Let $0<a<1$, and let $n$ be a positive integer. We have a factory that can process jobs. Each day, a job arrives - the number of days it takes to complete the job is $1\leq i\leq n$ with probability $p_i$, where $\sum_{i=1}^np_i= 1$. Also, the value (per day) of the job is drawn from a distribution $F$ over $[0,1]$ which is independent of the job length.
If the job has value $v\geq a$ and the factory is free when it arrives, the factory takes the job, is busy for the next $i$ days (where $i$ is the job length), and earns $iv$ dollars. If there are $D$ days in total, let $M_D$ denote the expected amount of money earned per day.
Is it true that the sequence $\{M_D\}_{D=1}^N$ converges?
Intuitively, this should be true because we reach a sort of "stable state" when the amount of time is long enough. But how can we show this formally? Does it follow from some general result?
First consider the average money made over days on which the factory is running. The fraction of the time that the factory spends on jobs of time $i$ is $\frac{i p_i}{\sum_{j=1}^n j p_j}$ (note that the probability that a job will be accepted is not dependent on how long it will take). For each such job the factory earns an average of $E[V \mid V \geq a]$ money, with each value being independent of the rest. So the average earnings per day for jobs of length $i$ is $\frac{E[V \mid V \geq a]}{i}$. Multiplying that by the probability that we are working on a job of length $i$ on a given day gives $\frac{E[V \mid V \geq a] p_i}{\sum_j j p_j}$. We then sum that over $i$ to get $\frac{E[V \mid V \geq a]}{\sum_j j p_j}$.
Now we just multiply that by the probability that the factory is running on any given day. I conjecture that will just be $\frac{\sum_i i p_i}{\left ( \frac{1}{P(V \geq a)} - 1 \right ) + \sum_j j p_j}$, i.e. the average time of a job divided by the average time of a job plus the average time in between jobs. Try to prove or disprove that.
This gives you the "ensemble" average of the earnings per day; you'll need some kind of a strong law of large numbers to ensure that the time average converges to this.