Algorithms for Higher-Order Derivatives of Erlang C Function JORGE S ´ A ESTEVES University of Aveiro Department of Mathematics Campus de Santiago, 3810-193 Aveiro PORTUGAL saesteves@ua.pt Abstract: In this paper we analyze the partial derivatives of any order of the continued Erlang C function in the number of servers. For the numerical computation of those derivatives, several algorithms are proposed and compared in terms of stability, efficiency and precision. This study concludes that a recursive matrix relation presented in a previous work [4, 5], may be used for the establishment of a simple and reliable algorithm having the best performance considering the trade-off of the different criteria. Some computational results are presented and discussed. A conjecture about the convexity of the first derivative is supported by these computational results. Key–Words: Performance Evaluation, Queueing Systems, Erlang C Formula, Stable Recursions. 1 Introduction The derivatives of Erlang C function are useful in the optimum design of stochastic service systems (queues or networks of queues). Actually, one of the most common models to support performance evaluation of those systems is the M/M/x/∞ (Erlang C model). Moreover, the Erlang C formula plays an important role in approximations for more general systems. Ex- amples of such systems are telephone switching net- works, computer systems (jobs submitted to parallel processors, dynamic shared memory), satellite com- munication systems (allocation of transmission band- width). Recently, Erlang-C model has been subjected to intensive study in the context of the dimensioning and workforce management of call centers [6]. Non-linear programming problems encountered in optimizations of those systems can be solved by using the performance functions and their derivatives, namely for Newton-Raphson and gradient type solu- tions. Moreover, modelers in areas such design opti- mization and economics are often interest in perform- ing post-optimal sensitivity analysis. This analysis consists in determining the sensitivity of the optimum to small perturbations in the parameter or constraint values. Additionally, the obtainment of an efficient method, with good accuracy, for calculating higher or- der derivatives of Erlang C function may be used for local approximations of the function and its deriva- tives by using a convenient Taylor or Hermite polyno- mial. Those approximations are specially important for iterative algorithms in small neighborhoods of the solution. A method for calculating the derivatives of order n of the Erlang B function in the number of servers was proposed by the author in [4]. In the sequel, a second paper [5] showed that for high values of the ar- guments it is possible to obtain a significant improve- ment in the method efficiency, without jeopardizing the required precision, by defining a reduced recur- sion starting from a point closer to the desired value of the number of servers(see [5]). Following the above mentioned works on the Er- lang B function derivatives, a method for calculating directly higher-order derivatives of Erlang C function, is developed in the present work. 2 The Erlang’s B and C Formulas The Erlang B and C formulas are true probability clas- sics. Indeed, much of the theory was developed by A. K. Erlang and his colleagues prior to 1925 [2]. The subject has been extensively studied and applied by telecommunications engineers and mathematicians ever since. A nice introductory account, including some of the telecommunications subtleties, is pro- vided by [3]. The Erlang B (or loss) formula gives the (steady-state) blocking probability in the Erlang loss model, i.e., in the M/M/x/0 model (see for ex- ample [3, pp. 5 and 79]): B(a, x) . = a x /x! ∑ x j =0 a j /j ! , x ∈ N 0 , a ∈ R + . (1) The numerical studies regarding this formula are usu- ally based on its analytical continuation, ascribed to Proceedings of the 13th WSEAS International Conference on COMMUNICATIONS ISSN: 1790-5117 72 ISBN: 978-960-474-098-7