Customers walk by a certain stall with one server according to a Poisson process with rate 20 per minute.

If there is either only one person being served or no one being served, then a customer goes to the stall with probability p/2 for some p (0, 1].

Once there are k 1 people waiting to be served (so (k + 1) in the system), then arriving customers join the queue with probability pk/(k + 1).

The time from when a customer orders to when they receive their food is exponentially distributed with mean 90 seconds.

For t 0, let Xt be the number of customers in the system (that is, those waiting for service or in service) t minutes into the lunch rush.

(a) Model (Xt)t0 as a continuous time Markov chain and specify its generator.

(b) Find the values of p where (Xt)t0 has a stationary distribution = (n)n0, and for those values of p, write down a simple formula (i.e., no infinite sums) for n.