Simulating gym waiting times.

1. Simulating gym capacity based on
Unisport Töölö arrival rates
30E00400 Simulation project
Okko Hakola

2. Introduction
 Töölö Unisport is very crowded from time to time
 There is a lot of randomness and uncertainty involved
 Suitable subject for simulating system capacity and
waiting times
 The model gets complicated fast

3. Not all machines are used in this simulation

4. The Model
 Inter-arrival times follow exponential distribution
 They are simulated with inverse method
 𝑥 = −
1
𝜆
ln 1 − 𝑢0 , where U ≈ 0,1
 λ is derived from Unisport crowdedness data
 changes as hour changes
Arrive Movement
1
Movement
10
Movement
2 ......
Machine options
Pick first one free
Machine options
Pick first one free
Machine options
Pick first one free
Waiting Waiting Waiting

5. http://www.helsinki.fi/unisport/Kuntosali_ka
vijamaarat/t%f6%f6l%f6_syyskuu14.pdf
Time Mo Tue Wed Thu Fri Mean Stdev Arrival rate λ Inter arrival time
7.00-8.00 11 11 13 9 9 10,6 1,673320053 11 0,090909091
8.00-9.00 7 3 4 7 6 5,4 1,816590212 6 0,166666667
9.00-10.00 9 16 12 8 10 11 3,16227766 11 0,090909091
10.00-11.00 14 11 11 14 12 12,4 1,516575089 13 0,076923077
11.00-12.00 9 11 14 10 7 10,2 2,588435821 11 0,090909091
12.00-13.00 8 7 11 9 9 8,8 1,483239697 9 0,111111111
13.00-14.00 14 12 16 11 12 13 2 13 0,076923077
14.00-15.00 20 13 14 8 8 12,6 4,979959839 13 0,076923077
15.00-16.00 12 21 12 12 13 14 3,937003937 14 0,071428571
16.00-17.00 14 1 12 12 21 12 7,176350047 12 0,083333333
17.00-18.00 21 16 16 14 9 15,2 4,324349662 16 0,0625
18.00-19.00 18 17 16 15 13 15,8 1,923538406 16 0,0625
19.00-20.00 23 17 20 14 10 16,8 5,069516742 17 0,058823529
20.00-21.00 9 18 13 14 3 11,4 5,683308895 12 0,083333333
21.00-22.00 3 4 2 2 2,75 0,957427108 3 0,333333333

6. Imaginary workout plan
Movement # Whole Body Sets Repetitions Time/repetition Possible machines
1 Warmup Whole 1 1 U(1/12,1/3) Cross-trainer 1 Cross-trainer 2 Cross-trainer 3 Treadmill 1 Treadmill 2
2 Squat 4 U(8,12) U(1/3600,4/3600) Rack 1 Rack 2
3 Leg press 3 U(8,12) U(1/3600,4/3600) Leg press 1 Leg press 2
4 Deadlift 4 U(8,12) U(1/3600,4/3600) Barbells 1 Barbells 2
5 Chinups 3 U(8,12) U(1/3600,4/3600) Chin-up 1 Chin-up 2 Lat pull
6 Cable seated row/shoulder push 3 U(8,12) U(1/3600,4/3600) Lower back Shoulders push up
7 Bench press 4 U(8,12) U(1/3600,4/3600) Bench press 1 Bench press 2
8 Bicep curls 4 U(8,12) U(1/3600,4/3600) Bench 1 Bench 2 Bench 3
9 Cable pushdown 4 U(8,12) U(1/3600,4/3600) Cables 1 Cables 2
10 Abs 3 U(8,12) U(1/3600,4/3600) Abs 1 Abs 2 Abs 3
Time between sets U(1/60,2/20)

7. ”Service times” for movements
 Aerobic movement: U~(5min, 20min)
 20 min is the Unisport max
 Other movements: U~(1sec, 4sec) per repetition
 Repetitions: U~(8,12)
 Times between sets: U~(1min, 2min)
 Uniforms are used for repetition times, because I assume repetition
times vary a lot from person to person
 For repetition numbers it was also convenient
 Easy to change to normal distribution etc.

