Resource Optimization with Flexible Numerology and Frame Structure for Heterogeneous Services

Lei You, Qi Liao, Nikolaos Pappas, Di Yuan

I Introduction

The fifth generation (5G) of wireless communications systems is required to support a large variety of services . A promising solution for higher resource efficiency while providing lower latency is the scalable transmission time intervals . These works fall within the general notion of flexible resource allocation in the time-frequency domain, Optimization along the frequency dimension yields similar structures to problems such as multi-dimensional Knapsack or weighted Matching . Resource optimization adopting flexibility in both dimensions regarding frequency and time, named 2-dimensional (2-D) resource allocation, poses new challenges . Although flexible resource allocation along both the time and frequency dimensions is not new , from an integer programming point of view, frequency selective resource allocation with flexible sizes of resource units along both the frequency and time dimensions, has not yet been addressed to the best of our knowledge.

Based on 3GPP release for scalable numerologies and frame structures , we consider the resulting 2-D resource allocation problem. We address tractability and propose an algorithm with scalability, utilizing both the primal space and dual space of optimization. We then provide numerical results for performance assessment.

II System Model

In 5G new radio (NR), a numerology is defined by sub-carrier spacing (SCS) and cyclic prefix (CP) length (a.k.a. the “guard interval” between the symbols). The radio frame structure is characterized by number of slots within a frame. A TTI can consist of one mini-slot with 1-13 symbols supported, or one slot with 14 symbols (or 12 symbols in case of extended CP), or multiple slots if slot aggregation is supported. One resource allocation to a service involves a set of adjacent SCSs and TTIs in the frequency and time domain respectively with a configured CP length. For simplicity, hereafter we refer to the resource configuration of numerology and frame structure as blocks, and consider a candidate set $\mathcal{B}$ of blocks, see Figure 1. For each $b\in\mathcal{B}$ , the achieved throughput on block $b$ if $b$ is assigned to service $k$ ( $k\in\mathcal{K}$ ) is denoted by $r_{b,k}$ .

Given the channel profile, the transmission power, and the noise power, $r_{b,k}$ depends on the configuration of block $b$ , including the time span and frequency range (characterized by SCS and TTI duration), CP length, and symbol duration. Moreover, this rate shall take into account the effect of guardband. To compute the achieved throughput per block, we assume a total number of nine multipath channel profiles [16, Table B.2.1-4], and we predefine the mapping from the configuration parameters to the throughput based on the model in . This model takes into account the inter-symbol-interference (ISI) depending on CP, and approximates the inter-channel interference (ICI) between the neighboring subbands with the same type of numerology (the ICI between subbands with different types of numerologies is not modeled in this paper due to the high complexity). In addition, we also consider the control overhead as one or more consecutive symbols per TTI. Due to limited space, we omit the details but provide the tutorial and source code in IEEE DataPort .

III Problem Formulation and Tractability

IV Problem Solving

We propose a sub-optimal but low-complexity algorithm, consisting in performing assignment of blocks to services, based on utility values generated from linear programming (LP) relaxation and the Lagrangian dual (LD).

IV-B Utility Estimation by LP Relaxation

One way to compute the utility matrix $\mathbf{u}$ is to solve the LP relaxation of P0 and to use the LP optimum $\mathbf{x}_{\text{LP}}$ .

We denote by $\mathbf{u}_{\text{LP}}=\mathbf{x}_{\text{LP}}$ the LP-based utility. Also, $\mathbf{x}_{\text{LP}}$ can be used for initialization: $\mathcal{S}=\{(b,k):u_{\text{LP},b,k}\geq\rho,b\in\mathcal{B},k\in\mathcal{K}\}$ with $\rho$ being a threshold.

IV-C Utility Estimation by LD

By relaxing the constraints (1c) of P0 with Lagrangian multiplier $\lambda_{i}$ ( $i\in\mathcal{I}$ ), the Lagrangian is defined as follows:

Note constraints (5b) are not present in P0, though these are implied by (1c) for services in $\mathcal{K}^{(c)}$ . Computing the optimum of P2 is straightforward. Each block $b$ is allocated to the service $\operatorname{\arg\max}_{k}{r_{b,k}-\alpha_{b}}$ with $r_{b,k}-\alpha_{b}>0$ .

