the function $f_{\rm KN}(x)$ is the correction factor to include the Klein-Nishina (KN) effect, which approaches to unity when $x$ is much smaller than unity. The mathematical expression of $f_{\rm KN}$ can be found in Moderski et al. (2005).

As shown in the last section, in general MSPs have a very long lifetime ($\tau \sim 5\times 10^{10}{\rm yr}$) which is comparable with the cosmic age. In such a case that a point-like source with a finite duration, taking into account the time-dependence of an injection rate ($Q_0(\varepsilon_e)\rightarrow Q_0(\varepsilon_e, \tilde{t})$), the spectrum can be calculated by integrating $G(t,r,\varepsilon_e;\tau)$ for $\tau$:

$$f_1(t,r,\varepsilon_e; t_i)=\int_{t_i}^{t} d\tilde{t}~G\left( t,r,\varepsilon_e; \tilde{t} \right) ,$$

where $t_i$ is the time when the particle injection from a source has started. Here we assume that the electron/positron injection spectrum can be described as mono-energetic distribution $$\label{mono}
Q_0(\varepsilon_e,\tilde{t})\propto \left( 1+\frac{\tilde{t}-t_i}{\tau} \right)^{-2},$$ or power-law distribution $$\label{pl}
Q_0(\varepsilon_e,\tilde{t})\propto \varepsilon_e^{-\alpha}\exp \left( -\frac{\varepsilon_e}{\varepsilon_{\rm cut}}\right) \left( 1+\frac{\tilde{t}-t_i}{\tau} \right)^{-2},$$ where $\alpha$ is the intrinsic power-law index of an electron/positron spectrum and $\varepsilon_{\rm cut}$ is the maximum electron/positron energy from a source.

Now we can calculate the average electron/positron spectrum by considering the birth rate of MSPs as follows:

$$f_{\rm ave}(\varepsilon_e)=\int_0^{t_0} dt_i \int_0^{d_{\rm diff}(\varepsilon_e,\varepsilon_{e,i})} 2\pi r dr f_1(t_0,r,\varepsilon_e; t_i)R,$$

