How to model this optimal control problem?

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Suppose there is a unit and entities enter with the rate $\lambda_t$. Each entity entering the unit at time $t$ can leave the unit within the interval $[t-\tau,t]$ otherwise, the entity remains in the unit. Moreover, each entity may leave the unit with probability $\alpha$. I am wondering how to write the dynamic of the number of entities in the unit and the number of entities that have left the unit over time.