To Re-transmit or Not to Re-transmit for Freshness

We consider a time slotted communication network with a base station (BS) and a user. At each time slot a fresh update packet arrives at the BS with probability $p>0$. When the BS transmits an update packet for the first time, it goes through with a success probability of $q_1$. In all subsequent re-transmissions, the packet goes through with a success probability of $q_2$ where $q_2>q_1$, due to the accumulation of observations at the receiver used to decode the packet. When the packet goes through the first time, the age of the user drops to 1, while when the packet goes through in subsequent transmissions, the age of the user drops to the age of the packet since its generation. Thus, when the BS is in the process of re-transmitting an old packet, if it receives a new packet, it has to decide whether to re-transmit the old packet with higher probability of successful transmission but resulting in higher age, or to transmit the new packet which will result in a lower age upon successful reception but this will happen with lower probability. In this paper, we provide an optimal algorithm to solve this problem.