105 research outputs found

### On Sojourn Times in the Finite Capacity $M/M/1$ 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

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