CALL CENTERS (CENTRES) - Faculty of Industrial Engineering and ...

over a finite interval, based on the number of counts in measurement subintervals. Such a linear
arrival-rate function can serve as a component of a piecewise-linear approximation to a general
arrival-rate function. We consider ordinary least squares (OLS), iterative weighted least squares
(IWLS) and maximum likelihood (ML), all constrained to yield a nonnegative rate function. We
prove that ML coincides with IWLS. As a reference point, we also consider the theoretically optimal
weighted least squares (TWLS), which is least squares with weights inversely proportional
to the variances (which would not be known with data). Overall, ML performs almost as well as
TWLS. We describe computer simulations conducted to evaluate these estimation procedures.
None of the procedures differ greatly when the rate function is not near 0 at either end, but
when the rate function is near 0 at one end, TWLS and ML are significantly more effective than
OLS. The number of measurement subintervals (with fixed total interval) makes surprisingly
little difference when the rate function is not near 0 at either end. The variances are higher
with only two or three subintervals, but there usually is little benefit from going above ten. In
contrast, more measurement intervals help TWLS and ML when the rate function is near 0 at
one end. We derive explicit formulas for the OLS variances and the asymptotic TWLS variances
(as the number of measurement intervals increases), assuming the nonnegativity constraints are
not violated. These formulas reveal the statistical precision of the estimators and the influence
of the parameters and the method. Knowing how the variance depends on the interval length
can help determine how to approximate general arrival-rate functions by piecewise-linear ones.
We also develop statistical tests to determine whether the linear Poisson model is appropriate.
Keywords: Digital simulation, Iterative methods, Least-squares approximations, Maximum likelihood
estimation, Parameter estimation, Piecewise linear techniques, Queueing theory, Stochastic
processes, Telecommunication traffic, Nonhomogeneous Poisson process, Piecewise linear approximation,
Linear arrival-rate function, Ordinary least squares, Iterative weighted least squares,
Computer simulations, Statistical precision, Traffic model
(Appears also in Section IX.)
10. Chlebus, E. Empirical validation of call holding time distribution in cellular communications
systems. Teletraffic Contributions for the Information Age. Proceedings of the 15th International
Teletraffic Congress, ITC-15. Elsevier, Amsterdam, The Netherlands, 1997, 1179–1188.
Abstract. Various probability distributions are fitted to empirical call holding time data collected
in cellular communications systems. Their parameters are determined through maximum
likelihood estimation. A visual plots examination of empirical and fitted cumulative distribution
functions enables qualitative comparison. Goodness-of-fit techniques based on supremum
and quadratic empirical distribution function statistics, namely the Kolmogorov-Smirnov and
Anderson-Darling tests, respectively are implemented to compare quantitatively the produced
fits.
Keywords: Empirical validation, Call holding time distribution, Cellular communications systems,
Probability distributions, Empirical call holding time data, Maximum likelihood estimation,
Cumulative distribution functions, Goodness-of-fit techniques, Supremum empirical distribution
function statistics, Quadratic empirical distribution function statistics, Kolmogorov-
Smirnov test, Anderson-Darling test, Exponential distribution, Gamma distribution, Lognormal
distribution
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