Because time is not scaled, limiting results for many-server scaling retains dependence on the service time distribution, as we saw in the scaling of M/GI/1. We extend these infinite server results to general time-dependent arrival streams.

We discuss the classic Jackson network with general i.i.d. interarrivals and service times, the generalized Jackson network. Like the GI/GI/1 system, the generalized Jackson network cannot be analyzed in detail, and we discuss fluid and diffusion approximations to the network process.

We consider Brownian problems of scheduling and admission control, where we force congestion to be kept at the least costly nodes, and use admission control to regulate congestion.

We study matching in parallel specialized service systems.

We define fluid limits and show that stability of the fluid limits implies stability of the stochastic queueing system. This enables us to study stability of MCQN under various policies.

We introduce Brownian motion and related processes.

We discuss control of transient MCQN. We introduce queues with infinite supply of work that can achieve stability with traffic intensity rho = 1.

We define the single queue, introduce notation and some relations and properties, and present simple examples of queues. We also discuss simulation of queues.

Queueing networks are all pervasive; they occur in service, manufacturing, communication, computing, the internet, and transportation. Much of queueing theory is aimed at performance evaluation of stochastic systems. Extending the methods of deterministic optimization to stochastic models so as to achieve both performance evaluation and control is an important and notoriously hard area of research. In this book our aim is to familiarize the reader with recent techniques for scheduling and control of queueing networks, with emphasis on both evaluation and optimization.

We study a queueing system with memoryless Poisson arrivals and generally distributed processing times, the so-called M/G/1 system. Performance measures of this system can be derived exactly, using the principle of work conservation and the property of PASTA (Poisson arrivals see time averages).

We look at control of networks in balanced heavy traffic. We consider routing to parallel servers; this illustrates our aim to always pool system resources. We also observe state space collapse and biriefly discuss the diffusion limit for MCQN.

We consider scheduling of batches of jobs and scheduling of stationary streams of jobs. We discuss priority queues and other service policies.

Heavy traffic M/M/s has full utilization of servers at the cost of congestion, while M/M/1 has no waiting but poor utilization. These are termed efficiency driven (ED) and quality driven (QD) regimes, respectively. A golden middle road of quality and efficiency driven (QED) is the Halfin–Whitt regime, studied and extended to G/GI/n here.

We discuss weak convergence, the functional strong law of large numbers, and the central limit theorem.