Asymptotic analysis of a large closed queueing network with discriminatory processor sharing

In this paper the steady-state behavior of a closed queueing network with multiple classes and large populations is investigated. One of the two nodes of the network simply introduces random delays and the discipline in the other node is discriminatory processor sharing. The network is not product-form, so not even the steady-state behavior is known. We assume that the usage is moderately heavy, and obtain two-term asymptotic approximations to the mean number of jobs, and the mean sojourn time, of each class of jobs in the processor node. We also obtain the leading term in the asymptotic approximation to the joint distribution of the number of jobs in the processor node, which is a zero-mean multivariate Gaussian distribution around a line through the origin.     

Asymptotic analysis of Emden–Fowler type equation with an application to power flow models

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider.     

The Becker-Döring equations at large times and their connection with the LSW theory of coarsening

A systematic procedure is given for obtaining the asymptotic late-time behavior of the Becker-Döring equations describing the time evolution of a population of clusters of particles. In lowest order of approximation, the distribution of the sizes of the largest clusters satisfies the equations of the Lifshitz-Slyozov-Wagner theory of coarsening.     

Asymptotics of the 1/2-dimension,Discrete Airy Equation and its Approximating Differential Equation

A discrete finite difference model is constructed for the Airy equation using a nonstandard scheme formulated by Mickens and Ramadhani. The method of dominant balance is then applied to obtain a first-order difference equation for the solution that increases sufficiently fast as k→∞. We then calculate the corresponding approximating differential equation and obtain its exact solution as well as its “exact” discrete finite difference representation. The application of various symmetry operations allows the determination of the related rapidly decreasing solution and the oscillatory solutions for negative values of x k>=hk, where h=?x.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

We consider weak solutions of an elliptic equation of the form ? _i ? _i (a _ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a _ij (x) ? δ _ij ) on the annulus B _2r \ B _r is bounded by a function Ω(r), where Ω²(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L ^p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L ^p -mean as r → 0.     

Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

The authors consider a free interface problem which stems from a gas-solid model in combustion with pattern formation. A third-order, fully nonlinear, self-consistent equation for the flame front is derived. Asymptotic methods reveal that the interface approaches a solution to the Kuramoto-Sivashinsky equation. Numerical results which illustrate the dynamics are presented.     

Non-standard asymptotics in an inhomogeneous gamma process

Nonhomogeneous Poisson process (NHPP) is a commonly used stochastic model that is utilized to describe the pattern of repeated occurrence of certain events or conditions. Aninhomogeneous gamma process evolves as a generalization to NHPP, where the observed failure epochs correspond to every successive κ-th event of the underlying Poisson process, κ being an unknown parameter to be estimated from the data. This article focuses on a special class of inhomogeneous gamma process, calledmodulated power law process (MPLP) that assumes the Weibull form of the intensity function. The traditional power law process is a popular stochastic formulation of certain empirical relationships between the time to failure and the cumulative number of failures, often observed in industrial experiments. The MPLP retains this underlying physical basis and provides a more flexible modeling environment potentially leading to a better fit to the failure data at hand. In this paper, we investigate inference issues related to MPLP. The maximum likelihood estimators (MLE’s) of the model parameters are not in closed form and enjoy the curious property that they are asymptotically normal with a singular variance-covariance matrix. Consequently, the derivation of the large-sample results requires non-standard modifications of the usual arguments. We also propose a set of simple closed-form estimators that are asymptotically equivalent to the MLE’s. Extensive simulation results are carried out to supplement the theoretical findings. Finally, we implement our inference results to a failure dataset arising from a repairable system.     

Asymptotics for a Population Size Estimator of a Partially Uncatchable Population

We present the asymptotic distribution for an estimator of the population size for the case of s partially catchable populations. Our approach is useful for capture–recapture studies with photo-identification data where part of the population does not have any distinctive characteristic which allows unique identification of the individuals. This work represents an extension of Theorem 4 in Sanathanan (1972, The Annals of Mathematical Statistics, 43, 142–152).