3.6 Combination of Monte Carlo Method with Direct Calculation

The MC part simulates only the frequent state space, which gives the occupation probabilities of frequent states. The occupation probability

The essential assumption is that the rare states cause only a small
perturbation to the frequent state probabilities.

where
*T*_{j,rare} is the time spent in the rare state *j*, which can be
directly calculated from the stationary ME.

is the number of times the rare state *j* would
be in average visited from state *i*,
is the
exit rate of state *j* and thus
is the average
time spent in state *j* for one visit. We are using time averages for the
direct calculation. Actually the durations are distributed with a Poisson
distribution. However, the directly calculated rare state space is only a small
perturbation to the frequent state space which is calculated with a MC
method where the Poisson distribution of the tunnel durations is fully
incorporated.
The occupation probability for a rare state is therefore

The algorithm follows all possible events starting
at frequent states. If a frequent state is encountered the algorithm
terminates that branch, because the state probability is already known.
Once the
probability of a state is lower than a predefined limit no further descent
into following branches from this state is made (see Fig. 3.10).