Total Flow Analysis (TFA) is a method for the worst-case analysis of time-sensitive networks. It uses service curve characterizations of the network nodes and arrival curves of flows at their sources; for tractability, the latter are often taken to be linear functions. For periodic flows, which are common in time-sensitive networks, linear arrival curves are known to provide less good bounds than ultimately pseudo-periodic (UPP) arrival curves, which exactly capture the periodic behaviours. However, in existing tools, applying TFA with many flows and UPP curves quickly becomes intractable because when aggregating several UPP curves, the pseudo-period of the aggregate might become extremely large. We propose a solution to this problem, called Finite-Horizon TFA. The method computes finite horizons over which arrival and service curves can be restricted without affecting the end-results of TFA. It can be applied to networks with cyclic dependencies. We numerically show that, while remaining computationally feasible, the method significantly improves the bounds obtained by TFA when using linear curves.