In this talk I will discuss our recent formulation aimed at mixing classical queuing theory with open quantum dynamics, in terms of the so-called collision models, which describe the sequential interactions of a system with independent ancillas. Our theory is motivated by recent advances in neutral atom arrays, which showcase the possibility of having classical controllers governing the quantum dynamics. We study a model where a system interacts with a sequence of ancillas governed by a classical controller that allows them to queue up while they wait for their turn to interact with the system. The ancillas can undergo individual open dynamics while they wait, which may cause them to decohere. The system, which plays the role of the server in the queue, can also undergo its own open dynamics whenever it is idle. We first show that this framework generalizes existing approaches for quantum collision models, recovering the deterministic and stochastic formulations in the appropriate limits. Next, we show how the classical queueing dynamics introduces non-trivial effects in the quantum collisions, that can lead to different phases in the system-ancilla response. We illustrate the idea with a model of coherence transfer under noisy waiting dynamics.