Joint Learning and Control in Stochastic Queueing Networks with Unknown Utilities

We study the optimal control problem in stochastic queueing networks with a set of job dispatchers connected to a set of parallel servers with queues. Jobs arrive at the dispatchers and get routed to the servers following some routing policy. The arrival processes of jobs and the service processes of servers are stochastic with unknown arrival rates and service rates. Upon the completion of each job from dispatcher un at server sm, a random utility whose mean is unknown is obtained. We seek to design a control policy that makes routing decisions at the dispatchers and scheduling decisions at the servers to maximize the total utility obtained by the end of a finite time horizon T. The performance of policies is measured by regret, which is defined as the difference in total expected utility with respect to the optimal dynamic policy that has access to arrival rates, service rates and underlying utilities. We first show that the expected utility of the optimal dynamic policy is upper bounded by T times the solution to a static linear program, where the optimization variables correspond to rates of jobs from dispatchers to servers and the feasibility region is parameterized by arrival rates and service rates. We next propose a policy for the optimal control problem that is an integration of a learning algorithm and a control policy. The learning algorithm seeks to learn the optimal extreme point solution to the static linear program based on the information available in the optimal control problem. The control policy, a mixture of priority-based and Joint-the-Shortest-Queue routing at the dispatchers and priority-based scheduling at the servers, makes decisions based on the graphical structure induced by the extreme point solutions provided by the learning algorithm. We prove that our policy achieves logarithmic regret whereas application of existing techniques to the optimal control problem would lead to Ω(√T)-regret. The theoretical analysis is further complemented with simulations to evaluate the empirical performance of our policy.