Low-traffic limit and first-passage times for a simple model of the continuous double auction

We consider a simplified model of the continuous double auction where prices are integers varying from 1 to N with limit orders and market orders, but quantity per order limited to a single share. For this model, the order process is equivalent to two M/M/1 queues. We study the behavior of the auction in the low-traffic limit where limit orders are immediately matched by market orders. In this limit, the distribution of prices can be computed exactly and gives a reasonable approximation of the price distribution when the ratio between the rate of order arrivals and the rate of order executions is below 1/2. This is further confirmed by the analysis of the first-passage time in 1 or N.