Unformatted text preview: Barber Shop Problem Barber Shop Problem
Another question to consider:
4. A person who is 30 minutes away wants
to know the probability of an immediate
service. When calling the shop he is
informed that:
(a) The shop is currently empty
(b) There are currently 2 people in the
shop Questions to consider:
1. What is the probability of a customer
walking in and getting immediate
service?
2. What is the probability of walking In
and leaving because the shop is full?
3. What does the barber make on an 8hour day, if the barber makes (after
expenses) $10.00 per haircut? Markov Chains
l l
l l l Example Modeling Procedures Barber Shop Model Deffne the Epoch to be used in the
model
Deffne all possible states of the system
Determine the transition probabilities
for all states to all other states
Complete the onestep transition
matrix
Perform necessary computations to
answer questions l l l Deftne the Epoch for the barber shop
problem as one 1 Smlnute haircut
Let the states of the system be equal to
the total number of customers In the
barber shop (zero if the shop is empty,
thru four If the shop full to capacity)
Determine the transition probabllltles transitions occur whenever either a
customer arrives or leaves Barber Shop Model Example Transition Matrix Barber Shop Model Cust in shop at time = (t The barber studied the system,
counting the customers entering the
shop in each l&minute period
l The resutts are tabulated below:
5+
1
2
n
0
3
4
P(n)[ 0.70 0.20 0.07 0.02 0.01 0 l n = number of arriving customers
P(n) = probability of n customers arriving 1
2
n
0
P(n)] 0.70 0.20 0.07 EM602 I QM710 (NJ) Lecture I.0
Page 106 3
0.02 l 16 min) 4
0.01 5+
0 .’  Barber Shop Model Barber Shop Model Transition Matrix Transition Matrix Cust in shop at time = (t n
0
1
2
P(n)1 0.70 0.20 0.07 3
0.02 l 16 min) 4
0.01 Curt in shop at time = (t n
0
1
2
P(n)/ 0.70 0.20 0.07 5+
0 16 min) l 3
0.02 4
0.01 Barber Shop Model Barber Shop Model Transition Matrix 5+
0 Transition Matrix Cust in shop at time = (t + 16 min) Cust in shop at time = (t 0 Cud in 0 1
1 0.70 0.20 2
0.07 16 min) l 3
0.02 4
0.011  n
0
1
2
3
P(n)1 0.70 0.20 0.07 _ 0.02 a
0.0...
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 Spring '94
 DonaldC.Johnson
 Probability theory, barber shop, Barber Shop Model, barber shop problem

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