In this paper, we propose two Model Predictive Control algorithms, whose on-line computational demands are significantly smaller than that for conventional Model Predictive Control algorithms, for control of large-scale constrained linear systems. We show that closed-loop stability can be guaranteed under some conditions. We also propose an optimal anti-windup scheme for approximating Model Predictive Control (thus eliminating the need for solving an on-line optimization problem) and derive a quantitative condition under which Model Predictive Control can be approximated effectively. These results make Model Predictive Control a very attractive candidate to be applied to systems with small sampling times and/or with a large number of inputs, and address achievable constrained performance by any anti-windup design.