Question

Prob. 7 (12.5 pts) A stochastic matrix for three separate rental-return auto locations is given below. The company wishes to find a steady state vector to apportion parking spaces at each location. Their rental fleet is 2000 cars. Unfortunately, the matrix was somewhat obscured by spilled cocoa during transference to you, so account. Cocoa drops are indicated by xx in the matrix. you will need to take this into X 0.30 0.30 xx xx 0.35 Lo.20 0.55 0.25 xx

Answer 1

ene et the lehedties ot a stechastie matis is ether Sunn of the elements in every ow cALuas elemats in eery Glumn equals OSun o the ne One O.30 XX. O.30 O.55 O.35 A = O 25 XX O.20 n this mati , Sum ot decend sow elemant - o.35+0.55+0.3 1.2 1 this is not equalk Columa must be equal to one stechasti matir Column 1 Celumn 2 Sum ot elemente in eey inoides it to be o ene hence X= 0.45 XX+0. 35+0.2 = | XX+0.55-+0.25= 1L .4 Celumn 3: XX O. 30+0.30+XX=1

Hena the mabie is o. 45 A = 0.2 O.30 0.55 O.35 0.30 O.25 0.4 LO.2 Steady state Vectol A 2 X2 3 X3 tstion of akig tloe , 1X213epecont at three lecations uhere the 2 3= } O. 3 0.45 O.2 X2 2 6.30 O.55 0.35 ス3 X 3 6.4 O.25 .2 O.45x,+0. 2 X2 + 0.3 3= X1 O. 35 X1 + 0.5522 +0 3 3= X2 0.2 X, + 0.25 X2+0.4X3= X3 O.IXI-O 35 X2= X-X2 0.65×2=0.9く」 18 メ1 2 13 Subcti tubing tin 1 8 0.4X3= X3 O.2X,+0. 25 T3 O.2X,+ 0.25 18,O. 6 X3 73 X3=

We have = 1 (1+ ) (75+ 103-11) X1 18 3 7 1 78 l08 1 257 257 Cnd 3 7 1 257 78 Steady state Vect = 25 I08 257 IL 257 Bental 2000 Cars 78 257 Number of Cary at Cach lecation= 2000 108 257 251 [607 840 553