In this paper, we employ a cooperative game to study coalition performance of a parallel service system with nonexponential service times, which is comprised of n independent parallel M/Ek/1 queues, where the service times of the coalitions are designed as a closed-form framework. For such a queueing cooperative game, we use the Erlang service time distributions to make analysis of various coalitions more feasible, and specifically, it is easy to set up a characteristic function for the coalitions under an Erlang closed-form framework. Once the characteristic function is given in the queueing cooperative game, we compute the Shapley value, the Owen value and the core whose aim is to suitably provide the cost allocation for the coalitions. Furthermore, we use numerical examples to give useful observation on the cost allocations for the coalitions through the Shapley value, the Owen value and the core.
AksinO.Z., De VericourtF. and KaraesmenF., Call center outsourcing contract design and choice, Management Science54 (2008), 354–368.
2.
AlbizuriM.J., Axiomatizations of the Owen value without efficiency, Mathematical Social Sciences55 (2008), 78–79.
3.
AnilyS. and HavivM., Cost-allocation for the first order interaction joint replenishment model, Operations Research55 (2007), 292–302.
4.
AnilyS. and HavivM., Cooperation in service systems, Operations Research58 (2010), 660–673.
5.
AnilyS. and HavivM., Subadditive and homogeneous of degree one games are totally balanced, Operations Research62 (2014), 788–793.
6.
BenjaafarS., CooperW.L. and KimJ.S., On the benefits of pooling in production-inventory systems, Management Science51 (2005), 548–565.
7.
BranzeiR., DimitrovD., TijsS., Models in Cooperative Game Theory, Springer-Verlag, New York, 2008.
8.
Cano-BerlangaS., Giménez-GómezJ. and VilellaC., Enjoying cooperative games: The R package Game Theory, Applied Mathematics and Computation305 (2017), 381–393.
9.
ChunY., A new axiomatization of the Shapley value, Games and Economic Behavior1 (1989), 119–130.
10.
Garcia-SanzM.D., FernandezF.R., Fiestras-JaneiroM.G., Garcia-JuradoI. and PuertoJ., Cooperation in markovian queueing models, European Journal of Operational Research188 (2008), 485–495.
11.
GonzalezP. and HerreroC., Optimal sharing of surgical costs in the presence of queues, Mathematical Methods of Operations Research59 (2004), 435–446.
12.
HamiacheG., A new axiomatization of the Owen value for games with coalition structures, Mathematical Social Sciences37 (1999), 281–305.
13.
HoppW.J., TekinE. and VanM.P., Oyen, Benefits of skill chaining in production lines with cross-trained workers, Management Science50 (2004), 83–98.
14.
KarstenF., SlikkerM. and HoutumG., Domain extensions of the Erlang loss function: Their scalability and its applications to cooperative games, Probability in the Engineering and Informational Sciences58 (2014), 473–488.
15.
LiQ.L., Constructive Computation in Stochastic Models with Applications: The RG-Factorizations, Springer and Tsinghua Press, New York, 2010.
16.
MandelbaumA. and ReimanM.I., On pooling in queueing networks, Management Science44 (1998), 971–981.
17.
ManiqueF., A characterization of the Shapley value in queueing problems, Journal of Economic Theory109 (2003), 90–103.
18.
NeutsM.F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Maryland, 1981.
19.
OwenG., Multilinear extensions of games, Management Science18 (1972), 64–79.
20.
OwenG., Values of games with a priori unions. In, Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar MorgensternR.Henn and O.Moeschlin eds., Springer-Verlag, New York, 28, 1977, pp. 76–88.
21.
PelegB., SudhölterP., Introduction to the Theory of Cooperative Games, Springer-Verlag, New York, 2007.
22.
PrabhuN.U., Foundations of Queueing Theory, Kluwer Academic Publishers, New York, 1997.
23.
RasmusenE., An Introduction to Game Theory, Blackwell Publishing, New York, 2007.
24.
RothA.E., The Shapley value as a von Neumann-Morgenstern utility, Econometrica45 (1977), 657–664.
25.
ShapleyL.S., A value for n-person games, in contributions to the theory of games II, Annals of Mathematics Studies, H.W.Kuhn and A.W.Tucker eds., Princeton University Press, Princeton, 28, 1953, 307–317.
26.
TimmerJ., ScheinhardtW., How to share the cost of cooperating queues in a tandem network?IEEE Conference proceedings of the 22nd International Teletraffic Congress, New York, 2010, 1–7.
27.
Vidal-PugaJ. and BergantinosG., An implementation of the Owen value, Games and Economic Behavior44 (2003), 412–427.
28.
YangZ. and ZhangQ., Resource allocation based on DEA and modified Shapley value, Applied Mathematics and Computation263 (2015), 280–286.
