A lightweight model predictive controller for a class of discrete event systems is developed. The target is first in first out (FIFO) systems, in which the precedence relations are represented by a directed acyclic graph. Using max-plus algebra, a system's behavior can be described by a set of linear equations referred to as the state space representation. Using this representation, we address the problem of adjusting system parameters and determining optimal control inputs for given reference signals. To determine system parameters, we formulate and solve an optimization problem using a branch-and-bound method, by which the number of constraints is reduced. Then, the optimal control inputs are determined. Since this part is often the bottleneck in online control, we propose two methods for the efficient computation thereof. The first method is beneficial for systems with sparse adjacency matrices and small-scale systems with a greater prediction horizon. In contrast, the second approach is beneficial for larger-scale systems with a smaller prediction horizon. The effectiveness of the proposed framework is verified through numerical simulations.