Customers arrive at a bus stop according to a Poisson process with rate $3$ per hour. Buses arrive at the stop according to a Poisson process with rate $1.5$ per hour.
(a) What is the probability that exactly $n$ customers arrive between consecutive buses?
(b) Assume each bus can hold at most $2$ customers. Also assume at most $4$ people wait for the bus (subsequent customers are turned away). Model this process as a continuous-time Markov chain. Give the transition-rate matrix Q.
My attempted solution Part (a) was solved thanks to Did's help!
(b) Let $X(t) =$ # of people at the stop at time $t$. Then $X(t) = 0,1,2,3,4$ (since subsequent customers are turned away). Thus, the transition rate matrix is $$Q =\pmatrix{-3 & 3 & 0 & 0 & 0\\ 1.5 & -4.5 & 3 & 0 & 0\\ 1.5 & 0 & -4.5 & 3 & 0\\ 0 & 1.5 & 0 & -4.5 & 3\\ 0 & 0 & 1.5 & 0 & -1.5}$$