I'm not sure you have demonstrated what you are trying to show.
If I understand correctly, you've calculated the chance a manager is promoted to director as 1/15 * average number of directors that leave over the relevant time period.
But you haven't made any argument about the chance an engineer is promoted to distinguished engineer besides stating promotion non rivalrous. A system in which no one is promoted to distinguished engineer is non rivalrous, for example.
If I understand correctly, you've calculated the chance a manager is promoted to director as 1/15 * average number of directors that leave over the relevant time period.
But you haven't made any argument about the chance an engineer is promoted to distinguished engineer besides stating promotion non rivalrous. A system in which no one is promoted to distinguished engineer is non rivalrous, for example.