# Performance of Dynamic Time Division Broadcast Protocol with Rateless Coding
## 2 System Models  
### System Model of DTDBC-RC  
Considering a delay-constrained transmission application with two sources (denoted by \(S_{1}\) and \(S_{2}\)) exchanging information with the aid of a relay (denoted by \(R\)), one-round information exchange with maximum allowable transmission time \(T\) consists of at most three time slots and each source's message is transmitted for at most two time slots (i.e., one time slot for its own transmission and the other for the relay). The first and second time slots are with maximum allowable transmission time \(\alpha T\), and the third time slot is with minimum transmission time \((1-2\alpha)T\), where \(0<\alpha<\frac{1}{2}\) if retransmission at the third time slot is possible. The message is transmitted in the form of codeword. For simplicity of analysis and without loss of generality, it is assumed that both sources transmit with the same rate and the same power (When the scenario with different data rates and/or different power is considered, the transmission lengths of each time slot in DTDBC-RC are different although with channel reciprocity assumption (see the system model in Section 2). This is a general scenario which can be extended directly from the simple scenario with same data rates and same power in the paper. Moreover, the performance analysis in Section 3 can be directly applied to the general scenario. Since the simple scenario captures the essence of utilizing the dynamics of the direct link transmission by rateless coding, we do not consider the general scenario here.). The channels keep unchanged in one-round information exchange and changes independently across different rounds. Moreover, the channels between any two nodes are assumed to be reciprocal. The codeword lengths of \(S_{1}\) and \(S_{2}\) corresponding to the two-time-slot transmission are denoted by \(I_{1}\) and \(I_{2}\), i.e., the transmitted codewords of \(S_{1}\) and \(S_{2}\) are denoted by \([X_{1}^{1},X_{2}^{1},\ldots,X_{I_{1}}^{1},X_{I_{1}+1}^{1},\ldots,X_{I_{1}+I_{2}} ^{1}]\) and \([X_{1}^{2},X_{2}^{2},\ldots,X_{I_{1}}^{2},X_{I_{1}+1}^{2},\ldots,X_{I_{1}+I_{2}} ^{2}]\), respectively. The proposed DTDBC-RC protocol works in the following way. In the first time slot (with given maximum allowable transmission time \(\alpha T\)), \(S_{1}\) broadcasts its first part of codeword (i.e., \([X_{1}^{1},X_{2}^{1},\ldots,X_{I_{1}}^{1}]\)) continuously to \(S_{2}\) and \(R\) in the rateless coding way. \(S_{2}\) decodes the received message before or until \(\alpha T\). If \(S_{2}\) decodes successfully, it feeds back one-bit acknowledgement (ACK). If \(S_{2}\) cannot decode successfully until \(\alpha T\), it feeds back one-bit non-acknowledgement (NACK). In the second time slot (also with given maximum allowable transmission time \(\alpha T\)), \(S_{2}\) works in the same way as \(S_{1}\) in the first time slot, i.e., it broadcasts the first part of codeword (i.e., \([X_{1}^{2},X_{2}^{2},\ldots,X_{I_{1}}^{2}]\)) continuously to \(S_{1}\) and \(R\) in the rateless coding way. Due to channel reciprocity, \(S_{2}\) has the same decoding results as \(S_{1}\), and it feeds back NACK or ACK accordingly. After the first two time slots, the following working process of DTDBC-RC depends on the decoding results of both \(S_{1}\) and \(S_{2}\), and there are only two decoding states listed as follows. **State 1: both \(S_{1}\) and \(S_{2}\) cannot decode successfully. ** In this state, depending on the decoding result at \(R\), there are two cases. In _Case I_, \(R\) successfully decodes the messages from both \(S_{1}\) and \(S_{2}\). It feeds backs an ACK to both \(S_{1}\) and \(S_{2}\), re-encodes the received message from both sources and network-encodes the two sources' message before retransmission (e.g., through bit-level XOR operation), and retransmits the network-encoded message to both \(S_{1}\) and \(S_{2}\) in the third time slot (i.e., until the transmission length \(T\) is reached).