Resource pooling and cost allocation among independent service providers

We study a situation where several independent service providers collaborate by complete pooling of their resources and customer streams into a joint service system. These service providers may represent such diverse organizations as hospitals that pool beds, call centers that share telephone operators, or maintenance firms that pool repairmen. We model the service systems as Erlang delay systems (M/M/s queues) that face a fixed cost rate per server and homogeneous delay costs for waiting customers. We examine rules to fairly allocate the collective costs of the pooled system amongst the participants by applying concepts from cooperative game theory. We consider both the case where players' numbers of servers are exogenously given and the scenario where any coalition picks an optimal number of servers. By exploiting new analytical properties of the continuous extension of the classic Erlang delay function, we provide suffcient conditions for the games under consideration to possess a core allocation (i.e., an allocation that gives no group of players an incentive to split of and form a separate pool) and to admit a population monotonic allocation scheme (whereby adding extra players does not make anyone worse off). This is not guaranteed in general, as illustrated via examples.