The personnel department of a large corporation is interested in constructing a mathematical model that will help them predict work force reduction due to worker illness. After researching employee attendence records over a five year period, they reached the following conclusions: Of the workers who are healthy and show up for work on a given day, 96% will be healthy and come to work the following (work) day. On the other hand, of the workers who call in sick on a given day, 60% will call in sick the next (work) day. (We'll assume that each worker is either healthy or sick on any given day and that healthy workers go to work. )

(a) Find the stochastic matrix N that represents this situation. (Let the percent of healthy workers each day be the first component of the state vectors and the percent of sick workers be the second component.)

(b) We've seen that we must be careful with strange answers in Maple at times, so before we use it, justify why the system will stabilitze by addressing regularity (do the columns add to 1 and are the entries all positive?).

(c)In Maple, find the steady-state vector by solving (I-N)x=0 for x, and be sure to add a bottom row of 1s to the augmented matrix for this system to represent the extra equation that the workers must add to 100% (ie no one quits or dies).

(d) Suppose that on one Monday in the middle of winter 15% of the workers call in sick. According to the assumptions of the model, show in Maple what percent will be sick on Tuesday? on Wednesday? What percent will be sick after ten weeks?