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.

We extend the methods developed in Chapter 16 to routing control and demonstrate significant savings that result from pooling efforts and balancing the contents of the nodes.

We present surprising examples of MCQN with traffic intensity rho < 1 that are unstable under some policies and prepare the background for rigorous treatment of stability.

We study the supermarket model and show that choosing the shortest of just a few randomly chosen queues is almost as good as JSQ. Another issue with many-server systems is specialization, with several customer and server types, and limited compatibility between them

We continue the discussion of control of transient MCQN. We formulate a fluid optimization problem that we can solve using a separated continuous linear programming (SCLP) algorithm.We then describe a method of tracking the optimal fluid solution, using virtual infinite queues and maximum pressure policy. We show that this procedure is asymptotically optimal for high-volume systems, as exemplified by semiconductor wafer fabs.