Paradoxes and Counter-Intuitive Examples in Analysis of Queues

Example 1 Consider a queue where customers arrive one by one. There is a single server that serves each customer on a first-come-first-served basis. Define the sojourn time experienced by a customer as the time he/she spends in the system: waiting (if any) for service plus the duration of the service at the server. The server toggles between being idle for some time when there are no customers in the system and being busy for a while serving customers one after the other. Define the busy period of the server as the time spent continuously serving customers. Notice that an arbitrary busy period is never smaller than the sojourn time of any of the customers that was served during that busy period. Yet, the average busy period could be smaller than the average sojourn time!