Asymptotic distribution of rewards accumulated by alternating renewal processes

Consider an alternating renewal process on the states 'broken' vs 'working'. Suppose that during any interval [0,τ], the process is rewarded at rate g(t∕τ) if it is working at time t. Let Qτ be the reward that is accumulated during [0,τ]. We calculate μQτ and σQτ2 such that (Qτ−μQτ)∕σQτ converges in distribution to a standard normal distribution as τ→∞.