Analyzing Queuing Systems with Coupled Processors through Semidefinite Programming

We consider queuing systems with coupled processors, where the service rate at each queue varies depending on the set of queues in the system with non-zero queue lengths. In general, such queuing systems are very difficult to analyze and steady state queue length distributions are known only for two-queue systems. The coupled-processors model arises naturally in the study of several systems where a resource is shared by several classes of customers. We study the stochastic recursive equations that govern such systems, and obtain lower and upper bounds on the moments of the queue length by formulating a moments problem and solving a semidefinite relaxation of the original problem. The resulting bounds are very close even at lower-order relaxations, and can be made progressively tighter at the expense of increasing the complexity of the associated semidefinite program. We demonstrate the effectiveness of our proposed methodology in studying and optimizing dynamic systems such as interference-coupled wireless networks and GPS servers.