Under many circumstances event-triggered scheduling outperforms time-triggered schemes for control applications when resources such as communication, computation, or energy are sparse. This paper investigates another benefit of event-triggered control concerning the ability of adaptation that enables the implementation of distributed scheduling mechanisms. The system under consideration comprises multiple heterogeneous control systems that share a common resource for accomplishing their control tasks. Each subsystem is modeled as a discrete-time stochastic linear system. The design problem is formulated as an average cost Markov decision process (MDP) problem with unknown global system parameters that are to be estimated during execution. Techniques from distributed optimization and adaptive MDPs are used to develop distributed self-regulating event-triggers that adapt their request rate to accommodate a global resource constraint. Stability and convergence issues are addressed by using methods from stochastic stability for Markov chains and stochastic approximation. Numerical simulations show the effectiveness of the approach and illustrate the convergence properties.