We calculate the expected lateness for a late job in an M/M/1/K queue (i.e., the expected waiting time in the system after a threshold lead time l, given that the waiting time is greater than l). The expected lateness for any given job can then be deduced immediately. Applications of our result include the calculation of the lateness penalty cost when a given lead time is quoted to the customers, such as in call centers or in some production systems.