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DOMAIN EXTENSIONS OF THE ERLANG LOSS FUNCTION: THEIR SCALABILITY AND ITS APPLICATIONS TO COOPERATIVE GAMES

Published online by Cambridge University Press:  05 September 2014

Frank Karsten ,
Marco Slikker  and
Geert-Jan van Houtum
Frank Karsten
Affiliation:
School of Industrial Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, EindhovenThe Netherlands E-mails: f.j.p.karsten@tue.nl; m.slikker@tue.nl; g.j.v.houtum@tue.nl
Marco Slikker
Affiliation:
School of Industrial Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, EindhovenThe Netherlands E-mails: f.j.p.karsten@tue.nl; m.slikker@tue.nl; g.j.v.houtum@tue.nl
Geert-Jan van Houtum
Affiliation:
School of Industrial Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, EindhovenThe Netherlands E-mails: f.j.p.karsten@tue.nl; m.slikker@tue.nl; g.j.v.houtum@tue.nl
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Abstract

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We prove that several extensions of the classic Erlang loss function to non-integral numbers of servers are scalable: the blocking probability as described by the extension decreases when the offered load and the number of servers s are increased with the same relative amount, even when scaling up from integral s to non-integral s. We use this to prove that when several Erlang loss systems pool their resources for efficiency, various corresponding cooperative games have a non-empty core.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 4 , October 2014 , pp. 473 - 488
Copyright
Copyright © Cambridge University Press 2014 

References

1.Anily, S. & Haviv, M. (2010). Cooperation in service systems. Operations Research 58(3): 660–673.CrossRefGoogle Scholar
2.Calabrese, J. (1992). Optimal workload allocation in open networks of multi-server queues. Management Science 38(12): 1792–1802.Google Scholar
3.Cooper, R. (1981). Introduction to queueing theory. New York: North-Holland.Google Scholar
4.Erlang, A. (1917). Løsning af nogle problemer fra sandsynlighedsregningen af betydning for de automatiske telefoncentraler. Electroteknikeren 13: 5–13. Translation: Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. In Brockmeyer, E., Halstrøm, H.L., and Jensen, A., editors, The Life and Works of A.K. Erlang, pages 138–155. Transactions of the Danish Academy of Technical Sciences, 1948.Google Scholar
5.Fredericks, A. (1980). Congestion in blocking systems — a simple approximation technique. Bell System Technical Journal 59(6): 805–827.Google Scholar
6.Harel, A. (1990). Convexity properties of the Erlang loss formula. Operations Research 38(3): 499–505.Google Scholar
7.Jagerman, D. (1974). Some properties of the Erlang loss function. Bell System Technical Journal 53(3): 525–551.Google Scholar
8.Jagers, A. & Van Doorn, E. (1986). On the continued Erlang loss function. Operations Research Letters 5(1): 43–46.Google Scholar
9.Karsten, F., Slikker, M. & Van Houtum, G. (2011). Analysis of resource pooling games via a new extension of the Erlang loss function. BETA Working Paper 344, Eindhoven University of Technology.Google Scholar
10.Karsten, F., Slikker, M. & Van Houtum, G. (2011). Resource pooling and cost allocation among independent service providers. BETA Working Paper 352, Eindhoven University of Technology.Google Scholar
11.Karsten, F., Slikker, M. & Van Houtum, G. (2012). Inventory pooling games for expensive, low-demand spare parts. Naval Research Logistics 59(5): 311–395.Google Scholar
12.Kortanek, K., Lee, D. & Polak, G. (1981). A linear programming model for design of communications networks with time varying probabilistic demands. Naval Research Logistics Quarterly 28(1): 1–32.Google Scholar
13.Özen, U., Reiman, M. & Wang, Q. (2011). On the core of cooperative queueing games. Operations Research Letters 39(5): 385–389.Google Scholar
14.Smith, D. & Whitt, W. (1981). Resource sharing for efficiency in traffic systems. Bell System Technical Journal 60(1): 39–55.CrossRefGoogle Scholar
15.Sprumont, Y. (1990). Population monotonic allocation schemes for cooperative games with transferable utility. Games and Economic Behavior 2(4): 378–394.Google Scholar
16.Van Houtum, G. & Zijm, W. (2000). On the relationship between cost and service models for general inventory systems. Statistica Neerlandica 54(2): 127–147.Google Scholar