After reset of AD7484, when CONVST signal is given by single time, the BUSY is always low. But when it is given a continuous pulse to CONVST pin, it works well. I found out that the problem is the RESET signal provided for the chip is too short to make it work normally. The specifications you provided on datasheet dosen't specify how long it is, when I correct it, all is well.
A reset pulse is vital to ensure correct operation of the AD7484. I would believe that this would fix the issue that the customer previously described. It is the edge of the reset signal that is used to reset the internals of the part so the pulse width shouldn't matter too much once it is a reasonable length (>5ns or so) and happens after the power to the device is stable. Since the customer didn't provide us with any plots showing the reset pulse widths before and after the fix, it's difficult to figure out whether it was truly related to the width of the pulse or the position of the pulse with respect to the VDD line becoming stable. I agree that a minimum pulse width should be included in the datasheet though.