## 1 Introduction
In CSMA/CA wireless networks, the sender of a data packet waits for the arrival of an acknowledgment (ACK) frame for the data packet from the receiver. If the ACK frame is received in a given time interval, the data packet is considered to be delivered successfully (i.e., reward is 1). Otherwise, the sender assumes the packet is lost (i.e., reward is 0) and retries sending the packet until the maximum limit (e.g., seven times for the short-retry limit case) is reached. Graphical-optimal rate sampling (G-ORS) [20], modified Thompson sampling (MTS) [21], and constrained Thompson sampling (CoTS) [22] are some of the MAB algorithms proposed to adaptively control the data transmission rates by adjusting the MCS levels. Most existing MAB algorithms for wireless communications use the expected throughputs of available MCS levels as a criterion to select the best MCS level for the next data transmission. As an example, if we consider a CSMA/CA wireless communication standard IEEE 802.11g [23], there are eight possible MCS levels that the sender can select for its data transmission to the receiver, i.e., MCS level \(i\in\{0,\ldots,7\}\). The (physical) data transmission rate \(r_{i}\) of each MCS level \(i\) is one of \(\{6,9,12,18,24,36,48,54\}\) Mbps, respectively, when we consider only the extended-rate physical-orthogonal frequency-division multiplexing (ERP-OFDM) in the standard. If we assume that the sender adopts the MTS algorithm to control its transmission rate, the sender uses the transmission rate \(r_{i}\) of each MCS level \(i\) and its success probability estimate \(p_{i}(t)\in[0,1]\), which is obtained from the history of the transmission experiences up to time \(t\), and selects the MCS level \(i\) that maximizes the expected throughput \(r_{i}\cdot p_{i}(t)\) as follows:\[i=\operatorname*{argmax}_{i}(r_{i}\cdot p_{i}(t)). \tag{1}\]  
However, we notice that there is strong preference to high MCS levels because the data transmission rates of high MCS levels are significantly higher than those of the low MCS levels if the estimated success probabilities are similar. Although packets can be transmitted faster using higher MCS levels, the strong preference to use high MCS levels might result in a low utilization of network resources because high MCS levels require a high signal-to-interference and noise ratio (SINR) at the receiver of the packets. Therefore, we argue that the utilities of the MCS levels may not be proportional to their nominal transmission rates. To mitigate the strong preference to the high MCS levels, as a remedy, we propose the use of the logarithmic (base e) values of the transmission rates when computing the expected throughputs as follows:  
\[i=\operatorname*{argmax}_{i}(\log(r_{i})\cdot p_{i}(t)). \tag{2}\]  
Here, we assume that \(r_{i}>1\). To demonstrate the effectiveness of the proposal, we apply the logarithmic transmission rates to two types of well-known MAB algorithms for the transmission rate control; one is based on KL-UCB and the other on Thompson sampling. We used the event-driven network simulator ns-3 [24] to evaluate the proposed MAB algorithms in various wireless communication scenarios including stationary and non-stationary cases, and with different standards such as IEEE 802.11g and 802.11 ax [25]. The experimental results show that the use of the logarithmic values of the transmission rates improves performance of both types of the MAB algorithms. The remainder of this paper is organized as follows: In Section 2, we explain works related to the MAB algorithms for the transmission rate control in CSMA/CA wireless networks. In Section 3, we propose two MAB algorithms that adopt the logarithmic transmission rates and embrace frame aggregations. The algorithms also show a sliding window scheme in detail to adapt to dynamically changing wireless environments. In Section 4, we present experimental results with the IEEE 802.11g standard in both of the stationary and non-stationary scenarios.