Consider an M/M/1 queue and suppose that for each $x \geq 0$ we accrue operating costs at rate $a + bx^2$ whenever there are $x$ customers in the system. Suppose that each customer pays us $c$ upon arriving to the system. What is largest value of $b$ at which we do not lose money in the long run?
I think if average earning rate $\geq$ average operating rate, then we will not lose money.
Then assumer the arrival rate is $\lambda$ and service rate is $\mu$ and define $\rho = \frac{\lambda}{\mu}$, long-run steady probabilities $\pi_i=$ probability that there are $i$ customers in the system.
average earning rate = $\lambda c$;
average operatig rate = $\sum\limits_{i=0}^{\infty}(a+bi^2)\pi_i = \sum\limits_{i=0}^{\infty}(a+bi^2)\rho^i(1-\rho) = a + b\frac{\rho^2 + \rho}{(1-\rho)^2}$.
But I am not sure if I am right since I do not consider the time that $i$ customers in the system.