Corrected Diffusion Approximations Research Articles

Managing capacity and inventory jointly for multi-server make-to-stock queues

We consider joint capacity---inventory management for multi-server make-to-stock queues operating under a base stock policy. The number of servers corresponds to the capacity decision, and the base stock level is the inventory decision. Our goal is to minimize a combination of capacity, inventory, and backordering costs. We develop a square-root rule for the joint decision and justify the rule analytically in a many-server queue asymptotic framework. We demonstrate the accuracy of the square-root rule, first via our derivation and numerical assessment of a novel corrected diffusion approximation and then more directly by conducting extensive numerical experiments. Finally, we provide operational insights into the trade-offs involved in such joint management problems, through various analysis based on the square-root rule as well as a comparison with analogous results for single-server make-to-stock queues.

Asymptotic solution of light transport problems in optically thick luminescent media

We study light transport in optically thick luminescent random media. Using radiative transport theory for luminescent media and applying asymptotic and computational methods, a corrected diffusion approximation is derived with the associated boundary conditions and boundary layer solution. The accuracy of this approach is verified for a plane-parallel slab problem. In particular, the reduced system models accurately the effect of reabsorption. The impacts of varying the Stokes shift and using experimentally measured luminescence data are explored in detail. The results of this study have application to the design of luminescent solar concentrators, fluorescence medical imaging, and optical cooling using anti-Stokes fluorescence.

Convolution model of the diffuse reflectance for layered tissues

We present an explicit model for the diffuse reflectance due to a collimated beam of light incident normally on layered tissues. This model is derived using the corrected diffusion approximation applied to a layered medium, and it takes the form of a convolution with an explicit kernel and the incident beam profile. This model corrects the standard diffusion approximation over all source-detector separation distances provided the beam is sufficiently wide compared to the scattering mean free path. We validate this model through comparison with Monte Carlo simulations. Then we use this model to estimate the optical properties of an epithelial layer from Monte Carlo simulation data. Using measurements at small source-detector separations and this model, we are able to estimate the absorption coefficient, scattering coefficient, and anisotropy factor of epithelial tissues efficiently with reasonable accuracy.

FINITE HORIZON RUIN PROBABILITIES FOR RANDOM WALKS WITH HEAVY TAILED INCREMENTS

We study the asymptotic behavior of finite horizon ruin probabilities for random walks with heavy tailed increment via corrected diffusion approximation. We follow the main idea in [4] of inverting Fourier transformation, and the Fourier transformation is calculated through optimal stopping and a central limit theorem for renewal process.

Modeling boundary measurements of scattered light using the corrected diffusion approximation

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation.

Refining Square-Root Safety Staffing by Expanding Erlang C

We apply a new corrected diffusion approximation for the Erlang C formula to determine staffing levels in cost minimization and constraint satisfaction problems. These problems are motivated by large customer contact centers that are modeled as an M/M/s queue with s the number of servers or agents. The proposed staffing levels are refinements of the celebrated square-root safety-staffing rule and have the appealing property that they are as simple as the conventional square-root safety-staffing rule. In addition, we provide theoretical support for the empirical fact that square-root safety-staffing works well for moderate-sized systems.

In This Issue

In This Issue

Correcting the diffusion approximation at the boundary

The diffusion approximation to the radiative transport equation applies for light that has propagated deeply into an optically thick medium, such as biological tissue. It does not accurately model light near boundaries where measurements of scattered light are often taken. Here, we compute a correction to the diffusion approximation at the boundary. This correction requires only small modifications to the standard diffusion approximation used in biomedical optics. In particular, one needs only to compute the coefficients in the boundary condition for the diffusion approximation and an additive correction. We give explicit procedures for these computations. Using numerical results for the steady-state plane-parallel slab problem, we show that this corrected diffusion approximation is a much better approximation than the standard diffusion approximation for modeling the reflectance and transmittance.

