Recent fluorescence spectroscopy measurements of the turnover time distribution of single-enzyme turnover kinetics of beta-galactosidase provide evidence of Michaelis-Menten kinetics at low substrate concentration. However, at high substrate concentrations, the dimensionless variance of the turnover time distribution shows systematic deviations from the Michaelis-Menten prediction. This difference is attributed to conformational fluctuations in both the enzyme and the enzyme substrate complex and to the possibility of both parallel- and off-pathway kinetics. Here, we use the chemical master equation to model the kinetics of a single fluctuating enzyme that can yield a product through either parallel- or off-pathway mechanisms. An exact expression is obtained for the turnover time distribution from which the mean turnover time and randomness parameters are calculated. The parallel- and off-pathway mechanisms yield strikingly different dependences of the mean turnover time and the randomness parameter on the substrate concentration. In the parallel mechanism, the distinct contributions of enzyme and enzyme substrate fluctuations are clearly discerned from the variation of the randomness parameter with substrate concentration. From these general results, we conclude that an off-pathway mechanism, with substantial enzyme substrate fluctuations, is needed to rationalize the experimental findings of single-enzyme turnover kinetics of beta-galactosidase.