On computing the distribution function of the sum of independent random variables
Section snippets
The convolution of random variables
In the mathematical analysis of activity networks (ANs) one is usually faced with the problem of evaluating the sums and maxima of independent (or approximately independent) random variables (r.v's). It is well known that the maximum of two r.v's is easily secured by multiplying their probability cumulative distribution functions (c.d.f.). The sum of two independent r.v's, however, leads in a natural way to the evaluation of their convolution, which is a rather demanding operation, and
Polynomial approximation
The polynomial approximations we propose are for finite and closed domains of the mutually independent variables. We recognize that the d.f. fX(t) of the r.v. X, or simply when there is no ambiguity about the reference to the r.v., is often defined for t∈[0,∞).2 Typically, approaches zero asymptotically. For modeling purposes of almost all real-life projects we take the range to be , where T̄ is the upper bound on the duration
The convolution of polynomials
The proposed algorithm using polynomial approximations can be used successfully for small-size ANs. For medium to large-size networks, the computational burden places a heavy penalty on its use. This increase in the computational complexity is due to two reasons. The first is the growth of the order of the polynomial with each convolution operation. And the second is the exponential increase, O(3n−1), in the number of subintervals n over which the convolution is defined. Then one must resort to
Approximation through discretization
Discretization of a continuous d.f. is a popular method for implementing project planning algorithms. The popularity is attributed to the ease of convolution (and maximum) operations albeit at the cost of reduced accuracy. Nonetheless, the computational ease enables one to model large-scale systems.
The most common methods used to discretize a continuous density function, along with their relative pro's and con's, are the following (for a more extensive discussion of these approaches, see [6]):
Mani K. Agrawal is the Chief Scientist at Manugistics, Inc., stationed in Englewood, Colorado. He has earned his baccalaureate degree from the Indian Institute of Technology in Kanpur, and his master's degree from Vanderbilt University, 1989, both in Civil Engineering and his doctorate degree at NCSV, 1999. He has worked as consultant before joining the University of Illinois, Urbana, IL, for one semester, after which he joined Manugistics.
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Mani K. Agrawal is the Chief Scientist at Manugistics, Inc., stationed in Englewood, Colorado. He has earned his baccalaureate degree from the Indian Institute of Technology in Kanpur, and his master's degree from Vanderbilt University, 1989, both in Civil Engineering and his doctorate degree at NCSV, 1999. He has worked as consultant before joining the University of Illinois, Urbana, IL, for one semester, after which he joined Manugistics.
Salah E. Elmaghraby is University Professor of Industrial Engineering and Operations Research at NCSU. He earned his baccalaureate degree in Mechanical Engineering from the University of Cairo, his master's degree from Ohio State University and his doctorate from Cornell University, in Industrial Engineering. His areas of interest are Capacity Planning and Job Scheduling, Activity Networks, and Dynamic Programming. He is author or co-author of six books and over 86 papers in scientific journals.