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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 pj 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, .. EM602 I QM710 (NJ) Lecture 10
Page 107 , Barber Shop Model
l Question #4 SteadyState 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 Shour
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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 Spring '94
 DonaldC.Johnson
 Probability theory, barber shop, Barber Shop Model, barber shop problem

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