One of the requirements of network slicing in 5G networks is RAN (radio access network) scheduling with rate guarantees. We study a three-time-scale algorithm for maximum sum utility scheduling, with minimum rate constraints. As usual, the scheduler computes an index for each UE in each slot, and schedules the UE with the maximum index. This is at the fastest, natural time-scale of channel fading. The next time-scale is of the exponentially weighted moving average (EWMA) rate update. The slowest time scale in our algorithm is an "index-bias" update by a stochastic approximation algorithm, with a step-size smaller than the EWMA. The index-biases are related to Lagrange multipliers, and bias the slot indices of the UEs with rate guarantees, promoting their more frequent scheduling. We obtain a pair of coupled ordinary differential equations (o.d.e.) such that the unique stable points of the two o.d.e.s are the primal and dual solutions of the constrained utility optimization problem. The UE rate and index-bias iterations track the asymptotic behaviour of the o.d.e. system for small step-sizes of the two slower time-scale iterations. Simulations show that, by running the index-bias iteration at a slower time-scale than the EWMA iteration and using the EWMA throughput itself in the index-bias update, the UE rates stabilize close to the optimum operating point on the rate region boundary, and the index-biases have small fluctuations around the optimum Lagrange multipliers. We compare our results with a prior two-time-scale algorithm and show improved performance.

翻译：5G网络切片的要求之一是具有速率保障的无线接入网络（RAN）调度。本文研究一种用于带最小速率约束的最大总和效用调度的三时标算法。与常规方法相同，调度器在每个时隙为每个用户设备（UE）计算一个指数，并调度具有最大指数的UE。这发生在信道衰落的快速自然时标上。下一个时标是指数加权移动平均（EWMA）速率更新。我们算法中最慢的时标是通过随机逼近算法进行的“指数偏置”更新，其步长小于EWMA更新步长。指数偏置与拉格朗日乘子相关，并对具有速率保障的UE的时隙指数进行偏置调整，从而促进其更频繁地被调度。我们得到一对耦合的常微分方程（o.d.e.），使得这两个o.d.e.的唯一稳定点即为约束效用优化问题的原问题解与对偶问题解。当两个较慢时标迭代的步长较小时，UE速率和指数偏置的迭代过程跟踪该o.d.e.系统的渐近行为。仿真结果表明，通过在比EWMA迭代更慢的时标上运行指数偏置迭代，并在指数偏置更新中使用EWMA吞吐量本身，UE速率稳定在速率区域边界的最优工作点附近，且指数偏置在最优拉格朗日乘子附近呈现小幅波动。我们将本文结果与先前的双时标算法进行比较，并展示了性能的提升。