Hi,
Suppose arrivals come as a rate lambda Poisson process. Let Au be
the time since the last arrival up to time u. Precisely, if
0 = T0 < T1 < T2 < T3 < ... where {Ti : i >= 1} are the arrival times
and Ti <= u < Ti+1, then Au = u-Ti which is a number in [0, u]. In
particular,
if there are no arrivals in [0, u] then we set At = t. The notation Au
refers to
age, namely the age of the most recent arrival.
Find the probability distribution of Au, by giving the cumulative
distribution function Gu(t) = P[Au <= t] for all real t.
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The following is my solution.
P[A(u)>t] = P[u-A(u)<u-t] = P[T(i) < u-t]
= P[no arriavals in (u-t, u)]
= P[N(u)-N(u-t) = 0]
= exp(-lamba*t)
Then P[A(u)<=t] = 1-exp(-lamba*t)
Is it right? When u goes to infinite, it doesn't influence P[A(u)<=t].
Anything wrong?
Thanks,
Fan
.
.
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