The virtual waiting time of the M/G/1 queue with customers of n types of impatience

We consider the $M/G/1$ queue in which the customers are classified into $n+1$ classes by their impatience times. First, we analyze the model with two types of customers; one is the customer with constant impatience time $k$ and the other is the patient customer whose impatience time is $\infty$. The expected busy period of the server and the limiting distribution of the virtual waiting time process are obtained. Then, the model is generalized to the one in which the impatience time of each customer is anyone in $\{k_1,k_2,\dots ,k_n,\infty \}$.