In multiple-user communications, the bursty nature of the packet arrival times cannot be divorced from the analysis of the transmission process. However, in traditional information theory the random arrival times are smoothed out by appropriated source coding and no consideration is made for the end-to-end delay. In this thesis, using tools from network theory, we investigate simple models that consider the end-to-end delay and/or the variability of the packet arrivals as important parameters, while staying in a information theoretic framework. First, we simplify the problem and focus on the transmission of a bursty source over a single-user channel. We introduce a new measure of channel features that enable us to incorporate the possibility to code among several packets in a scheduling problem. In this setup, we look for policies that minimize the average packet delay. Assuming that the packets are independent and sufficiently large to perform capacity achieving coding, we then consider the problem of allocating rates among a finite number of users communicating through a multiple-user channel. Following the previous work in the context of multiple-access channel, we formulate a scheduling problem in which the rate of each user is chosen from the capacity region of the multiple-user channel. Here, the goal is to find a scheduling policy that minimizes the sum of the transmitter queue lengths, such a policy is called delay optimal. In particular settings, for the additive Gaussian multiple-access channel we show the delay optimality of the Longer Queue Higher Rate policy introduced by Yeh. And, when the users communicate through a symmetric broadcast channel, we propose and show the delay optimality of a Best User Highest Possible Rate policy, among a large class of admissible policies. Finally, in the last part of this thesis, we look at the multiple-user channel coding problem from the perspective of the receivers. By measuring the transmission rates at the receivers, we are able to define variable length codes and to characterize the region of achievable rates when the receivers can decode their intended messages at different instants of time. For the two-user degraded broadcast channel and for the two-user multiple-access channel, we show that the gain in using variable length codes essentially comes from the possibility for the receivers to decode the transmitted messages in non-overlapping periods of time.
seeding'' perfect information on the replicas at the boundaries of the coupling chain. This extra information makes decoding easier near the boundaries, and this effect is then propagated into the coupling chain upon iterations of the decoding algorithm. Spatial coupling was also applied to various other problems that are governed by low-complexity message-passing algorithms, such as random constraint satisfaction problems, compressive sensing, and statistical physics. Each system has an associated algorithmic threshold and an optimal threshold. As with coding, once the underlying graphs are spatially coupled, the algorithms for these systems exhibit optimal performance. In this thesis, we analyze the performance of iterative low-complexity message-passing algorithms on general spatially coupled systems, and we specialize our results in coding theory applications. To do this, we express the evolution of the state of the system (along iterations of the algorithm) in a variational form, in terms of the so-called potential functional, in the continuum limit approximation. This thesis consists of two parts. In the first part, we consider the dynamic phase of the message-passing algorithm, in which iterations of the algorithm modify the state of the spatially coupled system. Assuming that the boundaries of the coupled chain are appropriately seeded'', we find a closed-form analytical formula for the velocity with which the extra information propagates into the chain. We apply this result to coupled irregular LDPC code-ensembles with transmission over general BMS channels and to coupled general scalar systems. We perform numerical simulations for several applications and show that our formula gives values that match the empirical, observed velocity. This confirms that the continuum limit is an approximation well-suited to the derivation of the formula. In the second part of this thesis, we consider the static phase of the message-passing algorithm, when it can no longer modify the state of the system. We introduce a novel proof technique that employs displacement convexity, a mathematical tool from optimal transport, to prove that the potential functional is strictly displacement convex under an alternative structure in the space of probability measures. We hence establish the uniqueness of the state to which the spatially coupled system converges, and we characterize it. We apply this result to the (l,r)-regular Gallager ensemble with transmission over the BEC and to coupled general scalar systems.