This paper considers the problem of obtaining bounded time-average expected queue sizes in a single-queue system with a partial-feedback structure. Time is slotted; in slot $t$ the transmitter chooses a rate $V(t)$ from a continuous interval. Transmission succeeds if and only if $V(t)\le C(t)$, where channel capacities $\{C(t)\}$ and arrivals are i.i.d. draws from fixed but unknown distributions. The transmitter observes only binary acknowledgments (ACK/NACK) indicating success or failure. Let $\varepsilon>0$ denote a sufficiently small lower bound on the slack between the arrival rate and the capacity region. We propose a phased algorithm that progressively refines a discretization of the uncountable infinite rate space and, without knowledge of $\varepsilon$, achieves a $\mathcal{O}\!\big(\log^{3.5}(1/\varepsilon)/\varepsilon^{3}\big)$ time-average expected queue size uniformly over the horizon. We also prove a converse result showing that for any rate-selection algorithm, regardless of whether $\varepsilon$ is known, there exists an environment in which the worst-case time-average expected queue size is $Ω(1/\varepsilon^{2})$. Thus, while a gap remains in the setting without knowledge of $\varepsilon$, we show that if $\varepsilon$ is known, a simple single-stage UCB type policy with a fixed discretization of the rate space achieves $\mathcal{O}\!\big(\log(1/\varepsilon)/\varepsilon^{2}\big)$, matching the converse up to logarithmic factors.

翻译：本文研究了在具有部分反馈结构的单队列系统中获得有界时间平均期望队列大小的问题。时间被划分为时隙；在时隙$t$，发送方从一个连续区间中选择一个速率$V(t)$。当且仅当$V(t)\le C(t)$时传输成功，其中信道容量$\{C(t)\}$和到达过程均为来自固定但未知分布的独立同分布抽样。发送方仅观察到指示成功或失败的二进制确认（ACK/NACK）。令$\varepsilon>0$表示到达速率与容量区域之间裕度的足够小的下界。我们提出了一种分阶段算法，该算法逐步细化不可数无限速率空间的离散化，并且在不知道$\varepsilon$的情况下，在整个时间范围内一致地实现了$\mathcal{O}\!\big(\log^{3.5}(1/\varepsilon)/\varepsilon^{3}\big)$的时间平均期望队列大小。我们还证明了一个逆结果：对于任何速率选择算法，无论是否知道$\varepsilon$，都存在一个环境，使得最坏情况下的时间平均期望队列大小为$Ω(1/\varepsilon^{2})$。因此，虽然在不知道$\varepsilon$的设置中仍存在差距，但我们证明了如果$\varepsilon$已知，一种采用固定速率空间离散化的简单单阶段UCB类型策略能够实现$\mathcal{O}\!\big(\log(1/\varepsilon)/\varepsilon^{2}\big)$，与逆结果在至多对数因子内匹配。