where $t_0$ is the cosmic age (i.e. the present time), and $R$ is the local pulsar birth rate (${\rm yr}^{-1}~{\rm kpc}^{-2}$). Here $\varepsilon_{e,i}$ is the energy at the time $t_i$ of CR electrons/positrons which are cooled down to $\varepsilon_e$ at $t$.

RESULTS AND DISCUSSIONS
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First, we calculate the cosmic ray electron/positron spectra from the pair starved MSPs. We set the pair multiplicity $\kappa= 1$, the lifetime $\tau= 5\times 10^{10}$yr, the total energy $E_{\rm rot}= 10^{52}$ erg, the local birth rate $R = 3\times 10^{-9}$ yr$^{-1}$ kpc$^{-2}$ and the fraction of the lost energy due to synchrotron emission 30% for each MSP. We assume that each MSP has the same value of the parameters ($B_0=10^{8.5}{\rm G}, \Omega=10^3{\rm s}^{-1}, R=10^6{\rm cm}$), because most MSPs have the almost same spin-down luminosity. For the injection distribution function, we assume mono-energetic distribution eq.(\[mono\]) with the energy $\varepsilon_e=50$ TeV. Even if we consider the power-law distribution eq.(\[pl\]), the energy range of the distribution is small because the allowed range of the cutoff energy should be only 50 - 80 TeV due to the observed constraint by KASKADE/GRAPES/CASA-MIA [@KY09]. Thus the distribution should be nearly mono-energetic distribution. Note that our local MSP birth rate is based 