Joint Optimization of Signal Design and Resource Allocation in Wireless D2D Edge Computing

Several existing works have considered resource allocation management in MEC systems. In particular, the problems of wireless scheduling [14, 15], interference management through allocation of different subchannels [16], power allocation [12, 17, 13], and multiple-input multiple-output (MIMO) signal design [18, 19, 20] have each been studied in this context. All of these works, however, have considered the network and communication resources separately, typically focusing on one or the other independently. In reality, these all are interdependent, and there is a need to understand how MEC systems should be designed to jointly optimize link selection and resource management for contemporary edge computing scenarios. Given the single direction offloading to the edge server, MEC systems can be viewed as special cases of the D2D networks that we study in this paper (see Figure 1).

In terms of D2D networks, some research on D2D networks has addressed resource management problems such as subchannel allocation [21], power allocation [22], or both of these jointly [23, 24, 25]. Also, MIMO signal design has been considered in scenarios of D2D communications underlaying cellular networks, in order to mitigate inter-channel interferences [26, 27] or to maximize utility models for cellular users communicating in such scenarios [28]. However, for the above works, the link selection problem is not considered, which is important for network efficiency maximization. Although some recent works addressed link selection [29] together with device power allocation [30] in D2D networks, they in turn have not addressed wireless signal design and subchannel allocation problems.

We develop an optimization methodology for network and communication resource management in wireless D2D networks, considering the problems of link selection, subchannel allocation, power allocation, and MIMO signal design jointly. Our approach is driven by two design principles:

The contributions of this paper are as follows:

• •

  We formulate a novel optimization problem for D2D edge computing networks that minimizes the total energy consumption required to process given data samples at the D2D edge nodes, based on a framework for joint wireless MIMO signal design, resource allocation (subchannel and power allocation), and link selection (Section II).

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  We decompose the integrated framework into two subproblems to make it computationally tractable: one for network resource allocation and the other for MIMO signal design. This facilitates the network resource optimization with integer variables and MIMO signal design optimization with non-integer variables (Section II).

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  In the optimization of network resource allocation, we jointly determine the link with subchannel allocation. For solving this problem, we modify our objective function and exploit an integer programming. We propose a greedy algorithm as another solution to reduce the computational complexity (Section III).

• •

  With respect to MIMO signal design optimization, we determine the transmit beamformer, transmit power and receive combiner by exploiting interference coordination techniques for MIMO systems. (Section III). The energy minimization problem is converted to the weighted minimum mean square error (WMMSE) minimization problem for tractability (Section III).

• •

  Extensive simulations are presented to evaluate the integrated framework and our proposed algorithms (Section IV).

Figure 2 demonstrates a wireless D2D network model of $K$ nodes that can transmit using multiple antennas.

We assume that each task should be processed as a whole, i.e., the task is divisible. In other words, If $a_{k,j}=1$, then $a_{k,k^{\prime}}=0$, since the node $k$ will not transmit its data to two or more nodes. We denote the set of link pairs attempting to transmit data as

with $(k,k^{\prime})_{n}=(k_{n},k^{\prime}_{n})$ denoting the $n$-th pair of transmit and receive node where $n=1,...,L$, and $L=|\mathcal{A}|$ being the total number of link pairs. For example, if we have the connected link from node 3 to node 1 and it is the first link pair $(n=1)$, then $(k_{1},k^{\prime}_{1})=(3,1)$.

Ideally, the total energy consumption with data offloading in wireless D2D network should be minimized. To illustrate this, consider a three node example where at most one link can be connected. For $k=1,2,3$, each node is assumed to have the same length of data $I_{k}=10$ Mbits, computing power $F_{k}=1$ Watt, transmission power $P_{k}=1$ Watt. The transmission data rate is assumed as $R_{k,k^{\prime}}=2$ Mbits/sec for $k\neq k^{\prime}$. However, we consider different computing speeds, $C_{1}=10,C_{2}=2$, and $C_{3}=1$ Mbits/sec. Using these definitions, the total energy consumption for local computation should be $E_{P,local}=F_{1}(I_{1}/C_{1})+F_{2}(I_{2}/C_{2})+F_{3}(I_{3}/C_{3})=16$ Joules.

