37 research outputs found

### The impact of a network split on cascading failure processes

Cascading failure models are typically used to capture the phenomenon where
failures possibly trigger further failures in succession, causing knock-on
effects. In many networks this ultimately leads to a disintegrated network
where the failure propagation continues independently across the various
components. In order to gain insight in the impact of network splitting on
cascading failure processes, we extend a well-established cascading failure
model for which the number of failures obeys a power-law distribution. We
assume that a single line failure immediately splits the network in two
components, and examine its effect on the power-law exponent. The results
provide valuable qualitative insights that are crucial first steps towards
understanding more complex network splitting scenarios

### Asymptotically Optimal Load Balancing Topologies

We consider a system of $N$ servers inter-connected by some underlying graph
topology $G_N$. Tasks arrive at the various servers as independent Poisson
processes of rate $\lambda$. Each incoming task is irrevocably assigned to
whichever server has the smallest number of tasks among the one where it
appears and its neighbors in $G_N$. Tasks have unit-mean exponential service
times and leave the system upon service completion.
The above model has been extensively investigated in the case $G_N$ is a
clique. Since the servers are exchangeable in that case, the queue length
process is quite tractable, and it has been proved that for any $\lambda < 1$,
the fraction of servers with two or more tasks vanishes in the limit as $N \to
\infty$. For an arbitrary graph $G_N$, the lack of exchangeability severely
complicates the analysis, and the queue length process tends to be worse than
for a clique. Accordingly, a graph $G_N$ is said to be $N$-optimal or
$\sqrt{N}$-optimal when the occupancy process on $G_N$ is equivalent to that on
a clique on an $N$-scale or $\sqrt{N}$-scale, respectively.
We prove that if $G_N$ is an Erd\H{o}s-R\'enyi random graph with average
degree $d(N)$, then it is with high probability $N$-optimal and
$\sqrt{N}$-optimal if $d(N) \to \infty$ and $d(N) / (\sqrt{N} \log(N)) \to
\infty$ as $N \to \infty$, respectively. This demonstrates that optimality can
be maintained at $N$-scale and $\sqrt{N}$-scale while reducing the number of
connections by nearly a factor $N$ and $\sqrt{N} / \log(N)$ compared to a
clique, provided the topology is suitably random. It is further shown that if
$G_N$ contains $\Theta(N)$ bounded-degree nodes, then it cannot be $N$-optimal.
In addition, we establish that an arbitrary graph $G_N$ is $N$-optimal when its
minimum degree is $N - o(N)$, and may not be $N$-optimal even when its minimum
degree is $c N + o(N)$ for any $0 < c < 1/2$.Comment: A few relevant results from arXiv:1612.00723 are included for
convenienc

### Mixing Properties of CSMA Networks on Partite Graphs

We consider a stylized stochastic model for a wireless CSMA network.
Experimental results in prior studies indicate that the model provides
remarkably accurate throughput estimates for IEEE 802.11 systems. In
particular, the model offers an explanation for the severe spatial unfairness
in throughputs observed in such networks with asymmetric interference
conditions. Even in symmetric scenarios, however, it may take a long time for
the activity process to move between dominant states, giving rise to potential
starvation issues. In order to gain insight in the transient throughput
characteristics and associated starvation effects, we examine in the present
paper the behavior of the transition time between dominant activity states. We
focus on partite interference graphs, and establish how the magnitude of the
transition time scales with the activation rate and the sizes of the various
network components. We also prove that in several cases the scaled transition
time has an asymptotically exponential distribution as the activation rate
grows large, and point out interesting connections with related exponentiality
results for rare events and meta-stability phenomena in statistical physics. In
addition, we investigate the convergence rate to equilibrium of the activity
process in terms of mixing times.Comment: Valuetools, 6th International Conference on Performance Evaluation
Methodologies and Tools, October 9-12, 2012, Carg\`ese, Franc

### Achievable Performance in Product-Form Networks

We characterize the achievable range of performance measures in product-form
networks where one or more system parameters can be freely set by a network
operator. Given a product-form network and a set of configurable parameters, we
identify which performance measures can be controlled and which target values
can be attained. We also discuss an online optimization algorithm, which allows
a network operator to set the system parameters so as to achieve target
performance metrics. In some cases, the algorithm can be implemented in a
distributed fashion, of which we give several examples. Finally, we give
conditions that guarantee convergence of the algorithm, under the assumption
that the target performance metrics are within the achievable range.Comment: 50th Annual Allerton Conference on Communication, Control and
Computing - 201

### Slow transitions, slow mixing and starvation in dense random-access networks

We consider dense wireless random-access networks, modeled as systems of
particles with hard-core interaction. The particles represent the network users
that try to become active after an exponential back-off time, and stay active
for an exponential transmission time. Due to wireless interference, active
users prevent other nearby users from simultaneous activity, which we describe
as hard-core interaction on a conflict graph. We show that dense networks with
aggressive back-off schemes lead to extremely slow transitions between dominant
states, and inevitably cause long mixing times and starvation effects.Comment: 29 pages, 5 figure

### Data Dissemination Performance in Large-Scale Sensor Networks

As the use of wireless sensor networks increases, the need for
(energy-)efficient and reliable broadcasting algorithms grows. Ideally, a
broadcasting algorithm should have the ability to quickly disseminate data,
while keeping the number of transmissions low. In this paper we develop a model
describing the message count in large-scale wireless sensor networks. We focus
our attention on the popular Trickle algorithm, which has been proposed as a
suitable communication protocol for code maintenance and propagation in
wireless sensor networks. Besides providing a mathematical analysis of the
algorithm, we propose a generalized version of Trickle, with an additional
parameter defining the length of a listen-only period. This generalization
proves to be useful for optimizing the design and usage of the algorithm. For
single-cell networks we show how the message count increases with the size of
the network and how this depends on the Trickle parameters. Furthermore, we
derive distributions of inter-broadcasting times and investigate their
asymptotic behavior. Our results prove conjectures made in the literature
concerning the effect of a listen-only period. Additionally, we develop an
approximation for the expected number of transmissions in multi-cell networks.
All results are validated by simulations

### Delay performance in random-access grid networks

We examine the impact of torpid mixing and meta-stability issues on the delay
performance in wireless random-access networks. Focusing on regular meshes as
prototypical scenarios, we show that the mean delays in an $L\times L$ toric
grid with normalized load $\rho$ are of the order $(\frac{1}{1-\rho})^L$. This
superlinear delay scaling is to be contrasted with the usual linear growth of
the order $\frac{1}{1-\rho}$ in conventional queueing networks. The intuitive
explanation for the poor delay characteristics is that (i) high load requires a
high activity factor, (ii) a high activity factor implies extremely slow
transitions between dominant activity states, and (iii) slow transitions cause
starvation and hence excessively long queues and delays. Our proof method
combines both renewal and conductance arguments. A critical ingredient in
quantifying the long transition times is the derivation of the communication
height of the uniformized Markov chain associated with the activity process. We
also discuss connections with Glauber dynamics, conductance and mixing times.
Our proof framework can be applied to other topologies as well, and is also
relevant for the hard-core model in statistical physics and the sampling from
independent sets using single-site update Markov chains