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The arrival and departure processes are obviously not independent: suppose that, with some very bad luck, no customers arrived to the system up to now; then (after the completion of the service of tho …

Consider an $M/M/1$ queue with the arrival rate $\lambda>0$ and the service rate $\mu>\lambda$ (so that it is stable), in the stationary regime. Let $A_t$ be the number of arrivals in the time interva …

Let $\eta_t$ be the number of customers in the system at time $t$ and $\rho=\lambda/\mu<1$ be the load. It holds that $\eta_0+A_t-D_t = \eta_t$, so $A_t-D_t = \eta_t-\eta_0$. Write $$ A_t D_t = \frac{ …

Let $\lambda:=1-\epsilon_1$, $\mu:=1-\epsilon_2$; also, denote $p_L:=\frac{\lambda}{L}(1-\frac{\mu}{L})$ and $q_L:=\frac{\mu}{L}(1-\frac{\lambda}{L})$. Consider a fixed queue (one of those $L$), then …

Heuristically, this probability should behave as $O(\sqrt{L/n})$, I guess. Observe that each queue, when not empty, is a random walk with zero drift, that actually moves once every $O(L^{-1})$ instanc …