We analyzed the initial rising behaviors of X-ray outbursts from two transient low-mass X-ray binaries (LMXBs) containing a neutron-star (NS), Aql X-1 and 4U 1608-52, which are continuously being monitored by MAXI/GSC in 2--20 keV, RXTE/ASM in 2--10 keV, and Swift/BAT in 15--50 keV. We found that the observed ten outbursts are classified into two types by the patterns of the relative intensity evolutions in the two energy bands below/above 15 keV. One type behaves as the 15--50 keV intensity achieves the maximum during the initial hard-state period and drops greatly at the hard-to-soft state transition. On the other hand, the other type does as both the 2--15 keV and the 15--50 keV intensities achieve the maximums after the transition. The former have the longer initial hard-state ($gtrsim$ 9 d) than the latters ($ltsim$5 d). Therefore, we named them as slow-type (S-type) and fast-type (F-type), respectively. These two types also show the differences in the luminosity at the hard-to-soft state transition as well as in the average luminosity before the outburst started, where the S-type are higher than the F-type in the both. These results suggest that the X-ray radiation during the pre-outburst period, which heats up the accretion disk and delays the disk transition (i.e., from a geometrically thick disk to a thin one), would determine whether the following outburst becomes S-type or F-type. The luminosity when the hard-to-soft state transition occurs is higher than $sim 8 times10^{36}$ erg s$^{-1}$ in the S-type, which corresponds to 4% of the Eddington luminosity for a 1.4 Mo NS.
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