Ultra-Reliable and Low Latency Communication in mmWave-Enabled Massive MIMO Networks

I Introduction

Currently, millimeter wave (mmWave) and massive multiple-input multiple-output (MIMO) techniques are investigated to provide reliable communication with an over-the-air latency of few milliseconds and extreme throughput . While massive MIMO with large degrees of freedom provides high energy and spectral efficiency , mmWave frequency bands provide large bandwidth . In addition, due to the short wavelength of mmWaves, large antenna array can be packed into highly directional beamforming, which makes massive MIMO practically feasible . Thus far, most of existing works on mmWave-enabled massive MIMO systems focus mainly on providing capacity improvement , while latency and reliability are not addressed. Although latency and reliability are applicable to many scenarios (e.g. mission-critical applications), in this work, we are interested in the integration of mmWave communication and massive MIMO techniques, which holds the promise of providing great enhancements of the overall system performance . Specifically, this letter is concerned with addressing the fundamental question in mmWave-enabled massive MIMO systems: “how to simultaneously provide order of magnitude capacity improvements and latency reduction?” By invoking the Lyapunov optimization framework, an utility-optimal solution is obtained to maximize network throughput subject to queuing stability . This solution establishes a utility-delay tradeoff, which achieves utility-optimality at the price of large queuing delays . To cope with this shortcoming, in this letter the Lyapunov framework is extended to incorporate probabilistic latency and reliability constraints, which takes into account queue length, arrival rate, and channel variations with a guaranteed probability. To do so, the problem is formulated as a network utility maximization (NUM). By applying the drift-plus-penalty technique, the problem is decoupled into a dynamic latency control and rate allocation. Here, the latency control problem is a difference of convex (DC) programming problem, which is solved efficiently by the convex-concave procedure (CCP) . Finally, a performance evaluation is carried out to show the latency reduction and the tradeoff between reliability, traffic intensity, and user density.

II System Model

Here, we invoke results from random matrix theory in order to get the deterministic equivalence for (2) . In particular, as $N\geq M$ and $N\gg 1$, for small $\alpha$, the ergodic DL rate (2) almost surely converges to

where $[x]^{+}\triangleq\max\{x,0\}$, and $a_{m}(t)$ is the data arrival rate of UE $m$. Further, we assume that $a_{m}(t)$ is i.i.d. over time slots with mean arrival rate $\lambda_{m}$ and upper bounded by $a_{m}^{\max}$ .

III Problem Formulation

In (5), ${d}_{m}^{\text{th}}$ reflects the UE delay requirement. Here, $\epsilon_{m}\ll 1$ is the target probability for reliable communication.

which represents the minimum rate requirement in slot $t$ for UE $m$ for reliable communication. Here, we transform the probabilistic latency and reliability constraint (5) into one linear constraint (8) of instantaneous rate requirements, which helps to analyse and optimize the URLLC problem. In particular, if the delay requirement/reliability constraint is looser (i.e., larger $d^{\rm th}_{m}$ or $\epsilon_{m}$), the instantaneous rate requirement is reduced. In contrast, if we have a tighter constraint for reliable communication or delay requirement, then the instantaneous rate requirement is higher. Combining (6) and (8), we rewrite OP as follows:

IV Lyapunov Optimization Framework

To tackle (9), we resort to Lyapunov optimization techniques . Firstly, for each DL rate $r_{m}(t)$, we introduce the auxiliary variable vector $\boldsymbol{\varphi}(t)=(\varphi_{m}(t)|\forall\,m\in\mathcal{M})$ that satisfies

In order to ensure the inequality constraint (10), a virtual queue vector $\mathbf{Y(t)}=(Y_{m}(t)|\forall\,m\in\mathcal{M})$ is introduced, where each element evolves according to

Subsequently, we express the conditional Lyapunov drift-plus-penalty for each time slot $t$ as:

Due to space limitation, we omit the details of the constant value $C$ which does not influence the system performance . We note that the solution to LP is acquired by minimizing the right-hand side (RHS) of (14a) and (14b) in every slot $t$. Further, (14a) is related to the reliability and QoS requirements while (14b) reflects optimal power allocation to UEs.

Considering the logarithmic fairness utility function, i.e., $f(x)=\log(x)$, minimizing the RHS of (14a) for each $m\in\mathcal{M}$ is formulated as:

Before proceeding with problem (15), we rewrite $-\nu_{m}(t)\log({\varphi}_{m}(t))$ in (15a), for any $\varphi_{m}(t)>0$ and $\nu_{m}(t)>0$, as

in which both $h_{0}({\varphi_{m},\nu_{m}})$ (i.e., relative entropy function) and $g_{0}({\nu_{m}})$ (i.e., negative entropy function) are convex functions. Since (15a) is the difference of convex functions while constraints (15b) and (15c) are affine functions, problem (15) belongs to DC programming problems , which can be efficiently and iteratively addressed by the CCP . The CCP algorithm to obtain the solution to problem (15) is detailed in Algorithm 1. We note that the CCP provably converges to the local optima of DC programming problems . However, due to space limitation, we omit the convergence proof of Algorithm 1 (please refer to for the formal proof).

