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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
![$\displaystyle \mathcal{T}^{G}_\beta = \sum_{\xi \in A\leftarrow\beta} \mathcal{P}_{\xi} \mathfrak{t}_{\xi}.$](img703.png) | (6.20) |
If we multiply every branching probability,
, that appears in
by
then the mean escape time can be obtained as: ![\begin{displaymath}\begin{array}{lll} \mathcal{T}^{G}_\beta &=& \displaystyle \l...
...ftarrow\beta} \mathcal{P}_{\xi} \mathfrak{t}_{\xi}. \end{array}\end{displaymath}](img706.png) | (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 ![\begin{displaymath}\begin{array}{rll} \mEt _{\alpha}^{G} &=& \dsum _{a\in A} \ti...
...\in G} \mEt _{\alpha}^{G} \mSt _{\alpha,\beta}^{G}. \end{array}\end{displaymath}](img712.png) | (6.22) |
Note that
etc., while the mean escape time can now be written as ![$\displaystyle \mathcal{T}^{G}_\beta = \left[ \frac{d\tilde{\Sigma}_\beta^{G}}{d\zeta} \right]_{\zeta=0}.$](img714.png) | (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