Question

Batman has gone out for a night of crime fighting to an alley frequented by criminals. Criminals arrive to the alley according to a Poisson Process with rate of 1 per 5 minutes. Using standard bad guys techniques, they only attack the Dark Knight one at a time. It takes our hero an exponential amount of time to knock each one out with an average of 1 minute. Due to the size of the alley, there can be at most 5 conscious criminals. Develop a CTMC model for number of conscious bad guys.

 (a) Give the Q matrix, draw the TPD, and solve for α.

 (b) What is the expected number of conscious criminals in the alley? 

(c) If Batman is in the alley for 2 hours, how long does he spend actually fighting?

 (d) Is this a birth-and-death process? Why or why not? If it is, recalculate the steady-state vector utilizing the special equations derived for a BAD process. Show all relevant formulas for full credit.