Functional Limit Theorems for a Simple Auction

We consider a continuous transparent auction with one asset. Buyers and sellers arrive at the auction according to independent renewal processes, willing to transact a random number of units (shares) of the asset for one of N possible prices. Upon arrival, they buy/sell as many units of the asset as they can for the best price available at that time. However, if the prices offered for sale/purchase are worse than their initial preferences, they do not transact, but instead they stay at the auction, posting buying/selling orders and waiting for a suitable counterparty. We model the numbers of outstanding orders of each type, together with the numbers of assets exchanged up to date by customers of different types, by stochastic processes and find the limiting distributions of these (suitably scaled) processes.