Asymptotic analysis of the M/G/1 queue with a time-dependent arrival rate

Authors:
Yongzhi (Peter) Yang;Charles Knessl
Affiliations:
Department of Mathematics, University of St. Thomas, St. Paul, MN;Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA
Venue:
Queueing Systems: Theory and Applications
Year:
1997

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Abstract

We consider the M/G/1 queue with an arrival rate \lambda that depends weakly upon time, as \lambda=\lambda (\varepsilon t) where \varepsilon is a small parameter. In the asymptotic limit \varepsilon \rightarrow 0, we construct approximations to the probability p_n (t) that n customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale \tau=\varepsilon t. We employ singular perturbation techniques and relate the various time scales by asymptotic matching.