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Half-Width (Or e Value) - Welch and Autocorrelation

1) Two techniques will be used to analyze the output of the agent-based simulation model: half-width and Welch's autocorrelation. 2) The warm period can be determined using the Welch moving average approach. For the sample data on mean queue size, the moving average stabilizes after period 8 or 400 observations, indicating this number of initial observations should be removed before further analysis. 3) Welch's moving average was calculated over a window of 6 periods for 20 periods of data to determine when the moving average stabilized, identifying the warm-up period.

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88 views2 pages

Half-Width (Or e Value) - Welch and Autocorrelation

1) Two techniques will be used to analyze the output of the agent-based simulation model: half-width and Welch's autocorrelation. 2) The warm period can be determined using the Welch moving average approach. For the sample data on mean queue size, the moving average stabilizes after period 8 or 400 observations, indicating this number of initial observations should be removed before further analysis. 3) Welch's moving average was calculated over a window of 6 periods for 20 periods of data to determine when the moving average stabilized, identifying the warm-up period.

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Ans 6: For output analysis of the agent based simulation model, following two techniques will be

used.
Half-Width ( or e value)
Welch and Autocorrelation

Consider a sample dataset (for example, mean number of people waiting in queue to get a
parking spot for bixi). Let us assume that the information provided in column 3 in the table 1
(replication 1) is our dataset. Let the confidence interval = 0.05, e= 2, s=5.6687 (based on
replication 1, column 3 in table below), n=20, then from the z-table, we can find z / 2 = z.025 =
1.96

z / 2 * s 2 1.96 * 5.6687 2
n =[ ] =[ ] = 30.8616 31 observations are required.
e 2
Assuming it to be a non-terminating system, the warm period can be determined using the Welch
moving average approach. Let us consider the hypothetical data for mean number of people
waiting in queue to get a parking spot for bixi. 5 replications are conducted.

Observed mean queue Welch


(replications) Moving
Period(i) Time 1 2 3 4 5 Total Average Average
5.4
1 50 3 5 9 8 2 27 5.4
8.8
2 100 2 20 11 12 9 54 10.8
9.2
3 150 15 5 10 17 4 51 10.2
-
4 200 14 17 19 16 5 71 14.2
-
5 250 17 2 5 1 2 27 5.4
9.69
6 300 3 18 13 10 9 53 10.6
9.8
7 350 20 16 3 17 10 66 13.2
10.38
8 400 6 1 16 8 6 37 7.4
10.24
9 450 14 6 10 18 3 51 10.2
-
10 500 2 19 10 1 7 39 7.8
-
11 550 6 18 2 18 13 57 11.4
-
12 600 4 20 11 15 14 64 12.8
-
13 650 6 11 1 4 18 40 8

11
10.75
14 700 8 16 14 10 17 65 13
15 750 13 1 10 4 17 45 9
16 800 12 2 19 8 10 51 10.2
17 850 17 7 13 15 10 62 12.4
18 900 10 4 20 20 18 72 14.4
19 950 4 19 9 19 2 53 10.6
20 1000 11 7 19 6 20 63 12.6

m(total number of periods) =20, w(window of moving average) =6,

y1 5.4
y1 (6) = = = 5 .4
1 1
y1 + y 2 + y 3
y 2 ( 6) = = 8 .8
3
y1 + y 2 + y 3 + y 4 + y 5
y 3 ( 6) = = 9 .2
5
..

y1 + y 2 + y 3 + y 4 + y 5 + y 6 + y 7 + y8 + y 9 + y10 + y11
y 6 ( 6) = = 9.69
11
y1 + y 2 + y 3 + y 4 + y 5 + y 6 + y 7 + y8 + y 9 + y10 + y11 + y12 + y13
y 7 ( 6) = = 9 .8
13
y 2 + y 3 + y 4 + y 5 + y 6 + y 7 + y8 + y 9 + y10 + y11 + y12 + y13 + y14
y 8 ( 6) = = 10.384
13
y 3 + y 4 + y5 + y 6 + y 7 + y8 + y 9 + y10 + y11 + y12 + y13 + y14 + y15
y 9 ( 6) = = 10.246
13
..

y8 + y 9 + y10 + y11 + y12 + y13 + y14 + y15 + y16 + y17 + y18 + y19 + y 20
y14 (6) = = 10.753
13
It can be seen from above that the moving average value stabilizes after period 8 or 400
observations. This is the warm-period or this is the number of observations that need to be
removed before applying traditional output analysis techniques on this system.

12

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