Uniform renewal theory with applications to expansions of random geometric sums

Consider a sequenceX= (Xn:n≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variableMwith parameterp. The random variableSM=X1+ ∙ ∙ ∙ +XMis called a geometric sum. In this paper we obtain asymptotic expansions for the distribution ofSMasp↘ 0. If EX1> 0, the asymptotic expansion is developed in powers ofpand it provides higher-order correction terms to Renyi's theorem, which states that P(pSM>x) ≈ exp(-x/EX1). Conversely, if EX1= 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

Uniform renewal theory with applications to expansions of random geometric sums

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

Calculation of finite time ruin probabilities for some risk models

In this paper we discuss the numerical calculation of finite time ruin probabilities for two particular insurance risk models. The first model allows for the investment at a fixed rate of interest of the surplus whenever this is above a given level. This is related to a model studied by Embrechts and Schmidli [Embrechts, P., Schmidli, H., 1994. Ruin estimation for a general insurance risk model. Adv. Appl. Probability 26 (2), 404–422] and by Schmidli [Schmidli, H., 1994a. Corrected diffusion approximations for a risk process with the possibility of borrowing and investment. Schweizerische Vereinigung der Versicherungsmathematiker. Mitteilungen (1), 71–82; Schmidli, H., 1994b. Diffusion approximations for a risk process with the possibility of borrowing and investment. Commun. Stat. Stochastic Models 10 (2), 365–388]. Our second model is the classical risk model but with the insurer’s premium rate depending on the level of the surplus. In our final section, we discuss the extension of the these models to allow for the parameters to change over time in a deterministic way. Our methodology for calculating finite time ruin probabilities is to bound the surplus process by discrete-time Markov chains; the average of the bounds gives an approximation to the ruin probability. This approach was used by the authors in a previous paper, Cardoso and Waters [Cardoso, R.M.R., Waters, H.R., 2003. Recursive calculation of finite time ruin probabilities under interest force. Insurance Math. Econ. 33 (3), 659–676], which considered a risk process with interest earned on the surplus.

The SPRT control chart for the process mean with samples starting at fixed times

In contrast to traditional fixed sampling interval (FSI) control charts, variable sampling interval (VSI) charts vary the sampling interval as a function of the process data, thus detecting many process changes substantially faster than their FSI counterparts. However, an administrative complication in the practical implementation of VSI charts is that future sampling times are unpredictable. This paper develops a control chart for monitoring the process mean that is based on applying a sequential probability ratio test (SPRT) to the observations at each sampling point. It is assumed that the time required to take each individual observation is too long to be neglected, and that the possible times for starting an SPRT are equally spaced and determined (fixed) by administrative or process requirements. This chart is called the SPRT fixed-times (SPRTFT) chart because the SPRTs are started at the specified fixed times. Exact expressions in terms of integral equations are developed for the performance measures of the SPRTFT chart. A simple design method for the SPRTFT chart is developed based on corrected diffusion approximations to its statistical properties. Comparisons are made between the performancy of the SPRTFT chart and the FSI and VSI X̄ and cumulative sum (CUSUM) charts, including VSI versions of the X̄ and CUSUM charts constrained to sample also at fixed times. Compared to these charts, the SPRTFT control chart is shown to be highly efficient and have significant administrative advantages.

Approximations for Finite Horizon Ruin Probabilities in the Renewal Model

In Asmussen (1984), 'corrected diffusion approximations' for finite horizon ruin probabilities in the classical Cramér-Lundberg model is given. In this paper we generalize this method to the renewal model, first introduced by Andersen (1957). We calculate the approximations in case of phase type distributed claims and interarrival times. A comparison with simulated values indicates a good fit in situations of main interest in risk theory.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

Let (X, S) = {(Xn, Sn); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | Xo= i}, and a two-barrier probability P{ST ≧b | Xo = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

Let (X, S) = {(Xn , Sn ); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn ∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | X o = i}, and a two-barrier probability P{ST ≧b | X o = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

Corrected diffusion approximations for a risk process with the possibility of borrowing and investment. : Schmidli H., Arthus, Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, Heft 1, 1994, pp. 71–81.

Corrected diffusion approximations for a risk process with the possibility of borrowing and investment. : Schmidli H., Arthus, Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, Heft 1, 1994, pp. 71–81.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

Corrected diffusion approximations in certain random walk problems

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Corrected diffusion approximations in certain random walk problems

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.