Queueing systems with many servers: Null controllability in heavy traffic

A queueing model has $J\ge2$ heterogeneous service stations, each consisting of many independent servers with identical capabilities. Customers of $I\ge2$ classes can be served at these stations at different rates, that depend on both the class and the station. A system administrator dynamically controls scheduling and routing. We study this model in the central limit theorem (or heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model on $\mathbb {R}^I$ with a singular control term that describes the scaling limit of the queueing model. The singular term may be used to constrain the diffusion to lie in certain subsets of $\mathbb {R}^I$ at all times $t>0$. We say that the diffusion is null-controllable if it can be constrained to $\mathbb {X}_-$, the minimal closed subset of $\mathbb {R}^I$ containing all states of the prelimit queueing model for which all queues are empty. We give sufficient conditions for null controllability of the diffusion. Under these conditions we also show that an analogous, asymptotic result holds for the queueing model, by constructing control policies under which, for any given $0<\epsilon <T<\infty$, all queues in the system are kept empty on the time interval $[\epsilon, T]$, with probability approaching one. This introduces a new, unusual heavy traffic ``behavior'': On one hand, the system is critically loaded, in the sense that an increase in any of the external arrival rates at the ``fluid level'' results with an overloaded system. On the other hand, as far as queue lengths are concerned, the system behaves as if it is underloaded.Comment: Published at http://dx.doi.org/10.1214/105051606000000358 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic

We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n\approx r+\beta \sqrtr for some scalar \beta. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.

Small Signals’ Study of Thermal Induced Current in Nanoscale SOI Sensor

A new nanoscale SOI dual-mode modulator is investigated as a function of optical and thermal activation modes. In order to accurately characterize the device specifications towards its future integration in microelectronics circuitry, current time variations are studied and compared for “large signal” constant temperature changes, as well as for “small signal” fluctuating temperature sources. An equivalent circuit model is presented to define the parameters which are assessed by numerical simulation. Assuring that the thermal response is fast enough, the device can be operated as a modulator via thermal stimulation or, on the other hand, can be used as thermal sensor/imager. We present here the design, simulation, and model of the next generation which seems capable of speeding up the processing capabilities. This novel device can serve as a building block towards the development of optical/thermal data processing while breaking through the way to all optic processors based on silicon chips that are fabricated via typical microelectronics fabrication process

Coalescence of skew Brownian motions

The purpose of this short note is to prove almost sure coalescence of two skew Brownian motions starting from different initial points, assuming that they are driven by the same Brownian motion. The result is very simple but we would like to record it in print as it has already become the foundation of a research project of Burdzy and Chen (1999). Our theorem is a by-product of an investigation of variably skewed Brownian motion, see Barlow et al. (1999). We use exccursion theory in a manner similar to that in a paper on "perturbed Brownian motion" by Perman and Werner (1997).Barlow's research partially supported by an NSERC (Canada) grant. Burdzy's research partially supported by NSF grant DMS-9700721. Kaspi and Mandelbaum's research partially supported by the Fund for the Promotion of Research at the Technion

Variably skewed Brownian motion

Given a standard Brownian motion B, we show that the equation X [subscript] t = x [subscript] 0 + B [subscript] t + [beta](L [to the power of X] [subscript] t ); t [is greater than or equal to] 0 ; has a unique strong solution X. Here L [to the power of X] is the symmetric local time of X at 0, and [beta] is a given differentiable function with [beta](0) = 0, -1 < [beta prime](.) < 1. (For linear [beta](.), the solution is the familiar skew Brownian motion).Barlow's research partially supported by an NSERC (Canada) grant. Burdzy's research partially supported by NSF grant DMS-9700721. Kaspi and Mandelbaum's research partially supported by the Fund for the Promotion of Research at the Technion

OPTIMAL SWITCHING BETWEEN A PAIR OF BROWNIAN MOTIONS

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Variably Skewed Brownian Motion

Given a standard Brownian motion B, we show that the equation X t = x 0 +B t + fi(L X t ) ; t 0 ; has a unique strong solution X. Here L X is the symmetric local time of X at 0, and fi is a given continuously differentiable function with fi(0) = 0, \Gamma1 ! fi 0 (\Delta) ! 1. (For linear fi(\Delta), the solution is the familiar skew Brownian motion). 1 Research partially supported by an NSERC (Canada) grant. 2 Research partially supported by NSF grant DMS-9700721. 3 Research partially supported by the Fund for the Promotion of Research at the Technion. 1 Introduction In this paper we solve the following stochastic differential equation X t = x 0 +B t + fi(L X t ); t 0: (1.1) Here X = fX t ; t 0g is an unknown semimartingale with continuous sample paths, and L X = fL X t ; t 0g is the symmetric local time of X at 0; B = fB t ; t 0g is a given standard Brownian motion starting at 0, and fi is a given function. We assume that fi is continuously differentiable, ..