To be more precise, the mathematical models usually assume that people arrive at one rate regardless of how recently someone arrived, and work finishes at another rate regardless of how long you have been working.

This makes the time between arrivals, and the time to complete a job both exponential distributions. It also makes the number of arrivals while a job is being worked on into a Poisson distribution.

For a computer simulation you can simplify this even further by forgetting about time and focusing on events. If the worker is busy, the next event is either a new arrival or a job completion with a probability based on the rates.