8. Simulation
1. Simulate inter-arrival times
2. Simulate serving times
3. Assign customers to movements and their machines
4. Calculate waiting times
5. Take statistics
6. Repeat average and others using Data-Table
7. Analyze

9. Count Random number Inter-arrival time (clock dependent) Clock Customer
U0
1 0,213860122 0,021874595 0,021874595 1
2 0,076785067 0,007263019 0,029137613 2
3 0,141063724 0,013823686 0,042961299 3
4 0,791809732 0,142663897 0,185625196 4
5 0,124160522 0,012052041 0,197677237 5
6 0,755301686 0,127975382 0,325652619 6
7 0,228690652 0,023605978 0,349258597 7
8 0,751227297 0,126474149 0,475732746 8
9 0,694683421 0,107855098 0,583587843 9
10 0,252516019 0,0264584 0,610046244 10
11 0,355398578 0,039920282 0,649966526 11
12 0,140766122 0,013792193 0,663758719 12
13 0,35897997 0,04042678 0,704185499 13
14 0,083770064 0,007953448 0,712138946 14
15 0,718186797 0,115137348 0,827276294 15
16 0,414159184 0,048609743 0,875886037 16
𝑥 = −
1
ln 1 − 𝑢0
Arrivals and the clock

10. M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
1 2 3 4 5 6 7 8 9 10
0,029989777 0,0770951 0,06133366 0,109158 0,073613 0,022371 0,051848 0,068327379 0,060346148 0,119546
0,244347934 0,0367107 0,00796867 0,082061 0,046053 0,074788 0,134178 0,015762168 0,121111201 0,080075
0,142724326 0,059199 0,02481298 0,04204 0,089312 0,038412 0,089061 0,038808887 0,079443855 0,071337
0,013571523 0,1195982 0,02765518 0,10736 0,109697 0,064208 0,141597 0,034988672 0,08560737 0,096991
0,190478819 0,0435704 0,03648402 0,106151 0,020305 0,061118 0,101462 0,115126363 0,021336734 0,07239
0,209284824 0,0526354 0,10633744 0,069674 0,072222 0,091682 0,127928 0,072301592 0,161284426 0,048574
0,047615072 0,1025319 0,06380841 0,040493 0,058178 0,088049 0,054817 0,076393119 0,047120976 0,042954
0,236091033 0,0789709 0,00494475 0,148444 0,082516 0,072184 0,068096 0,056490396 0,039219776 0,014211
0,079635471 0,1222331 0,11155271 0,07798 0,05847 0,051606 0,02732 0,043626728 0,163059948 0,043799
0,045201639 0,1389008 0,02404688 0,063069 0,083521 0,042981 0,049036 0,114661239 0,106551808 0,026662
0,102925007 0,1338951 0,02937927 0,120258 0,020132 0,112778 0,131637 0,053576304 0,116917556 0,033495
0,055844532 0,1387096 0,10749427 0,122575 0,096913 0,056982 0,093383 0,139393232 0,046331876 0,031763
0,112378313 0,049691 0,04878447 0,076016 0,105992 0,097687 0,085216 0,157670407 0,121054535 0,102636
0,117206952 0,1555321 0,11996196 0,132641 0,023895 0,047648 0,058517 0,116603726 0,127051332 0,118757
0,093474215 0,0729606 0,03484523 0,06976 0,099498 0,048886 0,074042 0,042827501 0,014267256 0,067407
0,151749687 0,0773996 0,08923596 0,102989 0,033463 0,068493 0,105161 0,133330839 0,110845341 0,079223
Serving times of movements=(sets*(repetitions*time/repetition+time between sets).
”Serving times”

