37. A hospital accepts k different types of patients, where type i patients arrive according to a Poisson proccess with rate , with these k Poisson processes being independent. Type i patients spend an exponentially distributed length of time with rate in the hospital, . Suppose that each type i patient in the hospital requiresI mage units of resources, and that the hospital will not accept a new patient if it would result in the total of all patient's resource needs exceeding the amount C. Consequently, it is possible to have type 1 patients, type 2 patients, … , and type k patients in the hospital at the same time if and only if

(a) Define a continuous-time Markov chain to analyze the preceding.
For parts (b), (c), and

(d) suppose that .

(b) If is the number of type i customers in the system at time t, what type of process is Image? Is it time reversible?

(c) What can be said about the vector process ?

(d) What are the limiting probabilities of the process of part (c).
For the remaining parts assume thatI mage.

(e) Find the limiting probabilities for the Markov chain of part (a).

(f) At what rate are type i patients admitted?
(g) What fraction of patients are admitted?