A department in a university has several full-time faculty positions. Whenever a faculty member leaves the department, he or she is replaced by a new member at the assistant professor level. Approximately 20% of the assistant professors leave (or are asked to leave) at the end of the fourth year (without being considered for tenure), 30% of the assistant professors are considered for tenure at the end of the fifth year, 30% at the end of the sixth year, and the remaining at the end of the seventh year. The probability of getting a tenure at the end of 5 years is .4, at the end of 6 years is .5, and at the end of 7 years is .6. If a tenure is granted, the assistant professor becomes an associate professor with tenure, otherwise he/she leaves the department. An associate professor spends a minimum of 3 years and a maximum of 7 years in that position before becoming a full professor. At the end of each of the years 3, 4, 5, 6, and 7, there is a 20% probability that the associate professor is promoted to full professor, independent of everything else. If the promotion does not come through in a given year, the associate professor leaves with probability .2, and continues for another year with probability .8. If no promotion comes through even at the end of the seventh year, the associate professor leaves the department. A full professor stays with the department for 6 years on average and then leaves. Let X(t) be the position of a faculty member at time t (1 if assistant professor, 2 if associate professor, and 3 if full professor). Note that if the faculty member leaves at time t, then X(t) = 1, since the new faculty member replacing the departing one starts as an assistant professor. Show that {X(t), t≥ 0} is an SMP. Compute its P matrix and w vector.
