The data transmission in wireless networks is usually analyzed under the assumption of non-stationary rates. Nevertheless, they strictly depend on the time of day, that is, the intensity of arrival and daily workload profiles confirm this fact. In this article, we consider the process of downloading a file within a single network segment and unsteady speeds—for arrivals, file sizes, and losses due to impatience. To simulate the scenario, a queuing system with elastic traffic with non-stationary intensity is used. Formulas are given for the main characteristics of the model: the probability of blocking a new user, the average number of users in service, and the queue. A method for calculating the boundaries of convergence of the model is proposed, which is based on the logarithmic norm of linear operators. The boundaries of the rate of convergence of the main limiting characteristics of the queue length process were also established. For clarity of the influence of the parameters, a numerical analysis was carried out and presented.