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$\dots$)


1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.