Technical Note—The Infinitely-Many-Server Queue with Semi-Markovian Arrivals and Customer-Dependent Exponential Service Times

This note studies an infinitely-many-server queue with N types of customers arriving according to a semi-Markov process. The service times are independent exponential random variables with means 1/μ_i, where i is the type of customer being served. The stationary distribution of the embedded Markov chain and the limiting distribution of the continuous-time process are obtained.