Some Asymptotic Results for the M/M/∞ Queue with Ranked Servers

Authors:
Charles Knessl
Affiliations:
Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA knessl@uic.edu
Venue:
Queueing Systems: Theory and Applications
Year:
2004

Citing 1
Cited 2

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Storage allocation under processor sharing I: exact solutions and asymptotics

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On some simple single server models of dynamic storage

Proceedings of the 6th International Conference on Queueing Theory and Network Applications

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Abstract

We consider an M/M/∞ model with m primary servers and infinitely many secondary ones. An arriving customer takes a primary server, if one is available. We derive integral representations for the joint steady state distribution of the number of occupied primary and secondary servers. Letting ρ=λ/μ be the ratio of arrival and service rates (all servers work at rate μ), we study the joint distribution asymptotically for ρ→∞. We consider both m=O(1) and m scaled to be of the same order as ρ. We also give results for the marginal distribution of the number of secondary servers that are occupied.