state probability vector associated with the stochastic matrix G satisfying G=e(0-A+AG) (6.48) [KSC98] proposed a simple, empirical real-time approximation approach for calculating the complementary distribution of the virtual waiting time for the MMPP(2)/D/1 system. In this approach, W(x) can be asymptotically approximated as a two-term exponential function: (6.49) whose Laplace transform is given by (6.50) Note that S1 and S2 are defined as asymptotic decay rates, while a1 and a2 are the asymptotic constants. The estimation of the asymptotic decay rate S and asymptotic constant a can be obtained by matching the poles, the residue of the dominant pole, and the mean waiting time between the Laplace transform of the exact distribution and the approximation distribution. The detailed computation procedure for obtaining the asymptotic approximation of the complementary distribution of the virtual waiting time Wappr(x) in equation (6.49) is explained as follows. (1) Compute the asymptotic decay rate S1 and S2: S1 and S2 are chosen as the negative poles of W(s), i.e. the roots of the equation from the denominator of W(s): det[s!+Q-A+Ae"]=0 (6.51) 6 Dimensioning for Single ATM-Based lub Equivalently, (6.52) Among the negative roots of (6.52), the parameters SI and S2 are chosen from the first two close to 0 negative values (S2 (s,<0). (2) Compute the vector g: The vector g is given by equation (6.53), derived from the stochastic matrix G: (6.53) The calculation of g is introduced in [FMH92] and [Luc91]. The detailed computation steps for G are: (a) Start with G = 0, (b) Compute cyclically (6.54) until GI and G2 become stable, i.e. GI and G2 shall converge. Otherwise exchange indices and start the procedure again. (c) Then, g can be directly computed by equation (6.53). (3) Compute the asymptotic constants a1 and a2: It is proposed here to approximate the asymptotic constants a and a2 by matching the mean waiting time. So we need to calculate the mean waiting time, denoted as w , which is calculated from [Luc91, eq. (47