We consider a queuing system with coupled processors (CPS), in which the service rate at each queue varies over time in function of the set of active queues in the system. Performance analysis of CPS has so far been based on simulations or on complex Markov chains under restricting assumptions on input traffic statistics. In contrast, we propose a fully analytical approach to CPS, based on a worst case analysis of system dynamics, and applicable to a large family of traffic characterizations. We derive sufficient conditions for stability for traffic characterized stochastically as well as for traffic constrained by arrival curves, and we show how to compute bounds on backlog and delay. We illustrate our approach and assess our results by means of an example of coupling of wireless transmissions.