Sampling from conditional distiribution

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I want to sample from condtional distribution given by formula: $$ P[T_{n+1} =t| T_n=t_n] = \frac{\alpha_t+m_(t_n)}{\alpha_t+\beta_t+s(t_n)}\prod^{t-1}_{j=0}\frac{\beta_j+r_j(t_n)}{\alpha_j+\beta_j+s_j(t_n)}$$ $t_n$ is a vector of given data,$s,r$-sums depending on vector $t_n$