This paper studies the complexity of pi-calculus processes with respect to the quantity of transitions caused by an incoming message. First we propose a typing system for integrating Bellantoni and Cook's characterisation of polynomially-bound recursive functions into Deng and Sangiorgi's typing system for termination. We then define computational complexity of distributed messages based on Degano and Priami's causal semantics, which identifies the dependency between interleaved transitions. Next we apply a syntactic flow analysis to typable processes to ensure the computational bound of distributed messages. We prove that our analysis is decidable for a given process; sound in the sense that it guarantees that the total number of messages causally dependent of an input request received from the outside is bounded by a polynomial of the content of this request; and complete which means that each polynomial recursive function can be computed by a typable process.