Abstract
A theory of traffic-measurement errors for loss systems with renewal input is developed. The results provide an accurate approximation for the variance of any differentiable function of one or more of the following basic traffic measurements taken during a given time-interval:
-
(i)
the total number of attempts (peg count)
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(ii)
the number of unsuccessful attempts (overflow count)
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(iii)
the usage based on discrete samples (TOR measurement) or on continuous scan.
The approximation is given in terms of the individual variances and covariance functions of the three measurements. Asymptotic approximations for these moments are obtained using the concept of a generalized renewal process, and are shown to be sufficiently accurate for telephone traffic-engineering purposes.
As an application of the theory, we examine the variances of the standard estimates of the load and peakedness (variance-to-mean-ratio) of an input traffic stream for a time interval of one hour. Other possible applications to Bell System trunking problems are discussed.
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References
Wilkinson, R. I., “Theories for Toll Traffic Engineering in the U.S.A.”, Bell Sys. Tech. J., Vol. 35, No. 2, 1956, pp. 421–514.
Benes, V. E., “The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement,” Bell Sys. Tech. J., Vol. 40, No. 1, 1961, pp. 117–148.
Descloux, A., “On the Accuracy of Loss Estimates,” Bell Sys. Tech. J., Vol. 44, No. 6, 1965, pp. 1139–1164.
Kuczura, A. and S. R. Neal, “The Accuracy of Call-Congestion Measurements for Loss Systems with Renewal Input,” to appear, B.S.T.J., Dec. 1972.
Descloux, A., “On Markovian Servers with Recurrent Input,” Proc. Sixth Intl. Teletraffic Cong., Munich, West Germany, 1970, pp. 331/1–331/6.
Kendall, M. G. and A. Stuart, The Advanced Theory of Statistics Vol. 1, Hafner, New York, 1958, Chapter 10.
Hammersley, J. M., Discussion of paper by W. L. Smith, “Renewal Theory and Its Ramifications,” J. Roy. Statist. Soc., Series B, Vol. 20, No. 2, 1958, pp. 289–291.
Riordan, J., Stochastic Service Systems Wiley, New York, 1962, pp. 36–40.
Takács, L., Introduction to the Theory of Queues Oxford University Press, New York, 1962, Chapter 4.
Eisenberg, M., “A Program for the Analysis of a Class of Electronic Switching Systems,” unpublished work, April 15, 1971.
Kuczura, A., “The Interrupted Poisson Process as an Overflow Process,” to appear, B.S.T.J., March, 1973.
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© 1974 Springer-Verlag Berlin · Heidelberg
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Neal, S.R., Kuczura, A. (1974). A Theory of Traffic-Measurement Errors for Loss Systems with Renewal Input. In: Clarke, A.B. (eds) Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems, vol 98. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80838-8_11
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DOI: https://doi.org/10.1007/978-3-642-80838-8_11
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