The simulation of chemical reacting systems is one of the most challenging topics in Systems Biology, due to their complexity and inherent randomness. The Gillespie's Stochastic Simulation Algorithm (SSA) is a standard algorithm to simulate well-stirred biochemical systems, butthe computational burden makes this algorithm slow to compute for many realistic problems. Recent programmability improvements allow non-graphics applications to leverage the Graphics Processing Units' (GPUs) computational power. This paper describes practical issues arising by a parallel implementation on GPU technology, shows how to reduce the memory space required by one of the most known versions of SSA, and presents the application of the implemented algorithm to a test model.