
September 17, 2005
Carleton Applied Probability Day
Carleton University, Ottawa


Speaker Abstracts

Barbara Gonzalez, Mathematics Department,
University of Louisiana at Lafayette
Modeling teletraffic arrivals by a Poisson cluster process.(.pdf
format)
In this paper we consider a Poisson cluster process $N$ as a
generating process for the arrivals of packets to a server. This
process generalizes in a more realistic way the infinite source
Poisson model which has been used for modeling teletraffic for a
long time. At each Poisson point $\Gamma_j$,a flow of packets is
initiated which is modeled as a partial iid sum process $\Gamma_j+\sum_{i=1}^kX_{ji}$,
$k\le K_j$,with a random limit $K_j$ which is independent of $(X_{ji})$
and the underlying Poisson points $(\Gamma_j)$. We study the covariance
structure of the increment process of $N$. In particular, the covariance
function of the increment process is not summable if the right tail
$P(K_j>x)$ is regularly varying with index $\alpha\in (1,2)$,
the distribution of the $X_{ji}$'s being irrelevant. This means
that the increment process exhibits longrange dependence. If ${\rm
var}(K_j)<\infty$ longrange dependence is excluded. We study
the asymptotic behavior of the process $(N(t))_{t\ge 0}$ and give
conditions on the distribution of $K_j$ and $X_{ji}$ under which
the random sums $\sum_{i=1}^{K_j}X_{ji}$ have a regularly varying
tail. Using the form of the distribution of the interarrival times
of the process $N$ under the Palm distribution, we also conduct
an exploratory statistical analysis of simulated data and of Internet
packet arrivals to a server. We illustrate how the theoretical results
can be used to detect distributional characteristics of $K_j$, $X_{ji}$,
and of the Poisson process.

Haipeng Shen, Department of Statistics
and Operations Research, University of North Carolina at Chapel
Hill
A QueueingScience Look at a Telephone Call Center
A call center is a service network in which agents provide telephonebased
services. Customers that seek these services are delayed in telequeues.
This talk summarizes an analysis of a unique record of call center
operations. The data comprise a complete operational history of
a small banking call center, call by call, over a full year.
We look at call centers through the perspective of queueing science,
which combines mathematically elegant queueing theory with empirically
datadriven statistical analys. The service process is decomposed
into three fundamental
components: arrivals, customer patience, and service durations.
Each component involves different basic mathematical structures
and requires a different style of statistical analysis. Some of
the key empirical results are sketched, along with descriptions
of the varied techniques required. The talk then surveys how the
characteristics deduced from the statistical analyses form the building
blocks for theoretically interesting and practically useful mathematical
models for call center operations.
The papers associated with this talk are downloadable at
http://www.unc.edu/~haipeng/.

Wojciech Szpankowski, Department
of Computer Science, Purdue University
Analytic Algorithmics, Combinatorics and Information Theory
Analytic information theory aims at studying problems of information
theory using analytic techniques of computer science and combinatorics.
Following Hadamard's and Knuth's precept, we tackle these problems
by complex analysis methods such as generating functions, Mellin
transform, Fourier series, saddle point method, analytic poissonization
and depoissonization, and singularity analysis. This approach lies
at the crossroad of computer science and information theory. In
this talk, we concentrate on one facet of information theory (i.e.,
source coding better known as data compression), namely the redundancy
rate problem and types. The redundancy rate problem for a class
of sources is the determination of how far the actual code length
exceeds the optimal (ideal) code length. The method of types is
a powerful technique in information theory, large deviations, and
analysis of algorithms. It reduces calculations of the probability
of rare events to a combinatorial analysis. Two sequences are of
the same type if they have the same empirical distribution. We shall
argue that counting types can be accomplished efficiently by enumerating
Eulerian paths (Markov types) or binary trees with a given path
length (universal types). On the other hand, analysis of the redundancy
rate problem for memoryless and Markov sources leads us to tree
generating functions (e.g., arising in counting labeled rooted trees)
studied extensively in computer science.


Damon Wischik, Department of Computer
Science University College London (UCL)
Queueing theory for TCP/IP
The current Internet is not very good for transferring large
files at high speeds, nor for transmitting live audio and video.
This is not because of hardware limitations; it is because of problems
with the architecture of the Internet, and especially with TCP,
the algorithm for controlling congestion, implemented in every computer
on the Internet.
TCP controls congestion by adjusting the transmission rate of a
traffic flow in response to the congestion it perceives in the network.
Queueing theorists, on the other hand, are used to using traffic
models which do not respond to congestion. For example, queueing
theorists are used to the idea that the larger the buffer the less
likely it is to overflow. But TCP thinks that the absence of overflow
means the network is underutilized, and so it increases its transmission
rate until the buffer overflows, no matter how large it is.
In this talk I will describe some recent work on queueing theory
for TCP. This theory uses the predictability of aggregate traffic
flows, both at a fluid level and a stochastic level. The theory
suggests ways to enhance TCP and build cheaper Internet routers.
For further information, http://www.cs.ucl.ac.uk/staff/D.Wischik/Talks/iparch.html
http://www.cs.ucl.ac.uk/staff/D.Wischik/Research/tcptheory.html

