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.

翻译：我们考虑一个由基站和用户组成的时隙通信网络。在每个时隙，基站以概率 $p>0$ 接收到一个新鲜更新数据包。当基站首次发送更新数据包时，其成功传输概率为 $q_1$。在所有后续重传中，由于接收端用于解码数据包的观测积累，数据包的成功传输概率提升至 $q_2$（其中 $q_2>q_1$）。若数据包在首次传输成功，用户年龄降至1；若在后续传输成功，用户年龄降至数据包自生成时刻起的年龄。因此，当基站正在重传旧数据包时收到新数据包时，需决策：是继续以更高成功概率重传旧数据包（但会导致较高年龄），还是转而发送新数据包（成功接收后年龄较低但成功概率较低）。本文针对该问题提出了一种最优算法。