Maurits de Graaf

Optimal deployment of caches in the plane

We consider wireless caches placed in the plane according to a homogeneous Poisson process. A data file is stored at the caches, which have limited storage capabilities. Clients can contact the caches to retrieve the data. The caches store the data according to one of the two data allocation strategies: partitioning & coding. We consider the Pareto front of the expected deployment cost of the caches and the expected cost of a client retrieving the data from the caches. We show that there is a strong trade-off between the expected retrieval and the expected deployment cost under the partitioning and the coding strategies. We also show that under coding, it is optimal to deploy a high number of caches, each with low storage capacity.

Probabilistic analysis of power assignments

A fundamental problem for wireless ad hoc networks is the assignment of suitable transmission powers to the wireless devices such that the resulting communication graph is connected. The goal is to minimize the total transmit power in order to maximize the life-time of the network. Our aim is a probabilistic analysis of this power assignment problem. We prove complete convergence for arbitrary combinations of the dimension d and the distance-power gradient p. Furthermore, we prove that the expected approximation ratio of the simple spanning tree heuristic is strictly less than its worst-case ratio of 2. Our main technical novelties are two-fold: First, we find a way to deal with the unbounded degree that the communication network induced by the optimal power assignment can have. Minimum spanning trees and traveling salesman tours, for which strong concentration results are known in Euclidean space, have bounded degree, which is heavily exploited in their analysis. Second, we apply a recent generalization of Azuma-Hoeffdings inequality to prove complete convergence for the case p>=d for both power assignments and minimum spanning trees (MSTs). As far as we are aware, complete convergence for

Data retrieval time for energy-harvesting wireless sensor networks

p > d

Energy-efficient data collection in wireless sensor networks with time constraints

has not been proved yet for any Euclidean functional.

Deployment versus data retrieval costs for caches in the plane

We consider the problem of retrieving a reliable estimate of an attribute monitored by a wireless sensor network, where the sensors harvest energy from the environment independently, at random. Each sensor stores the harvested energy in batteries of limited capacity. Moreover, provided they have sufficient energy, the sensors broadcast their measurements in a decentralized fashion. Clients arrive at the sensor network according to a Poisson process and are interested in retrieving a fixed number of sensor measurements, based on which a reliable estimate is computed. We show that the time until an arbitrary sensor broadcasts has a phase-type distribution. Based on this result and the theory of order statistics of phase-type distributions, we determine the probability distribution of the time needed for a client to retrieve a reliable estimate of an attribute monitored by the sensor network. We also provide closed-form expression for the retrieval time of a reliable estimate when the capacity of the sensor battery or the rate at which energy is harvested is asymptotically large. In addition, we analyze numerically the retrieval time of a reliable estimate for various sizes of the sensor network, maximum capacity of the sensor batteries and rate at which energy is harvested. These results show that the energy harvesting rate and the broadcasting rate are the main parameters that influence the retrieval time of a reliable estimate, while deploying sensors with large batteries does not significantly reduce the retrieval time.

Content-Based Routing in Networks with Time-Fluctuating Request Rates

We consider the problem of retrieving a reliable estimate of an attribute from a wireless sensor network within a fixed time window and with minimum energy consumption for the sensors. The sensors are located in the plane according to some random spatial process. They perform energy harvesting and follow an asleep/awake cycle. A sink, at a random location in the plane, requests measurements from the awake sensors in order to retrieve an estimate of an attribute. The sink has to collect a sufficient number of measurements within a fixed time window. Moreover, the sink aims to minimize the energy that the sensors use to transmit their measurements. We determine a closed-form expression for the expected energy consumption of the sensors when measurements are retrieved according to a Greedy schedule. We also provide an upper bound on the maximum expected distance over which a sensor transmits under this Greedy schedule. Furthermore, we formulate a Markov Decision Process (MDP) to determine a sensor transmission schedule with general time constraints. We also develop a heuristic that schedules the sensors for transmission. We compare numerically the performance of the MDP schedule with the heuristic and with an offline, optimal schedule, where the asleep/awake state of the sensors is assumed to be known ahead of time. We show that the energy consumption under the MDP schedule converges to the energy of the offline schedule as the size of the time window for measurement collection increases. We also show that the heuristic performs close to the MDP schedule in terms of energy consumption.

On a tandem queue with batch service and its applications in wireless sensor networks

We consider the problem of finding the Pareto front of the expected deployment cost of wireless caches in the plane and the expected retrieval cost of a client requesting data from the caches. The data is stored at the caches according to partitioning and coding strategies. We show that under coding, it is optimal to deploy many caches with low storage capacity. For partitioning, we derive a simple relation between the cost of the cache deployment and the cost of retrieving the data from the caches. We quantify the improvements offered by optimal coding in comparison to partitioning, i.e., we derive a relation for the difference in deployment cost required to achieve a given retrieval cost. Finally, we show that even non-optimal coding is better than partitioning in the sense that no coded deployment can be dominated by a partitioning strategy.

Average Case Analysis of the MST-heuristic for the Power Assignment Problem: Special Cases

In large-scale distributed applications, a loosely-coupled event-based style of communication as in publish-subscribe systems eases the integration of autonomous, heterogeneous components. In a publish-subscribe system, content-based routing - where routing is based on the content of the messages - is an alternative to address-based delivery. In this paper we provide a time-dependent analysis of the identity-based routing scheme. Our analytical approach is based on continuous-time Markov chains and extends the steady-state approach by Jaeger and Muhl [7] to systems with time-fluctuating parameters. For m -ary trees with k levels, with a single publisher at the root and subscribers at the leaves, we obtain explicit closed form solutions for the time-dependent distribution of the traffic rates in the network. The results allow us to investigate, for example, the impact of time-fluctuating request rates versus time-independent request rates, and the switching point between optimality of flooding and identity-based routing.

Probabilistic Analysis of Power Assignments

We present a tandem network of queues