A general formula for the distribution of sojourn time in a Markov chain bas been derived by Takacs [2]. Some special cases of that distribution are considered in this paper. The methods of deriving the distributions are elementary and quite different from that of Takacs. The cases considered are a two state and a three state Markow chain with cyclic transitions which are Poisson type events with different parameters. A simple generalization top states is also discussed in Section 4.