In this article we introduce multiple stuttering-Poisson distributions and discuss their genesis, properties and applications. We consider a special case of triple stuttering-Poisson distribution with probability generating functionG(t) = Exp [a(t− 1) +b(t2− 1) +c(t3− l)] and investigate (1) maximum likelihood estimation, (2) moment estimation, and (3) mixed moment estimation of the parametersa,b,c. First order terms in the expressions for the biases and covariances of moment estimators are presented. The joint asymptotic efficiencies of moment estimators are tabulated for a few selected parameter points. The first order terms in the biases of maximum likelihood estimators and moment estimators are also given for a few selected parameter points. As judged from the contribution of the first order term, the possible bias of maximum likelihood estimators can be substantial over a considerable region of the parameter space. The moment estimators have low joint asymptotic efficiency over a large part of the parameter space. However, the moments of moment estimators are always finite. Estimation in four parameter distribution is also considered. An illustrative example is given and some concluding remarks concerning estimators are made.