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 1. 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 2, where 1  2. Customers arrive according to a Poisson process with mean rate .

(a) Formulate this model as a continuous time Markov chain by defining the states and constructing the rate diagram accordingly.

(b) Develop the balance equations.

(c) Suppose that numerical values are specified for 1, 2, and , and that   2 (so that a steady-state distribution exists).

Since this model has an infinite number of states, the steadystate distribution is the simultaneous solution of an infinite number of balance equations (plus the equation specifying that the sum of the probabilities equals 1). Suppose that you are unable to obtain this solution analytically, so you wish to use a computer to solve the model numerically. Considering that it is impossible to solve an infinite number of equations numerically, briefly describe what still can be done with these equations to obtain an approximation of the steady-state distribution. Under what circumstances will this approximation be essentially exact?

(d) Given that the steady-state distribution has been obtained, give explicit expressions for calculating L, Lq, W, and Wq.

(e) Given this steady-state distribution, develop an expression for P{ t} that is analogous to Eq. (1) in Prob. 17.6-17.