interarrival time of uniform arrivals

We know that if customer arrival to a service has a poisson distribution with rate A, the interarrival time between customers are exponentially distributed with mean 1/A.
So, please tell me about interarrival time;
if customer arrival has a uniform distribution with parameters B and C?

Can you write the down the PMF for the number of customers who have arrived by time t, as unless you are very careful i think you will end up with an impossible one.

Also state any limits on the system (eg, is t going to run to infinity or not).