Using stochastic geometry, downlink (DL) coverage probability in slot-based two-tier heterogeneous cellular networks (HCNs) is studied. Two traffic patterns namely slotted-asynchronous and exponential-interarrival are introduced to capture user activity, and the DL coverage probability based on max-SINR association is derived. In contrast to many works, the transmission slots of base stations (BS) are not synchronous. This causes a packet transmitted in one slot to experience varying levels of interference during the time slot interval. The SINR statistical distribution during one slot is derived for both traffic patterns. The numerical results are presented to illustrate the analytical ones. The comparison between asynchronous and conventional synchronous HCNs is provided to illustrate the asynchronous effect on DL performance. It is shown that to meet high coverage requirements, synchronizing slots from different BSs help to boost the coverage performance. Furthermore, it is shown that densifying networks with low-power BSs improves coverage performance and the improvement is more pronounced at higher path-loss exponents.