Partial sharing allows providers to possibly pool a fraction of their resources when full pooling is not beneficial to them. Recent work in systems without sharing has shown that redundancy can improve performance considerably. In this paper, we combine partial sharing and redundancy by developing partial sharing models for providers operating multi-server systems with redundancy. Two M M N queues with redundant service models are considered. Copies of an arriving job are placed in the queues of servers that can serve the job. Partial sharing models for cancel-on-complete and cancel-on-start redundancy models are developed. For cancel-on-complete, it is shown that the Pareto efficient region is the full pooling configuration. For a cancel-on-start policy, we conjecture that the Pareto frontier is always non-empty and is such that at least one of the two providers is sharing all of its resources. For this system, using bargaining theory the sharing configuration that the providers may use is determined. Mean response time and probability of waiting are the performance metrics considered.
Complete resource pooling between independent service systems has been studied from a cooperative game theory standpoint; see, for example, @cite_8 @cite_13 @cite_1 . The goal of this literature is to analyze stable mechanisms for sharing the surplus (and costs) of the grand coalition among the various agents. Single server as well as multi-server settings have been considered, for queueing as well as loss systems; see @cite_1 for a comprehensive survey of this literature. A complementary view of resource sharing comes from an optimization standpoint. Here the organization is interested in optimally provisioning (potentially heterogenous) resources and or sharing its service resources between various activities; see, for example, @cite_14 @cite_2 @cite_15 .
@cite_14 @cite_8 @cite_1 @cite_2 @cite_15 @cite_13
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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 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. We consider a number of servers that may improve the efficiency of the system by pooling their service capacities to serve the union of the individual streams of customers. This economies-of-scope phenomenon is due to the reduction in the steady-state mean total number of customers in the system. The question we pose is how the servers should split among themselves the cost of the pooled system. When the individual incoming streams of customers form Poisson processes and individual service times are exponential, we define a transferable utility cooperative game in which the cost of a coalition is the mean number of customers (or jobs) in the pooled system. We show that, despite the characteristic function is neither monotone nor concave, the game and its subgames possess nonempty cores. In other words, for any subset of servers there exist cost-sharing allocations under which no partial subset can take advantage by breaking away and forming a separate coalition. We give an explicit expression for all (infinitely many) nonnegative core cost allocations of this game. Finally, we show that, except for the case where all individual servers have the same cost, there exist infinitely many core allocations with negative entries, and we show how to construct a convex subset of the core where at least one server is being paid to join the grand coalition. Fragmentation of expensive resources, e.g., spectrum for wireless services, between providers can introduce inefficiencies in resource utilisation and worsen overall system performance. In such cases, resource pooling between independent service providers can be used to improve performance. However, for providers to agree to pool their resources, the arrangement has to be mutually beneficial. The traditional notion of resource pooling, which implies complete sharing, need not have this property. For example, under full pooling, one of the providers may be worse off and hence have no incentive to participate. In this paper, we propose partial resource sharing models as a generalization of full pooling, which can be configured to be beneficial to all participants. We formally define and analyze two partial sharing models between two service providers, each of which is an Erlang-B loss system with the blocking probabilities as the performance measure. We show that there always exist partial sharing configurations that are beneficial to both providers, irrespective of the load and the number of circuits of each of the providers. A key result is that the Pareto frontier has at least one of the providers sharing all its resources with the other. Furthermore, full pooling may not lie inside this Pareto set. The choice of the sharing configurations within the Pareto set is formalized based on bargaining theory. Finally, large system approximations of the blocking probabilities in the quality-efficiency-driven regime are presented. Humanity faces numerous problems of common-pool resource appropriation. This class of multi-agent social dilemma includes the problems of ensuring sustainable use of fresh water, common fisheries, grazing pastures, and irrigation systems. Abstract models of common-pool resource appropriation based on non-cooperative game theory predict that self-interested agents will generally fail to find socially positive equilibria---a phenomenon called the tragedy of the commons. However, in reality, human societies are sometimes able to discover and implement stable cooperative solutions. Decades of behavioral game theory research have sought to uncover aspects of human behavior that make this possible. Most of that work was based on laboratory experiments where participants only make a single choice: how much to appropriate. Recognizing the importance of spatial and temporal resource dynamics, a recent trend has been toward experiments in more complex real-time video game-like environments. However, standard methods of non-cooperative game theory can no longer be used to generate predictions for this case. Here we show that deep reinforcement learning can be used instead. To that end, we study the emergent behavior of groups of independently learning agents in a partially observed Markov game modeling common-pool resource appropriation. Our experiments highlight the importance of trial-and-error learning in common-pool resource appropriation and shed light on the relationship between exclusion, sustainability, and inequality.