Analysis of resource pooling games via a new extension of the Erlang loss function

We study a situation where several independent service providers collaborate by pooling their resources into a joint service system. These service providers may represent such diverse organizations as hospitals that pool intensive care beds and ambulances, airline companies that share spare parts, or car rental agencies that pool rental cars. We model the service systems as Erlang loss systems that face a fixed cost rate per server and penalty costs for lost customers. We examine the allocation of costs of the pooled system amongst the participants by formulating a cooperative cost game in which each coalition optimizes the number of servers. We identify a cost allocation that is in the core of this game, giving no subset of players an incentive to split off and form a separate pooling group. Moreover, we axiomatically characterize this allocation rule and show that it can be reached through a population monotonic allocation scheme. To obtain these results, we introduce a new extension of the classic Erlang loss function to non-integral numbers of servers and establish several of its structural properties.