Class dependent departure process from multiclass phase queues: Exact and approximate analyses

In this paper we study class dependent departure processes from phase type queues. When the arrival process for a subset of the classes is a Poisson process, we determine the Laplace-Stieltjes transform of the stationary inter-departure times of the combined output of all the other classes. We also propose and test approximations for the squared coefficient of variation of the stationary inter-departure times of each customer class. The approximations are based on the detailed structure of the second order measures of the aggregate departure process. Finally, we propose renewal approximations for the class dependent departure process that take into account the utilization of the queue that customers next visit.     

On moment measures of departure from the normal and exponential laws

For scale mixtures of distributions it is possible to prescribe simple moment measures of distance. In the case of departure from the normal and exponential laws of scale mixtures of the normal and exponential, these distances may be taken as the kurtosis and half the squared coefficient of variation minus one respectively. In this paper these measures of distance are exhibited as bounds on the uniform metric for the distance between distribution functions. The results considerably sharpen earlier results of a similar character in [2].     

Comparison of Inhomogeneous Poisson Processes

This note illustrates the use of various partial orderings for point processes by comparing inhomogeneous Poisson processes, and studies the relationships among these orderings mainly in the ease of inhomogeneous Poisson processes. Whitt has detailed several definitions of partial orderings which have been applied mainly to renewal processes (see, e. g., Whitt [14] and Miller[4]) and semi-Markov proeesse (see, e. g., Sonderman [10]).     

Renewal Approximations of the Switched Poisson Processes and Their Applications to Queueing Systems

The switched Poisson process (SPP), also known as the doubly stochastic Poisson process, has been widely used in the modelling of point processes whose rates vary subject to some random mechanism. The class of SPP includes a wide range of both renewal and non-renewal processes with squared coefficients of variation being larger than one. In this paper, we survey various approaches to approximate a non-renewal process by a renewal process. We derive the expressions for the first two moments of the inter-renewal time of a renewal process that approximates the SPP. We illustrate the quality of these approximations with numerical results in queueing applications. We believe that our approximations have potential applications in areas such as reliability, inventory control, telecommunications and maintenance.     

A comparative study of Gaussian geostatistical models and Gaussian Markov random field models

Gaussian geostatistical models (GGMs) and Gaussian Markov random fields (GMRFs) are two distinct approaches commonly used in spatial models for modeling point-referenced and areal data, respectively. In this paper, the relations between GGMs and GMRFs are explored based on approximations of GMRFs by GGMs, and approximations of GGMs by GMRFs. Two new metrics of approximation are proposed : (i) the Kullback-Leibler discrepancy of spectral densities and (ii) the chi-squared distance between spectral densities. The distances between the spectral density functions of GGMs and GMRFs measured by these metrics are minimized to obtain the approximations of GGMs and GMRFs. The proposed methodologies are validated through several empirical studies. We compare the performance of our approach to other methods based on covariance functions, in terms of the average mean squared prediction error and also the computational time. A spatial analysis of a dataset on PM_2.5 collected in California is presented to illustrate the proposed method.     

Targeted smoothing parameter selection for estimating average causal effects

The non-parametric estimation of average causal effects in observational studies often relies on controlling for confounding covariates through smoothing regression methods such as kernel, splines or local polynomial regression. Such regression methods are tuned via smoothing parameters which regulates the amount of degrees of freedom used in the fit. In this paper we propose data-driven methods for selecting smoothing parameters when the targeted parameter is an average causal effect. For this purpose, we propose to estimate the exact expression of the mean squared error of the estimators. Asymptotic approximations indicate that the smoothing parameters minimizing this mean squared error converges to zero faster than the optimal smoothing parameter for the estimation of the regression functions. In a simulation study we show that the proposed data-driven methods for selecting the smoothing parameters yield lower empirical mean squared error than other methods available such as, e.g., cross-validation.     

Designing a call center with an IVR (Interactive Voice Response)

A call center is a service operation that caters to customer needs via the telephone. Call centers typically consist of agents that serve customers, telephone lines, an Interactive Voice Response (IVR) unit, and a switch that routes calls to agents. In this paper we study a Markovian model for a call center with an IVR. We calculate operational performance measures, such as the probability for a busy signal and the average wait time for an agent. Exact calculations of these measures are cumbersome and they lack insight. We thus approximate the measures in an asymptotic regime known as QED (Quality and Efficiency Driven) or the Halfin–Whitt regime, which accommodates moderate to large call centers. The approximations are both insightful and easy to apply (for up to 1000’s of agents). They yield, as special cases, known and novel approximations for the M/M/N/N (Erlang-B), M/M/S (Erlang-C) and M/M/S/N queue.     

Stabilized approximation to degenerate transport equations via filtering

We analyze a stabilization technique for degenerate transport equations. Of particular interest are coupled parabolic/hyperbolic problems, when the diffusion coefficient is zero in part of the domain. The unstabilized, computed approximations of these problems are highly oscillatory, and several techniques have been proposed and analyzed to mitigate the effects of the sub-grid errors that contribute to the oscillatory behavior. In this paper, we modify a time-relaxation algorithm proposed in [1] and further studied in [10]. Our modification introduces the relaxation operator as a post-processing step. The operator is not time-dependent, so the discrete (relaxation) system need only be factored once. We provide convergence analysis for our algorithm along with numerical results for several model problems.     

