On the rate of convergence to stationarity of the M/M/N queue in the Halfin-Whitt regime

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant $B^*\approx1.85772$ s.t. when a certain excess parameter $B\in(0,B^*]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^2}{4}$. For $B>B^*$, the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed n by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer]. We also prove explicit bounds on the distance to stationarity for the M/M/n queue in the Halfin-Whitt regime, when $B<B^*$. Our bounds scale independently of $n$ in the Halfin-Whitt regime, and do not follow from the weak-convergence theory.Comment: Published in at http://dx.doi.org/10.1214/12-AAP889 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Comment on "Superinstantons and the Reliability of Perturbation Theory in Non-Abelian Models"

In a recent letter (hep-lat/9311019) A. Patrascioiu and E. Seiler argued that when taking into account "superinstantons configurations" the perturbative expansion and the beta-function of the two-dimensional non-linear sigma-model are modified at two loops order. I point out that: (1) perturbation theory in a superinstanton background is infra-red singular beyond three loops; (2) the new infra-red singular terms, which change the two loop terms, come from singular operators - describing superinstanton insertions - in the OPE; (3) taking into account these operators, the beta-function is not modified. Therefore the results of Patrascioiu and Seiler do not contradict perturbation theory.Comment: 1 page, REVTeX, no figure

A Critical Examination of the FDA’s Efforts to Preempt Failure-to-Warn Claims

This article explores the legality and wisdom of the FDA’s effort to persuade courts to find most failure-to-warn claims preempted. The article first analyzes the FDA’s justifications for reversing its long-held views to the contrary and explains why the FDA’s position cannot be reconciled with its governing statute. The article then examines why the FDA’s position, if ultimately adopted by the courts, would undermine the incentives drug manufacturers have to change labeling to respond to newly-discovered risks. The background possibility of failure-to-warn litigation provides important incentives for drug companies to ensure that drug labels reflect accurate and up-to-date safety information. The article next explains why the agency’s view that it is capable of singlehandedly regulating the safety of drugs is unrealistic. The agency does not have the resources to perform the Herculean task of monitoring the performance of every drug on the market. Both the Institute of Medicine and the Government Accountability Office have explained the shortcomings in the FDA’s recent performance, and they express doubt that the FDA is in capable of facing an increasingly challenging future. The article then explains how state damages litigation helps uncover and assess risks that are not apparent to the agency during a drug’s approval process, and why this “feedback loop” enables the agency to better do its job. FDA approval of drugs is based on clinical trials that involve, at most, a few thousand patients and last a year or so. These trials cannot detect risks that are relatively rare, affect vulnerable sub-populations, or have long latency periods. For this reason, most serious adverse effects do not become evident until a drug is used in larger population groups for periods in excess of one year. Time and again, failure-to-warn litigation has brought to light information that would not otherwise be available to the FDA, to doctors, to other health care providers, and to consumers. And failure-to-warn litigation often has preceded and clearly influenced FDA decisions to modify labeling, and, at times, to withdraw drugs from the market

Extended Laguerre inequalities and a criterion for real zeros

Let $f(z)=e^{-bz^2}f_1(z)$ where $b \geq 0$ and $f_1(z)$ is a real entire function of genus 0 or 1. We give a necessary and sufficient condition in terms of a sequence of inequalities for all of the zeros of $f(z)$ to be real. These inequalities are an extension of the classical Laguerre inequalities.Comment: The paper is based on a talk given at the 7th ISAAC Congress held at Imperial College in London in July 200