Cooperation in Service Systems

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 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 albeit the fact the characteristic function is neither monotone nor concave, the game (and its subgames) possess non-empty 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) non-negative 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 in order to join the grand coalition.

Resource Pooling and Cost Allocation Among Independent Service Providers

“…Conversely, Anily and Haviv (2010) study a model in which each player has its own capacity endowment, modeled as a potential service rate. To reduce congestion costs, each coalition may cooperate by pooling these endowments and their individual customer streams into a single M/M/1 system whose service rate is the sum of the potential service rates of its members.…”

“…Costs consist of linear resource costs for servers and linear delay costs for customers that have to wait before being served. Our modeling approach differs from the approach taken in most of the previous work on cooperative queueing games (e.g., González and Herrero, 2004;Anily and Haviv, 2010). These authors consider cooperative games where each coalition operates an M/M/1 queue.…”

“…Although such a model is applicable if service capacity can be easily consolidated into a single server (e.g., by choice of material or technology), the M/M/1 model is not appropriate when service facilities consist of multiple servers whose service speeds are given, as in all our real-life examples mentioned above. Despite that, some of these examples have been previously used to motivate the study of M/M/1 games: for instance, González and Herrero (2004) are motivated by shared medical services, and Anily and Haviv (2010) mention pooled service technicians in their introduction. By more accurately modeling these settings as M/M/s queues, we obtain more precise results and insights for these settings.…”

“…It is also applicable for existing situations if the number of servers can be easily adjusted against negligible costs. In both cases, we assume that customers are homogenous in waiting time costs and in service requirements across players (in line with Anily and Haviv, 2010). Altogether, our analysis of cooperative games for both cases provides insights for a variety of practically relevant collaborative arrangements.…”

Domain Extensions of the Erlang Loss Function: Their Scalability and Its Applications to Cooperative Games

“…So, these functions are described on their domain by Figure 1 on page 476 provides an illustration: there, the graph of X (1,1) corresponds to the dashed curve on [1,2) and the graph of L (1,1) corresponds to the middle curve on [1,2). The following lemma states two useful properties of X.…”