Continuous and reliable service support is crucial for emerging latency-sensitive and computation-intensive applications in UAV-assisted edge networks (UENs) due to operational dynamics and environmental uncertainty. Although conventional designs can improve coverage and computing efficiency, they often rely on instantaneous resource optimization or reactive handover, rendering ongoing services vulnerable to non-negligible interruptions when the serving UAV degrades due to mobility, energy depletion, or channel dynamics. To avoid such post-failure recovery, a promising approach is to prepare a successor UAV in advance, i.e., a standby UAV that reserves minimal resources and synchronizes service context for possible takeover. Thus, we consider a dynamic UEN architecture where each mobile user carries an ongoing computing mission requiring persistent service support, while UAVs provide wireless access and computing services under time-varying network dynamics and stringent onboard energy constraints. To facilitate proactive and continuous service provisioning, we propose a forecasting-driven proactive reservation-based continuous service scheduling framework, termed Fresco. In Fresco, an LSTM-based module is first used to predict short-term disruption risks of ongoing missions from historical network observations. Guided by these predictions, an online risk-aware successor matching scheme selects suitable standby UAVs for high-risk missions under delay, resource, and energy constraints, while incorporating minimal communication/computation reservation and lightweight service-context synchronization for efficient takeover preparation. Experiments show that Fresco significantly reduces service interruptions and improves mission continuity over reactive and non-predictive baselines, with only modest reservation overhead.

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