### presentation 3

```An Investment Planning
Paxton Zhou
Paul Kim
Basic parameters
• Customers arrives according to a Poisson distribution
– Male and female has the same arrival rate of 3 customers
per hour
• Processing time for each customer is uniformly
distributed
– Male: [15, 75] min per customer
– Female: [21, 81] min per customer
• 5 barbers; 2 only handle male hair, 2 only handle
female hair, the fifth barber (the master) can handle
both male and female hair and is 2x faster than the
other barbers
The no-queue scenario
Male
Baber 1
Male
Barber
2
Master
Female
Barber
3
Female
Barber
4
Customers arrive
Poisson (λ=3) for both
male and female
•We assume that customers have no patience—if no barber is
free they will just leave and go to another barber shop near by
•As a result, we assume immediate blocking when all
“machines” are busy
List scheduling (L-S) method:
• FCFS
• When a customer arrives
– If male, assigned to male only barbers (B1, B2) first
• If no male barber available, assign to master barber (B5)
• If master barber is busy, the customer leaves for another barber
shop
– If Female, assign to female only barbers first (B3, B4)
• If no female barber available, assign to master barber
• If master barber is busy, the customer leaves for another barber
shop
• The barber shop opens for 8 hours. Any customer who arrives after 8 is
not admitted. But barbers still need to work overtime to serve, for
example, a customer who arrives at 7:50 and still need an hour to
process
Run simulation 10,000 times and take
average
Result of List-scheduling (L-S)
• The Average make span is (in hours):
8.666184804397565
• The Average # of Male customer served:
17.578
• The Average # Female Customer Served:
16.978
• The Average # of Total Customer Served:
34.556
Let’s think of how to improve the
system to serve more customers, while
minimizing the makespan
• The list-scheduling method assign customers
to gender constrained barbers (barbers who
only handle male or female)
• Can we improve the system by prioritizing the
master barber, who is 2x faster and can handle
both male and female?
Fastest Barber First (FBF) method:
• FCFS
• When a customer arrives
– If male, assigned to B5
• If B5 is busy, assign to B1 or B2, whoever is available
• If B1 and B2 are busy, the customer leaves for another barber
shop
– If Female, assigned to B5
• If B5 is busy, assign to B3 or B4, whoever is available
• If B3 and B4 are busy, the customer leaves for another barber
shop
• Same as above, the barber shop opens for 8 hours only
Run FBF 10,000 times and take
average, compare with L-S method
Method
Cmax
# male
# female
Total
L-S
8.6662
17.578
16.978
34.556
FBF
8.6342
17.4713
16.8489
34.3202
•There seems to be a very minor improvement
•We introduce another master barber B6 and
see whether prioritizing fastest machines
would improve the system
Compare Fastest Barber First with 2 master
barbers(FBF-2) to List Scheduling with 2 master
barbers (L-S-2)
• FCFS
• FBF-2: Assign new arrivals to B5 or B6 first, then to
gender constraint barbers B1~4
• L-S-2: Assign new arrivals to B1~4, if these gender
constraint barbers are busy, assign to B5 or B6
• Same as above, the barber shop opens for 8 hours
only
Run L-S-2, FBF-2 10,000 times and take
average, compare with previous methods
Method
Cmax
L-S
8.6662
% change
# male
17.578
% change
# female
16.978
% change
Total
% change
34.556
FBF
L-S-2
FBF-2
8.6342 8.6731 8.6052
-0.4%
-0.8%
17.4713 20.3233 20.2719
-0.6%
-0.3%
16.8489 19.8989 19.8443
-0.8%
-0.3%
34.3202 40.2222 40.1162
-0.7%
-0.3%
•There is no improvement in terms of # of customers served,
actually, fewer (very little) customers are served
•There very limited reduction in makespan
•Hiring more barbers does not reduce makespan!
The queue scenario
Male
Baber 1
Customers arrive
Poisson (λ=3) for both
male and female
Waiting
seats
Male
Barber
2
Master
Female
Barber
1
Female
Barber
2
•We assume that customers are willing to wait for at most 30
minutes—if no barber is free then they will just leave and go to
another barber shop near by
Introducing a Queue/ waiting area
• We compare the performance of L-S, FBF. L-S-2, FBF-2
L-S
FBF
L-S-2
FBF-2
8.8487 8.8413 8.8366 8.73794
2.1%
-0.1%
-1.1%
21.9175 21.9203 23.7944 23.4529
0.0%
-1.4%
21.2255 21.2782 23.7611 23.2181
0.2%
-2.3%
43.143 43.1985 47.5555 46.671
0.1%
-1.9%
•Having the queue does not make FBF any better
compared to L-S
•FBF results in 2% less customers being served
Should we have a queue/ build a waiting area?
