Customers line up to be serviced according to a Poisson process at an average rate of five per hour. If the time it takes to serve one customer is a continuous uniform random variable on $[0,4]$, independent of arrival times, what is the probability that the second customer who arrives will have to wait to be served?
I'm really rather lost I think. I'm saying the time $T_n$ that the $n$th customer arrives is a sum of exponential variables with parameter 5, which ends up being a Gamma distribution with parameters $n, 5$. The time that it takes to serve the $n$th customer is $S\sim U(0,4)$. I figure that the second customer waits whenever $T_1+S>T_2$, so that's the probability that I'm looking for. The problem is, I don't know how to find that probability. I tried taking the convolution of $S$ and $T_1$ to get the distribution function of $T_1 +S$ but something seems to be going wrong with that:
$$F_{T_1+S}(a)=\int_{-\infty}^{\infty}F_{T_{1}}(a-y)f_{S}(y)dy=\frac{1}{4}\int_{0}^{4}(1-e^{-5(a-y)})dy \\ =\frac{1}{4}(4-\frac{1}{5}e^{-5a}[e^{20}-1])$$ which is clearly not a CDF (it's negative between $0$ and $4$), so I must have done that wrong. But even if I figure out what I did wrong there, I wouldn't know what to do next.
Best Answer
Hint: $T_2=T_1+R$ with $R$ independent of $T_1$ and $S$, and $[T_1+S\gt T_2]=[S\gt R]$.
Sub-hint: $\mathrm P(R\gt S)=\int\limits_0^4\mathrm P(R\gt s)\,\frac{\mathrm ds}4$ and $\mathrm P(R\gt s)=\mathrm e^{-5s}$ for every $s\geqslant0$.