implicit Runge-Kutta\nmethods with a particular set of properties favourable for stiff initial value\nproblems, such as L-stability. It can facilitate the analysis of fully discrete\napproximations to PDEs and is amenable to to multi-dimensional spcae-time\ndiscretizations, in which case the explicit connection to Runge-Kutta methods\nis often lost. A few examples of known and novel Runge-Kutta methods associated\nwith GSBP operators are presented. The novel methods, all of which are L-stable\nand BN-stable, include a four-stage seventh-order fully-implicit method, a\nthree-stage third-order diagonally-implicit method, and a fourth-order\nfour-stage diagonally-implicit method. The relative efficiency of the schemes\nis investigated and compared with a few popular non-GSBP Runge-Kutta methods.\n

Irrespective of whether the test for homogeneity is significant or not, most researchers assume time-homogeneity in analysing Markov chains due to scanty literature on the analysis of time-inhomogeneous Markov chains. Based on the assumption that, for each point in time in the future, a stochastic process will be subjected to a randomly selected transition matrix from an ergodic set of transition matrices the process was subjected to in the recent past, a methodology was proposed for analysing the long-run behaviours of time-inhomogeneous Markov chains. The proposed model was implemented to historical data consisting of the exchange rate of cedi-dollar, cedi-pound, and cedi-euro spanning over 6 years (January 2012 to December 2017). The results show that under certain “closeness” conditions, the long-run behaviours of the time-inhomogeneous case are almost identical to those of the time-homogeneous case. The paper asserted that even if the Markov chain exhibit time-inhomogeneity, analysing the Markov chain under the assumption of time-homogeneity is a step in the right direction under certain “closeness” conditions; otherwise, the proposed method is recommended. It was also found that investing in dollars yields better returns than the other currencies in Ghana.

The M/M/k/setup model, where there is a penalty for turning servers on, is common in data centers, call centers and manufacturing systems. Setup costs take the form of a time delay, and sometimes there is additionally a power penalty, as in the case of data centers. While the M/M/1/setup was exactly analyzed in 1964, no exact analysis exists to date for the M/M/k/setup with k>1. In this paper we provide the first exact, closed-form analysis for the M/M/k/setup and some of its important variants including systems in which idle servers delay for a period of time before turning off or can be put to sleep. Our analysis is made possible by our development of a new technique, Recursive Renewal Reward (RRR), for solving Markov chains with a repeating structure. RRR uses ideas from renewal reward theory and busy period analysis to obtain closed-form expressions for metrics of interest such as the transform of time in system and the transform of power consumed by the system. The simplicity, intuitiveness, and versatility of RRR makes it useful for analyzing Markov chains far beyond the M/M/k/setup. In general, RRR should be used to reduce the analysis of any 2-dimensional Markov chain which is infinite in at most one dimension and repeating to the problem of solving a system of polynomial equations. In the case where all transitions in the repeating portion of the Markov chain are skip-free and all up/down arrows are unidirectional, the resulting system of equations will yield a closed-form solution.

Markov chain sampling has recently received considerable attention, in particular in the context of Bayesian computation and maximum likelihood estimation. This article discusses the use of Markov chain splitting, originally developed for the theoretical analysis of general state-space Markov chains, to introduce regeneration into Markov chain samplers. This allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance. The approach is applied to several samplers, including certain Metropolis samplers that can be used on their own or in hybrid samplers, and is illustrated in several examples.

A technique is presented, which enables the state space of a Harris recurrent Markov chain to be “split” in a way, which introduces into the split state space an “atom”. Hence the full force of renewal theory can be used in the analysis of Markov chains on a general state space. As a first illustration of the method we show how Derman's construction for the invariant measure works in the general state space. The Splitting Technique is also applied to the study of sums of transition probabilities.

Frequency evaluation for exponentially fitted Runge–Kutta methods