Abstract: In this talk, I will discuss methods to compute the efficiency of a microwave photon detector based on a current-biased Josephson junction (JJ), which is either connected to a (driven) microwave resonator or driven directly by a classical microwave source. Here I consider the evolution of the system in the presence of the environment and tunneling events of the JJ to the voltage state. I will then discuss how one can use the waiting time distribution to compute the average switching rate of the JJ and the efficiency of the photon detector. Finally, I will present a preliminary result on the condition for power matching between the detector, which I treat as a linear oscillator, and the microwave source.