Method
Cmax
% change
# male
% change
# female
% change
Total
% change
Total
Customers
generated/
arrived
% served
No queue/immediate blocking
L-S
FBF
L-S-2
FBF-2
8.6662
8.6342 8.6731 8.6052
-0.4%
-0.8%
17.578 17.4713 20.3233 20.2719
-0.6%
-0.3%
16.978 16.8489 19.8989 19.8443
-0.8%
-0.3%
34.556 34.3202 40.2222 40.1162
-0.7%
-0.3%
Queue with 30min patience limit
L-S
FBF
L-S-2
FBF-2
8.8487 8.8413 8.8366 8.73794
2.1%
-0.1%
-1.1%
21.9175 21.9203 23.7944 23.4529
0.0%
-1.4%
21.2255 21.2782 23.7611 23.2181
0.2%
-2.3%
43.143 43.1985 47.5555 46.671
0.1%
-1.9%
49
48
48
49
46
46
49
47
71%
72%
84%
82%
94%
94%
97%
99%
•Having a queue does not substantially increases makespan, but we can serve more
customers  keep the queue!
•Do not hire one more master barber: only to serve 3~4 more customers is not
worth it as an average barber serves more than 8 under the 5 barber system
•Having a queue improves the system by serving more customers, while not
dramatically increasing the makespan!.
• If the Poission arrival rate is very low, then whether or not
do we have a queue does not make a big difference
• When a customer came in, one barber maybe idle and
probably have been waiting for a long time
• But if the arrival rate is high (much higher than λ=3 per
hour, i.e a holiday),:
– Having a queue will allow the barber shop to serve more
customers (more arrivals will be “stored “ in the queue for later
processing)
– The only drawback is that the makespan could be higher
(customers in the queue may force barbers to work longer)
• We run a simulation under λ=10, and see if having a queue
will benefit us. We also compare L-S to FBF
λ=10
Method
Cmax
% change
# male
% change
# female
% change
Total
% change
Total
Customers
generated/
arrived
% served
No queue/immediate blocking
L-S
FBF
L-S-2
FBF-2
8.7609
8.758
8.7707
8.762
0.0%
-0.1%
25.7788 25.841 31.9343 32.0519
0.2%
0.4%
24.0903 23.975 30.5485 30.3575
-0.5%
-0.6%
49.8691 49.816 62.4828 62.4094
-0.1%
-0.1%
153
33%
153
33%
153
41%
153
41%
Queue with 30min patience limit
L-S
FBF
L-S-2
FBF-2
9.1916 9.1944 18.8693 9.2394
0.0%
-51.0%
30.6416 30.5901 68.4953 36.0811
-0.2%
-47.3%
28.3694 28.3569 68.2495 34.0217
0.0%
-50.2%
59.011 58.947 136.7448 70.1028
-0.1%
-48.7%
153
39%
153
38%
153
89%
153
46%
•
Having a queue allows us to serve 5 more customers per day by increasing
30mins makespan this is desirable as each customer has an average
processing time of more than 30min
•
•
•
L-S-2 is not desirable
A growing queue!
Hiring an extra master barber allows us to serve 12 more customers. This may
be desirable, as originally, a master barber processes more than 12
customers we now serve more customers without suffering form
diminishing return
Under low arrival rate
• This is intuitive
• A queue is not required
• No need to hire an extra master barber not
many customers out there to serve
• FBF is optimal (example: the shop is quite
empty and most of the barbers idle, a
customer arrives at t=7:50 you want to
assign him/her to the available fastest barber
to reduce makespan)
Our conclusion:
• Under normal arrival rate; we need to implement
a queue; we do not hire an extra master barber;
and either L-S or FBF is optimal
• Under high arrival rate, we need a queue; we may
hire an extra barber; FBF-2 is optimal
• Under low arrival rate, a queue is not necessary;
we do not need to hire an extra master barber;
FBF is optimal
```