# Poisson arrivals, poisson departures with rate proportional to queue length

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)?

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

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 $\frac{\lambda}{\lambda_d}$.