Queueing Dynamics and State Space Collapse in Fragmented Limit Order Book Markets

In modern equity markets, participants have a choice of many exchanges at which to trade. Exchanges typically operate as electronic limit order books under a price-time priority rule and, in turn, can be modeled as multiclass first-in-first-out queueing systems. A market with multiple exchanges can be thought as a decentralized, parallel queueing system. Heterogeneous traders that submit limit orders select the exchange (i.e., the queue), in which to place their orders by trading off financial considerations against anticipated delays until their orders may fill. These limit orders can be thought of as jobs waiting for service. Simultaneously, traders that submit market orders select which exchange (i.e., queue) to direct their order. These market orders trigger instantaneous service completions of queued limit orders. In this way, the server is the aggregation of self-interested, atomistic traders submitting market orders. Taking into account the effect of investors’ order-routing decisions across exchanges, we find that the equilibrium of this decentralized market exhibits a state space collapse property whereby (a) the queue lengths at different exchanges are coupled in an intuitive manner; (b) the behavior of the market is captured through a one-dimensional process that can be viewed as a weighted aggregate queue length across all exchanges; and (c) the behavior at each exchange can be inferred via a mapping of the aggregated market depth process that takes into account the heterogeneous trader characteristics. The key driver of this coupling phenomenon is anticipated delay as opposed to the queue lengths themselves. Analyzing a trade and quote data set for a sample of stocks over a one-month period, we find empirical support for the predicted state space collapse. Separately, using the data before and after NASDAQ’s natural fee-change experiment from 2015, we again find agreement between the observed market behavior and the model’s predictions around the fee change.