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Cooperation and sharing costs in a tandem queueing network

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  • Bendel, Dan
  • Haviv, Moshe

Abstract

We consider a tandem network of queues with a Poisson arrival process to the first queue. Service times are assumed to be exponential. In cases where they are not, we additionally assume a processor sharing service discipline in all servers. Consecutive servers may cooperate by pooling resources which leads to the formation of a single combined server that satisfies the aggregated service demands with a greater service rate. On this basis we define a cooperative game with transferable utility, where the cost of a coalition is the steady-state mean total number of customers in the system formed by its members. We show that the game is subadditive, leading to full cooperation being socially optimal. We then show the non-emptiness of the core, despite the characteristic function being neither monotone, nor concave. Finally, we derive several well-known solution concepts, including the Shapley value, the Banzhaf, value and the nucleolus, for the case where servers have equal mean service demands. In particular, we show that all three values coincide in this case.

Suggested Citation

  • Bendel, Dan & Haviv, Moshe, 2018. "Cooperation and sharing costs in a tandem queueing network," European Journal of Operational Research, Elsevier, vol. 271(3), pages 926-933.
    • Handle: RePEc:eee:ejores:v:271:y:2018:i:3:p:926-933
    DOI: 10.1016/j.ejor.2018.04.049
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    1. Kar, Anirban & Mitra, Manipushpak & Mutuswami, Suresh, 2009. "On the coincidence of the prenucleolus and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 16-25, January.
    2. Gerichhausen, Marloes & Hamers, Herbert, 2009. "Partitioning sequencing situations and games," European Journal of Operational Research, Elsevier, vol. 196(1), pages 207-216, July.
    3. Haviv, Moshe, 1995. "Consecutive amalgamations and an axiomatization of the Shapley value," Economics Letters, Elsevier, vol. 49(1), pages 7-11, July.
    4. Peter Borm & Herbert Hamers & Ruud Hendrickx, 2001. "Operations research games: A survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 139-199, December.
    5. Sigrún Andradóttir & Hayriye Ayhan, 2005. "Throughput Maximization for Tandem Lines with Two Stations and Flexible Servers," Operations Research, INFORMS, vol. 53(3), pages 516-531, June.
    6. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Frank Karsten & Marco Slikker & Geert-Jan van Houtum, 2015. "Resource Pooling and Cost Allocation Among Independent Service Providers," Operations Research, INFORMS, vol. 63(2), pages 476-488, April.
    8. Guajardo, Mario & Jörnsten, Kurt, 2015. "Common mistakes in computing the nucleolus," European Journal of Operational Research, Elsevier, vol. 241(3), pages 931-935.
    9. (*), Gerard van der Laan & RenÊ van den Brink, 1998. "Axiomatizations of the normalized Banzhaf value and the Shapley value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(4), pages 567-582.
    10. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    11. Roth,Alvin E. (ed.), 1989. "The Shapley Value," Cambridge Books, Cambridge University Press, number 9780521361774, November.
    12. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-281.
    13. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    14. Yimin Yu & Saif Benjaafar & Yigal Gerchak, 2015. "Capacity Sharing and Cost Allocation among Independent Firms with Congestion," Production and Operations Management, Production and Operations Management Society, vol. 24(8), pages 1285-1310, August.
    15. Chris Snijders, 1995. "Axiomatization of the Nucleolus," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 189-196, February.
    16. Shoshana Anily & Moshe Haviv, 2017. "Line Balancing in Parallel M/M/1 Lines and Loss Systems as Cooperative Games," Production and Operations Management, Production and Operations Management Society, vol. 26(8), pages 1568-1584, August.
    17. Ruiz, Luis M & Valenciano, Federico & Zarzuelo, Jose M, 1996. "The Least Square Prenucleolus and the Least Square Nucleolus. Two Values for TU Games Based on the Excess Vector," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 113-134.
    18. Winter, Eyal, 2002. "The shapley value," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 53, pages 2025-2054, Elsevier.
    19. Frank Karsten & Marco Slikker & Geert‐Jan van Houtum, 2012. "Inventory pooling games for expensive, low‐demand spare parts," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(5), pages 311-324, August.
    20. Shoshana Anily & Moshe Haviv, 2010. "Cooperation in Service Systems," Operations Research, INFORMS, vol. 58(3), pages 660-673, June.
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    1. Luo, Chunlin & Zhou, Xiaoyang & Lev, Benjamin, 2022. "Core, shapley value, nucleolus and nash bargaining solution: A Survey of recent developments and applications in operations management," Omega, Elsevier, vol. 110(C).

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