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

Abstract

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.

Bulk characterization of topological crystalline insulators: Stability under interactions and relations to symmetry enriched U (1) quantum spin liquids

“…As shown in Ref. 31 and 32 (see also Appendix B), indeed, M 2 = −1 on m 2 . This is consistent with the above result obtained by studying the symmetric gapless surface.…”

“…As a consistency check, we can also derive the quantum numbers of the monopole from a symmetric STO of this TCI. This TCI can have a symmetric Z 4 STO similar to the one in (9) [31,32]. In particular, in this case e carries charge-1/2 and m is neutral, too.…”

“…Very recently, the stability of various such TCIs under interactions has been considered and some of them are shown to be unstable, by studying the anomalous surface properties of the TCIs, studying certain dimensionally reduced versions of the TCIs, or studying the formal field theories of the TCIs. [21][22][23][24][25][26][27][28][29][30][31][32] However, an important and interesting question has remained unanswered: how to characterize nontrivial interacting TCIs directly by their bulk properties in a physical way?…”