Performance paradox of dynamic matching models under greedy policies

Abstract

AbstractWe consider the stochastic matching model on a non-bipartite compatibility graph and analyze the impact of adding an edge to the expected number of items in the system. One may see adding an edge as increasing the flexibility of the system, for example, asking a family registering for social housing to list fewer requirements in order to be compatible with more housing units. Therefore, it may be natural to think that adding edges to the compatibility graph will lead to a decrease in the expected number of items in the system and the waiting time to be assigned. In our previous work, we proved this is not always true for the First Come First Matched discipline and provided sufficient conditions for the existence of the performance paradox: despite a new edge in the compatibility graph, the expected total number of items can increase. These sufficient conditions are related to the heavy-traffic assumptions in queueing systems. The intuition behind this is that the performance paradox occurs when the added edge in the compatibility graph disrupts the draining of a bottleneck. In this paper, we generalize this performance paradox result to a family of so-called greedy matching policies and explore the type of compatibility graphs where such a paradox occurs. Intuitively, a greedy matching policy never leaves compatible items unassigned, so the state space of the system consists of finite words of item classes that belong to an independent set of the compatibility graph. Some examples of greedy matching policies are First Come First Match, Match the Longest, Match the Shortest, Random, Priority. We prove several results about the existence of performance paradoxes for greedy disciplines for some family of graphs. More precisely, we prove several results about the lifting of the paradox from one graph to another one. For a certain family of graphs, we prove that there exists a paradox for the whole family of greedy policies. Most of these results are based on strong aggregation of Markov chains and graph theoretical properties.