On Effectiveness of Backlog Bounds Using Stochastic Network Calculus in 802.11

Network calculus is a powerful methodology of characterizing queueing processes and has wide applications, but few works on applying it to 802.11 by far. In this paper, we take one of the first steps to analyze the backlog bounds of an 802.11 wireless LAN using stochastic network calculus. In particular, we want to address its effectiveness on bounding backlogs. We model a wireless node as a single server with impairment service based on two best-known models in stochastic network calculus: Jiang's and Ciucu's. Interestingly, we find that the two models can derive equivalent stochastic service curves and backlog bounds in our studied case. We prove that the network-calculus bounds imply stable backlogs as long as the average rate of traffic arrival is less than that of service, indicating the theoretical effectiveness of stochastic network calculus in bounding backlogs. From A. Kumar's 802.11 model, we derive the concrete stochastic service curve of an 802.11 node and its backlog bounds. We compare the derived bounds with ns-2 simulations and find that the former are very loose and we discuss the reasons. And we show that the martingale and independent case analysis techniques can improve the bounds significantly. Our work offers a good reference to applying stochastic network calculus to practical scenarios

An Approach using Demisubmartingales for the Stochastic Analysis of Networks

Stochastic network calculus is the probabilistic version of the network calculus, which uses envelopes to perform probabilistic analysis of queueing networks. The accuracy of probabilistic end-to-end delay or backlog bounds computed using network calculus has always been a concern. In this paper, we propose novel end-to-end probabilistic bounds based on demisubmartingale inequalities which improve the existing bounds for the tandem networks of GI/GI/1 queues. In particular, we show that reasonably accurate bounds are achieved by comparing the new bounds with the existing results for a network of M/M/1 queues.Comment: 4 pages, 2 figur

Algebraic Approach for Performance Bound Calculus on Transportation Networks (Road Network Calculus)

We propose in this article an adaptation of the basic techniques of the deterministic network calculus theory to the road traffic flow theory. Network calculus is a theory based on min-plus algebra. It uses algebraic techniques to compute performance bounds in communication networks, such as maximum end-to-end delays and backlogs. The objective of this article is to investigate the application of such techniques for determining performance bounds on road networks, such as maximum bounds on travel times. The main difficulty to apply the network calculus theory on road networks is the modeling of interaction of cars inside one road, or more precisely the congestion phase. We propose a traffic model for a single-lane road without passing, which is compatible with the network calculus theory. The model permits to derive a maximum bound of the travel time of cars through the road. Then, basing on that model, we explain how to extend the approach to model intersections and large-scale networks.Comment: 22 page

Higgsed network calculus

We introduce a formalism for describing holomorphic blocks of 3d quiver gauge theories using networks of Ding-Iohara-Miki algebra intertwiners. Our approach is very direct and gives an explicit identification of the blocks with Dotsenko-Fateev type integrals for q-deformed quiver W-algebras. We also explain how quiver theories corresponding to Dynkin diagrams of superalgebras arise, write down the corresponding partition functions and W-algebras, and explain the connection with supersymmetric Macdonald-Ruijsenaars commuting Hamiltonians.Comment: 22 pages, typos corrected, references adde

Analysis of Stochastic Service Guarantees in Communication Networks: A Basic Calculus

A basic calculus is presented for stochastic service guarantee analysis in communication networks. Central to the calculus are two definitions, maximum-(virtual)-backlog-centric (m.b.c) stochastic arrival curve and stochastic service curve, which respectively generalize arrival curve and service curve in the deterministic network calculus framework. With m.b.c stochastic arrival curve and stochastic service curve, various basic results are derived under the (min, +) algebra for the general case analysis, which are crucial to the development of stochastic network calculus. These results include (i) superposition of flows, (ii) concatenation of servers, (iii) output characterization, (iv) per-flow service under aggregation, and (v) stochastic backlog and delay guarantees. In addition, to perform independent case analysis, stochastic strict server is defined, which uses an ideal service process and an impairment process to characterize a server. The concept of stochastic strict server not only allows us to improve the basic results (i) -- (v) under the independent case, but also provides a convenient way to find the stochastic service curve of a serve. Moreover, an approach is introduced to find the m.b.c stochastic arrival curve of a flow and the stochastic service curve of a server

A Formal Model for Programming Wireless Sensor Networks

In this paper we present new developments in the expressiveness and in the theory of a Calculus for Sensor Networks (CSN). We combine a network layer of sensor devices with a local object model to describe sensor devices with state. The resulting calculus is quite small and yet very expressive. We also present a type system and a type invariance result for the calculus. These results provide the fundamental framework for the development of programming languages and run-time environments.Comment: 14 pages, 0 figures, Submitted for Publicatio

Stability and performance guarantees in networks with cyclic dependencies

With the development of real-time networks such as reactive embedded systems, there is a need to compute deterministic performance bounds. This paper focuses on the performance guarantees and stability conditions in networks with cyclic dependencies in the network calculus framework. We first propose an algorithm that computes tight backlog bounds in tree networks for any set of flows crossing a server. Then, we show how this algorithm can be applied to improve bounds from the literature fir any topology, including cyclic networks. In particular, we show that the ring is stable in the network calculus framework

The differential calculus of causal functions

Causal functions of sequences occur throughout computer science, from theory to hardware to machine learning. Mealy machines, synchronous digital circuits, signal flow graphs, and recurrent neural networks all have behaviour that can be described by causal functions. In this work, we examine a differential calculus of causal functions which includes many of the familiar properties of standard multivariable differential calculus. These causal functions operate on infinite sequences, but this work gives a different notion of an infinite-dimensional derivative than either the Fr\'echet or Gateaux derivative used in functional analysis. In addition to showing many standard properties of differentiation, we show causal differentiation obeys a unique recurrence rule. We use this recurrence rule to compute the derivative of a simple recurrent neural network called an Elman network by hand and describe how the computed derivative can be used to train the network

Technical Report The Stochastic Network Calculator

In this technical report, we provide an in-depth description of the Stochastic Network Calculator tool. This tool is designed to compute and automatically optimize performance bounds in queueing networks using the methodology of stochastic network calculus

Decoding of Subspace Codes, a Problem of Schubert Calculus over Finite Fields

Schubert calculus provides algebraic tools to solve enumerative problems. There have been several applied problems in systems theory, linear algebra and physics which were studied by means of Schubert calculus. The method is most powerful when the base field is algebraically closed. In this article we first review some of the successes Schubert calculus had in the past. Then we show how the problem of decoding of subspace codes used in random network coding can be formulated as a problem in Schubert calculus. Since for this application the base field has to be assumed to be a finite field new techniques will have to be developed in the future.Comment: To appear in Mathematical System Theory - Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, CreateSpace, 2012, ISBN 978-147004400