In this paper, we study the time-dependent shortest paths problem for two types of time-dependent FIFO networks. First, we consider networks where the availability of links, given by a set of disjoint time intervals for each link, changes over time. Here, each interval is assigned a non-negative real value which represents the travel time on the link during the corresponding interval. The resulting shortest path problem is the time-dependent shortest path problem for availability intervals ( ), which asks to compute all shortest paths to any (or all) destination node(s) d for all possible start times at a given source node s . Second, we study time-dependent networks where the cost of using a link is given by a non-decreasing piece-wise linear function of a real-valued argument. Here, each piece-wise linear function represents the travel time on the link based on the time when the link is used. The resulting shortest paths problem is the time-dependent shortest path problem for piece-wise linear functions ( ) which asks to compute, for a given source node s and destination d , the shortest paths from s to d , for all possible starting times. We present an algorithm for the problem that runs in time O (( F d + γ )(| E |+| V |log | V |)) where F d is the output size (i.e., number of linear pieces needed to represent the earliest arrival time function to d ) and γ is the input size (i.e., number of linear pieces needed to represent the local earliest arrival time functions for all links in the network). We then solve the problem in O ( λ (| E |+| V |log | V |)) time by reducing it to an instance of the problem. Here, λ denotes the total number of availability intervals in the entire network. Both methods improve significantly on the previously known algorithms.