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 operating under a “price-time” priority rule and, in turn, can be modeled as multi-class FIFO 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 delays until their order may fill against financial considerations. These limit orders can be thought as jobs waiting for service. Simultaneously, traders that submit market orders select the exchange, i.e., the 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. This predicted dimension reduction is the result of high-frequency order routing decisions that essentially couple the dynamics across exchanges. We derive a characterization of the market equilibrium and the associated aggregated depth process. Analyzing a TAQ dataset for a sample of stocks over a one month period, we find empirical support for the predicted state space collapse