tokens: x min in our research. Figure 1. Cumulative distributions of fluence, peak counts, rise time and duration for 4U 1728-34. The best-fitting results are shown by the red lines. The best-fitting parameters are given in each panel. Based on the criteria proposed by Aschwanden ( 2011 , 2014 ), which are the statistical independence of events, non-linear growth phase, random rise times and power-law distribution of these observed parameters, type I X-ray bursts can be explained by a SOC process. The SOC theory predicts that subsystems self-organize, owing to some driving force, to a critical state, at which a slight perturbation can cause a chain reaction of any size within the system (Bak, Tang Wiesenfeld 1987 ). The SOC theory also predicts a power-law distribution for the waiting times (Aschwanden 2011 ; Marković Gros 2014 ). The WTD has been widely discussed in astrophysics (Wheatland, Sturrock McTiernan 1998 ; Aschwanden McTiernan 2010 ; Wang Dai 2013 ; Li et al. 2014 ; Wang et al. 2015 ; Guidorzi et al. 2015 ; Wang Yu 2017 ). In the SOC theory, the WTD is usually described as a (non-)stationary Poisson process. The probability function is expressed by (Wheatland et al. 1998 ) where Δ t is the waiting time, T is the period and λ( t ) is the burst occurrence rate. Below, we model the occurrence rate λ. We propose that the burst rate can be modelled by λ( t ) ∝ t −γ . The reason is given below. Type I X-ray bursts are induced by a thin-shell instability. In LMXBs, the accreted H/He fuel accumulates on the stellar surface and forms a shell. The typical time for nuclear energy release is τ N ∝ Δ M / L N , where Δ M is the mass of the shell and L N is the nuclear energy generation rate (Hōshi 1968 ; Joss 1977 ). In the quiescent stage, L N remains almost unchanged. The energy will be diffused in a typical time τ d ∝ Δ M 2 , which also depends on the stellar temperature and the opacity of the shell (Hōshi 1968 ; Ayasli Joss 1982 ). The condition for triggering a burst is τ N /τ d = Δ M crit /Δ M ≤ 1, where the critical shell mass Δ M crit is obtained by τ N = τ d (Hōshi 1968 ; Ayasli Joss 1982 ). Alternative descriptions of this critical condition are also used (Fujimoto et al. 1981 , 1987 ; Narayan Heyl 2003 ). We assume that Δ M = Δ M crit happens at a typical time t 0 . When the shell grows thick enough at a time T , any perturbation affects the stellar hydrostatic structure significantly and leads to stable nuclear burning (Schwarzschild Harm 1965 ). Therefore, a burst can be triggered only in a time range t 0 ≤ t T . Since these time-scales are mainly determined by the shell mass, it is plausible to assume that the occurrence rate also mainly depends on the shell mass. Because the accreted mass is proportional to time, we can adopt for the burst rate The occurrence rate is taken as λ = 0 when t t 0 . We put all the time-dependent effects into the power-law index γ, which contains the dependence on shell mass and the possible time evolution of
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