## 4 Dynamic BWP Allocation: Dynamic Multiplexing vs. Orthogonal Slicing  
### ThrouInput Analysis for Dynamic Multiplexing and Orthogonal Slicing  
In DM-based URLLC multiplexing, the eMBB AV flushes out the corresponding resources where the URLLC overlaps. Owing to the effect of flushing, the bits of the corresponding eMBB CBG are partially erased. To compensate for the erased bits, we assumed that eMBB traffic is encoded with an MDS code at a coding rate of \(r=k/n\), which has the error-correction ability to fix \(l\) erased errors. Because the MDS code achieves the Singleton bound, it has a minimum distance of \(d=n-k+1\), which can correct \(l=d-1=n-k\) erased bits [16]. Furthermore, it is assumed that ideal scrambling is applied to the RBs assigned to the CBGs that belong to one TB of eMBB AVs; therefore, the erased bits in each CBG are uniformly distributed, and the locations of the erased bits can be known. Because the MDS code for CBG can fix \(l\) erased bits, we can model the CBG error rate (CBGER) function with a shifted unit step function. In NR specification, CBG is constructed  
Figure 2: Two different eMBB/URLLC multiplexing: DM vs. OS. by low density parity check (LDPC) code for shared channel. As LDPC code is a kind of MDS code; \((n,k)\) code can correct \(n-k\) bits [16]. If the number of bits in error incurred by puncturing is less than \(n-k\) bits, therefore, the CBG error rate is 0. Otherwise, the CBG error rate is 1. Therefore, the CBG error rate can be modeled as a unit step function. In order to obtain our switching threshold, \(D_{threshold}\), for dynamic BWP allocation scheme, however, we expect the CBG error rate to be a continuous function. To maintain continuity in the CBGER function, we approximate it as follows:  
\[P_{CBGER}(x)= \begin{cases}\begin{array}{c}0,\ x<l-\varepsilon\\ \frac{1}{2\varepsilon}(x-l)+\frac{1}{2},\ l-\varepsilon\leq x\leq l+\varepsilon \\ 1,\ l+\varepsilon<x\end{array}\end{cases}  \tag{2}\]  
where \(x\) denotes a number of erased bits in CBG and \(\varepsilon\) is zero or positive integer value indicating a margin for the bit-error-correction capability \(l\). Figure 3 illustrates CBGER, which approaches a shifted unit step function as \(\varepsilon\) tends to zero. Once \(D_{threshold}\) is computed with the CBRER function with a parameter \(\varepsilon\) in Figure 3, an approximate closed-form expression for near-optimal threshold, \(D_{threshold}\), for LDPC code can be obtained by setting \(\varepsilon\) to zero, making the CBRER function close to a unit step function, as required for LDPC code. Furthermore, if we assume that all eMBB AVs are assigned the same number of RBs, the number of bits erased by URLLC traffic in each eMBB CBG is given as  
\[\delta_{\text{C}BG}=\frac{\phi_{u}\cdot b(I_{\text{MCS}}^{(u)})}{N_{\text{C}BG }}\cdot\frac{\Phi_{\text{URLLC}}}{\Phi}=\frac{\phi_{u}\cdot b(I_{\text{MCS}}^{ (u)})}{N_{\text{C}BG}}\cdot\frac{D}{\Phi\cdot b(I_{\text{MCS}}^{\text{URLLC}})} \tag{3}\]  
where \(b(I_{\text{MCS}}^{(u)})\) and \(b(I_{\text{MCS}}^{\text{URLLC}})\) are the number of bits per RB when the modulation and coding scheme (MCS) index for AV \(u\) is given by \(I_{\text{MCS}}^{(u)}\) and the MCS index for all the URLLC AVs is given as \(I_{\text{MCS}}^{\text{URLLC}}\). Considering (2), the throughput of an eMBB AV is given as  
\[\sum_{u\in\mathcal{U}_{\text{eMBB}}}\phi_{u}\cdot b(I_{\text{MCS}}^{(u)})\cdot (1-P_{\text{C}BGER}(\delta_{\text{C}BG})) \tag{4}\]  
From (4), the throughput of one CBG in an eMBB AV \(u\) in the DM is given as  
\[\frac{\phi_{u}b(I_{\text{MCS}}^{(u)})}{N_{\text{C}BG}}\cdot(1-P_{\text{C}BGER }(\delta_{\text{C}BG})) \tag{5}\]  
In OS-based URLLC multiplexing, on the other hand, the resource is orthogonally sliced in proportion to the URLLC load \(D\). If all URLLC AVs have an MCS of \(I_{\text{MCS}}^{\text{URLLC}}\), the number of RBs assigned to URLLC \(\Phi_{\text{URLLC}}\) is given by \(\Phi_{\text{URLLC}}=D/b(I_{\text{MCS}}^{\text{URLLC}})\).