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6 - Taylor Series Expansion and Application in Error Estimate
Chunyan Li, Louisiana State University
Book:

Time Series Data Analysis in Oceanography

Published online:

21 April 2022

Print publication:

05 May 2022, pp 110-129
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Summary

The objective of this chapter is to review the Taylor series expansion and discuss its usage in error estimation. The unique value of Taylor series expansion is often neglected. The major assumption is that a function must be infinitely differentiable to use the Taylor series expansion. In real applications in oceanography, however, hardly there is a need to worry about a derivative higher than the 3rd order, although one may think of some exceptions. The point is, there is rarely a need in oceanography and other environmental sciences to actually consider calculating a very high order derivative, unless for theoretical investigations or under special situations. So the application of Taylor series expansion usually only involves the first two derivatives. In this chapter, some simple examples are included for a better understanding of the applications.

The mean waiting time of a GI/G/1 queue in light traffic via random thinning
Part of
Soracha Nananukul, Wei-Bo Gong
Journal:

Journal of Applied Probability / Volume 32 / Issue 1 / March 1995

Published online by Cambridge University Press:

14 July 2016, pp. 256-266

Print publication:

March 1995
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In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

A new approach to the G/G/1 queue with generalized setup time and exhaustive service
Part of
- Special processes
- Operations research and management science
Huan Li, Yixin Zhu
Journal:

Journal of Applied Probability / Volume 31 / Issue 4 / December 1994

Published online by Cambridge University Press:

14 July 2016, pp. 1083-1097

Print publication:

December 1994
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We consider a class of G/G/1 queueing models with independent generalized setup time and exhaustive service. It is shown that a variety of single-server queueing systems with service interruption are special cases of our model. We give a simple computational scheme for the moments of the stationary waiting time and sojourn time. Our numerical investigations indicate that the algorithm is quite accurate and fast in general. For the M/G/1 case, we are able to derive a recursive formula for the moments of the stationary waiting time, which includes the Takács formula as a special case. It immediately results in the stochastic decompòsition property which can be found in the literature.