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

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

A Mean-Field Model for Multiple TCP Connections through a Buffer Implementing RED

Active queue management schemes like RED (Random Early Detection) have been suggested when multiple TCP sessions are multiplexed through a bottleneck buffer. The idea is to detect congestion before the buffer overflows and packets are lost. When the queue length reaches a certain threshold RED schemes drop/mark incoming packets with a probability that increases as the queue size increases. The objectives are an equitable distribution of packet loss, reduced delay and delay variation and improved network utilization. Here we model multiple connections maintained in the congestion avoidance regime by the RED mechanism. The window sizes of each TCP session evolve like independent dynamical systems coupled by the queue length at the buffer. We introduce a mean-field approximation to one such RED system as the number of flows tends to infinity. The deterministic limiting system is described by a transport equation. The numerical solution of the limiting system is found to provide a good description of the evolution of the distribution of the window sizes, the average queue size, the average loss rate per connection and the total throughput. TCP with RED or tail-drop may exhibit limit cycles and this causes unnecessary packet delay variation and variable loss rates. The root cause of these limit cycles is the hysteresis due to the round trip time delay in reacting to a packet loss

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.Comment: 26 page

Role of CBL Mutations in Cancer and Non-Malignant Phenotype

CBL plays a key role in different cell pathways, mainly related to cancer onset and progres-sion, hematopoietic development and T cell receptor regulation. Somatic CBL mutations have been reported in a variety of malignancies, ranging from acute myeloid leukemia to lung cancer. Growing evidence have defined the clinical spectrum of germline CBL mutations configuring the so-called CBL syndrome; a cancer-predisposing condition that also includes multisystemic involvement char-acterized by variable phenotypic expression and expressivity. This review provides a comprehensive overview of the molecular mechanisms in which CBL exerts its function and describes the clinical manifestation of CBL mutations in humans

Equilibria of a Class of Transport Equations Arising in Congestion Control

This paper studies a class of transport equations arising from stochastic models in congestion control. This class contains two cases of loss point process models: the rate-independent Poisson case where the packet loss rate is independent of the throughput of the flow and the rate-dependent case where the point process of losses has an intensity which is a function of the instantaneous rate. This class of equations covers both the case of persistent and of non-persistent flows. We give a direct proof of the fact that there is a unique density solving the associated differential equation and we provide a closed form expression for this density and for its mean value

Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.Comment: 20 pages, 1 figur