Downscaling site rainfall from daily to sub-daily resolution is often approached using the multiplicative discrete random cascade (MDRC) class of models, with mixed success. Questions in any application – for MDRCs or indeed other classes of downscaling model - is to what extent and in what way are model parameters functions of rainfall event type and/or large scale climate controls. These questions underlie the applicability of downscaling models for analysing rainfall and hydrological extremes, in particular for synthesising long-term historical or future sub-daily extremes conditional on historic or projected daily data. Using fine resolution data from two gauges in central Brisbane, Australia, covering the period 1908-2015, microcanonical MDRC models are fitted using data from 1 day to 11.25 minute resolutions in seven cascade levels, each level dividing the time interval and its rainfall volume into two sub-intervals. Each cascade level involves estimating: the probabilities that all the rainfall observed in a time interval is concentrated in the first and the second of the two sub-intervals; and also two Beta distribution parameters that define the probability of a given division of the rainfall into both sub-intervals. These parameters are found to vary systematically with time of day, month of year, decade, rainfall volume, event temporal structure and ENSO anomaly. Reasonable downscaling performance is achieved in an evaluation period - in terms of replicating extreme values and autocorrelation structure of 11.25-minute rainfall given the observed daily data - by including the parameter dependence on the rainfall volume and event structure, which involves 16 parameters per cascade level. Using only a volume dependence and assuming symmetrical probability distributions reduces the number of parameters to two per level with only a small loss of performance; and empirical relationships between parameter values and cascade level reduces the total number of parameters to four, with indetectable further loss of performance. Improving the parameterisation of the volume dependence is considered the most promising opportunity for improving at-site performance.