Jo 13 M, =30 = My 3. Let's look at a linked series of M/M/1 computers: (i) 10 jobs a minute are being sent to computer 1 which takes 2 seconds for each job. 50% of the completed jobs are sent to computer 2 (the rest are done and go out of the sys (ii) Computer 2 also takes 2 seconds for each job. All of the finished jobs are sent to computer 3. (iii) 10 jobs a minute are being sent directly to computer 3. Computer 3 takes 1 second per job. 50% of the finished jobs are sent to computer 1 and 30% are sent to computer 2 (the rest are done). a) (4 pts) Set up the system of equations and solve for A1, 12, and 13. M3 = 60 * , = 10 + .5 x3 1 , = 10 + . 5 ( 107 12 ) 2 : 12 = . 5 * , +, 343 12 = 1 51, +13 (10+ dz ) 3 : to = 10 + 2 2 d , - 15 1 2 = 15 12 = 23, 33 = 703 + 2 ( - 15x,. 7. 7 /2 = 3 ) * , = 15 + 15 12 = 30 1 9 /2 = 21 13 = 100 / 3 099 b) (3 pts) Find E(N1), E(N2), and E(N3). . 173 E ( N ) = M - 1 1 ' Po = T PI=$1 - E ( N. ) = 8 1. 2 , P1 9 9 Cor PI = 9 , p2 = / 9 E ( N2.) = 3.5 2: Po = 2 5.4 P. = 9 9 p3 = $ ... ] E ( N ) = 11250 3: Po = . 247 c) (3 pts) Find the probability that there is a total of 1 job in the system. Let ( a , b, c ) = ( # jobs in ), # in 2, # in J p ( total = 1 ) = p ( ), ,, 0 ) + 1 / 0 , 1 , 0 ) + P ( 0 , 0 , , ) 0 PR = ( 1 -p )p - po = Ip pic ( 1- p ) p 5 0244 d) (3 pts) Find E(T) (you can use Little's Theorem with A being the rate that jobs are entering the system, you will also need the sum of the mean number of jobs at each computer). E ( N ) = E ( N. ) + E ( NZ ) + E ( NB ) = 12.75 0 1 = 20 @ - E ( T ) = 30 ( 12 . 75 ) = 3 = . 6375 min3. Let's look at another type of queue where we have a sequence of M/M/ 1 computers: (i) 10 jobs a minute are being sent to computer 1 which takes 3 seconds for each job, then all of the completed jobs are sent to computer 2. (ii) along with the jobs from computer 1, 2 jobs a minute are being sent directly to computer 2. Computer 2 takes 2 seconds per job. 40% of the nished jobs are sent to computer 3 and 30% to computer 4 (the rest are done). (iii) Computer 3 takes 5 seconds per job. 50% of the nished jobs are sent to computer 4 (the rest are done). (iv) Computer 4 takes 3 seconds per job. 20% of the nished jobs are sent to computer 1 and 30% to computer 2 (the rest are done). a) (4 pts) Set up the system of equations and solve for A1, A2, A3, and A4. b) (3 pts) Find E(N1), E(N2), E(N3), and E(N4). c) (2 pts) Find the probability that there is a total of 1 job in the computers. d) (2 pts) Find E(T) (you can use Little's Theorem with A being the rate that jobs are entering the system, you will also need the sum of the mean number of jobs at each computer). Note for problem 3: Two theorem's (Burke's and Jackson's) give us a few properties once the systems are in steady-state that allow us to do the previous problem: (i) if the number entering is Poisson then the number leaving is Poisson with the same rate. (ii) we can solve for the pieces as if they are independent. So after we solve for the As we can use the usual formulas for M/M/l. (iii) Little's theorem still applies to the system if our A is the total number entering the system