  We obtain the state probabilities of various fractional versions of the
classical homogeneous Poisson process using an alternate and simpler method
known as the Adomian decomposition method (ADM). Generally these state
probabilities are obtained by evaluating probability generating function using
Laplace transform. A generalization of the space and time fractional Poisson
process involving the Caputo type Saigo differential operator is introduced and
its state probabilities are obtained using ADM.
