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where b (s) is the Laplace transform of the interoccurrence-time density b(t) Suppose now that this density is given by a lognormal density In this particular case it is not possible to give an explicit expression for b (s) and one has to handle an analytically intractable integral How do we handle this Suppose we wish to compute M(t) for a number of t-values in the interval [0, t0 ] The key observation is that, by the representation (E11), the renewal function M(t) for 0 t t0 uses the interoccurrence-time density b(t) only for 0 t t0 The same is true
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APPENDICES
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for the waiting-time distribution function Wq (t) in the M/G/1 queue with service in order of arrival Then it follows from the representation (8210) that Wq (t) for 0 t t0 requires the service-time density b(t) only for 0 t t0 If the Laplace transform b (s) of the density b(t) is analytically intractable, the idea is to approximate the density b(t) by a polynomial P (t) on the interval [0, t0 ] and by zero outside this interval Consequently, the intractable Laplace transform b (s) is approximated by a tractable expression
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bapp (s) = t0 0
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e st P (t) dt
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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
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APPENDIX G THE ROOT-FINDING PROBLEM
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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
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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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