IV-B Power Allocation

The optimal transmit power in (14b) is computed by

Here, the objective function is strictly convex for ${p}_{m}(t)\geq 0,\forall\,m\in\mathcal{M}$, and the constraints are compact. Therefore, the optimal solution of $\mathbf{P}^{\star}(t)$ exists and is efficiently reached by numerical methods.

After obtaining the optimal auxiliary variable and transmit power, we update the queues ${Q}_{m}(t+1)$ and ${Y}_{m}(t+1)$ as per (4) and (12), respectively.

V Numerical Results

We consider a single-cell massive MIMO systemThe multi-cell scenario raises a problem of additional delay due to the need of information exchange among base stations, which is required by either the coordination scheme or distributed approach. This problem is also a very interesting open topic for future work. in which the MBS, with $N=32$ antennas and $P=38$ dBm, is located at the center of the $0.5\times 0.5$ $\text{km}^{2}$ square area. UEs (from $8$ to $60$ UEs per $\text{km}^{2}$) are randomly deployed within the MBS’s coverage with a minimum MBS-UE distance of $35$ m. Data arrivals follow a Poisson distribution with different means, and the rate requirements are specified as $r_{m}^{\max}=1.2\lambda_{m},r_{m}^{\min}=0.8\lambda_{m},\forall\,m\in\mathcal{M}$. The system bandwidth is 1 GHz. The path loss is modeled as a distance-based path loss with the line-of-sight (LOS) modelWe assume that the probability of LOS communication is very high, while the impact of other channel models is left for future works. for urban environments at $28$ GHz . The maximum delay requirement ${d}^{\text{th}}$ and the target reliability probability $\epsilon$ are set to $10\,\text{ms}$ and $5\%$, respectively. The numerical results are obtained via Monte-Carlo simulations over $10000$ realizations with different channel realizations and UE locations. Furthermore, we compare our proposed scheme with the following baselines:

Baseline $1$ refers to the Lyapunov framework in which the probabilistic latency constraint $\eqref{delayconst1}$ is considered.

Baseline $2$ is a variant of Baseline $1$ without the probabilistic latency constraint $\eqref{delayconst1}$.

In Fig. 2, we report the tail distribution (complementary cumulative distribution function (CCDF)) of latency to showcase how often the system achieves a delay greater than target delay levels. In particular, at $\lambda=2.4$ Gbps, by imposing the probabilistic latency constraint $\eqref{delayconst1}$, our proposed approach and baseline $1$ ensure reliable communication with better guaranteed probabilities, i.e, $\text{Pr}(\text{delay}>7.5\text{ms})<10^{-4}$ and $\text{Pr}(\text{delay}>9.4\text{ms})<10^{-4}$, respectively. In contrast, baseline $2$ violates the latency constraint with a high probability, where $\text{Pr}(\text{delay}>10\text{ms})=74.75\%$.

V-B Impact of User Density

In Fig. 3, we compare the average user throughput (avgUT) and average latency of our proposed approach with the two baselines under the impact of user density. Additionally, we consider the weighted sum rate maximization (WSRM) case without considering queue dynamics, i.e., problem (6) without the constraints (5) and (6). The WSRM case is the conventional way to find the system throughput limit but suffers from higher latency. Since all users share the same resources, the average delay (“solid lines”) increases with the number of users M, whereas the avgUT (“dash lines”) decreases. Fig. 3 further shows that when $\emph{M}>24$, the delay of all schemes increases dramatically and is far-above the latency requirement. Hence, only a limited number of users can be served to guarantee the delay requirement, above which, a tradeoff between latency and network density exists. Our proposed approach achieves better throughput and higher latency reduction than baselines 1 and 2, while the WSRM case has the worst delay performance as expected. Compared with WSRM, our proposed approach maintains at least $87\%$ of the throughput limit, while achieving up to $80\%$ latency reduction. Moreover, our proposed approach reaches Gbps capacity, which represents the capacity improvement brought by the combination of mmWave and massive MIMO techniques. Numerical results show that our approach simultaneously provides order of magnitude capacity improvements and latency reduction.

VI Conclusion

In this letter, we have investigated the problem of mmWave-enabled massive MIMO networks from a latency and reliability standpoint. Specifically, the problem is modeled as a NUM problem subject to the probabilistic latency/reliability constraint and QoS/rate requirement. By incorporating these constraints, we have proposed a dynamic Lyapunov control approach, which adapts to channel variations and system dynamics. Numerical results show that our proposed approach reduces the latency by $28.41\%$ and $77.11\%$ as compared to current baselines.

References