We consider (max,+)-algebra products of random matrices, which arise from performance evaluation of acyclic fork-join queueing networks. A new algebraic technique to examine properties of the product and investigate its limiting behaviour is proposed based on an extension of the standard matrix (max,+)-algebra by endowing it with the ordinary matrix addition as an external operation. As an application, we derive bounds on the (max,+)-algebra maximal Lyapunov exponent which can be considered as the cycle time of the networks.