The quickest path problem deals with the transmission of a message of size ${sigma}$ from a source to a destination with the minimum end to end delay over a network with bandwidth and delay constraints on the links. We consider two basic modes and four variations for the message delivery at the nodes reflecting the mechanisms such as circuit switching. Internet protocol, and their combinations, For each of first five modes, we present O($m^2+mnlogn$) algorithm to compute the quickest path for a given message size ${sigma}$, For the last mode, the quickest path can be computed in O(m+nlogn) time.