Let's say I have $N$ deterministic departure processes from $N$ sources, each with period $T_i$. Assume no knowledge on the starting times for the processes, or assume that these starting times are initialized from time to time. (I think that this assumption is the same as saying that the times of the first departures are random variables distributed $U(0, T_i)$)
I am considering the combination of all these departure times from the point of view of a server receiving and serving these entities, regardless of their provenance.
The inter-arrival time expected value is easily calculated since the total rate of arrivals is known, but what about the variance?
Is there a way to calculate this variance, or getting an approximation under reasonable assumptions?
Thank you!
Noting $I$ an inter-arrival time, you have immediately:
$I\in [0,\min(T_i)]$.
hence $Var(I)\leq \frac{\min(T_i)}{4}$.
If your periods $T_i$ are submultiples of a bigger one $T$, you may let $n_i=T/T_i$ and use the information $E[I]=\frac{T}{\sum n_i}$ to improve your bound. Or proceed to numerical experiments (with Monte-Carlo's method) to get numerical approximations.
Using the density of $\mathbb{Q}$ it shouldn't be hard to show that:
$\displaystyle E[i]=\frac{1}{\sum \frac{1}{T_i}}$
holds for any $(T_i)$.