Conjectured form of the M/M/1 processor-sharing queue's average age of information

Let $\rho=\frac{\lambda}{\mu}<1$ be the traffic intensity of a stable M/M/1 queue with processor-sharing discipline, arrival rate $\lambda$, service rate $\mu$, and average age of information $\Delta_{M/M/1-PS}$. Let $C(\rho)$ be a function satisfying the bounds and limiting conditions below.

M/M/1-PS AAoI conjecture. The average age of information is

$$\Delta_{M/M/1-PS}=\frac{1}{\mu}\left(\frac{1}{\rho}+1+C(\rho)\right),$$

where

$$\lim_{\rho\to 0}C(\rho)=0,\qquad \lim_{\rho\to 1}C(\rho)=+\infty,$$

and

$$0\leq C(\rho)\leq\frac{\rho^2}{1-\rho}\quad\text{for all }\rho\in(0,1).$$

Moreover, when $\rho$ is large enough,

$$\frac{(\rho-0.5)^3}{1-\rho}\leq C(\rho)\leq\frac{0.75\rho}{(1-\rho)^{\frac{1}{2}}}.$$

The conjecture is motivated by numerical experiments and the known expression for the M/M/1 queue with first-generated-first-served discipline. An explicit characterization of the processor-sharing AAoI remains unavailable; the stated bounds and asymptotic conditions are likewise conjectural.