0
$\begingroup$

Suppose arrivals to a queue (/collection, more accurately) follow $Poisson(\lambda_a)$. In other words, the arrival rate is constant $\lambda_a$.

Departures, however, at time $t$ go at rate $n_t\lambda_d$ where $n_t$ is the number of items in the queue at time $t$.

If $n_0=0$, what is the distribution of $n_t$?


For additional clarity, an analogy can be made with radio-active particles: Suppose they arrive at a Poisson rate $\lambda_a$ but decay randomly with expected lifetime of $\tau=1/\lambda_d$; what is the distribution of the number of undecayed particles at time $t$ (after starting with zero)?

$\endgroup$
  • 1
    $\begingroup$ I hope the downvoter will at least tell me what's wrong with the question. $\endgroup$ – Museful Oct 7 '15 at 18:54
1
$\begingroup$

What you are describing sounds exactly like the $M/M/\infty$ queue. Much is known about this queue, e.g. the stationary distribution of the number of users in the system when $t\to\infty$ is Poisson with rate $\frac{\lambda}{\lambda_d}$.

$\endgroup$

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.