THURSDAY, April 7, 2011
STOCHASTIC ACTIVITY MONTH
LECTURE DAY
SUMMARY
This is the official opening the first Stochastic Activity Month (SAM)
of Eurandom and STAR. The first
SAM
is devoted to Stochastic Modelling and Analysis of Networks. Examples of
networks are computer networks like the internet, communication networks like
wireless, adhoc and sensor networks, and biological and social networks.
For organizational reasons, we kindly ask all participants to register online.
This will help us organize a perfect Lecture Day! Deadline for registration is
April 4.
Please follow the link:
REGISTRATION
(You will be redirected to the TU/e website for the registration
form)
SPEAKERS
Sergey Foss 
HeriotWatt 
Jesse Goodman 
EURANDOM 
Remco van der Hofstad 
EURANDOM  TU/e 
Gerard Hooghiemstra 
TU Delft 
Peter van de Ven 
EURANDOM 
Uri Yechiali 
Tel Aviv University 
PROGRAMME
ABSTRACTS
Sergey Foss (HeriotWatt
University)
Poisson Hail on a Hot Ground
We consider a queue where the server is the Euclidean space, and
the customers are random closed sets (RACS) of the Euclidean space.
These RACS arrive according to a Poisson rain and each of them has a random
service time (in the case of hail falling on the Euclidean
plane, this is the height of the hailstone, whereas the RACS is its footprint).
The Euclidean space serves customers at speed 1. The service
discipline is a hard exclusion rule: no two intersecting RACS can be served
simultaneously and service is in the First In First Out order: only the
hailstones in contact with the ground melt at speed 1, whereas the other ones
are queued; a tagged RACS waits until all RACS arrived before it and
intersecting it have fully melted before starting its own melting. We give the
evolution equations for this queue. We prove that it is stable for a
sufficiently small arrival intensity, provided the typical diameter of the RACS
and the typical service time have finite exponential moments. We also discuss
the percolation properties of the stationary regime of the RACS in the queue.
This is a joint work with Francois Baccelli (INRIA and ENS, Paris)
PRESENTATION
Jesse Goodman
Invasion percolation on networks and trees
Invasion percolation is an exploration process on a graph with
edge weights that grows a cluster by adding the boundary edge of minimal weight
at each time step. We will outline what is known about this process in the
context of trees and treelike graphs, such as the complete graph, including
connections to (standard) percolation, queueing theory, first passage
percolation, and the minimal spanning tree.
PRESENTATION
Remco van der Hofstad
Critical behavior in inhomogeneous random graphs
Empirical findings have shown that many realworld networks
share fascinating features. Indeed, many realworld networks are small worlds,
in the sense that typical distances are much smaller than the size of the
network. Further, many realworld networks are scalefree in the sense that
there is a high variability in the number of connections of the elements of the
networks. Therefore, such networks are highly inhomogeneous.
Spurred by these empirical findings, models have been proposed for such
networks. In this talk, we shall discuss some of the empirical findings of
realworld networks, and describe a particular class of random graphs, in which
edges are present independently but with unequal edge occupation probabilities
that are moderated by appropriate vertex weights. For such models, it is known
precisely when there is a giant component, meaning that a positive proportion of
the vertices is connected to one another. We discuss what happens at and close
to criticality, a problem having strong connections to statistical mechanics. We
identify the scaling limit of the connected components ordered by their size.
Our results show that, for inhomogeneous random graphs with highly variable
vertex degrees, the critical behavior admits a transition when the third moment
of the degrees turns from finite to infinite. When the third moment is finite,
the largest clusters in a graph of n vertices have size n^{2/3} and the scaling
limit equals that on the ErdoesRenyi random graph. When the third moment is
infinite, the largest clusters have size to n^{\alpha} for some \alpha\in
(1/2,2/3), and the scaling limit is rather different. Similar phase transitions
have been shown to occur for typical distances in such random graphs when
the variance of the degrees turns from finite to infinite.
[This is joint work with Shankar Bhamidi and Johan van Leeuwaarden.]
Gerard Hooghiemstra
First passage percolation on (random) graphs
Consider a (connected) random graph G with n vertices. Assign to each
of the edges an i.i.d. weight. Given two vertices A and B, taken
uniformly at random from the n vertices, we de ne the hopcount H_{n},
and the minimal weight W_{n}, respectively, as the number of
edges and the weight of the path between A and B with minimal
total path weight. Our interest is to prove limit theorems for H_{n}
and W_{n}, and in particular, to investigate the universality of
a central limit theorem for H_{n}. I will try to summarize
results in case the base graph G is the complete graph, the ErdösRényi
graph and the conguration model. The weights are i.i.d. copies of an exponential
random variable E, or a power of an exponential random variable.
(joint work with Shankar Bhamidi[UNC] and Remco van der Hofstad[TU/e])
PRESENTATION
Peter van de Ven
Throughput, fairness and stability of randomaccess networks
Randomaccess algorithms such as the
CarrierSense MultipleAccess (CSMA) protocol provide a popular mechanism for
distributed medium access control in largescale wireless networks.
Randomaccess networks may exhibit severe unfairness in throughput, in the sense
that some nodes receive consistently higher throughput than others. We study the
unfairness in saturated networks, and adapt the randomaccess CSMA protocol to
remove the unfairness completely.
These models primarily pertain to a saturated scenario where nodes always have
packets to transmit. In reality however, the buffers may occasionally be empty
as packets are randomly generated and transmitted over time. The resulting
interplay between the activity states and the buffer contents gives rise to
quite complicated queueing dynamics, and even establishing the stability
criteria is usually a serious challenge. We explicitly identify the stability
conditions in a few relevant scenarios, and illustrate the difficulties arising
in other cases.
PRESENTATION
Uri Yechiali
The Totally Asymmetric Inclusion Process (TASIP) or Tandem
Network Queues with Batch Service
Inspired by the biological process of gene translation and its
modeling as a flow of particles through a tandem array of limitedcapacity
‘servers’  known in biophysics as the Totally Asymmetric Exclusion Process (TASEP),
we devise and analyze a new model: the Totally Asymmetric Inclusion Process (TASIP).
The TASIP is a system of tandem Markovian queues with batch service. That is,
when service is completed at queue k all particles present at that queue move
simultaneously to queue k+1. Equations for the timedependent Probability
Generating Function (PGF) of the systemstate (occupancy) vector at arbitrary
time instants, as well as at ‘event epochs’ (arrival or service completion), are
derived. We show that, at steady state, the two PGFs coincide – implying a
PASTAtype phenomenon. We explicitly compute the mean of the systemstate vector
at steady state, and establish a Littletype law. Furthermore, we explicitly
compute the PGF of the TASIP’s total load in steady state, and obtain a
stochastic decomposition result for the total load. Finally, we show that TASIP
models with identical service rates are, under various perspectives, optimal.
Joint work with Shlomi Reuveni, Iddo Eliazar
Last updated
180711,
by
PK
