On the dynamics and optimization of spatial random systems
In this thesis, we model and study a collection of real-world systems – each of whose evolution or behavior depends on spatial relationships between locations of interacting entities that comprise the system. For each, we use the theory of Poisson point processes (PPPs) to model the random locations of constituent entities. All but one of the systems we study are wireless networks comprised of transmitters and receivers, whose interaction with the system at large is via the signal power they transmit and receive respectively. The other is an epidemic system – here the constituent entities are members of a population among whom a contagion is spreading. Each individual's interaction with the system at large is via their transmission of the contagion to other individuals. We are concerned with two lines of inquiry with respect to these systems - dynamics and optimization. The former line of inquiry is the study of the time-evolution of a time-varying system, the characterization of its steady-state behavior and engineering/design insights derived thereafter. The first two works presented in this thesis fall under this umbrella. In the first work, we consider a wireless queue in a field of moving transmitters. The queue's arrival rate is independent of the rest of the system, but its departure rate depends on the interference it experiences. This interference in turns depends on the locations of the moving transmitters and is therefore time-varying. We characterize stability conditions for the queue as a function of the mobility (showing the non-existence of stability guarantees in the absence of mobility and the existence of a stability guarantee in its presence). We then show that the queue length decreases (according to a stochastic ordering) as mobility increases, which is a consequence of the interference becoming less variable. The overall insight is that mobility in networks improves queuing delay-related performance metrics. The second work also studies the effect of mobility, albeit its effect on the spread and survival of an SIS epidemic. Individuals in a population can spread a contagion to neighbors that are close enough, and recover from the contagion independent of the rest of the system. Individuals move around via large i.i.d. displacements. We find exact expressions that characterize the steady-state fraction of infected individuals using the rate conservation principle, and approximate expressions for the same via moment measure closure methods. We then use them to conjecture a phase diagram (supported by simulations) that describes the values of system parameters for which the epidemic dies out. In doing so, we find that reducing mobility (eq., lockdowns) in the population need not always lead to the contagion dying out faster. We then shift our attention to the latter line of inquiry, optimization, which is concerned with studying an objective function (eq., a performance metric) of a wireless system, and maximizing it with respect to a particular system variable. In the third and fourth works in this thesis, the system variable we optimize over is the intensity measure of the PPP. In our third work, we attempt to analytically solve a specific optimization problem – given transmitters distributed according to a homogeneous PPP, our goal is to show (under general assumptions on the system) that the probability of coverage of an arbitrary fixed location is maximized for a unique value of the intensity of the PPP. We make partial progress by using the theory of Malliavin calculus to show that there must be at least one and at most a finite number of maxima, but fall short of our final goal. Our next work takes a numerical approach to solving a far more general problem – given a wireless performance metric (the mean of a functional of a PPP), we propose an algorithm to find the optimal mean number of transmitters and a corresponding optimized intensity measure that controls placement of transmitters. This algorithm also uses the theory of Malliavin calculus – here, to iteratively move along a steepest ascent direction and hence improve the intensity measure. Numerical results show that the algorithm produces wireless network models that adapt well to underlying system configurations. The proposed framework is hence a powerful tool for the dimensioning and planning of wireless networks, and to our knowledge, the first such tool based on point process theory. Our final work deals with the problem of optimizing a global proportional fairness metric for a (very large) wireless network in a distributed way using local power control. We do so via a heuristic that considers the nearest-neighbor graph of transmitter-receiver pairs and decomposes the graph into its individual connected components. We establish that each connected component is on average small, and hence the corresponding optimization problems are of low average complexity. Under specific assumptions on the system, the optimization problem can be further simplified and has a closed form solution. Simulations show that the resulting power control scheme results in significant performance gains.