# Transmission Delay-Based Uplink Multi-User Scheduling in IEEE 802.11 ax Networks +
## 3 System Model  
In this paper, we consider an uplink multi-user transmission scenario in an IEEE 802.11 ax-enabled network, which consists of one AP and non-AP stations. Let an IEEE 802.11 ax-enabled AP and non-AP stations associated with the AP be \(a_{\mathcal{S}}\) and \(\mathcal{S}=\{s_{1},s_{2},\cdots,s_{\mathcal{S}}\}\), respectively. In the network, with the help of MU-MIMO and OFDMA techniques, the AP \(a_{\mathcal{S}}\) can simultaneously receive multiple data streams transmitted from multiple stations in \(\mathcal{S}\). To perform uplink multi-user transmissions, the channel state information (CSI) between the AP and each station should be notified in advance. In the IEEE 802.11 ax standard, the channel-sounding process for obtaining CSI is specified as follows: First, AP broadcasts a null data packet (NDP) announcement followed by an NDP to initiate the channel sounding process. After, AP transmits a beamforming report (BFRP) trigger frame and the station's response to the BFRP trigger by sending the CSI to the AP. In the channel-sounding process, the BFRP trigger frame can be transmitted repeatedly to more than one sequence to obtain the CSI of all stations in the network. Note that the buffer status information of stations can also be notified to the AP jointly with the channel sounding process. In IEEE 802.11 ax standard, the buffer status response (BSR) can be implicitly reported in the QoS control field or BSR control field as well as explicitly reported following by the BSR poll (BSRP) trigger from the AP. We denote the channel between each station and AP estimated during the channel sounding process by \(\mathbf{H}_{a_{\mathcal{S}},s_{i}}\). Then, the signal-to-interference plus noise power of each uplink signal from a station to the AP is given as follows:  
\[\gamma_{a_{\mathcal{S}},s_{i}}=\frac{p_{s_{i}}|\mathbf{H}_{a_{\mathcal{S}},s_{i }}|^{2}}{\sum_{q_{j}\in\mathcal{S}}p_{a_{j}}|\mathbf{H}_{a_{\mathcal{S}},s_{j }}|^{2}+\sigma^{2}}, \tag{1}\]  
where \(p_{s_{i}}\) is the transmission power of \(s_{i}\) and \(\sigma^{2}\) is noise power. The maximum uplink transmission rate of each station depends on the expected SINR \(\gamma_{a_{\mathcal{S}},s_{i}}\), and modulation and coding scheme (MCS) specified in IEEE 802.11 ax. In this paper, we assume that each station tries to transmit data with the maximum transmission rate, and denote the mapping function that introduces the uplink transmission rate by \(R_{s_{i}}(\cdot)\), i.e., the transmission rate of station \(s_{i}\) is given as \(R_{s_{i}}(\gamma_{a_{\mathcal{S}},s_{i}})\). After the RTS and CTS procedure, the uplink channel can be estimated, and the appropriate MCS level is decided by the AP and notified to stations using triggering messages. For simplicity, in this paper, we assume that each uplink transmission has the same bandwidth, and denote the maximum capacity of the simultaneous uplink multi-user transmissions in the network by \(B_{a_{\mathcal{S}}}\). In other words, the AP \(a_{\mathcal{S}}\) can receive at most \(B_{a_{\mathcal{S}}}\) data streams at the same time. In the uplink multi-user transmission scenario in IEEE 802.11 ax, an AP can allow up to \(S\) stations to transmit their data simultaneously with rate \(R_{s_{i}}(\gamma_{a_{\mathcal{S}},s_{i}})\) for each station \(s_{i}\). An uplink multi-user transmission generally improves network throughput performance by avoiding spectral inefficiency caused by excessive competition among stations. However, if there are huge differences in transmission delays among the stations selected for uplinkmulti-user transmissions, the network throughput performance may degrade owing to low channel use. Figure 1 depicts the uplink multi-user transmission scenario with one AP and three non-AP stations in an IEEE 802.11 ax network [21]. First, AP \(a_{S}\) transmits a multi-user request-to-send (RTS) frame to stations \(s_{1},s_{2}\), and \(s_{3}\). Then, stations respond to the RTS by sending clear-to-send (CTS) frames to the AP.