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Serguei Popov
Serguei Popov
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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 $\mathop{\mathrm{Cov}}(A_t,D_t)$? (This has to beProbably, this is known, but I couldn't find the reference$\dots$)

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 $\mathop{\mathrm{Cov}}(A_t,D_t)$? (This has to be known, but I couldn't find the reference$\dots$)

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$\dots$)

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Serguei Popov
Serguei Popov
  • 1.9k
  • 12
  • 21

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 $N_t$$A_t$ be the number of arrivals in the time interval $[0,t]$ and $D_t$ isbe the number of departures in that time interval; then both $N_t$$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{Cov}(N_t,D_t)$$\mathop{\mathrm{Cov}}(A_t,D_t)$? (This has to be known, but I couldn't find the reference$\dots$)

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 $N_t$ be the number of arrivals in the time interval $[0,t]$ and $D_t$ is the number of departures in that time interval; then both $N_t$ 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 $\mathop{Cov}(N_t,D_t)$? (This has to be known, but I couldn't find the reference$\dots$)

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 $\mathop{\mathrm{Cov}}(A_t,D_t)$? (This has to be known, but I couldn't find the reference$\dots$)

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Serguei Popov
Serguei Popov
  • 1.9k
  • 12
  • 21

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 $N_t$ be the number of arrivals in the time interval $[0,t]$ and $D_t$ is the number of departures in that time interval; then both $N_t$ 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 $\mathop{Cov}(N_t,D_t)$? (This has to be known, but I couldn't find the reference$\dots$)