<p>This paper analyzes a basic queueing model of service platforms in digital sharing economy, in which seeker arrivals follow a Markovian arrival process and are therefore non-Poisson, and the matching rate between a seeker and an owner depends on the number of the idle owners. The queueing model of service platforms can be expressed as a level-independent quasi-birth-and-death process. We apply the matrix-geometric solution to provide a detailed analysis, including the system stability, the average stationary numbers of seekers and of idle owners, the average sojourn time of an arriving seeker, and the expected revenues for both the service platform and each owner. Specifically, we propose a new effective method for computing the average sojourn time of any seeker by means of the first passage times and the phase-type distributions. Finally, numerical examples verify the theoretical results and illustrate the impact of key parameters, arrival correlation, and full distributional information. We believe that the methodology and results developed in this paper not only can be applied to study more queueing models of service platforms, but also will open a series of promising innovative research topics, such as performance evaluation and queueing-game for digital sharing economy.</p>