25 points) A credit card company has a Refer-a-Friend program in which each member is assumed to successfully refer a friend at
an exponential rate of λ per member. Each member cancels membership at an exponential rate of μ. If the total number of members is less than N, the credit card company will advertise online, and additional customers will join at an exponential rate of θ due to the advertising. If the total number of members is greater or equal to N, the credit card company will stop advertising, and customers will only join through the referral program. (a) Set this up as a birth and death model. That is clearly define the states and transition rates. Draw rate diagram.
(b) Set up balance equations to be solved to find P's for 1 2 0. Do not solve them.
(C) Let N = 30, express the proportion of time that customers will get the credit card of the company through referral program in terms of P's.
