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Customers arrive at a shop following a homogeneous Poisson process $N(t), t ≥ 0$, rate $\lambda$. Each customer spends some amount of time, $t_i$ in the shop, with mean $E[t_i] = \mu_t$. If there is a customer in the shop already and a new customer arrives, the new customer leaves. Suppose that each customer spends some amount of money, $d_i$, with mean $E[d_i] = \mu_d$. Let $D(t)$ be the total sales of the shop up to time $t$. Find $\lim_{t \rightarrow \infty} \frac{D(t)}{t}$. (Hint: Let $r_i$ be the time between customers being served).

I'm not sure where to start with this problem. Any hints would be appreciated. Thanks!

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1 Answer 1

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We know how much money a customer spends on average. So we can get the rate of revenue if we can get the rate of customers being served, which is the reciprocal of the average time it takes to start serving a new customer. This in turn is the sum of the average time a customer spends in the shop and the average time it takes for a new customer to arrive when the previous one has left. Thus

$$ \lim_{t\to\infty}\frac{D(t)}t=\frac{\mu_d}{\mu_t+\frac1\lambda}\;. $$

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