11 research outputs found
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 toric
grid with normalized load are of the order . This
superlinear delay scaling is to be contrasted with the usual linear growth of
the order 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
Store-Forward and its implications for Proportional Scheduling
The Proportional Scheduler was recently proposed as a scheduling algorithm
for multi-hop switch networks. For these networks, the BackPressure scheduler
is the classical benchmark. For networks with fixed routing, the Proportional
Scheduler is maximum stable, myopic and, furthermore, will alleviate certain
scaling issued found in BackPressure for large networks. Nonetheless, the
equilibrium and delay properties of the Proportional Scheduler has not been
fully characterized.
In this article, we postulate on the equilibrium behaviour of the
Proportional Scheduler though the analysis of an analogous rule called the
Store-Forward allocation. It has been shown that Store-Forward has
asymptotically allocates according to the Proportional Scheduler. Further, for
Store-Forward networks, numerous equilibrium quantities are explicitly
calculable. For FIFO networks under Store-Forward, we calculate the policies
stationary distribution and end-to-end route delay. We discuss network
topologies when the stationary distribution is product-form, a phenomenon which
we call \emph{product form resource pooling}. We extend this product form
notion to independent set scheduling on perfect graphs, where we show that
non-neighbouring queues are statistically independent. Finally, we analyse the
large deviations behaviour of the equilibrium distribution of Store-Forward
networks in order to construct Lyapunov functions for FIFO switch networks
Temporal starvation in multi-channel CSMA networks: an analytical framework
In this paper we consider a stochastic model for a frequency-agile CSMA
protocol for wireless networks where multiple orthogonal frequency channels are
available. Even when the possible interference on the different channels is
described by different conflict graphs, we show that the network dynamics can
be equivalently described as that of a single-channel CSMA algorithm on an
appropriate virtual network. Our focus is on the asymptotic regime in which the
network nodes try to activate aggressively in order to achieve maximum
throughput. Of particular interest is the scenario where the number of
available channels is not sufficient for all nodes of the network to be
simultaneously active and the well-studied temporal starvation issues of the
single-channel CSMA dynamics persist. For most networks we expect that a larger
number of available channels should alleviate these temporal starvation issues.
However, we prove that the aggregate throughput is a non-increasing function of
the number of available channels. To investigate this trade-off that emerges
between aggregate throughput and temporal starvation phenomena, we propose an
analytical framework to study the transient dynamics of multi-channel CSMA
networks by means of first hitting times. Our analysis further reveals that the
mixing time of the activity process does not always correctly characterize the
temporal starvation in the multi-channel scenario and often leads to
pessimistic performance estimates.Comment: 15 pages, 4 figures. Accepted for publication at IFIP Performance
Conference 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
Crossover times in bipartite networks with activity constraints and time-varying switching rates
In this paper we study the performance of a bipartite network in which
customers arrive at the nodes of the network, but not all nodes are able to
serve their customers at all times. Each node can be either active or inactive,
and two nodes connected by a bond cannot be active simultaneously. This
situation arises in wireless random-access networks where, due to destructive
interference, stations that are close to each other cannot use the same
frequency band.
We consider a model where the network is bipartite, the active nodes switch
themselves off at rate 1, and the inactive nodes switch themselves on at a rate
that depends on time and on which half of the bipartite network they are in. An
inactive node cannot become active when one of the nodes it is connected to by
a bond is active. The switching protocol allows the nodes to share activity
among each other. In the limit as the activation rate becomes large, we compute
the crossover time between the two states where one half of the network is
active and the other half is inactive. This allows us to assess the overall
activity of the network depending on the switching protocol. Our results make
use of the metastability analysis for hard-core interacting particle models on
finite bipartite graphs derived in an earlier paper. They are valid for a large
class of bipartite networks, subject to certain assumptions. Proofs rely on a
comparison with switching protocols that are not time-varying, through coupling
techniques.Comment: 32 pages, 2 figur
Temporal starvation in multi-channel CSMA networks: an analytical framework
In this paper, we consider a stochastic model for a frequency-agile CSMA protocol for wireless networks where multiple orthogonal frequency channels are available. Even when the possible interference on the different channels is described by different conflict graphs, we show that the network dynamics can be equivalently described as that of a single-channel CSMA algorithm on an appropriate virtual network. Our focus is on the asymptotic regime in which the network nodes try to activate aggressively in order to achieve maximum throughput. Of particular interest is the scenario where the number of available channels is not sufficient for all nodes of the network to be simultaneously active and the well-studied temporal starvation issues of the single-channel CSMA dynamics persist. For most networks, we expect that a larger number of available channels should alleviate these temporal starvation issues. However, we prove that the aggregate throughput is a non-increasing function of the number of available channels. To investigate this trade-off that emerges between aggregate throughput and temporal starvation phenomena, we propose an analytic framework to study the transient dynamics of multi-channel CSMA networks by means of first hitting times. Our analysis further reveals that the mixing time of the activity process does not always correctly characterize the temporal starvation in the multi-channel scenario and often leads to pessimistic performance estimates