11. Assigning customers to machines
Arrival clock Cross-trainer 1 Cross-trainer 2 Cross-trainer 3 Treadmill 1 Treadmill 2
Begins Ends Occupied? Begins Ends Occupied? Begins Ends Occupied? Begins Ends Occupied? Begins Ends Occupied?
Clock at end of
movement 1
0,021874595 0,021874595 0,051864 1 0 0 0 0 0 0 0 0 0,051864372
0,029137613 0,051864 1 0,029137613 0,273486 1 0 0 0 0 0 0 0,273485547
0,042961299 0,051864 0 0,273486 1 0,042961299 0,185686 1 0 0 0 0 0,185685626
0,185625196 0,185625196 0,199197 1 0,273486 1 0,185686 0 0 0 0 0 0,199196719
0,197677237 0,199197 0 0,273486 0 0,197677237 0,388156 1 0 0 0 0 0,388156056
0,325652619 0,325652619 0,534937 1 0,273486 0 0,388156 1 0 0 0 0 0,534937443
0,349258597 0,534937 1 0,349258597 0,396874 0 0,388156 0 0 0 0 0 0,396873668
0,475732746 0,534937 0 0,475732746 0,711824 1 0,388156 0 0 0 0 0 0,711823778
0,583587843 0,583587843 0,663223 1 0,711824 1 0,388156 0 0 0 0 0 0,663223314
0,610046244 0,663223 1 0,711824 1 0,610046244 0,655248 1 0 0 0 0 0,655247882
0,649966526 0,663223 0 0,711824 1 0,655248 0 0,6499665 0,752892 1 0 0 0,752891532
0,663758719 0,663758719 0,719603 1 0,711824 1 0,655248 0 0,752892 1 0 0 0,719603251
0,704185499 0,719603 1 0,711824 0 0,704185499 0,816564 1 0,752892 1 0 0 0,816563811
0,712138946 0,719603 0 0,712138946 0,829346 1 0,816564 0 0,752892 0 0 0 0,829345898
0,827276294 0,827276294 0,920751 1 0,829346 0 0,816564 0 0,752892 0 0 0 0,92075051
0,875886037 0,920751 0 0,875886037 1,027636 1 0,816564 0 0,752892 0 0 0 1,027635725
0,934025236 0,934025236 1,0584 1 1,027636 1 0,816564 0 0,752892 0 0 0 1,058399614
0,942851777 1,0584 0 1,027636 0 0,942851777 1,01377 0 0,752892 0 0 0 1,013770377
1,16791075 1,16791075 1,438855 1 1,027636 0 1,01377 0 0,752892 0 0 0 1,438854916
1,376072059 1,438855 0 1,376072059 1,494566 0 1,01377 0 0,752892 0 0 0 1,494566357
1,526734813 1,526734813 1,573897 0 1,494566 0 1,01377 0 0,752892 0 0 0 1,573897211
1,770822038 1,770822038 1,91149 1 1,494566 0 1,01377 0 0,752892 0 0 0 1,911490116
Movement 1

12. System statistics from Data Table analysis (n=500)
Averages and statistics of the replications in seconds
Whole workout
in hours Arriv-M1 M1-M2 M2-M3 M3-M4 M4-M5 M5-M6 M6-M7 M7-M8 M8-M9 M9-M10
TOTAL
time
Mean 0,96066401 1,206182 194,6151 27,61721 89,68459 1,521112 34,23245 84,37241 6,35025 96,36117 1,413271 537,3738
Stdev 0,041192247 2,191366 71,28229 9,03782 36,76929 1,21792 9,705413 29,29266 3,793051 30,98273 1,211048 141,5925
Std error 0,001842173 0,098001 3,187841 0,404184 1,644373 0,054467 0,434039 1,310008 0,16963 1,38559 0,05416 6,332207
Min 0,869256609 0 57,50543 8,084163 24,75367 0 13,90527 30,9084 0,437752 39,60995 0 221,9124
Max 1,172037206 23,04431 580,3904 77,19827 291,3033 6,335351 70,19339 223,6238 38,11927 258,4095 7,914859 1301,925
2 Squat Rack 1 Rack 2
4 Deadlift Barbells 1 Barbells 2
7 Bench press Bench press 1 Bench press 2
9 Cable pushdown Cables 1 Cables 2

13. Adding machines to bottleneck movements
Averages and statistics of the replications i
Whole workout
in hours Arriv-M1 M1-M2
TOTAL
time
Mean 0,881496356 1,018278 38,231 255,3116
Stdev 0,021476988 1,694854 17,492 64,97161
Std error 0,00096048 0,075796 0,7822 2,905619
Min 0,827679181 0 3,9458 83,45129
Max 0,97422678 18,29542 123,45 584,5118

14. Peak hours

15. Sensitivity of inter-arrival rates (Lambdas and waiting)

17. Assumptions and limitations
 Most important assumptions and limitations:
 1 person at a time in a movement and no ”in between”
 Serving times are overly simplified
 Arrival rates are only from 1 month
 Only machines which I decided are used in the workout(most common at Unisport)
 1 workout for everyone and strict order
 No people in after 21.30
 People are allowed to finish their workouts
 In order to simulate you must simplify and assign rules and
assumptions in a way that still represents the system well
enough

18. Conclusion and recommendations for
future
 Simulating a gym gets complicated really fast
 The model gives an idea how much one has to wait between movements to
the most popular machines
 Recommendations: Unisport should add space and machines to increase
capacity
Future simulations:
 All machines
 Different workouts
 Better serving times