Random-access algorithms such as the Carrier-Sense Multiple-Access (CSMA) protocol provide a popular mechanism for distributed medium access control in large-scale wireless networks. In recent years, fairly tractable models have been shown to yield remarkably accurate throughput estimates for CSMA networks. These models typically assume that both the transmission durations and the back-off periods are exponentially distributed. We show that the stationary distribution of the system is in fact insensitive with respect to the transmission durations and the back-off times. These models primarily pertain to a saturated scenario where nodes always have packets to transmit. In reality however, the buffers may occasionally be empty as packets are randomly generated and transmitted over time. The resulting interplay between the activity states and the buffer contents gives rise to quite complicated queueing dynamics, and even establishing the stability criteria is usually a serious challenge. We explicitly identify the stability conditions in a few relevant scenarios, and illustrate the difficulties arising in other cases. Keywords: CSMA protocol; Insensitivity properties; Interference graphs; Random-access algorithms; Stability conditions; Wireless networks.