Bits through Time

In any communication system, there exist two dimensions through which the information at the source becomes distorted before reaching the destination: the noisy channel and time. Messages transmitted through a noisy channel are susceptible to modification in their content, due to the action of the noise of the channel. Claude E. Shannon, in his seminal paper of 1948 "A Mathematical Theory of Communication", introduces the bit as a unit of measure of information, and he lays down the theoretical foundations needed to understand the problem of sending bits reliably through a noisy channel. The distortion measure, which he used to quantify reliability, is the error probability. In his paper, Shannon shows that any channel is characterized by a number that he calls capacity: It represents the highest transmission rate that can be used to communicate information with, through this same channel, while guaranteeing a negligible error probability. Whereas, even if the messages are sent through a perfect channel, the time they take to reach their destination causes the receiver to acquire a distorted view of the status of the source that generated these messages. For instance, take the case of a monitor interested in the status of a distant process. A sender observes this process and, to keep the monitor up-to-date, sends updates to it. However, if, at any time t, the last received update at the monitor was generated at time u(t), then the information at the receiver reflects the status of the process at time u(t), not at time t. Hence, the monitor has a distorted version of reality. In fact, it has an obsolete version with an age of t-u(t). The concept of age as a distortion measure in communication systems was first used in 2011 by Kaul et al., in order to assess the performance of a given vehicular network. The aim of the authors was to come up with a transmission scheme that would minimize an age-related metric: the average age. Since then, a growing body of works has used this metric to evaluate the performance of multiple communication systems. The drive behind this interest lies in the importance that status-update applications are gaining in today's life (in vehicular networks, warehouse and environment surveillance, news feed,etc.). In this thesis, we choose age as a distortion measure and derive expressions for the average age and the average peak-age (another age-related metric) for different communication systems. Therefore, we divide this dissertation into two parts: In the first part, we assume that the the updates are transmitted through a noiseless channel that has a random service time. In the second part, we consider a special category of noisy channels, namely the erasure channel. In the first part of this thesis, in order to compute the age-related metrics, we employ queue-theoretic concepts. We study and compare the performance of various transmission schemes under different settings.We show that the optimal transmission scheme when the monitor is interested in a single source loses its optimality when another source of higher priority shares the system. In the second part of this thesis, we introduce, in our age calculations, the distortion caused by the erasure channel on the transmitted updates. In order to combat the erasures of the channel, we first consider two flavors of the hybrid automatic repeat request (HARQ). Finally, we focus on the optimal average age that could be achieved over an erasure channel.