Question

(20 points) Suppose a small local hospital has 3 nurses. On any given day, independent of the number of nurses that are working, either one of the working nurses becomes unavailable with probability 1 - q > 0, or all of the nurses are available with probability q. If a nurse becomes unavailable, with certainty s/he will be available next day (we make the simplifying assumption that if a nurse becomes unavailable, s/he becomes unavailable at the beginning of the day and remains unavailable for the rest of the day and will be available again at the beginning of the next day). The hospital earns $500 per day, per working nurse. Let Xn be the number of nurses working on day n. The transition matrix for the Markov chain {x,,, n 0. 1. 2.. J and its two states. 2 (i.e. two working nurses) and 3 (i.e. three working nurses), is then given by 1-q q 1-q q (a) (2 points) Is this Markov chain irreducible? (b) (3 points) Is this Markov chain aperiodic? (c) (15 points) In the long-run, how much is the hospital earning per day from the 3 nurses? (Hint: your answer should be in terms of q.)

Answer 1

La). Sina in merkou chin s communic with each ontkur· So that chau n irnducibda Cb). Transilion ma