Cooperation in Service Systems

TLDR

Pooling servers reduces the steady‑state mean number of customers, improving system efficiency by serving the union of individual customer streams. The study seeks to determine how servers should share the cost of the pooled system. Assuming Poisson arrivals and exponential service times, the authors formulate a transferable‑utility cooperative game where a coalition’s cost equals the mean number of customers in the pooled system. They show the game has a nonempty core, give explicit core allocations, and demonstrate that, except when all servers have equal cost, there are infinitely many core allocations with negative entries, including a convex subset where at least one server is paid to join the grand coalition.

Abstract

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.