# Collaborating queues: large service network and a limit order book

Thesis

We analyse the steady-state behaviour of two different models with collaborating queues: that is, models in which "customers" can be served by many types of "servers", and "servers" can process many types of "customers". The first example is a large-scale service system, such as a call centre. Collaboration is the result of cross-trained staff attending to several different types of incoming calls. We first examine a load-balancing policy, which aims to keep servers in different pools equally busy. Although the policy behaves order-optimally over fixed time horizons, we show that the steady-state distribution may fail to be tight on the diffusion scale. That is, in a family of ever-larger networks whose arrival rates grow as O(r) (where r is a scaling parameter growing to infinity), the sequence of steady-state deviations from equilibrium scaled down by sqrt(r) is not tight. We then propose a different policy, for which we show that the sequence of invariant distributions is tight on the r^(1/2+epsilon) scale, for any epsilon > 0. For this policy we conjecture that tightness holds on the diffusion scale as well. The second example models a limit order book, a pricing mechanism for a single-commodity market in which buyers (respectively sellers) are prepared to wait for the price to drop (respectively rise). We analyse the behaviour of a simplified model, in which the arrival events are independent of each other and the state of the limit order book. The system can be represented by a queueing model, with "customers" and "servers" corresponding to bids and asks; the roles of customers and servers are symmetric. We show that, with probability 1, the price interval breaks up into three regions. At small (respectively large) prices, only finitely many bid (respectively ask) orders ever get fulfilled, while in the middle region all orders eventually clear. We derive equations which define the boundaries between these regions, and solve them explicitly in the case of iid uniform arrivals to obtain numeric values of the thresholds. We derive a heuristic for the distribution of the highest bid (respectively lowest ask), and present simulation data confirming it.