  We design a new scheduling policy to minimize the general non-decreasing cost
function of age of information (AoI) in a multiuser system. In this system, the
base station stochastically generates time-sensitive packets and transmits them
to corresponding user equipments via an unreliable channel. We first formulate
the transmission scheduling problem as an average cost constrained Markov
decision process problem. Through introducing the service charge, we derive the
closed-form expression for the Whittle index, based on which we design the
scheduling policy. Using numerical results, we demonstrate the performance gain
of our designed scheduling policy compared to the existing policies, such as
the optimal policy, the on-demand Whittle index policy, and the age greedy
policy.
