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# Aber and Byft are two competing mobile ride-sharing companies serving one million customers jointly.Suppose

that each customer chooses from Aber and Byft according to the following rules:

• Each customer can only choose one of the two companies in each month, and must stay with that company for the entire month, before considering switching the next month.
• A new customer of Aber is someone who was not using Aber in the prior month. Such customer is equally likely to stay with Aber (a loyal customer), or to switch (becomes a new customer to Byft) next month. The same holds true for new customer of Byft.
• A loyal customer is someone who used the same compnay in the prior month. SUch customer has 3/4 probability to stay loyal, and a 1/4 probability to switch to the competitor.
• As shown in the following figure, a Markov chain can be used to model the usage behavior of customers in above setting. Here, AL stands for loyal customers of Aber, AN stands for new customers to Aber, BL stands for loyal customers of Byft, BN stands for new customers to Byft.

(a) At a given time, suppose that the system has run for a long time and is stabilized. What fraction of the one million users should one expect to be the new customers of Aber?

(b) For a loyal customer of Aber, what is the expected duration in months, that s/he stays as a loyal customer of Aber before switching?

(c) For a loyal customer to Aber, what is the expected time in months, until s/he becomes a loyal customer to Byft?

(d) After the system is stabilized, the CEO of Aber came up with a marketing idea to offer new customers of Aber a cash-bonus of \$10. How much should the CEO expect to pay for such cash bonuses in the first month of implementing the policy?

(e) With the cash-bonus policy implemented for a long time, Aber observes that more new customers are now becoming loyal to them (3/4 of them as opposed to 1/2 before the offer). To keep this going, how much Aber needs to pay on average on cash-bonus per month?

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