There are only two alternative transition for a person in
this system: Either a rise of income from one year to the
next (note that the income classes are on the log scale)
which has probability p; or death of the person, i.e.
transition to the zero class, which has probability q
(p + q = 1). In addition, there are entries from the zero
to
class^replenish the stock of income receiveif.
The essence of this- model is described by Feller 1 ^ in the
following terms: The state represents the age of the
system. When the system reaches age K, it either 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 a, is i 'q.
We are interested in the question: How many years have
passed, i.e. how many income steps have been mounted, since
the last rejuvenation? This is the "spent waiting time”
of the renex^al process. Choosing an arbitrary starting point
we can say that in the year n the system will be in state E/t
if and only if the last rejuvenation occured in year n-M,.
Letting n-K increase we obtain in the limit the steady state
probability of the'fepent waiting time" £ ^. It is proportionate
!) Z"7 7 Vol. I. 2?.3, p. 382.
2 V^T7 Chapter V.