3 
continues to age or it rejuvenates and starts afresh from 
age zero.The successive passages through the zero state 
represent a recurrent event. The probability that the 
recurrence time equals k is p^ ■ L q. 
We are interested in the question:How many years have 
passed,or rather how many steps in the hierarchy have been 
mounted,since the last rejuvenation? This is the "spent 
waiting time" of a renewal process. Choosing an arbitrary 
starting point we can say that in the year n the system 
will be in state if and only if the last re-juvenation 
occurred in year n-k. Letting n-k increase we obtain in 
the limit the steady state probability of the "spent 
waiting time" (Feller 1968 Chapter V ). It is 
proportionate to the tail of the recurrence time 
• • • • lc 
distribution i.e. to p . 
More directly the vector of steady state probabilities u^ 
can be derived from the following two conditions: 
u k = P u k-l* 
Uq = qu Q + qu x + qu 2 + (1) 
The first condition ensures the invariance of the steady 
state;the second ensures that entries to and exits from 
the population balance. 
It follows that 
u k = P ku o*