Mean time for the development of large workloads and large queue lengths in the GI/G/1 queue. (1996)

• Record Type: Journal Article
• Authors: Knessl, Charles; Tier, Charles
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 9:Number 2(1996)
• Page Start: 107
• Page End: 142
• Publication Date: 1996
• Subjects: queueing systems -- asymptotics -- singular perturbations; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953396000147
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : 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 .