Each P3[ $k$ ] can be reformulated as a Knapsack Problem, and optimally solved by dynamic programming.

The dual problem P0-LD can be solved using a sub-gradient method . Denote by $\mathbf{x}^{(h)}_{\text{LD}}$ the LD solution in the $h_{\text{th}}$ iteration of the sub-gradient method. We let $\mathbf{u}_{\text{LD}}=\sum_{h}\mathbf{x}^{(h)}_{\text{LD}}$ to be the LD-based utility.

IV-D Algorithm Implementation

In addition to BA( $\mathcal{S}$ , $\mathbf{u}_{\text{LP}}$ ) and BA( $\mathcal{S}$ , $\mathbf{u}_{\text{LD}}$ ), we consider algorithm “LP+LD” that returns the best solution of BA( $\mathcal{S}$ , $\mathbf{u}_{\text{LP}}$ ) and BA( $\mathcal{S}$ , $\mathbf{u}_{\text{LD}}$ ). We remark that BA( $\mathcal{S}$ , $\mathbf{u}_{\text{LD}}$ ) is quite flexible in terms of computational effort, as one can use accumulated $\mathbf{u}_{\text{LD}}$ before full convergence. Overall, the algorithm scales well. Moreover, if necessary, the service sets can be decomposed into subsets, and the algorithm can be applies to one subset at a time to further reduce complexity.

V Numerical Results

The use of flexible numerology is expected to outperform fixed numerology. The purpose of performance evaluation is to examine the amount of improvement, which is of significance in particular as the control channel overhead for supporting the flexible structure is accounted for. The result also tell how well the proposed algorithm is suited for the flexible structure.

Comparing to LTE that applies a fixed SCS of 15 kHz and TTI of $1.0$ ms, we consider four shapes, Shape 1, Shape 2, Shape 3, and Shape 4, with SCS being $15$ kHz, $30$ kHz, $60$ kHz, and $60$ kHz, CP $4.7$ $\mu$ s, $2.3$ $\mu$ s $1.2$ $\mu$ s, and $4.17$ $\mu$ s, and the number of symbols $7$ , $7$ , $7$ , and $6$ , respectively. The TTI durations of the four shapes are $0.5$ ms, $0.25$ ms, $0.125$ ms, and $0.125$ ms, respectively. The numerologies ( $\Delta f=2^{\mu}\times 15$ kHz, $\mu=0,1,2\ldots$ ) originate from Release 15 [20, Table 4.2-1]. Note that Release 15 also specifies subcarrier spacing up to $240$ kHz. However, by [21, Table I], a TTI of $0.125$ ms meets all the worst-case transmission latencies for the listed 5G ultra-reliable low-latency communication configurations.

Parameter settings are given in Table I. We test our algorithm for a set of candidate thresholds $\rho\in\{0.05,0.5,\ldots,0.95\}$ among which the one achieves the best objective is selected. The maximum sub-gradient iterations is set to $200$ . While calculating the block rates, the impact on capacity due to guardband is included by following the model in . The rate reduction due to control overhead follows that in , where two symbols per TTI constitute the overhead. We emphasize on accurate assessment in terms of optimality, that is, how much does the proposed algorithm perform with respect to global optimum. We use the global optimum obtained by solving the integer programming problem (1) via a solver. This is not a scalable method. The purpose here is for benchmarking, to demonstrate that our low-complexity algorithm has little loss in optimality. The number of users as well as the bandwidth is chosen such that the global optimum can be obtained with reasonable amount of computing effort. Similarly, in view of the computational effort of obtaining global optimum for benchmarking, we do not include all TTI sizes that are permitted by 5G NR .

Without showing by the figures, we remark that the problem feasibility of the three non-flexible schemes is very sensitive to the latency tolerance. This issue is alleviated by the flexible structure. In comparison to the related work considering advantages of flexible numerology, our results emphasize the significance of block-service assignment optimization.

VI Conclusion

We suggest that combining a flexible numerology and frame structure serves as a promising option for spectral efficiency. Utilizing LP and LD enables efficient problem solving.

Acknowledgement

This work has been partially supported by European Union H2020 MSCA projects ACT5G (643002) and DECADE (645705), and the Center for Industrial Information Technology (CENIIT). The work of the first author was partly accomplished while he was at Linköping University, Sweden.