Simulating the queue time for the skyline and luge

1. FNCE 645: Financial Modelling
Skyline and Luge
Ng Wen Ying | Sim Siang Huat
Zou Yue | Zhong Shi Yao

2. Background
What is the Skyline and Luge?

3. Issue
What issue are we modelling?
Skyride
Top
Skyride
Base
Dragon Trail
Jungle Trail
688m
628m
New Joiners
Re-joiners
Issue: Long queueing time
1 Widen Track
2 Create a third trail
Parameter:
1 Peak Hour: 10.30am – 12.30pm
2 Weekend

4. Base Case
Current situation
Methodology
Option 1
Trails widen
Option 2
Additional Luge trail added
Flowchart

5. Base CaseCurrent Situation

6. Inflow: New Joiners
Simulating 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 0
0.550853514 0.550853514 2 0.3
0.012227288 0.563080802 3 0.6
0.001761651 0.564842454 4 0.9
0.013120036 0.577962489 5 1.2
0.049912837 0.627875326 6 1.5 10
0.228993028 0.856868354 7 1.8
0.322342296 1.17921065 8 2.1
0.096762228 1.275972878 9 2.4
0.138550518 1.414523396 10 2.7
0.040588191 1.455111587 11 3.0 7
0.283460628 1.738572215 12 3.3
<1.5
COUNTIF

7. J is a longer trail so
the luges arrive later
than D
Inflow: Rejoiners
Simulating 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 D
Rejoin 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)

8. Skyride: Queue Length
Simulating the Queue Length for the Skyride
Formula:
Queue
Length (t)
Queue
Length (t - 1)
# Skyriders
leaving (t – 1)= + -Total Inflow
(t)
New Joiners
+
Rejoiners
Queue
Time =
Queue
Length (t)
Ave. #
Skyriders
leaving with
each chairlift
x
Frequency of
chairlift:
0.3mins
Skyriders can
choose to sit in
groups of 1, 2, 3 or
4 on the chairlift.
How to
simulate??

9. Skyride: Riders per Chairlift
Simulating 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)
# Skyriders
leaving (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???

10. Luge: Queues Length
Simulating 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 Riders
Choice of
1st Rider
Choice of
2nd Rider
Choice of
3rd Rider
Choice of
4th Rider
Choose 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 0
4 1 2 1 1 3 1
2 2 2 0 0 0 2
2 1 1 0 0 2 0
Illustration:

11. 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 0
4 1 2 1 1 3 1
2 2 2 0 0 0 2
2 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??

12. Outflow: Leave/Rejoin
Simulating the Queue Length for the Skyride
Same as Rejoiner at Inflow
Cycle REPEATS
(for 2-hours worth of data)
# riders
finishing J
Rejoin from J
# riders
finishing D
Rejoin 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

13. Base Case Results
What is the average queue time now?
Average Queue
Time
41.5 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’s
Queue Time= + MAX( Queue time for J, D)

14. Option 1Enlarge the trails

15. Option 1 Results
What 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 Time
28.4 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

16. Option 2Build an additional luge trail

17. Option 2 Results
What 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
Time
17.9 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 … … … … …. …. …. …. …. ….

18. Overall Results
Sensitivity Analysis and Summary of all Results
10
12
14
16
18
20
1.2 2.2 3.2 4.2 5.2
Ave.QueueTime
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:

19. Excel Demonstration
Ng Wen Ying | Sim Siang Huat
Zou Yue | Zhong Shi Yao

20. 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 Queue
33% 60%