  Enzyme kinetics are cyclic. A more realistic reversible three-step mechanism
of the Michaelis-Menten kinetics is investigated in detail, and three kinds of
waiting cycle times $T$, $T_{+}$, $T_{-}$ are defined. It is shown that the
mean waiting cycle times $<T>$, $<T_{+}>$, and $<T_{-}>$ are the reciprocal of
the steady-state cycle flux $J^{ss}$, the forward steady-state cycle flux
$J^{ss}_{+}$ and the backward steady-state cycle flux $J^{ss}_{-}$
respectively. We also show that the distribution of $T_{+}$ conditioned on
$T_{+}<T_{-}$ is identical to the distribution of $T_{-}$ conditioned on
$T_{-}<T_{+}$, which is referred as generalized Haldane equality. Consequently,
the mean waiting cycle time of $T_{+}$ conditioned on $T_{+}<T_{-}$ ($<T_{+}|
T_{+}<T_{-}>$) and the one of $T_{-}$ conditioned on $T_{-}<T_{+}$ ($<T_{-}|
T_{-}<T_{+} >$) are both just the same as $<T>$. In addition, the forward and
backward stepping probabilities $p^{+},p^{-}$ are also defined and discussed,
especially their relationship with the cycle fluxes and waiting cycle times.
Furthermore, we extend the same results to the $n$-step cycle, and finally,
experimental and theoretically based evidences are also included.
