On Sojourn Times in the Finite Capacity M/M/1M/M/1 Queue with Processor Sharing

We consider a processor shared $M/M/1$ queue that can accommodate at most a finite number $K$ of customers. We give an exact expression for the sojourn time distribution in the finite capacity model, in terms of a Laplace transform. We then give the tail behavior, for the limit $K\to\infty$, by locating the dominant singularity of the Laplace transform.Comment: 10 page

An Explicit Solution to the Chessboard Pebbling Problem

We consider the chessboard pebbling problem analyzed by Chung, Graham, Morrison and Odlyzko [3]. We study the number of reachable configurations $G(k)$ and a related double sequence $G(k,m)$. Exact expressions for these are derived, and we then consider various asymptotic limits.Comment: 12 pages, 7 reference

On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.Comment: 14 pages, 0 figure

Some Asymptotic Results for the Transient Distribution of the Halfin-Whitt Diffusion Process

We consider the Halfin-Whitt diffusion process $X_d(t)$, which is used, for example, as an approximation to the $m$-server $M/M/m$ queue. We use recently obtained integral representations for the transient density $p(x,t)$ of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter $\beta$ in the model is large, and the state variable $x$ and the initial condition $x_0$ (with $X_d(0)=x_0>0$) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if $x_0>0$ the probability mass migrates from $X_d(t)>0$ to the range $X_d(t)<0$, which is where it concentrates as $t\to\infty$, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.Comment: 43 pages and 8 figure

Asymptotic analysis by the saddle point method of the Anick-Mitra-Sondhi model

We consider a fluid queue where the input process consists of N identical sources that turn on and off at exponential waiting times. The server works at the constant rate c and an on source generates fluid at unit rate. This model was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an alternate representation of the joint steady state distribution of the buffer content and the number of on sources. This is given as a contour integral that we then analyze for large N. We give detailed asymptotic results for the joint distribution, as well as the associated marginal and conditional distributions. In particular, simple conditional limits laws are obtained. These shows how the buffer content behaves conditioned on the number of active sources and vice versa. Numerical comparisons show that our asymptotic results are very accurate even for N=20