Learning optimal admission control in partially observable queueing networks

We develop an efficient reinforcement learning algorithm that learns the optimal admission control policy in a partially observable queueing network. Specifically, only the arrival and departure times from the network are observable, optimality refers to the average holding/rejection cost in infinite horizon, and efficiency is with respect to regret performance. While reinforcement learning in partially-observable Markov Decision Processes (MDP) is prohibitively expensive in general, we show that the regret at time T induced by our algorithm is $${\tilde{O}} \left( \sqrt{T \log (1/\rho )}\right) $$ O ~ T log ( 1 / ρ ) where $$\rho \in (0,1)$$ ρ ∈ ( 0 , 1 ) is connected to the mixing time of the underlying MDP. In contrast with existing regret bounds, ours does not depend on the diameter (D) of the underlying MDP, which in most queueing systems is at least exponential in S, i.e., the maximal number of jobs in the network. Instead, the role of the diameter is played by the $$\log (1/\rho )$$ log ( 1 / ρ ) term, which may depend on S but we find that such dependence is “minimal”. In the case of acyclic or hyperstable queueing networks, we prove that $$\log (1/\rho )=O(S)$$ log ( 1 / ρ ) = O ( S ) , which overall provides a regret bound of the order of $${\tilde{O}} \left( \sqrt{T S}\right) $$ O ~ TS . In the general case, numerical simulations support the claim that the term $$\log (1/\rho )$$ log ( 1 / ρ ) remains extremely small compared to the diameter. The novelty of our approach is to leverage Norton’s theorem for queueing networks and an efficient reinforcement learning algorithm for MDPs with the structure of birth-and-death processes.