The Freshness Game: Timely Communications in the Presence of an Adversary

We consider a communication system where a base station (BS) transmits update packets to N users, one user at a time, over a wireless channel. We investigate the age of this status updating system with an adversary that jams the update packets in the downlink. We consider two system models: with diversity and without diversity. In the model without diversity, in each time slot, the BS schedules a user from N users according to a user scheduling algorithm. The constrained adversary blocks at most a given fraction, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>, of the time slots over a horizon of T slots, i.e., it can block at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha T$ </tex-math></inline-formula> slots of its choosing out of the total T time slots. We show that if the BS schedules the users with a stationary randomized policy, then the optimal choice for the adversary is to block the user which has the lowest probability of getting scheduled by the BS, at the middle of the time horizon, consecutively for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha T$ </tex-math></inline-formula> time slots. The interesting consecutive property of the blocked time slots is due to the cumulative nature of the age metric. In the model with diversity, in each time slot, the BS schedules a user from N users and chooses a sub-carrier from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{sub}$ </tex-math></inline-formula> sub-carriers to transmit update packets to the scheduled user according to a user scheduling algorithm and a sub-carrier choosing algorithm, respectively. The adversary blocks <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha T$ </tex-math></inline-formula> time slots of its choosing out of T time slots at the sub-carriers of its choosing. We show that for large T, the uniform user scheduling algorithm together with the uniform sub-carrier choosing algorithm is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {2 N_{sub}}{N_{sub}-1}$ </tex-math></inline-formula> optimal. Next, we investigate the game theoretic equilibrium points of this status updating system. For the model without diversity, we show that a Nash equilibrium does not exist, however, a Stackelberg equilibrium exists when the scheduling algorithm of the BS acts as the leader and the adversary acts as the follower. For the model with diversity, we show that a Nash equilibrium exists and identify the Nash equilibrium. Finally, we extend the model without diversity to the case where the BS can serve multiple users and the adversary can jam multiple users, at a time.