In this paper, we present the conditions under which the traffic processes in a pure jump Markov process with a general state space are Poisson processes, and give a simple proof of PASTA type theorem in Melamed (1982) and Walrand (1988).Furthermore, we consider a generalized network with phase type negative arrivals and show that the network has a product-form invariant distribution and its traffic processes which represent the customers exiting from the network are Poisson processes.