Probability of get a kidney donated 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?
 A: Can we not think like this that probability of a patient getting rejected first time is (M-I)/M which is less than 1.
Probability of a person not getting a kidney second time is (M'-I')/M' which also less than 1.
Now, the probability of a person not getting a kidney in first attempt and then in second attempt is (M-I)/M*(M'-I')/M.
As at any attempt probability of a person not getting a kidney will be always  less than 1,the probability of a person not getting a kidney in n attempts will keep on decreasing and if n tends to infinity, it will tends to zero.
So, if probability of a person not getting a kidney in infinite attempts is zero than its inverse, probability of a person getting a kidney in infinite attempts should be 1.
I am not sure about this, just trying to think in the  opposite way.
A: Let me de the calculations.
IF $M_t$ measures the patients at $t^{th}$ period, given that $\mu_I<m$, $\lim_{t\to \infty} M_t=\infty $
But what's the probability for a specific patient to recieve a kidney in the period $t$ (event $K_t$)?
$\begin{align}
P(K_t)&=P(I\geq M_t)\times 1+P(I<M)\times\frac{I}{M_t}\\
\lim_{t\to \infty}P(K_t)&= 0\times 1 + 1\times \frac{I}{\infty}\\
&=0
\end{align}$
As you see, $P(K_t)>P(K_{t+1})$. As time goes by, probability gets nearer to $0$, making the expected waiting of periods $\infty$ for all the patients.
