I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to tell the emitters apart. I would like to estimate the probability that signals from these emitters overlap at the receiver. Because signal overlap will result in spurious data, I need an idea of how many receivers I can use and as well as realistic values for signal lengths and the time delay between emissions.
Therefore, I would like to derive a general expression for the probability of signal collisions in terms of the number of emitters $N$, signal length $t_{sig}$, and time delay between emissions $T$.
$N$ will be ~ 10 - 100, $t_{sig}$ will be on the order of milliseconds, and $T$ will be between 10s and 60s.
Each emitter will have a random start time uniformly distributed in the interval $[0,T]$. Each emitter will have equal values for $t_{sig}$ and $T$. The start time of signal emissions from different emitters are independent of one another.