# The input and output processes in a single-server queue

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 interval $$[0,t]$$ and $$D_t$$ be the number of departures in that time interval; then both $$A_t$$ and $$D_t$$ have Poisson distribution with mean $$\lambda t$$ (the second one due to Burke's theorem), but they are of course not independent.

Question: what is $$\mathop{\mathrm{Cov}}(A_t,D_t)$$? (Probably, this is known, but I couldn't find the reference$$\dots$$)

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 $$A_t D_t = \frac{1}{2}(A_t^2 + D_t^2 - (A_t-D_t)^2),$$ from which we obtain (recall that both $$A_t$$ and $$D_t$$ are Poisson($$\lambda t$$)) $$\mathop{\mathrm{Cov}}(A_t,D_t) = \lambda t - \frac{1}{2}\mathbb{E}(\eta_t-\eta_0)^2,$$ thus expressing $$\mathop{\mathrm{Cov}}(A_t,D_t)$$ in terms of $$\mathop{\mathrm{Cov}}(\eta_0,\eta_t)$$.
Note that the second term in the above expression is bounded, and moreover converges to $$\mathop{\mathrm{Var}} \eta_0 = \frac{\rho}{(1-\rho)^2}$$ as $$t\to\infty$$ due to asymptotic independence of $$\eta_0$$ and $$\eta_t$$.
For a fixed $$t$$, the explicit expression for $$\mathop{\mathrm{Cov}}(\eta_0,\eta_t)$$ can be found in e.g. Section 2 of the following paper: Reynolds, J. F. (1975). The covariance structure of queues and related processes – a survey of recent work. Advances in Applied Probability, 7(02), 383–415, DOI:10.2307/1426082