This paper considers linear discrete-time systems with additive bounded disturbances subject to hard control input bounds and constraints on the expected number of state-constraint violations averaged over time, or, equivalently, constraints on the probability of a state-constraint violation averaged over time. This specification facilitates the exploitation of the information on the number of past constraint violations, and consequently enables a significant reduction in conservatism. For the type of constraint considered we develop a recursively feasible receding horizon scheme, and, as a simple modification of our approach, we show how a bound on the average number of violations can be enforced robustly. The computational complexity (online as well as offline) is comparable to existing model predictive control schemes. The effectiveness of the proposed methodology is demonstrated by means of a numerical example.