Distributed computing networks enable the deployment of a wide range of digital services in the form of interconnected functions instantiated over computing resources distributed across the network. We consider the problem of optimal delivery of next generation information services over a distributed computing network, in which nodes are equipped with both communication and computing resources. We address the design of distributed control solutions that drive local routing, processing, and resource allocation decisions, while providing global objective guarantees subject to physical and network layer constraints.  We characterize the computing network capacity region and design a joint flow scheduling and resource allocation algorithm that stabilizes the underlying queuing system while achieving arbitrarily close to minimum network cost, with a trade off in network delay.