The objective is to minimize total energy to find best link pair among 6 possible links: $(1,2)$, $(1,3)$, $(2,1)$, $(2,3)$, $(3,1)$, and $(3,2)$. It is noteworthy that the link must be properly selected to minimize the total energy. Suppose we have link $(3,1)$. Then the total energy including transmission energy should be $E_{P,31}=F_{1}(I_{1}/C_{1})+F_{2}(I_{2}/C_{2})+(P_{3}(I_{3}/R_{3,1})+F_{1}(I_{3}/C_{1}))=12$ Joules, which is less than the local computation energy $E_{P,local}$. This implies that if transmission channel is available and energy-efficient, we can exploit the communication to minimize total energy for data processing. If we consider link $(2,1)$ instead of link $(3,1)$, the total energy would be $E_{P,21}=F_{1}(I_{1}/C_{1})+(P_{2}(I_{2}/R_{2,1})+F_{1}(I_{2}/C_{1}))+F_{3}(I_{3}/C_{3})=17$ Joules, which is larger than local computation energy $E_{P,local}$. Further, we are aware that link $(3,1)$ is the best link pair for minimizing total energy consumption with the relationship $E_{P,31}<E_{P,local}<E_{P,kk^{\prime}}$ for any $(k,k^{\prime})\neq(3,1)$, where $E_{P,kk^{\prime}}$ is the total energy consumption with the link pair $(k,k^{\prime})$. Therefore, the link selection optimization is essential for minimizing total energy consumption. Although this example assumed only one link and identical data rates of the nodes, we are aware that minimizing the inter-channel interferences in many links environment are critical to data rate maximization for selected links, which will lead to total energy minimization.

We are interested in the combination of MIMO signal design parameters – i.e., transmit powers $\{P_{k}\}$, transmit beamformers $\{{\bf{f}}_{k}\}$, and receive combiners $\{{\bf{z}}_{k^{\prime}}\}$ – and network resource allocation parameters – i.e., transmission link pairs $\{a_{k,k^{\prime}}\}$ and subchannel allocations $\{m_{n,i}\}$ – that will lead to processing the network data $I_{1},...,I_{K}$ with minimum energy consumption. We can formulate the following optimization problem:

Constraints (7-9) relate to the MIMO signal design variables, as described in Sec. II-A. In (7), since $R_{k,k^{\prime}}$ is only defined when $a_{k,k^{\prime}}=1$, the equation (7) can be considered as $R_{k_{n},k^{\prime}_{n}}$ for $n=1,...,L$ as in (2). In (9), we impose an overall transmission power budget $P$ that the network cannot exceed. Constraints (10-13), by contrast, restrict the link pair and subchannel allocations, as follows:

Assuming all nodes have $N$ transmit and receive antennas, i.e., $N=N_{k}^{\text{Tx}}=N_{k}^{\text{Rx}}$ for all $k$, the optimization (6-13) is a mixed integer program (MIP) with $(2N+1)K$ non-integer and $K(K-1)(1+S)$ integer variables. Also, since the term $P_{k}\cdot I_{k}/R_{k,k^{\prime}}$ as a function of $P_{k}$ has the form $x/\log(x)$, which is non-convex, the problem is a non-convex MIP. Existing solvers for non-convex MIPs do not scale well with the number of variables [33], and even with $K=10$ nodes equipped with $N=5$ antennas and $S=10$ subchannels, our problem already has more than 1000 variables. One of the challenges we address in Sec. III is developing an effective algorithm to solve this problem. Shown in Figure 3, as a first step, we decompose it into two interrelated subproblems, one for the network resource allocation variables (integer) and the other for the MIMO signal design variables (non-integer) as follows:

We propose two different algorithms to solve for $\{a_{k,k^{\prime}}\}$ and $\{m_{n,i}\}$ with $\{{\bf{f}}_{k}\}$, $\{{\bf{z}}_{k^{\prime}}\}$ and $\{P_{k}\}$ fixed.

