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A naive approach uses a single polynomial approximation P (t) for the whole interval [0, t0 ] A polynomial approximation that is easy to handle is the Chebyshev approximating polynomial Gauss Legendre integration is then recommended to evaluate the required function values of bapp (s) A code to compute the function values of the Chebyshev approximating polynomial at the points used in the numerical integration procedure can be found in the sourcebook by Press et al (1992) One has a smooth function P (t) when using a single Chebyshev polynomial approximation P (t) for the whole interval [0, t0 ] However, a better accuracy is obtained by a more re ned approach in which the function b(t) on the interval [0, t0 ] is replaced by a piecewise polynomial approximation on each of the subintervals of length with as in (F2) Den Iseger (2002) suggests approximating b(t) on each of the subintervals [k , (k + 1) ) by a linear combination of Legendre polynomials of degrees 0, 1, , 2n 1 with n as in (F2) This leads to an approximating function with discontinuities at the points k However, this dif culty can be resolved by the modi cation (F7) for non-smooth functions Details can be found in Den Iseger (2002) A simpler approach seems possible when the analytically intractable Laplace transform b (s) is given by b (s) = E(e sX ) for a continuous random variable X with a strictly increasing probability distribution function F (x) Then b (s) = E[g(U, s)] for a uniform (0, 1) random variable 1 U , where g(u, s) = exp( sF 1 (u)) The (complex) integral 0 g(u, s) du can be evaluated by Gauss Legendre integration The required numerical values of the inverse function F 1 (u) may be obtained by using bisection.

The analysis of many queueing problems can be simpli ed by computing rst the roots of a certain function inside or on the unit circle in the complex plane It is a myth that the method of nding roots in the complex plane is dif cult to use for practical purposes In this appendix we address the problem of nding the roots of the equation (G1) 1 zc e D{1 (z)} = 0

inside or on the unit circle Here c is a positive integer, (z) = j =1 j zj is the generating function of a discrete probability distribution { j , j 1} and the real

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numbers and D are positive constants such that D /c < 1 with = j =1 j j X /D/c queue This root- nding problem arises in the analysis of the multi-server M with batch arrivals The equation (G1) has c roots inside or on the unit circle The proof is not given here, but is standard in complex analysis and uses the so-called Rouch theorem; see for example Chaudry and Templeton (1983) Moreover, all e the c roots of (G1) are distinct This follows from the following general result in Dukhovny (1994): if K(z) is the generating function of a non-negative, integervalued random variable such that K (1) < c and zK (z) K (1) |K(z)| for |z| 1, then all the roots of the equation zc = K(z) in the region |z| 1 are distinct Apply this result with K(z) = e D{1 (z)} and note that K(z) is the generating function of the total number of arrivals in a compound Poisson arrival process; see Section 12 To obtain the roots of (G1) it is not recommended to directly apply Newton Raphson iteration to (G1) In this procedure numerical dif culties arise when roots are close together This dif culty can be circumvented by a simple idea The key to the numerical solution of equation (G1) is the observation that it can be written as (G2) zc e D{1 (z)} = e 2 ik

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where k is any integer The next step is to use logarithms The general logarithmic function of a complex variable is de ned as the inverse of the exponential function and is therefore a many-valued function (as a consequence of e z+2 i = e z ) It suf ces to consider the principal branch of the logarithmic function This principal branch is denoted by ln(z) and adds to each complex number z = 0 the unique complex number w in the in nite strip < Im(w) such that e w = z The principal branch of the logarithmic function of a complex variable can be expressed in terms of elementary functions by ln(z) = ln(r) + i using the representation z = re i with r = |z| and < Since e ln(z) = z for any z = 0, we can write (G2) as e c ln(z)+ D{1 (z)} = e 2 ik with k is any integer This suggests we should consider the equation c ln(z) + D{1 (z)} = 2 ik (G3)

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