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FNCE 645: Financial Modelling
Skyline and LugeNg Wen Ying | Sim Siang Huat
Zou Yue | Zhong Shi Yao
BackgroundWhat is the Skyline and Luge?
IssueWhat issue are we modelling?
Skyride Top
Skyride Base
Dragon Trail
Jungle Trail
688m
628m
New Joiners
Re-joiners
Issue: Long queueing time1 Widen Track2 Create a third trail
Parameter:1 Peak Hour: 10.30am – 12.30pm2 Weekend
Base CaseCurrent situation
MethodologyOption 1Trails widen
Option 2Additional Luge trail added
Flowchart
Base CaseCurrent Situation
Inflow: New JoinersSimulating the number of new joiners flowing into the system
Assumptions:- The inter-arrival time between visitors follows an
exponential distribution- The average inter-arrival time is 0.21mins- Ushers will let a batch of new joiners in every 1.5mins
Inter-arrival Time = – Average * LN (1-Rand())Formula:
Illustration:
Inter-arrival Time
Arrival Time Time Intervals # Time New Joiners
0 1 00.550853514 0.550853514 2 0.30.012227288 0.563080802 3 0.60.001761651 0.564842454 4 0.90.013120036 0.577962489 5 1.20.049912837 0.627875326 6 1.5 100.228993028 0.856868354 7 1.80.322342296 1.17921065 8 2.10.096762228 1.275972878 9 2.40.138550518 1.414523396 10 2.70.040588191 1.455111587 11 3.0 70.283460628 1.738572215 12 3.3
<1.5COUNTIF
J is a longer trail so the luges arrive later than D
Inflow: RejoinersSimulating the number of rejoiners flowing into the system
Assumptions:- Riders’ decision to rejoin follows a Binomial distribution- The probability of rejoin is 0.6- Interval between each luge release is 1.8mins
Formula:
Illustration:
Rejoins from each trail = CRITBINOM (# riders finishing trails, 0.6, Rand())
# riders finishing J
Rejoin from J# riders
finishing DRejoin from D
1 0
3 0
5 4
5 3
1 0
5 2
1.8mins
1.8mins
Total Rejoin (t) = SUM (Rejoin frm J & D at t)
Skyride: Queue Length Simulating the Queue Length for the Skyride
Formula:
Queue Length (t)
Queue Length (t - 1)
# Skyridersleaving (t – 1)= + -Total Inflow
(t)
New Joiners +
Rejoiners
Queue Time =
Queue Length (t)
Ave. # Skyriders
leaving with each chairlift
xFrequency of
chairlift: 0.3mins
Skyriders can choose to sit in
groups of 1, 2, 3 or 4 on the chairlift.
How to simulate??
Skyride: Riders per ChairliftSimulating the number of skyriders on each chairlift
Assumptions:- 1 Chairlift leaves every 0.3mins- Maximum no. of riders on each chairlift – 4 pax- It is equal likely for riders to board in groups of 1,2,3 or 4- Chairlift takes 5.1mins to travel from base to peak
Formula:
If Queue Length < 4
RANDBETWEEN (1, Queue Length)
# Skyridersleaving (t ) =
If Queue Length > 4
RANDBETWEEN (1,4)
# Riders arriving at peak and choosing which
luge trail to take
5.1mins later…
How many riders would choose Jungle trail /
Dragon trail???
Luge: Queues LengthSimulating the Queue Length for the Luge Trails
Assumptions:- Riders are equally likely to choose Jungle or Dragon trail- Every 0.3mins, one chairlift will carry a max of 4 to peak- Let “1” represent Jungle and “2” represent Dragon
Formula:
# Luge RidersChoice of
1st Rider
Choice of
2nd Rider
Choice of
3rd Rider
Choice of
4th RiderChoose J Choose D
X
Those arriving on the chairlift
Rider 1: IF(X>0, RANDBETWEEN(1,2),0)
Rider 2: IF(X>1, RANDBETWEEN(1,2),0)
Rider 3: IF(X>2, RANDBETWEEN(1,2),0)
Rider 4: IF(X>3, RANDBETWEEN(1,2),0)
J: Count if the choice is “1”
OR
D: Count if the choice is “2”
# Luge Riders
Arriving
Choice of 1st Rdier
Choice of 2nd Rdier
Choice of 3rd Rdier
Choice of 4th Rdier
Choose Jungle
Choose Dragon
0 0 0 0 0 0 04 1 2 1 1 3 12 2 2 0 0 0 22 1 1 0 0 2 0
Illustration:
Formula:
Queue Length for
J (t)
Queue Length for J
(t - 1)
# riders released (t – 1)
= + -# Riders Chose J (t)
Luge: Queues Length (ctd’)Simulating the Queue Length for the Luge Trails
# Luge Riders
Arriving
Choice of 1st Rdier
Choice of 2nd Rdier
Choice of 3rd Rdier
Choice of 4th Rdier
Choose Jungle
Choose Dragon
0 0 0 0 0 0 04 1 2 1 1 3 12 2 2 0 0 0 22 1 1 0 0 2 0
Max 5 Riders will be released in batches
every 1.8mins, may be less than 5…
How to simulate??
Outflow: Leave/RejoinSimulating the Queue Length for the Skyride
Same as Rejoiner at InflowCycle REPEATS
(for 2-hours worth of data)
# riders finishing J
Rejoin from J# riders
finishing DRejoin from D
1 0
3 0
5 4
5 3
1 0
5 2
1.8mins
1.8mins
Recall:
If there are less than 5 people in the queue, all the people will be released.
