Joint tail distribution conjecture for overlap times in identical tandem queues

Let $j$ be the number of stations, and let $O^{(i)}_{n,n+k}$ denote the overlap time at station $i$ for the $n^{th}$ and $(n+k)^{th}$ customers in an $(M/M/\infty)^j$ tandem queue whose stations have identically distributed service times. For parameters $\lambda$ and $\mu$, and thresholds $x_1,\dots,x_j$, Joint overlap-tail conjecture. The joint tail distribution is

$$\mathbb{P} \left( O^{(1)}_{n,n+k} > x_1, O^{(2)}_{n,n+k} > x_2, \dots, O^{(j)}_{n,n+k} > x_j \right) = \frac{e^{-2\mu \left(\sum^j_{i=1} x_i \right)}}{2^{j-1}} \left( \frac{\lambda}{\lambda+\mu} \right)^k.$$

This conjecture extends the paper's explicit two-station overlap-time analysis to an arbitrary number of identically distributed stations. The proposed $j$-dimensional formula is motivated by the need to generalize the results beyond two stations; the source does not provide a resolution, so its status remains open.