Efficient algorithms for higher-order derivatives of the continued Erlang delay function

Authors:
Jorge Sá Esteves
Affiliations:
University of Aveiro, Department of Mathematics, Aveiro, Portugal
Venue:
WSEAS Transactions on Mathematics
Year:
2009

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Abstract

In this paper we analyze the partial derivatives of any order of the continued Erlang delay function in the number of servers. Several properties with strong analytical relations between the high-order derivatives of Erlang's B and C functions are established. Using these relations, three algorithms are proposed for the numerical computation of the cited derivatives. For comparison purposes, it is also generalized a numerical method based on a quadrature procedure suggested by D. L. Jagerman [16]. All the computational methods are compared in terms of stability, efficiency and precision. Our study concludes that a recursive matrix relation presented in a previous work [10, 11], may be used for the establishment of a simple and reliable algorithm having the best performance considering the trade-off of the different criteria. Extensive computational results are presented and discussed. In the sequel, a conjecture about the strict convexity of the first derivative of Erlang delay function is presented and supported by numerical evidence.