Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field

Why Max-min Fairness Is Not Suitable For Multi-Hop Wireless Networks

We consider the issue of which criteria to use when evaluating the design of a wireless multihop network. It is known, and we illustrate in this paper, that maximizing the total capacity, or transport capacity, leads to gross imbalance and is not suitable. An alternative, which is often used in networking, is to consider the max-min fair allocation of rates, or of transport rates per node. We apply max-min fairness to the class of wireless, multi-hop networks for which the rate of a wireless link is an increasing functions of signal-to-noise ratio. This class includes CDMA and UWB. We show that, for a network in this class, the max-min fair allocation of bit or transport rates always gives the same rate to all flows. We show on one example that such an allocation is highly undesirable when the network is asymmetric. Another form of fairness, utility fairness, does not appear to have the same problem

A General Method for Finding the Most Economical Distributed Router Architecture

In this work we present a novel method to determine the optimal parameters of a router architecture when certain router performance constraints are given. The total financial expense, or cost, is the optimality criterion. We introduce a general, essentially distributed, router architecture model, consisting of locally or remotely located forwarding engines or processing units gathered around a switch of variable speed. Given the following constraints: number of inputs, maximum line interface bandwidth, and maximum packet delay in a router, the presented method finds the optimal amount and distribution of processing power among the various available processing units and the optimal parameters for the switching element. The optimization employs an estimated market-based cost function per element and finds the most economical system solution. The results show that the optimal solutions gather around two extreme points of the solution space, distinguishable by the distribution of the processing power mass and corresponding switch speed. We discuss when, depending on the customer input, one or the other solution is appropriate

On Downlink Capacity of Cellular Data Networks with WLAN/WPAN Relays

We consider the downlink of a cellular network supporting data traffic. In addition to the direct traffic from the base-station, each user is equipped with the same type of 802.11-like WLAN or WPAN interface that is used to relay packets to further users and hence to improve the performance of the overall network. We are interested in analyzing what are the design guidelines for such networks and how much capacity improvements can the additional relay layer bring, in comparison to cellular networks. We consider a realistic dynamic setting where users randomly initiate downloads and leave the system upon transfer completion. A first objective is to provide a scheduling/relay strategy that maximizes the network capacity, which is the traffic in bit/s/cell that the network can support. We find that, regardless of the spatial traffic distribution, when the cell approaches saturation (the number of active users is very large), the capacity-achieving strategy divides the cell into two areas: one closer to the base-station where the relay layer is always saturated and some nodes receive traffic through both direct and relay links, and the further one where the relay is never saturated and the direct traffic does not exist. We further show that it is approximately optimal to use fixed link lengths, and we derive this length. We give a simple algorithm to calculate the cell capacity. The obtained capacity is shown to be independent of the cell size (unlike in traditional cellular networks), and it is 20%-60% higher than already proposed relay architectures when the number of users is large. Finally, we provide guidelines for future protocol design

On Performance of Event-to-Sink Transport in Transmit-Only Sensor Networks

We consider a hybrid wireless sensor network with regular and transmit-only sensors. The transmit-only sensors do not have receiver circuit, hence are cheaper and less energy consuming, but their transmissions cannot be coordinated. Regular sensors, also called cluster-heads, are responsible for receiving information from transmit-only sensors and forwarding it to sinks. The main goal of such a hybrid network is to reduce the cost of deployment while achieving some performance constraints (minimum coverage, sensing rate, etc). In this paper we are interested in the communication between transmit-only sensors and cluster-heads. We develop a detailed analytical model of the physical and MAC layer using tools from queuing theory and stochastic geometry. (The MAC model, that we call Erlang's loss model with interference, might be of independent interest as adequate for any non-slotted; i.e., unsynchronized, wireless communication channel.) We give an explicit formula for the frequency of successful packet reception by a cluster-head, given sensors' locations. We further define packet admission policies at a cluster-head, and we calculate the optimal policies for different performance criteria. Finally we show that the proposed hybrid network, using the optimal policies, can achieve substantial cost savings as compared to conventional architectures

M/D/1/0 loss system with interference and applications to transmit-only sensor networks

We propose and analyze a probabilistic model of packet reception in the steady state regime of a non-slotted wireless communication channel. It is an extension of the classical M/D/1/0 Erlang's loss model where the {\em interference} created by different packet emissions is introduced by means of a shot-noise process. More precisely, we assume that a given {\em packet is admitted} by the single receiver if this latter is idle at the packet arrival epoch and {\em successfully received} if, in addition, its signal-to-interference-and-noise ratio averaged over the reception period is large enough. As the main result we prove an analog of the Erlang's formula for the ergodic rate of the successfully received packets. Our work is motivated by some applications to transmit-only sensor networks

Communication complexity of approximate maximum matching in the message-passing model

We consider the communication complexity of finding an approximate maximum matching in a graph in a multi-party message-passing communication model. The maximum matching problem is one of the most fundamental graph combinatorial problems, with a variety of applications. The input to the problem is a graph G that has n vertices and the set of edges partitioned over k sites, and an approximation ratio parameter α. The output is required to be a matching in G that has to be reported by one of the sites, whose size is at least factor α of the size of a maximum matching in G. We show that the communication complexity of this problem is Ω(α2kn)information bits. This bound is shown to be tight up to a log n factor, by constructing an algorithm, establishing its correctness, and an upper bound on the communication cost. The lower bound also applies to other graph combinatorial problems in the message-passing communication model, including max-flow and graph sparsification