Consider a generalization of the M/M/1 model where the server needs to “warm up” at the beginning of a busy period, and so serves the first customer of a busy period at a slower rate than other customers. In particular, if an arriving customer finds the server idle, the customer experiences a service time that has an exponential distribution with parameter μ1. However, if an arriving customer finds the server busy, that customer joins the queue and subsequently experiences a service time that has an exponential distribution with parameter μ2, where μ1 < μ2. Customers arrive according to a Poisson process with mean rate λ.
(a) Formulate this model in terms of transitions that only involve exponential distributions by defining the appropriate states and constructing the rate diagram accordingly.
(b) Develop the balance equations.