The document describes the steps to perform a simulation of an M/M/1 queue with varying buffer sizes (K) from 0 to 10. Key steps include downloading simulation software, running the simulation for each K value to generate output files, importing results to Excel, and calculating performance metrics like load, loss ratio, and queuing ratio along with 90%, 95%, and 99% confidence intervals and lag-1 autocorrelation. Plots show load increases and loss decreases with higher K while queuing initially increases then levels off, and confidence intervals and autocorrelation values vary across K values.

1. ET2534 Simulation VT17 LP4
Group 10
Mamidi Sai Prakash
Ivvala Venkata Sai Krishna Chaitanya
Peteti Sridhar

2. How to perform the task?
Steps to perform the task
To perform the task we need to follow certain steps as follows
STEP 1
Download the simulation software “mm1.exe”.
• Seed : 2017
• Arrival Process type (M/D/U) : D
• Mean Interval-arrival time E[Ta] : 100
• Queue Size (K) : {0,1,2,3,4,5,6,7,8,9,10}
• Server process type (M/D/U) : M
• Mean server time E[Ts] : 60
• Warm-up batches(0/1) : 1
• No. of batches : 100
• No. of arrivals per batch : 10000

3. How to perform the task?
STEP 2
Open the command prompt and then run the Simulation by
following command
mm1.exe >outK.txt.
STEP 3
By changing the Queue size K by keeping all the other parameters
constant and repeat this process for K=0 to K=10 and save the output
for each value of k in a different excel sheet.
STEP 4
The command will generate an output text file which contains the
simulated output.

4. How to perform the task?
STEP 5
Import the output results from the text files to the excel sheet.
STEP 6
By default software gives the average Load, Loss ,Queuing ratio, for
95%-confidence interval.
STEP 7
By using the confidence function in the excel sheet we can calculate
the 90%-99% confidence interval.
Command for confidence (=CONFIDENCE(α;STDEV; Size))
– α = 0.1 for 90%
α = 0.05 for 95
α = 0.01 for 99%
– Standard deviation =STDEV(B18:B117)
– Size= 100

5. How to perform the task?
STEP 8
By dividing the absolute confidence value with the average load value we
get the relative half size of the confidence intervals.
STEP 9
Relative half size’s command(=CONFIDENCE(alpha;STDEV; size)/Load
Ratio or Loss Ratio or Queuing Ratio)
STEP 10
By using the correl function in the excel sheet the lag-1 autocorrelation
values can be calculated
lag-1autocorrelation (=CORREL(array1; array2))
• Array1: B18:B116; Array2: B19:B117 (for load)
• Array1: C18:C116; Array2: C19:C117 (for loss ratio)
• Array1: D18:D116; Array2: D19:D117 (for queuing ratio)

6. K
Load
estimation
90 % CI
rel. half-size
95 % CI
rel. half-size
99 % CI
rel. half-size
Lag-1 auto-
correlation
0 0.48 0.12% 0.15% 0.19% -0.60%
1 0.56 0.13% 0.15% 0.20% 7.35%
2 0.59 0.14% 0.17% 0.22% 3.70%
3 0.59 0.15% 0.18% 0.24% 6.57%
4 0.59 0.15% 0.18% 0.24% 7.70%
5 0.59 0.16% 0.19% 0.25% 5.26%
6 0.59 0.16% 0.19% 0.25% 3.58%
7 0.59 0.16% 0.19% 0.25% 3.90%
8 0.59 0.16% 0.19% 0.25% 4.46%
9 0.59 0.16% 0.19% 0.25% 4.41%
10 0.59 0.16% 0.19% 0.25% 4.55%
Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000

7. K
Loss ratio
estimation
90 % CI
rel. half-size
95 % CI
rel. half-size
99 % CI
rel. half-size
Lag-1 auto-
correlation
0 0.188 0.33% 0.40% 0.52% 1.20%
1 0.051 0.93% 1.11% 0.14% 5.99%
2 0.015 1.81% 2.16% 2.84% 12.67%
3 0.005 3.29% 3.92% 5.15% 7.01%
4 0.001 5.86% 6.99% 9.18% 8.15%
5 0.0005 10.62% 12.66% 16.64% -2.40%
6 0.0001 18.19% 21.67% 28.48% -3.03%
7 0.00004 32.53% 38.76% 50.94% -9.44%
8 0.00001 62.52% 74.50% 97.91% -3.49%
9 0.000005 108.40% 129.17% 169.75% -2.35%
10 0.000002 164.48% 195.49% 257.58% -1.02%
Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000

