621 research outputs found
The Stationary Behaviour of Fluid Limits of Reversible Processes is Concentrated on Stationary Points
Assume that a stochastic processes can be approximated, when some scale
parameter gets large, by a fluid limit (also called "mean field limit", or
"hydrodynamic limit"). A common practice, often called the "fixed point
approximation" consists in approximating the stationary behaviour of the
stochastic process by the stationary points of the fluid limit. It is known
that this may be incorrect in general, as the stationary behaviour of the fluid
limit may not be described by its stationary points. We show however that, if
the stochastic process is reversible, the fixed point approximation is indeed
valid. More precisely, we assume that the stochastic process converges to the
fluid limit in distribution (hence in probability) at every fixed point in
time. This assumption is very weak and holds for a large family of processes,
among which many mean field and other interaction models. We show that the
reversibility of the stochastic process implies that any limit point of its
stationary distribution is concentrated on stationary points of the fluid
limit. If the fluid limit has a unique stationary point, it is an approximation
of the stationary distribution of the stochastic process.Comment: 7 pages, preprin
On Time Synchronization Issues in Time-Sensitive Networks with Regulators and Nonideal Clocks
Flow reshaping is used in time-sensitive networks (as in the context of IEEE
TSN and IETF Detnet) in order to reduce burstiness inside the network and to
support the computation of guaranteed latency bounds. This is performed using
per-flow regulators (such as the Token Bucket Filter) or interleaved regulators
(as with IEEE TSN Asynchronous Traffic Shaping). Both types of regulators are
beneficial as they cancel the increase of burstiness due to multiplexing inside
the network. It was demonstrated, by using network calculus, that they do not
increase the worst-case latency. However, the properties of regulators were
established assuming that time is perfect in all network nodes. In reality,
nodes use local, imperfect clocks. Time-sensitive networks exist in two
flavours: (1) in non-synchronized networks, local clocks run independently at
every node and their deviations are not controlled and (2) in synchronized
networks, the deviations of local clocks are kept within very small bounds
using for example a synchronization protocol (such as PTP) or a satellite based
geo-positioning system (such as GPS). We revisit the properties of regulators
in both cases. In non-synchronized networks, we show that ignoring the timing
inaccuracies can lead to network instability due to unbounded delay in per-flow
or interleaved regulators. We propose and analyze two methods (rate and burst
cascade, and asynchronous dual arrival-curve method) for avoiding this problem.
In synchronized networks, we show that there is no instability with per-flow
regulators but, surprisingly, interleaved regulators can lead to instability.
To establish these results, we develop a new framework that captures industrial
requirements on clocks in both non-synchronized and synchronized networks, and
we develop a toolbox that extends network calculus to account for clock
imperfections.Comment: ACM SIGMETRICS 2020 Boston, Massachusetts, USA June 8-12, 202
Counting algebraic points in expansions of o-minimal structures by a dense set
The Pila-Wilkie theorem states that if a set is
definable in an o-minimal structure and contains `many' rational
points, then it contains an infinite semialgebraic set. In this paper, we
extend this theorem to an expansion of by a dense set , which is either an elementary
substructure of , or it is independent, as follows. If is
definable in and contains many rational points, then
it is dense in an infinite semialgebraic set. Moreover, it contains an infinite
set which is -definable in ,
where is the real field
On Mean Field Convergence and Stationary Regime
Assume that a family of stochastic processes on some Polish space
converges to a deterministic process; the convergence is in distribution (hence
in probability) at every fixed point in time. This assumption holds for a large
family of processes, among which many mean field interaction models and is
weaker than previously assumed. We show that any limit point of an invariant
probability of the stochastic process is an invariant probability of the
deterministic process. The results are valid in discrete and in continuous
time
Analysis of a Reputation System for Mobile Ad-Hoc Networks with Liars
The application of decentralized reputation systems is a promising approach
to ensure cooperation and fairness, as well as to address random failures and
malicious attacks in Mobile Ad-Hoc Networks. However, they are potentially
vulnerable to liars. With our work, we provide a first step to analyzing
robustness of a reputation system based on a deviation test. Using a mean-field
approach to our stochastic process model, we show that liars have no impact
unless their number exceeds a certain threshold (phase transition). We give
precise formulae for the critical values and thus provide guidelines for an
optimal choice of parameters.Comment: 17 pages, 6 figure
Application of Network Calculus To Guaranteed Service Networks
We use recent network calculus results to study some properties of lossless multiplexing as it may be used in guaranteed service networks. We call network calculus a set of results that apply min-plus algebra to packet networks. We provide a simple proof that shaping a traffic stream to conform with a burstiness constraint preserves the original constraints satisfied by the traffic stream We show how all rate based packet schedulers can be modeled with a simple rate latency service curve. Then we define a general form of deterministic effective bandwidth and equivalent capacity. We find that call acceptance regions based on deterministic criteria (loss or delay) are convex, in contrast to statistical cases where it the complement of the region which is convex. We thus find that, in general, the limit of the call acceptance region based on statistical multiplexing when the loss probability target tends to 0 may be strictly larger than the call acceptance region based on lossless multiplexing. Lastly, we consider the problem of determining the optimal parameters of a variable bit rate (VBR) connection when it is used as a trunk, or tunnel, given that the input traffic is known. We find that there is an optimal peak rate for the VBR trunk, essentially insensitive to the optimization criteria. For a linear cost function, we find an explicit algorithm for the optimal remaining parameters of the VBR trunk
Some Properties Of Variable Length Packet Shapers
The min-plus theory of greedy shapers has been developed after Cruz`s results on the calculus of network delays. An example of greedy shaper is the buffered leaky bucket controller. The theory of greedy shapers establishes a number of properties such as the series decomposition of shapers or the conservation of arrival constraints by re-shaping. It applies in all rigor either to fluid systems, or to packets of constant size such as ATM. For variable length packets, the distortion introduced by packetization affects the theory, which is no longer valid. In this paper, we elucidate the relationship between shaping and packetization effects. We show a central result, namely, the min-plus representation of a packetized greedy shaper. We find a sufficient condition under which series decomposition of shapers and conservation of arrival constraints still hold in presence of packetization effects. This allows us to demonstrate the equivalence of implementing a buffered leaky bucket controller based on either virtual finish times or on bucket replenishment. However, we show on some examples that if the condition is not satisfied, then the property may not hold any more. This indicates that, for variable size packets, unlike for fluid systems, there is a fundamental difference between constraints based on leaky buckets, and constraints based on general arrival curves, such as spacing constraints. The latter are used in the context of ATM to obtain tight end-to-end delay bounds. In this paper, we use a min-plus theory, and obtain results on greedy shapers for variable length packets which are not readily explained with the max-plus theory of Chang
Optimal Smoothing for Guaranteed Service
We consider a scenario where multimedia data is sent over a network offering a guaranteed service such as ATM VBR or the guaranteed service of the IETF. A smoothing device writes the stream into a networking device for transmission, possibly with some pre-fetchin
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