  We present a comprehensive steady-state analysis of threshold-ALOHA, a
distributed age-aware modification of slotted ALOHA proposed in recent
literature. In threshold-ALOHA, each terminal suspends its transmissions until
the Age of Information (AoI) of the status update flow it is sending reaches a
certain threshold $\Gamma$. Once the age exceeds $\Gamma$, the terminal
attempts transmission with constant probability $\tau$ in each slot, as in
standard slotted ALOHA. We analyze the time-average expected AoI attained by
this policy, and explore its scaling with network size, $n$. We derive the
probability distribution of the number of active users at steady state, and
show that as network size increases the policy converges to one that runs
slotted ALOHA with fewer sources: on average about one fifth of the users is
active at any time. We obtain an expression for steady-state expected AoI and
use this to optimize the parameters $\Gamma$ and $\tau$, resolving the
conjectures in \cite{doga} by confirming that the optimal age threshold and
transmission probability are $2.2n$ and $4.69/n$, respectively. We find that
the optimal AoI scales with the network size as $1.4169n$, which is almost half
the minimum AoI achievable with slotted ALOHA, while the loss from the maximum
throughput of $e^{-1}$ remains below $1\%$. We compare the performance of this
rudimentary algorithm to that of the SAT policy that dynamically adapts its
transmission probabilities.
