Leading by the development of Starlink, the low earth orbit (LEO) satellite networks are expected to be the customer-grade Internet infrastructure that will co-work with the terrestrial networks for carrying Internet traffic. In this paper, focusing on network usage fluctuations that are caused by human activity variations in work and rest time slots, we introduce the concept of time zones into the global satellite network and develop a time-zone-aware traffic model. Accordingly, we propose a relay-based routing algorithm to balance the workload of the satellites and the inter-satellite links over different time zones. Simulation results show that the proposed model and algorithm can reduce the blocking ratio by up to 34.8% and increase the bandwidth utilization ratio by up to 45.6%, with a limited cost in average connection latency.