Properties of a doubly dynamic simulation assignment model were investigated. The model involves specifying a day-to-day route choice model as a discrete time stochastic process, combining a between-day driver learning and adjusting model with a continuous-time, within-day dynamic network loading. Such a simulation model may be regarded as the realization of a stochastic process, which under certain mild conditions admits a unique stationary probability distribution (i.e., an invariant probability distribution over time of network flows and travel times). Such a stationary state of the stochastic process is of interest to transport modelers, as one can then describe the stochastic process by its moments such as the means, variances, and covariances of the flow and travel time profiles. The results of a simulation experiment are reported in which the process of individual drivers' day-to-day route choices are based on the aggregate learning of the experienced within-day route costs by all drivers departing in the same period. Experimental results of the stationarity of the stochastic process are discussed, along with an analysis of the sensitivity of autocorrelations of the route flows to the route choice model parameters. The results illustrate the consistency of the link flow model with properties such as first in-first out, and a simple network is used to illustrate the properties.