1 EE 549 — Queueing Theory Instructor : Michael Neely

A modern data communication system consists of a network of information processing stations or nodes interconnected by data transmission channels or links. Information flows from node to node in packetized data units. These packets can originate from a variety of sources, and the paths over which they traverse the network can intersect, so that multiple packets are often intended for delivery over the same transmission link. When the number of packets in a particular node exceeds the service capabilities of the outgoing links of the node, some packets must either be dropped (resulting in a loss of information) or placed in a storage buffer or queue for future service. Queueing theory is the study of congestion and delay in such systems. The dynamics of queueing systems are event driven, that is, the state of a queue changes based on discrete events such as packet arrivals or departures. Such systems are fundamentally different from the linear or nonlinear systems studied in classical control theory, where inputs are continuous functions of time (such as voltage or current waveforms) and where state dynamics are described by differential equations. Queueing systems are also very different from the point-to-point transmission models of physical layer communication, where inputs consist of band-limited signals that are corrupted by continuous background processes such as additive Gaussian noise. Thus, queueing systems require a different and unique set of mathematical tools. An understanding of these tools is essential in the analysis, design, and control of data networks. In this course, we study queueing systems from both a deterministic and probabilistic perspective. In our deterministic analysis, we illuminate several simple properties and conservation laws exhibited by queueing systems with any arbitrary set of arrival times and packet sizes. We also develop several fundamental bounds on the worst case congestion and delay in networks when traffic is deterministically controlled so that the rate and burstiness of packet arrivals conforms to specified constraints. This deterministic analysis sets the stage for a probabilistic description of queueing systems. In this context, the arrival streams and service processes are assumed to be driven by simple probability laws, so that exact performance metrics can be calculated, such as the average delay in the network, or the probability distribution for the number of packets in a particular queue. The probabilistic study of event driven systems involves the theory of discrete stochastic processes. We shall find that several of the key results of discrete stochastic process theory, such as the ergodicity properties of Markovian networks as well as Little’s Theorem for average delay, establish a fundamental relationship between deterministic and probabilistic analysis of queueing networks.