This work studies the average Age of Information (AoI) of a remote monitoring setup in which a multi-sensor system observes independent sources and updates the status to a common monitor using orthogonal channels. Considering the limited buffer size at the sensors, we first model each sensor as a first-come-first-served M/M/1/1 queue. Leveraging tools from stochastic hybrid systems, we derive the average AoI of a homogeneous single-source multi-sensor system in which all sensors' arrival and service rates are the same. We then extend the results to the multi-source, multi-sensor system. For a multi-source dual-sensor system, we present an approximate optimal arrival rate for a given sum arrival rate at a light load. For heterogeneous cases with different arrival and service rates at sensors, the average AoI is derived for the single-source dual-sensor and more general multi-source systems. Our analysis shows that the average AoI decreases by 16.44% and 21.44% for the dual-sensor and three-sensor systems, respectively, compared to the single-sensor system when the service rate and the total arrival rate of the sensors are normalized. Numerical results confirm that the average AoI performance of the single-source dual-sensor system outperforms the M/M/2 system at high system load.