Exercise regarding Poisson processes and the uniform distribution This is Exercise 5.4.5 in Introduction to Stochastic Modeling (4th edition) page 257:

Customers arrive at a certain facility according to a Poisson process of rate lambda. Suppose that it is known that five customers arrived in the first hour. Each customer spends a time in the store that is a random variable, exponentially distributed with parameter alpha and independent of other customer times and then departs. What is the probability that the store is empty at the end of the first hour?

I have no clue how to model this situation or proceed. Help would be much appreciated. Thanks so much.

First: You probably meant to write $[1 - (1 - e^{ - \alpha } )/\alpha ]^5$ as the solution to this exercise, right? 
Hint 1: The arrival times of the five customers can be simulated as five i.i.d. uniform$[0,1]$ rv's. That is, if you place five i.i.d. uniform rv's on $[0,1]$, the points correspond to arrival times of five customers.
Hint 2: If $Z_1,\ldots,Z_5$ are i.i.d. rv's with distribution function $F$, how can you express the distribution function of $\max \{ Z_1 , \ldots ,Z_5 \}$ in terms of $F$?
Hint 3: You'll have to use the law of total probability, in order to calculate a certain probability. In this context, it may be useful to note that $1-U$ and $U$ are identically distributed, for $U$ a uniform$[0,1]$ random variable.
EDIT (further hints, in response to the OP's request). Suppose that $Z = U + Y$, where $U$ and $Y$ are independent uniform$[0,1]$ and exponential$(\alpha)$ random variables, respectively. First note that 
$$
{\rm P}(Z \le 1) = {\rm P}(U + Y \le 1) = {\rm P}(Y \le 1 - U) = {\rm P}(Y \le U),
$$
where the last equality (which is actually not essential) follows from the fact that $U$ and $1-U$ are identically distributed. Now, since $U$ has constant density $f(u)=1$ for $u \in [0,1]$, the law of total probability gives
$$
{\rm P}(Y \le U) = \int_0^1 {{\rm P}(Y \le U|U = u)1\,{\rm d}u}  = \int_0^1 {{\rm P}(Y \le u)\,{\rm d}u}  = \int_0^1 {(1 - e^{ - \alpha u} )\,{\rm d}u} .
$$
Calculate the integral on the right-hand side, and recall Hint 2.