If there are more than 5 people, a max of 5 will be released
Assumptions:- A max of 5 riders will be released every 1.8mins
Base Case ResultsWhat is the average queue time now?
Average Queue
Time41.5 mins
38.82 1 2 3 4 5 6 7 8 9 10
1 39.48 43.44 39.06 37.53 43.95 49.77 37.2 35.94 44.61 46.68
2 47.28 37.83 38.4 46.98 37.89 38.04 39.48 45.48 35.22 42.72
3 42 45.39 42 37.41 37.26 38.34 40.17 40.65 40.14 44.67
4 39.42 37.17 44.1 40.77 45.54 40.86 44.16 40.86 40.77 41.13
5 42.93 45.03 42.87 42.24 37.47 39.69 46.53 47.19 46.77 37.89
6 40.44 38.25 40.2 38.4 40.71 42.48 42.3 38.25 39.84 41.73
7 45.48 41.73 41.43 42.57 40.86 40.35 44.73 44.01 42.66 35.52
8 45.45 37.32 43.23 43.8 40.56 39 40.65 46.56 46.05 41.73
9 38.55 43.53 40.56 44.19 46.47 38.13 36.45 42.87 47.67 43.29
10 39.69 42.51 40.92 40.5 47.79 39.54 34.44 41.28 36.96 35.73
Total Queue Time
Skyride’sQueue Time= + MAX( Queue time for J, D)
Option 1Enlarge the trails
Option 1 ResultsWhat is the average queue time if we expand the trails?
If there are less than 6 people in the queue, all the people will be released.
Changed Assumption:- A max of 6 riders will be released every 1.8mins
If there are more than 6 people, a max of 6 will be released
1 2
Average
Queue Time28.4 mins
26.1 1 2 3 4 5 6 7 8 9 10
1 28.4 26.7 33.1 32.9 24.6 25.9 33.7 27.6 28.5 26.5
2 28 26.5 28.6 25.4 30.7 34.8 29 26.3 26.7 24.6
3 27.5 30.6 30.6 23.85 21.1 31.5 32.8 28.3 24.5 25.7
4 25.8 32.2 27.5 28.5 24.9 27.7 29.2 28.5 31.4 28.5
5 28.2 23.9 27 33.8 28.9 26 27.7 30.1 27.5 26.1
6 29.1 27.2 37.2 30.4 32.3 28.4 32.2 25.5 25.05 27.9
7 26.1 28.4 31.8 29 25 33.1 33.8 35.4 23.7 25.3
8 26.7 30.7 27.4 25.4 28.7 28.6 29 29.6 27.4 36.1
9 31.8 28.9 28.4 25.4 29.5 29 24 22.5 35.4 27
10 26 28.4 35.8 29.9 25.6 19.2 28.7 27.8 32 24.7
Option 2Build an additional luge trail
Option 2 ResultsWhat is the average queue time if we add one more trail?
Changed Assumption:
New Joiners
Rejoiners
Skyride Queue
Jungle Queue
Dragon Queue
Leaves
Rejoin
Leaves
Exponential Distribution
Uniformly Distributed 33%
40%
60%
60%
40%
# Riders / Chair
33%
Inflow Skyride Luge rides Outflow
Rainbow Queue33% 60%
Average Queue
Time17.9 mins
17.9 1 2 3 4 5 6 7 8 9 10
1 18.6 15.84 17.06 19.06 15.58 13.12 21.94 16.2 18.82 17.64
2 23.5 12.8 15.48 20.66 17.12 19.12 12.86 15.94 17.28 15.84
3 18.3 19.28 10.34 14.6 16.14 17.68 19.18 25.76 15.22 25.4
4 24.64 20.76 20 17.32 18.14 13.78 14.04 17.94 15.84 15.48
5 19.14 14.2 15.28 14.2 13.58 15.58 21.28 15.78 21.12 20.52
6 19.44 20.9 14.86 22.32 20.36 17.16 16.86 21.44 22.72 15.28
7 … … … … …. …. …. …. …. ….
Overall ResultsSensitivity Analysis and Summary of all Results
10
12
14
16
18
20
1.2 2.2 3.2 4.2 5.2
Ave
. Qu
eue
Tim
e
Luge Time
Sensitivity AnalysisOption 2 – Sensitivity Analysis- Additional trail Long or Short?- Long trail = takes a longer time to travel
down to base “Luge Time”
- CONCLUSION: The length of the additional trail has little effect on the queue time
Scenario Average Queue Time (mins) % Improvement
Base Case 41.5 NA
Enlarged Trails 28.4 32%
Additional Trail – Shorter trail 17.9 57%
Additional Trail – Longer tail 18.5 55%
- Both options are effective in enhancing the queue time - An additional trail is more effective - A short or a long trail does not alter the results much- Considering that a longer trail would cost more to construct, our suggestion to the
management is to build an additional shorter trail to improve the queue time.
Conclusion:
Excel DemonstrationNg Wen Ying | Sim Siang Huat
Zou Yue | Zhong Shi Yao
New Joiners
Rejoiners
Skyride Queue
Jungle Queue
Dragon Queue
Leaves
Rejoin
Leaves
Exponential Distribution
Uniformly Distributed33%
40%
60%
60%
40%
# Riders / Chair
33%
Inflow Skyride Luge rides Outflow
Rainbow Queue33% 60%