Towards better multi-class parametric-decomposition approximations for open queueing networks

Methods are developed for approximately characterizing the departure process of each customer class from a multi-class single-server queue with unlimited waiting space and the first-in-first-out service discipline. The model is (GT _i /GI _i )/1 with a non-Poisson renewal arrival process and a non-exponential service-time distribution for each class. The methods provide a basis for improving parametric-decomposition approximations for analyzing non-Markov open queueing networks with multiple classes. For example, parametric-decomposition approximations are used in the Queueing Network Analyzer (QNA). The specific approximations here extend ones developed by Bitran and Tirupati [5]. For example, the effect of class-dependent service times is considered here. With all procedures proposed here, the approximate variability parameter of the departure process of each class is a linear function of the variability parameters of the arrival processes of all the classes served at that queue, thus ensuring that the final arrival variability parameters in a general open network can be calculated by solving a system of linear equations.     

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.     

Rough path limits of the Wong-Zakai type with a modified drift term

The Wong-Zakai theorem asserts that ODEs driven by “reasonable” (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study “non-reasonable” approximations and go beyond a well-known criterion of [Ikeda, Watanabe, North Holland, 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussmann ('91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates.     

Direct construction of code loops

Code loops were introduced by R.L. Griess Jr. [Code loops, J. Algebra 100 (1) (1986) 224-234] and T. Hsu [Explicit constructions of code loops as centrally twisted products, Math. Proc. Cambridge Philos. Soc. 128 (2) (2000) 223-232] gave methods to construct the corresponding code loop from any given doubly even binary code; both these methods used some kind of induction. In this paper, we present a global construction of the loop, where we apply the correspondence between the concepts of Moufang loops and groups with triality.     

The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.     

On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations

It is well known that methods for solving semidiscretized parabolic partial differential equations based on the second-order diagonal [1/1] Padé approximation (the Crank–Nicolson or trapezoidal method) can produce poor numerical results when a time discretization is imposed with steps that are “too large” relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher-order methods suffer a similar time step restriction as the [1/1] Padé. Next, various high-order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first-order implicit Euler method based on the [0/1] Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations. Numerical results confirm this when various test problems from the literature are implemented on a Multiple Instruction Multiple Data (MIMD) machine such as an Alliant FX/8. © 1993 John Wiley & Sons, Inc.     

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded \(G{/}GI{/}n+GI\) queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.     

Generalized multiresolution analysis on unstructured grids

Summary. Efficiency of high-order essentially non-oscillatory (ENO) approximations of conservation laws can be drastically improved if ideas of multiresolution analysis are taken into account. These methods of data compression not only reduce the necessary amount of discrete data but can also serve as tools in detecting local low-dimensional features in the numerical solution. We describe the mathematical background of the generalized multiresolution analysis as developed by Abgrall and Harten in [14], [15] and [3]. We were able to ultimately reduce the functional analytic background to matrix-vector operations of linear algebra. We consider the example of interpolation on the line as well as the important case of multiresolution analysis of cell average data which is used in finite volume approximations. In contrast to Abgrall and Harten, we develop a robust agglomeration procedure and recovery algorithms based on least-squeare polynomials. The efficiency of our algorithms is documented by means of several examples. Received April 4, 1998 / Revised version August 2, 1999 / Published online June 8, 2000     

Optimal dividends with incomplete information in the dual model

In [Gerber, H.U., Shiu, E.S.W., Smith, N., 2008. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance: Math. Econ. 42 (1), 243-254], methods were analyzed for estimating the optimal dividend barrier (in the sense of de Finetti). In particular, De Vylder approximations and diffusion approximations are discussed. These methods are useful when only the first few moments of the claim amount distribution are known.The purpose of this paper is to examine these and other methods (such as the gamma approximations and the gamproc approximations) in the dual model, see [Avanzi, B., Gerber, H.U., Shiu, E.S., 2007. Optimal dividends in the dual model. Insurance: Math. Econ. 41 (1), 111-123]. The dual model is obtained if the roles of premiums and claims are exchanged. In other words, the company has random gains, which constitute a compound Poisson process, and expenses occur continuously at a constant rate. The approximations can easily be implemented, and their accuracy is surprisingly good. Several numerical illustrations enhance the paper.     

Sensitivity analysis of permeability parameters for flows on Barcelona networks

We consider the problem of optimizing vehicular traffic flows on an urban network of Barcelona type, i.e. square network with streets of not equal length. In particular, we describe the effects of variation of permeability parameters, that indicate the amount of flow allowed to enter a junction from incoming roads.On each road, a model suggested by Helbing et al. (2007) [11] is considered: free and congested regimes are distinguished, characterized by an arrival flow and a departure flow, the latter depending on a permeability parameter. Moreover we provide a rigorous derivation of the model from fluid dynamic ones, using recent results of Bretti et al. (2006) [3]. For solving the dynamics at nodes of the network, a Riemann solver maximizing the through flux is used, see Coclite et al. (2005) [4] and Helbing et al. (2007) [11].The network dynamics gives rise to complicate equations, where the evolution of fluxes at a single node may involve time-delayed terms from all other nodes. Thus we propose an alternative hybrid approach, introducing additional logic variables. Finally we compute the effects of variations on permeability parameters over the hybrid dynamics and test the obtained results via simulations.     

Incremental unknowns for solving partial differential equations

Summary Incremental unknowns have been proposed in [T] as a method to approximate fractal attractors by using finite difference approximations of evolution equations. In the case of linear elliptic problems, the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization; the method is similar but different from the classical multigrid method.In this article we describe the application of incremental unknowns for solving Laplace equations in dimensions one and two. We provide theoretical results concerning two-level approximations and we report on numerical tests done with multi-level approximations.     

最小二乘估计精度及其界限

Puntanen[1]提出用均方误差来度量最小二乘估计的精度,以后Styan[2],Rao[3]等相继讨论了这种精度及其界限.本文考虑采用广义方差,从而引进了一种新的最小二乘估计精度的度量并讨论了它的界.