## 3 Design of Synchronous Multi-Link Distributed Coordination Functions  
As we mentioned in Section 2, if non-STR MLDs need to be supported, we need to consider synchronous transmissions where end times of transmission are aligned. Thus, we only focus on synchronous multi-link DCFs and model the Markov Chains based on them. We assume that an MLD has dual radios, each of which is equipped with PHY and MAC layers, and the radios of the MLDs are on the non-STR condition, which means they have constraints of simultaneous transmission and reception as seen in Figure 2a,b. In this section, we elaborate designs of the multi-link DCFs mentioned above. Simplistic operations of the DCFs are illustrated in Figure 4. From now on, a link in which transmission is initiated by winning a contention is called a primary link. Transmissions on other links that rely on the transmission on the primary link may be initiated, and the link on which the transmission is initiated is referred to as an auxiliary link. Specifically, we classify the mentioned multi-link DCFs according to the channel access methodology into four following four multi-link DCFs and compare them: (1) single-link access (SLA) [5, 6, 7], (2) multi-link access with the shortest backoff (MLA-S) [4, 5, 8, 9, 10], (3) multi-link access with the longest backoff (MLA-L) [12, 16], and (4) multi-link access with end-time alignment (MLA-A) [4, 9, 17]. In the case of SLA, as shown in Figure 4a, only one pre-defined link is eligible to access the channel, and therefore, the link shall be a primary link permanently. Once the primary link is permitted to access the channel through an expired backoff counter, the other link, which is the auxiliary link, can participate in transmission only if the channel on the links were idle during a fixed interval, which will be called by \(T\)th, immediately preceding the start of the transmission opportunity (TXOP). Otherwise, TXOP will set only on the primary link. In conclusion, the backoff operation only proceeds on the primary link, and there are no backoff operations on the auxiliary link. On the other hand, accessing the channel on all active links may be allowed, unlike SLA. That is to say, all the links can be operated as primary links, which we call by multi-link access (MLA) mechanism. We next classify MLA into three cases, MLA-S, MLA-L, and MLA-A. For MLA-S, as shown in Figure 4b, TXOP is obtained by an MLD when the backoff counter of either one of the MLD's links is expired. At this time, the link where the transmission is initiated is set to the primary link, and the other link is set to the auxiliary link. In addition, if the auxiliary link has been idle during \(T\)th before TXOP starts, the channel is accessed even if the backoff counter of the other link is remaining and in consequence, the link will become the auxiliary link. For example, suppose that backoff counters of links 1 and 2 in the initial stage are set to 2 and 5, respectively, as shown at the beginning in Figure 4b, then the backoff counter of link 1 will expire earlier, and link 1 will be the primary link. If the channel on link 2 is idle during \(T\)th right before TXOP on link 1, link 2 will get the opportunity to transmit the data frame. After transmission, these backoff counters are reset within the initial stage for successful transmission. In this example, they are set to 8 and 4, respectively. CW values are doubled if the transmission was failed, and the backoff counter for the next channel access will be selected with the revised CW. MLA-L, which is shown in Figure 4c, is slightly different with MLA-S. In MLA-L, a channel can be accessed when all backoff counters are expired for all links only. Since backoff counters of stations affiliated with the same MLD can be shared, channel access is initiated when the backoff counter of the last station becomes zero. For example, if backoff counters of links 1 and 2 are 5 and 2, respectively, as shown in Figure 4c, then acquiring the TXOP is available after all the backoff counters are expired.