Mean time for the development of large workloads and large queue lengths in the G I / G / 1 queue

We consider the G I / G / 1 queue described by either the workload U ( t ) (unfinished work) or the number of customers N ( t ) in the system. We compute the mean time until U ( t ) reaches excess of the level K , and also the mean time until N ( t ) reaches N 0 . For the M / G / 1 and G I / M / 1 models, we obtain exact contour integral representations for these mean first passage times. We then compute the mean times asymptotically, as K and N 0 → ∞ , by evaluating these contour integrals. For the general G I / G / 1 model, we 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 N 0 .