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Expected costs are thus:
:<math>\mathbb{E}C = \int_0^{\infty} \left ( \int_0^{t} x e^{-rs} ds \right ) \cdot pr(\hat{\tau_i} = t or \tau_i = t) dt + F\;</math>
:<math>\therefore \mathbb{E}C = \int_0^{\infty} \left ( \int_0^{t} x e^{-rs} ds \right ) \cdot (a+h) e^{-(a+h)t} dt + F\;</math>
:<math>\therefore \mathbb{E}C = \frac{x}{a+h+r} + F\;</math>
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