Summary: A classical result states that for a finite-state homogeneous continuous-time Markov chain \(\{X_t\}_{t\geq 0}\) with finite state space \(E=\{0,\dots,n- 1\}\) and intensity matrix \(Q= (q_{kj})\) the matrix of transition probabilities \(P_{kj}(t)= P(X_t= j\mid X_0=k)\) is given by \(P(t)= (P_{kj}(t))_{k,j\in E}=\exp(Qt)=\sum^\infty_{s= 0}Q^st^s/s!\). In contrast to this infinite series expression, we derive an explicit finite sum formula for this transition matrix for situations where \(Q\) is a circulant matrix. Our result is motivated by the compound Poisson process \(D_t= \sum^{N_t}_{j=0} Y_j\) (with discrete random i.i.d. variables \(Y_j\) and a Poisson counting process \(\{N_t\}_{t\geq 0}\)): It is shown that the stochastic process \(X_t\equiv D_t\text{ mod }n\) is a Markov process on \(E\) with a circulant intensity matrix \(Q\) and we apply the previous results to calculate, e.g., the distribution and the expectation of \(X_t\). The process \(\{X_t\}_{t\geq 0}\) provides a stochastic model for, e.g., channel assignment in telecommunication, bus occupancies, box packing etc.