  We consider a network with multiple sources and a base station that send
time-sensitive information to remote clients. The Age of Incorrect Information
(AoII) captures the freshness of the informative pieces of status update
packets at the destinations. We derive the closed-form Whittle Index
formulation for a push-based multi-user network over unreliable channels with
AoII-dependent cost functions. We also propose a new semantic performance
metric for pull-based systems, named the Age of Incorrect Information at Query
(QAoII), that quantifies AoII at particular instants when clients generate
queries. Simulation results demonstrate that the proposed Whittle Index-based
scheduling policies for both AoII and QAoII-dependent cost functions are
superior to benchmark policies, and adopting query-aware scheduling can
significantly improve the timeliness for scenarios where a single user or
multiple users are scheduled at a time.
