Probability that a task Is operating at a given instant = l/t.
Probability that a task isn’t operating at a given instant = (t-l)/t.
Probability that at least m of N tasks are operating at a given instant is C(N,i)[ (l/t)^i][(t-l)/t]^(N-i) summed for i=m to N where C(N,i) is the combination of N objects taken i at a time.
Integrate the sum over (0,t] . Result is (((t-l)^N)/t^(N-1))∑i=m to N [C(N,i)(l/(t-l))^i] which further simplifies to
(((t-l)^N)/t^(N-1))[(t/(t-l))^N – ((t/(t-l))^m)]
This further simplifies to t[1-((t-l)/t)^(N-m)]
We only have to compute the probability for one interval of length t since the start times for each task has fixed period t and task lasts for same interval l. So if the event of m tasks happening in an instant over an interval of length t doesn’t happen once, it never happens.