The interval Shapley value of an M/M/1 service system

Open access


Service systems and their cooperation are one of the most important and hot topics in management and information sciences. To design a reasonable allocation mechanism of service systems is the key issue in the cooperation of service systems. In this paper, we systematically introduce the interval Shapley value as cost allocation of cooperative interval games 〈N, V〉 arising from cooperation in a multi-server service system, and provide an explicit expression for the interval Shapley value of cooperative interval games 〈N, V〉. We construct an interval game 〈N, W〉 of a service system which shares the same value for the grand coalition with the original interval game, by using the characteristic function which is dominated by the function of the original interval game. Finally, we prove that the interval game 〈N, W〉 is concave, which means that the interval Shapley value of the interval game 〈N, W〉 is in the interval core of this interval game, and illustrate this conclusion by using numerical examples.

Aksin, O., De Vericourt, F. and Karaesmen, F. (2008). Call center outsourcing contract design and choice, Management Science 54(2): 354–368.

Alparslan-Gok, S., Branzei, O., Branzei, R. and Tijs, S. (2011). Set-valued solution concepts using interval-type payoffs for interval games, Journal of Mathematical Economics 47(4): 621–626.

Alparslan-Gok, S., Miquel, S. and Tijs, S. (2009). Cooperation under interval uncertainty, Mathematical Methods of Operations Research 69(1): 99–109.

Alparslan-Gok, S., Palancı, O. and Olgun, M. (2014). Cooperative interval games: Mountain situations with interval data, Journal of Computational and Applied Mathematics 259(6): 622–632.

Anily, S. and Haviv, M. (2007). Cost-allocation problem for the first order interaction joint replenishment model, Operations Research 55(2): 292–302.

Anily, S. and Haviv, M. (2010). Cooperation in service systems, Operations Research 58(3): 660–673.

Benjaafar, S. (1995). Performance bounds for the effectiveness of pooling in multi-processing systems, European Journal of Operational Research 87(2): 375–388.

Branzei, R., Branzei, O., Alparslan Gok, S. and Tijs, S. (2010). Cooperative interval games: A survey, Central European Journal of Operations Research 18(3): 397–411.

Buzacott, J. (1996). Commonalities in reengineered business processes: Models and issues, Management Science 42(5): 768–782.

Chun, Y. (1989). A new axiomatization of the Shapley value, Games and Economic Behavior 1(2): 119–130.

Garcia-Sanz, M., Fernandez, F., Fiestras-Janeiro, M., Garcia-Jurado, I. and Puerto, J. (2008). Cooperation in Markovian queueing models, European Journal of Operational Research 188(2): 485–495.

Gonzalez, P. and Herrero, C. (2004). Optimal sharing of surgical costs in the presence of queues, Mathematical Methods of Operations Research 59(3): 435–446.

Han, W., Sun, H. and Xu, G. (2012). A new approach of cooperative interval games: The interval core and Shapley value revisited, Operations Research Letters 40(6): 462–468.

Hart, S. and Mas-Colell, A. (1989). Potential, value and consistency, Econometrica 57(3): 589–614.

Hopp, W., Tekin, E. and Van Oyen, M. (2004). Benefits of skill chaining in serial production lines with cross-trained workers, Management Science 50(1): 83–98.

Hwang, Y. and Yang, W. (2014). A note on potential approach under interval games, Top 22(2): 571–577.

Karsten, F., Slikker, M. and 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, Eindhoven.

Li, S., Sun, W., E, C.-G. and Shi, L. (2016). A scheme of resource allocation and stability for peer-to-peer file-sharing networks, International Journal of Applied Mathematics and Computer Science 26(3): 707–719, DOI: 10.1515/amcs-2016-0049.

Mallozzi, L., Scalzo, V. and Tijs, S. (2011). Fuzzy interval cooperative games, Fuzzy Sets and Systems 165(1): 98–105.

Mandelbaum, A. and Reiman, M. (1998). On pooling in queueing networks, Management Science 44(7): 971–981.

Maniquet, F. (2003). A characterization of the Shapley value in queueing problems, Journal of Economic Theory 109(1): 90–103.

Mariano, P. and Correia, L. (2015). The Give and Take game: Analysis of a resource sharing game, International Journal of Applied Mathematics and Computer Science 25(4): 753–767, DOI: 10.1515/amcs-2015-0054.

Moulin, H. and Strong, R. (2002). Fair queuing and other probabilistic allocation methods, Mathematics of Operations Research 27(1): 1–30.

Nagarajan, M. and Sosic, G. (2008). Game-theoretic analysis of cooperation among supply chain agents: Review and extensions, European Journal of Operational Research 187(3): 719–745.

Roth, A. (1977). The Shapley value as a von Neumann–Morgenstern utility, Econometrica 45(3): 657–664.

Shapley, L. (1953). A value for n-person games, Annals ofMathematics Studies 28: 307–317.

Stidham, S. (1970). On the optimality of single-server queuing systems, Operations Research 18(4): 708–732.

Young, H. (1985). Monotonic solutions of cooperative games, International Journal of Game Theory 14(2): 65–72.

International Journal of Applied Mathematics and Computer Science

Journal of the University of Zielona Góra

Journal Information

IMPACT FACTOR 2017: 1.694
5-year IMPACT FACTOR: 1.712

CiteScore 2017: 2.20

SCImago Journal Rank (SJR) 2017: 0.729
Source Normalized Impact per Paper (SNIP) 2017: 1.604

Mathematical Citation Quotient (MCQ) 2017: 0.13


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 202 149 16
PDF Downloads 123 107 6