Lecture10

70 020 2 007 16 min l 3 002 4 0011 n 0 1 2 3 pn1

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Unformatted text preview: 1 5+ 0 Barber Shop Model Barber Shop Model Transition Matrix SteadyState Probabilities Curt in shop at time = (t 1 I 0.70 0.20 0 Cust in 0 2 0.07 l x = XP 16 min) 3 0.02 I.7 .2 .07 .02 .oq , .7 .2 .07 .02 .Ol p-j 0 4 0.01 1 .7 .2 .07 .03i 1 0 0 10 0 .7 .2 .l 0 .7 .3 1 n,, =.77[,i.7z,, +.7x I K =.2xn +.ta X, =.07R,+.07R,+.2Rlf.7R, R, =.02Ro+.02R,.07R,+.2R,+.7R, n 0 1 2 P(n)1 0.70 0.20 0.07 3 0.02 4 0.01 R, =.01Ro+.0~R,+.~R,+.1R,+.3R, 5+ 0 l=R,+R,+R'l+R,+R, .. EM-602 I QM-710 (NJ) Lecture 10 Page 10-7 , Barber Shop Model l Question #4 Steady-State Probabilities Barber Shop Problem After elimination of one redundant equation and solving simultaneously, we get the steady state vector: 1. What is the probabilfty of a customer walking in and getting immediate servlce? l Immediate service implies that the barber chair and all wafting chairs are empty l This condition only exists at state 0 l Probability of system being (in the long run) in state 0 is given by 4 (x,, = .566) l Therefore, this probability is 56.6% x=tno Xl x2 5 %I = [ .566 .243 .116 .OSl .024 ] x Question #2 Question #3 Barber Shop Problem Barber Shop Problem 2. What is the probability of walking in and leaving because the shop is full? l A full shop implies that the barber is busy and all waiting chairs are filled l This condition only exists at state 4 l Probability of system being (in the long run) in state 4 is given by x, (x4 = ,024) l Therefore, this probability is 2.4% ’ 3. What does the barber make on an S-hour l l l l day, if the barber makes $10 per haircut? One 6hour day = 32 ldminute periods $10 per haircut revenue is had when the system has at least one customer (whenever the shop is not empty) Probability of earning revenue is (x,+ x1+ lq n,) = (1 - x0) = 0.434 Therefore, the Expected Monetary Value EMV is ($10)(32)(0.434) = $136.66 I day Question #4 Question ##4 Barber Shop Problem Barber Shop Problem 4. A person who is 30 minutes away wants to know the probabiltty of an immediate haircut. In calling the shop they are told: (a) The shop is currently empty (b) The...
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