In this work, we consider the problem of jointly minimizing the average cost of sampling and transmitting status updates by users over a wireless channel subject to average Age of Information (AoI) constraints. Errors in the transmission may occur and a scheduling policy has to decide if the users sample a new packet or attempt for retransmission of the packet sampled previously. The cost consists of both sampling and transmission costs. The sampling of a new packet after a failure imposes an additional cost on the system. We formulate a stochastic optimization problem with the average cost in the objective under average AoI constraints. To solve this problem, we propose three scheduling policies; a) a dynamic policy, that is centralized and requires full knowledge of the state of the system, b) two stationary randomized policies that require no knowledge of the state of the system. We utilize tools from Lyapunov optimization theory in order to provide the dynamic policy, and we prove that its solution is arbitrary close to the optimal one. In order to provide the randomized policies, we model the system by utilizing Discrete Time Markov Chain (DTMC). We provide closed-form and approximated expressions for the average AoI and its distribution, for each randomized policy. Simulation results show the importance of providing the option to transmit an old packet in order to minimize the total average cost.
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