Age of Information of a Power Constrained Scheduler in the Presence of a Power Constrained Adversary

We consider a time slotted communication network consisting of a base station (BS), an adversary,
$N$
users and
$N_{s}$
communication channels. In the first part of the paper, we consider the setting where
$N_{s}$
communication channels
$N_{s}$
are heterogeneously divided among
$N$
users. The BS transmits an update to the ith user on a subset of the communication channels
$N_{s,j}$
where
$N_{s,i} n N_{s,j}$
is not necessarily an empty set. At each time slot, the BS transmits an update packet to a user through a communication channel and the adversary aims to block the update packet sent by the BS by blocking a communication channel. The BS has
$n$
discrete transmission power levels to communicate with the users and the adversary has
$m$
discrete blocking power levels to block the communication channels. The probability of successful transmission of an update packet depends on these power levels. The BS and the adversary have a transmission and blocking average power constraint, respectively. We provide a universal lower bound for the average age of information for this communication network. We prove that the uniform user choosing policy, the uniform communication channel choosing policy with any arbitrary feasible transmission power choosing policy is 4 optimal; and the max-age user choosing policy, the uniform communication channel choosing policy with any arbitrary feasible transmission power choosing policy is 2 optimal. In the second part of the paper, we consider the setting where the BS chooses a transmission policy and the adversary chooses a blocking policy from the set of randomized stationary policies and
$N_{s}$
, i =
$N_{s}$
for all
$i$
, i.e., all users can receive updates on all channels. We show that a Nash equilibrium mayor may not exist for this communication network, and identify special cases where a Nash equilibrium always exists.