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Take the M/M/1 queue (exponential inter-arrival times, exponential service times, one server).

Consider the queue to have initially n(0) customers.

The queue runs for a finite amount of time $T$.

While the queue is running, there are $n(t)$ customers in the queue at time $t \in \left[0,T\right]$.

Consider a hotdog vendor placed near the queue. This vendor makes money off the queue. The longer the queue, the more money he makes per minute. He makes precisely $k$ credits per minute when the queue contains $n(t)=k$ customers (including the customer in service).

How much money does the vendor make before the queue is shutdown (after time $T$ has passed) given the service rate $\mu$ and the arrival rate $\lambda$ are equal (i.e. the queue is critically loaded)? Note this means I seek $\mathbb{P}(\text{revenue in time $T$})=R(T)$.

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My understanding of this is that I need to consider the transient behaviour of the queue (which is known in terms of an infinite sum of Bessel functions), but on further investigation, the probability mass function of the vendors revenue involves some difficult number theory concerning solutions to a Diophantine equation in a Poissonly distributed number of variables. Are there known results on this problem? Can anyone point me to a solution?

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    $\begingroup$ Consider the amount of money earned before the system becomes empty again (and the process $\{n(t):t\geqslant 0\}$ regenerates), and the probability distribution of the number of regeneration epochs before time $T$. $\endgroup$ – Math1000 Mar 31 '16 at 21:39

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