There is an M/M/1 (getting a result for M/M/c is ideal) queue with arrival rate and service rate and participants can renege.
However, the difference between normal reneging settings (constant time $\tau$) and this setting is that the participants know its service time ahead of time and the time to renege is a function of its service time (sampling from exponential). Let's say the function is a linear function of its service time $ w(\cdot) = a \cdot s$, where $ s \sim Exp(\mu)$ and $ a $ is a constant.
There is no balking.
What would be the expression for the steady-state probability that a participant receives service?