Resource Pooling and Cost Allocation Among Independent Service Providers

Abstract

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 sufficient 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 off 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.

Cost allocation rules for elastic single-attribute situations

“…So, by Theorem 3, the proportional allocation is coalitionally rational. The corollary that the associated single‐attribute game has a non‐empty core was also obtained by Guo et al , and the result is in line with the finding of Karsten et al that when queues join forces under optimized real‐valued number of servers, the corresponding game in their model admits a proportional core allocation. ◊…”

Cost Allocation Rules for Elastic Single-Attribute Situations

“…So, by Theorem 2.3, the proportional allocation P(ϕ) is coalitionally rational. The corollary that the associated single-attribute game (N, c ϕ ) has a non-empty core was also obtained by Guo et al (2013), and the result is in line with the finding of Karsten et al (2015) that when M/M/s queues join forces under optimized real-valued number of servers, the corresponding game in their model admits a proportional core allocation. ♦…”

Pooling of critical, low-utilization resources with unavailability

“…Availability games are mostly overlapping with the last category. Recent publications in this category focus on economic order quantity situations (Meca et al 2004), economic lot sizing situations (Van Heuvel et al 2007;Drechsel and Kimms 2011), newsvendor situations (Özen et al 2008), truckload delivery situations (Hezarkhani et al 2016;Li et al 2016) and spare parts situations (Karsten et al 2012;Karsten and Basten 2014;Karsten et al 2015). Recently, Bachrach et al (2011Bachrach et al ( , 2012Bachrach et al ( , 2013 introduced and investigated a new class of operations research games, called cooperative reliability games, which comes closer to our work.…”