Combining two circulating normal distributions

Question

I am working in estimating the impact of location error on location based services. My question is listed below. If the error distribution of location estimation follows in general a normal distribution $N_1(0,\sigma_1)$. However, after a time interval $t_1$, the error follows another distribution $N_2(0,\sigma_2)$ for a time interval $t_2$. Then, it returns back to distribution $N_1$ for $t_1$, and so on. My question is how to combine both distributions to represent the final distribution of the location error? Everything is independent i.e. $N_1$ and $N_2$ is independent and $t_1$ and $t_2$ is independent as