Each member in Super Fancy VIP club is assumed to successfully refer a friend to join the club 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 club will advertise online, and additional customers will join the club at an exponential rate of θ due to advertisement. If the total number of members is greater or equal to N , the club will stop advertising, and customers will only join the club through reference. (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 Pj ’s for j ≥ 0. Do not solve them. (c) Let N = 25. Express the proportion of time that customers will only join the club through reference in terms of Pj ’s. (Do not need to solve)
Each member in Super Fancy VIP club is assumed to successfully refer a friend to join the
club 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 club will advertise online, and additional
customers will join the club at an exponential rate of θ due to advertisement. If the total number of
members is greater or equal to N , the club will stop advertising, and customers will only join the club
through reference.
(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 Pj ’s for j ≥ 0. Do not solve them.
(c) Let N = 25. Express the proportion of time that customers will only join the club
through reference in terms of Pj ’s. (Do not need to solve)
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