We calculate the exact tail asymptotics of stationary response times for open stochastic event graphs, in the irreducible and reducible cases. These networks admit a representation as (max,plus)-linear systems in a random medium. We study the case of renewal input and i.i.d service times with subexponential distributions. We show that the stationary response times have tail asymptotics of the same order as the integrated tail of service times. The mutiplicative constants only involve the intensity of the arrival process and the (max,plus)-Lyapunov exponents of certain sequences of (max,plus)-matrices associated to the event graph.