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Mean Escape Times
Since the mean first passage time between states and , or the escape time from a subgraph, is of particular interest, we first illustrate a means to derive formulae for these quantities in terms of pathway probabilities.
The average time taken to traverse a path is calculated as , where is the mean waiting time for escape from node , as above, identifies the th node along path , and is the length of path .The mean escape time from a graph if started from node is then
| || (6.20)|
If we multiply every branching probability, , that appears in by then the mean escape time can be obtained as:
| || (6.21)|
This approach is useful if we have analytic results for the total probability , which may then be manipulated into formulae for , and is standard practice in probability theory literature [208,209]. The quantity is similar to the ` probability' described in Reference GoldhirschG86. Analogous techniques are usually employed to obtain and higher moments of the first-passage time distribution from analytic expressions for the first-passage probability generating function (see, for example, References Raykin92,MurthyK89). We now define and the related quantities
| || (6.22)|
Note that etc., while the mean escape time can now be written as
| || (6.23)|
In the remaining sections we show how to calculate the pathway probabilities, , exactly, along with pathway averages, such as the waiting time. Chain graphs are treated in Section 4.2 and the results are generalised to arbitrary graphs in Section 4.3.
Next: Chain Graphs Up: Introduction Previous: KMC and DPS Averages Contents Semen A Trygubenko 2006-04-10