# Cross-Cycled Uplink Resource Allocation over NB-IoT
## 2 System Model and Problem Formulation
### Problem Formulation  
This paper investigates the uplink resource allocation problem considering cross-cycled radio resources in NB-IoT cellular networks. This paper targets the same problem as [12], while [12] does not consider cross-cycled resource allocation. This paper attempts to minimize the consumed subframes while each UE satisfies its data requests. The system model is described as follows:  
A cell must support \(D\) UEs. The number of NPs consumed to satisfy \(D\) UEs is denoted as \(P\). In an uplink radio frequency, a subframe has \(F\) subcarriers. The frequency space of a subcarrier is \(W\). There are \(r_{p}^{s}\) subcarriers occupied by NPRACHs in subframe \(s\) in the \(p\)-th NP. Device \(d\) has an uplink data requirement with data of size \(\psi_{d}\). The cell provides \(U\) resource unit types. Unit type \(u\) should use \(f_{u}\) subcarriers in the frequency domain and \(t_{u}\) subframes in the time domain, where \(f_{u+1}>f_{u}\). The set of resource unit numbers is \(I_{RUI}=\{I^{1},I^{2},..,I^{h},..,I^{h}\}\) in the resource assignment field, where \(H\) is the maximum index and \(I^{h}\) is the maximum number of resource units. We have \(M\) modulation and coding scheme indexes, each of which can refer to a TBS index. The cell can select a data repetition number for a UE to transmit data from the set of repetition numbers \(N_{Rep}\). The cell should choose a repetition number meeting \(N_{d}^{u,m}\geq\hat{N}_{d}^{u,m}\), \(N_{d}^{u,m}\in N_{Rep}\) to guarantee the predefined transmission reliability via a success probability model (This paper can be applied no matter which success probability model is considered. ), where \(\hat{N}_{d}^{u,m}\) is the repetition requirement of UE \(d\) using modulation \(m\) and unit type \(u\) under an SNR value. The calculation of \(\hat{N}_{d}^{u,m}\) values can be found in [12]. When \(I_{d}^{h}\) units with type \(u\) using modulation \(m\) are assigned to UE \(d\), the cell should allocate \(f_{u}\) subcarriers during \(I^{h}\times t_{u}\times N_{d}^{u,m}\) continuous NPUSCH subframes to UE \(d\) using \(N_{d}^{u,m}\) repetitions. The UE can transmit a data size of \(\eta(I^{h},m)\) according to the TBS table. We use an indicator function \(X_{p,d}^{u,m}=1\), if the cell decides UE \(d\) using unit type \(u\) and modulation \(m\) in the \(p\)-th NP. The parameter \(R_{max}\) value is the subframe number for the NPDCCH in an NP, and the parameter \(G\) value is the system parameter. The length of an NP is \(L=R_{max}\times G\). Both \(R_{max}\) and \(G\) values can be used to adjust the NP length for UEs and are given in this paper. This paper considers using the DCI repetition number \(R=R_{max}/8\) to transmit one DCI for each UE. The repetition number used to transmit a DCI should also meet the reliability (i.e., \(R\geq\hat{\mathcal{D}}_{d}\)), where \(\hat{\mathcal{D}}_{d}\) is the required DCI repetition number of UE \(d\). Specifically, when a UE needs a higher DCI repetition number, the cell should choose a higher \(R_{max}\) value to increase the \(R\) value. Given the repetition number \(R\), the DCI number is \(\xi\) (i.e., \(\xi=2\times(R_{max}/R)\)) in an NP, because DCI format N0 occupies one control channel element in an NPDCCH subframe, each of which has two control channel elements. We define \(\mathcal{DCC}_{p,d}^{c}=1\) when DCI \(c\) in the \(p\)-th NP is assigned for UE \(d\) and \(0\) otherwise. We define \(\mathcal{S}_{p}^{s}\) used to represent the subcarriers that have been assigned to UEs in subframe \(s\) of the \(p\)-th NP. \(K=\{I_{0}^{1},I_{0^{2}}^{2},..,I_{0^{h}}^{j},..\}\) is the set of scheduling delay values. The objective of our target problem is to decrease the total consumed subframes \(P\times L\) to  
Figure 3: Concept of cross-cycled resource allocation. (**a**) some subframes cannot be used by the current NP; (**b**) a cell can allocate the NPUSCH subframes of the next NP, which cannot be used in the next NP, for a UE without causing wireless interference. support \(D\) UEs for the uplink as much as possible.