236 ◾ Simple Statistical Methods for Software Engineering
From the above examples, it can be seen, inspite of simplicity, the uniform dis-
tribution can serve as a handy model to represent several real time processes.
Box 14.1 Airport tAxi-out time
Flight delay has been and continues to be one of the most critical problems
for airports. A large percentage of flight delays occur on the ground. Among
all the delays, historical data indicate that taxi-out times contribute to more
than 60% of the total. e taxi-out time is defined as the ground transit time
between the pushback time scheduled or updated by airlines and the takeoff
time when the aircraft is captured by the radar tracking system. A queu-
ing model was introduced to estimate the taxi-out time at Logan Airport.
e takeoff queue size was defined as the number of takeoffs that take place
between the aircraft pushback time and its takeoff time.
A histogram of taxi-out time is flat, suggesting uniform distribution. e
cumulative frequencies form a straight line, confirming the assumption of
uniform distribution. e straight line regresses, with an R
2
value of 0.98.
Extreme taxi-out times occur due to bad weather. ese are not included
in the construction of uniform PDF. Other models have an average predic-
tion error of three minutes, whereas uniform distribution PDF prediction
error is less than 1 minute [2].
e PDF is reconstructed in Figure 14.4.
By analogy, the same uniform distribution is relevant to software support
services. e response time in complex situations follows uniform distribution.
Taxi-out time (minutes)
5 10 15 20 25 30 35 40
Probability
Figure 14.4 PDF of taxi-out time.
Law of Compliance ◾ 237
Review Questions
1. What are pseudorandom numbers? How are they related to the uniform
distribution?
2. What is the shape of the CDF of uniform distribution?
3. What is the uncertainty in a digital measuring device if its resolution is
±1 mV? We assume uniform distribution here.
4. What is the famous German tank problem?
5. What is the formula for variance in uniform distribution?
Exercises
1. Calculate kurtosis in uniform distribution if A = 1 and B = 2.
2. Solve the German tank problem if 30 tanks were captured and the largest
serial number is 115. at means you have to estimate the number of tanks
produced in Germany. Clue: use Equation 14.3.
3. Calculate the median of the uniform distribution with A = 2 and B = 3.
References
1. D. Evans, e German Tank Problem, Rose-Hulman Institute of Technology, Terre Haute,
Indiana. Available at https://www.causeweb.org/webinar/activity/2009-09/2009-09.pdf.
2. T. V. Truong, e distribution function of airport taxi-out times and selected applica-
tions, Journal of the Transportation Research Forum, 50(2), 33–44, Summer 2011.
Suggested Readings
Adams, T. M., Guide for Estimation of Measurement Uncertainty in Testing, A2LA Guide for
Estimation of Measurement Uncertainty in Testing, July 2002. https://www.a2la.org
/guidance/est_mu_testing.pdf.
Bell, S., A Beginner’s Guide to Uncertainty of Measurement, Centre for Basic, ermal and
Length Metrology, National Physical Laboratory, Teddington, Middlesex, UK, August
1999. https://www.wmo.int/pages/prog/gcos/documents/gruanmanuals/UK_NPL
/mgpg11.pdf.
Castrup, H., A Critique of the Uniform Distribution, Integrated Sciences Group, 2000.
Hellekalek, P., Good random number generators are (not so) easy to find, Mathematics and
Computers in Simulation, 46(5–6), 485–505, 1998.
Larsen, R. J. and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications,
Prentice Hall, Upper Saddle River, NJ, 2006.
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