It is really a probability problem. I use the story of kidney donation because it is easier to describe.
Consider the following scenario:
Time is discrete.
At each period, the measure of patients in the hospital is $M$.
At each period, the measure of new kidney supply $I$ is i.i.d. positive, with mean $\mu_I$ and variance $\sigma_I^2$.
If the measure of kidneys $I$ is larger than measure of waiting patients $M$, then every patient will be matched. But unmatched kidneys cannot be used next time. If $I<M$, each will be matched by probability $I/M$. There is no first come first serve rule here.
A patient leaves the game if and only if he/she gets donated.
If one is not matched at the current period, he can go to next period. A patient never dies.
The number of patients next period will be $$M'=M-I+m$$ - a measure $m>0$ of new patients will come. $m$ is certain.
Now I assume $E(I)<m$ - the waiting pool has a positive probability of going to infinitely large.
Now my question is, whether a patient's probability of being matched with a kidney is strictly smaller than one or not at $t$ goes to infinity?