8. K
Queuing ratio
estimation
90 % CI
rel. half-size
95 % CI
rel. half-size
99 % CI
rel. half-size
Lag-1 auto-
correlation
0 0 0 0 0 0
1 0.22 0.28% 0.34% 0.44% 1.81%
2 0.29 0.34% 0.41% 0.54% -1.34%
3 0.31 0.39% 0.47% 0.62% 1.69%
4 0.32 0.39% 0.46% 0.61% 1.61%
5 0.32 0.40% 0.48% 0.64% 3.27%
6 0.32 0.41% 0.49% 0.64% 0.34%
7 0.32 0.40% 0.48% 0.64% 1.44%
8 0.32 0.40% 0.48% 0.64% 2.38%
9 0.32 0.40% 0.48% 0.64% 1.87%
10 0.32 0.40% 0.48% 0.63% 1.87%
Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000

9. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5 6 7 8 9 10
LoadEstimation
K
Average load vs Buffer size (K)

10. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000
0.1
1
0 1 2 3 4 5 6 7 8 9 10
Loadestimation
K
Average load vs Buffer size (K)

11. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000
-0.05
0
0.05
0.1
0.15
0.2
0 1 2 3 4 5 6 7 8 9 10
Lossratioestimation
K
Average loss vs Buffer size

12. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 6 7 8 9 10
Lossratioestimation
K
Average loss vs Buffer size (K)

13. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6 7 8 9 10
queuingratioestimation
K
Average Queuing vs Buffer size (K)

14. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000
0.1
1
0 1 2 3 4 5 6 7 8 9 10
Queuingratioestimation
K
Average queuing vs Buffer size (K)

15. Grp. Seed Arr.
proc. E[TA]
Serv.
proc. E[TS]
Warm-
up?
# bat.
n
# arr.
k
10 2017 D 100 M 60 1 100 10000

16. Observations
Load ratio:
• Load ratio increases as the buffer size increases, the significant
change can be observed for k=0, 1, 2 after k=3 the change will be
minimum.
• If buffer size increases the no of arrivals into the system increases
so that load on the system also increases.
Loss ratio:
• Loss ratio decreases as the buffer size increases, the significant
changes can be observed for k=0 to k=4 after k=5 the changes will
be minimum.
• If the buffer size increases the packet loss in the system will be
decreases so that loss ratio decreases

17. Observations
Queuing ratio:
• Queuing ratio increases with increase in the buffer size, the
significant change can be observed form k=0 to k=4 after k=5 the
change is minimum.
Confidence interval:
• Confidence interval for load increases with increase in the buffer
size.
• Confidence interval for loss decreases with increase in the buffer
size.
• Confidence interval for queuing increases with increase in the buffer
size.

18. Observations
Confidence interval Relative half size:
• For 90%CI , 95% CI, and 99% CI relative half-size are increasing
with increase in buffer size for k=1 to k=10, for load, loss and
queuing.
• When queue size is zero the 90%,95%,99% confidence levels of the
queuing ratio are not defined.
Lag-1 auto correlation:
• The lag-1 auto-correlation values for the load, loss and queuing ratio
are different for k=0 to k=10 they do not follow any pattern.

19. Observations
• The lag-1 auto-correlation for the load ratio with changing buffer size
is positive for k=1to k=10 and has negative correlation for k=0.
• The lag-1 auto correlation for the loss ratio with changing buffer size
is negative for k=5,6,7,8,9,10 and positive for k=0 to k=4.
• The lag-1 auto correlation for the queuing ratio with changing buffer
size is negative for k=2.It has positive correlation for buffer sizes
k=0,1,3,4,5,6,7,8,9,10.
• The lag-1 auto correlation of the queuing ratio is not defined for zero
queue size.

20. APPENDIX
Mathematical Formulas
Average: 𝑋 =
1
𝑛 𝑖=1
𝑛
𝑥𝑖
Standard Deviation: 𝑠 =
1
𝑛−1 𝑖=1
𝑛
(𝑥𝑖 − 𝑋)
2
Absolute Confidence Interval: =𝑋 ± 𝑍1−
∝
2
𝑠
𝑛
alpha=1- 0.90 for 90% CI
0.05 for 95% CI
0.01 for 99% CI
Relative Confidence Interval: = 1 ±
𝑍
1−
∝
2
𝑛
𝑆
𝑋
Lag-1 Auto Correlation: 𝒓 𝒌 = 𝒊=𝟏
𝑵−𝒌
(𝒀 𝒊−𝒀)(𝒀 𝒊+𝒌−𝒚)
𝒊=𝟏
𝑵 (𝒀 𝒊−𝒀)
𝟐