attractor

Using phase-type distribution results for Markov chains [10], we obtain an explicit formula for calculating the expected event rate [[lambda].sub.Y] to reach the absorbing state Y, given the initial probability row vector [pi] for the transient states [mathematical expression not reproducible], as

As seen in the figure, the system has an absorbing state as an output, and thus this is the discrete case.

R is (n - r) x r matrix denoting the probabilities of transitions from the absorbing states to transient states.

In the HMF setting, under PI dynamics, when [epsilon] = 0 the final state for the population is the absorbing state with a density of cooperators [rho] = 0 (full defection) except if the initial state is full cooperation.

4.3 Expected time [T.sub.w] from S(1, 0) to absorbing state S(w, w)

Let H be the index of a portfolio, which at time t has [S.sub.i]:i=-3,-2,-1,0,1,2,3 transient states and [R.sub.k]:k=1,2,3,4 charged-off or absorbing states. Here, H, over a given time interval 0, t, is given as

In this case, one can see that it takes 29.38866 or approximately 30 steps for a loan initially in state -3 to leave the transient states for any absorbing state. In other words, since the step is 1 month, a loan more than 3 months prepaid (state 3) could become sold or defaulted in approximately 30 months or 2.5 years, while a loan with 3 months past due could reach the same destiny in approximately 7 months.

Inactivity has become a much less absorbing state since the beginning of the Danish labour market reforms in 1994.

The example is shown below using standard Markov chain notation, for a Markov chain with a source term (recruits) and an absorbing state (leavers):

Thus, beginning with a single index case (person 1 without loss of generality), the system undergoes stochastic transitions until it reaches an absorbing state. Figure 2 provides an example of one such trajectory.

The sales model advanced below has both transient states (states that can be both entered and exited) and absorbing states (states that can be entered but not exited).

Both of these states are called absorbing states. The other states are called transient states, because they may either be entered or exited.