The authors consider the GI/G/1 queue described by either the workload U(t) (unfinished work) or the number of customers N(t) in the system. They compute the mean time until U(t) reaches excess of the level K, and also the mean time until N(t) reaches N0. For the M/G/1 and GI/M/1 models, the authors obtain exact contour integral representations for these mean first passage times. They then compute the mean times asymptotically, as K and N0⇒•, by evaluating these contour integrals. For the general GI/G/1 model, the authors obtain asymptotic results by a singular perturbation analysis of the appropriate backward Kolmogorov equation(s). Numerical comparisons show that the asymptotic formulas are very accurate even for moderate values of K and N0.
