Determining the exit time distribution for a closed cyclic network

AbstractConsider a closed, N-node, cyclic network, where each node has an independent, exponential single server. Using lattice-Bessel functions, we can explicitly solve for the transition probabilities of events that occur prior to one of the nodes becoming empty. This calculation entails associating with this absorbing process a symmetry group that is the semidirect product of simpler groups. As a byproduct, we are able to compute explicitly the entire spectrum for the finite-dimensional matrix generator of this process. When the number of nodes exceeds 1, such a spectrum is no longer purely real. Moreover, we are also able to obtain the quasistationary distribution or the limiting behavior of the network conditioned on no node ever being idle

Towards a generalisation of formal concept analysis for data mining purposes

In this paper we justify the need for a generalisation of Formal Concept Analysis for the purpose of data mining and begin the synthesis of such theory. For that purpose, we first review semirings and semimodules over semirings as the appropriate objects to use in abstracting the Boolean algebra and the notion of extents and intents, respectively. We later bring to bear powerful theorems developed in the field of linear algebra over idempotent semimodules to try to build a Fundamental Theorem for K-Formal Concept Analysis, where K is a type of idempotent semiring. Finally, we try to put Formal Concept Analysis in new perspective by considering it as a concrete instance of the theory developed

Analysis of stochastic fluid queues driven by local time processes

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a L\'evy process (a subordinator) hence making the theory of L\'evy processes applicable. Another important ingredient in our approach is the Palm calculus coming from the point process point of view.Comment: 32 pages, 6 figure

Distributed Resource Allocation for Stream Data Processing

Abstract. Data streaming applications are becoming more and more common due to the rapid development in the areas such as sensor net-works, multimedia streaming, and on-line data mining, etc. These ap-plications are often running in a decentralized, distributed environment. The requirements for processing large volumes of streaming data at real time have posed many great design challenges. It is critical to optimize the ongoing resource consumption of multiple, distributed, cooperating, processing units. In this paper, we consider a generic model for the gen-eral stream data processing systems. We address the resource alloca-tion problem for a collection of processing units so as to maximize the weighted sum of the throughput of different streams. Each processing unit may require multiple input data streams simultaneously and pro-duce one or many valuable output streams. Data streams flow through such a system after processing at multiple processing units. Based on this framework, we develop distributed algorithms for finding the best resource allocation schemes in such data stream processing networks. Performance analysis on the optimality and complexity of these algo-rithms are also provided

Branching processes, the max-plus algebra and network calculus

Branching processes can describe the dynamics of various queueing systems, peer-to-peer systems, delay tolerant networks, etc. In this paper we study the basic stochastic recursion of multitype branching processes, but in two non-standard contexts. First, we consider this recursion in the max-plus algebra where branching corresponds to finding the maximal offspring of the current generation. Secondly, we consider network-calculus-type deterministic bounds as introduced by Cruz, which we extend to handle branching-type processes. The paper provides both qualitative and quantitative results and introduces various applications of (max-plus) branching processes in queueing theory

Reachability problems for products of matrices in semirings

We consider the following matrix reachability problem: given $r$ square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? We define similarly the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied at left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any $r\geq 2$ is equivalent to the specialization to $r=2$. As an application of this result and of a theorem of Krob, we show that when $r=2$, the vector and matrix reachability problems are undecidable over the max-plus semiring $(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are ``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.Comment: 21 page

TCP is Max-Plus Linear and what it tells us on its throughput

Projet MCRWe give a representation of the packet-level dynamical behavior of the Reno and Tahoe variants of TCP over a single end-to-end connection. This representation allows one to consider the case when the connection involves a network made of several, possibly heterogeneous, deterministic or random routers in series. It is shown that the key features of the protocol and of the network can be expressed via a linear dynamical system in the so called max-plus algebra. This opens new ways of both analytical evaluation and fast simulation based on products of matrices in this algebra. This also leads to closed form formulas for the throughput allowed by TCP under natural assumptions on the behavior of the routers and on the detection of losses and timeouts; these new formulas are shown to refine those obtained from earlier models which either assume that the network could be reduced to a single bottleneck router and/or approximate the packets by a fluid

Determining the exit time distribution for a closed cyclic network

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An algorithm to describe the solution set of any tropical linear system A x=B x

An algorithm to give an explicit description of all the solutions to any tropical linear system A x=B x is presented. The given system is converted into a finite (rather small) number p of pairs (S,T) of classical linear systems: a system S of equations and a system T of inequalities. The notion, introduced here, that makes p small, is called compatibility. The particular feature of both S and T is that each item (equation or inequality) is bivariate, i.e., it involves exactly two variables; one variable with coefficient 1 and the other one with -1. S is solved by Gaussian elimination. We explain how to solve T by a method similar to Gaussian elimination. To achieve this, we introduce the notion of sub-special matrix. The procedure applied to T is, therefore, called sub-specialization