A counter meant to register successes is locked for exactly r-1
epochs following a registration so the occurrence of a success at
epoch
n would be registered if no registration of one occurred in the
preceding r-1 epochs. Upon a registration of a success at epoch
n the mechanism would then be locked for epochs n+1 through
n+r-1, so that a success could not be registered during any of
these. A success would be registered at the n+r th epoch if it
occurred since the counter would then be free. Let wn be the
probability the counter is free (will register a success if one
occurs ) at epoch n. If the
probability of success p lies in (0,1) then there is, for each M
, 0 < M < 1, greater than the reciprocal of the smallest magnitude of those
roots of 1- ( q z + p zr) = 0 outside the unit circle in the complex plane, a finite
K such that for all n
| wn - 1/(q+rp) | &le K Mn
while for for every M less than that reciprocal there is no
choice of real
K for which this holds.
This is but one example of the application of the results
on convergence rates of renewal sequences.