Waiting time distribution of solar energetic particle events modeled with a non-stationary Poisson process

We present a study of the waiting time distributions (WTDs) of solar energetic particle (SEP) events observed with the spacecraft $WIND$ and $GOES$. Both the WTDs of solar electron events (SEEs) and solar proton events (SPEs) display a power-law tail $sim Delta t^{-gamma}$. The SEEs display a broken power-law WTD. The power-law index is $gamma_{1} =$ 0.99 for the short waiting times ($<$70 hours) and $gamma_{2} =$ 1.92 for large waiting times ($>$100 hours). The break of the WTD of SEEs is probably due to the modulation of the corotating interaction regions (CIRs). The power-law index $gamma sim$ 1.82 is derived for the WTD of SPEs that is consistent with the WTD of type II radio bursts, indicating a close relationship between the shock wave and the production of energetic protons. The WTDs of SEP events can be modeled with a non-stationary Poisson process which was proposed to understand the waiting time statistics of solar flares (Wheatland 2000; Aschwanden $&$ McTiernan 2010). We generalize the method and find that, if the SEP event rate $lambda = 1/Delta t$ varies as the time distribution of event rate $f(lambda) = A lambda^{-alpha}exp(-beta lambda)$, the time-dependent Poisson distribution can produce a power-law tail WTD $sim Delta t^{alpha - 3}$, where $0 leq alpha < 2$.