  In potential flow networks, the equilibrium flow rates are usually not
proportional to the demands and flow control elements are required to regulate
the flow. The control elements can broadly be classified into two types -
discrete and continuous. Discrete control elements can have only two
operational states: fully open or fully closed. On the other hand, continuous
control elements may be operated in any intermediate position in addition to
the fully open and fully closed states. Naturally, with their increased
flexibility, continuous control elements can provide better network
performance, but $to~what~extent$?
  We consider a class of branched networks with a single source and multiple
sinks. The potential drop across edges ($\Delta H$) is assumed to be
proportional to the $n^{th}$ power of flow rate ($Q$), i.e., $\Delta H=kQ^n$ ,
($n>=1$). We define $\textbf{R}$ as the ratio of minimal operational times
required to transport a given quantum of material with either type of control
element and show that $1\leq \textbf{R}\leq m^{\left(1-1/n\right)}$, where $m$
is the maximum depth of the network. The results point to the role of network
topology in the variations in operational time. Further analysis reveals that
the selfish operation of a network with continuous control valves has the same
bounds on the price of anarchy.