We turn to the optimization of the MIMO signal design variables, $\{{\bf{f}}_{k}\}$, $\{{\bf{z}}_{k^{\prime}}\}$ and $\{P_{k}\}$ with $\{a_{k,k^{\prime}}\}$ and $\{m_{n,i}\}$ fixed. Instead of minimizing (2), we desire to convert the problem to familiar problem and find the sub-optimal solution for communication demand. We will use the relationship $\sum_{i}{(x_{i}y_{i})}\leq(\sum_{i}{x_{i}})(\sum_{i}y_{i})$ for $x_{i},y_{i}\geq 0$, and minimize the upper bound for the communication demand in (2), $E^{up}_{M}$, which is given as

where $P$ is the total transmission power budget in (9). The problem in (21) is simplified to a problem for minimizing the time delay, which has a similar form as maximizing harmonic-rate [36]. Combining the power $P_{k_{n}}$ and ${{\bf{f}}_{k_{n}}}$ into ${{\bf{g}}_{k_{n}}}$, i.e., ${{\bf{g}}_{k_{n}}}=\sqrt{{P_{k_{n}}}}{{\bf{f}}_{k_{n}}}$, leads to

where ${R_{k_{n},k^{\prime}_{n}}}$ is given in (3) and the constraint (23) comes from (8) and (9). This problem is non-convex and NP-hard. Therefore, suboptimal solution can be found by alternatively solving for the variables in (24).

For resource allocation, we can apply one of our two proposed schemes, the non-linear integer programming (NLIP) in III-A1 or greedy algorithm in III-A2. For MIMO signal design, the WMMSE algorithm in III-B is used. Therefore, we evaluate two combinations of algorithm, NLIP–WMMSE and Greedy–WMMSE, which will be denoted by NLIP and Greedy in Figs. 5-7. The decomposed optimization depends on initial values of $\{{\bf{f}}_{k}\}$, $\{{\bf{z}}_{k^{\prime}}\}$, $\{P_{k}\}$, $\{a_{k,k^{\prime}}\}$, and $\{m_{n,i}\}$. For our proposed algorithm, NLIP–WMMSE and Greedy–WMMSE, we run each of them 10 times with different initializations and keep the mean of the results and the minimizer, called the best in Figure 5-7. The minimizer is selected as the one minimizing the total energy the most. For initializations, the link pair $\{a_{k,k^{\prime}}\}$ is randomly chosen from link cadidate set $\mathcal{X}$, the subchannel $\{m_{n,i}\}$ is allocated to connected links $a_{k,k^{\prime}}=1$ randomly, $P_{k}=P/L_{max}$, and $\{{\bf{f}}_{k}\}$, $\{{\bf{z}}_{k^{\prime}}\}$ are generated to be uniformly distributed on the complex unit sphere [40], where $L_{max}=\lfloor K/2\rfloor$ is the maximum number of link pairs with total $K$ nodes.

For all our simulations, we generated random Gaussian channel ${\bf{H}}_{k,k^{\prime}}$ for $k,k^{\prime}=1,...,K$, and used the bandwidth $W=1$ MHz for each subchannel, and $P=5$ Watt with noise power $\sigma^{2}=1$. As for computational speed $C_{k}$, the general unit Hz (or cycles/s) can be converted to bits/sec according to the application it processes. For example, in the work of the audio signal detection [41], 500 cycles are required to process 1 bit, i.e., 500 cycles/bit. In other words, 500 MHz CPU can be converted to 1 Mbits/sec. As an example, for computation power $F_{k}$ the mobile devices with $500$ MHz CPU require about 0.6 Watt for CPU processing [42]. Based on trying to emulate heterogeneous devices with different energy efficiency, we generate the length of data $I_{k}\sim U(1,20)$ Mbits, computational speed $C_{k}\sim U(0.1,2)$ Mbits/sec, and computing power $F_{k}\sim U(0.5,1)$ Watt for $k=1,...,K$, where $U(a,b)$ is continuous uniform distribution ranging from $a$ to $b$. All nodes are assumed to have $N$ transmit and receive antennas, i.e., $N=N_{k}^{\text{Tx}}=N_{k}^{\text{Rx}}$ for all $k$. For all our simulations, we select the number of antennas based on properness of interference alignment [43], $2N\geq L_{max}+1$.

Figure 4 depicts the relationship among the total energy $E_{P}$, communication energy $E_{M}$, and computation energy $E_{F}$, along the number of links $L$ where $E_{F}$ can be calculated as $E_{F}=E_{P}-E_{M}$. Interestingly, the total energy $E_{P}$ decreases below the local computation energy as the available communication resources (links) are more heavily exploited, while the energy for communication increases and that for computation decreases. In other words, since more nodes participate in offloading, we can further reduce energy consumption upto approximately 17 $\%$, reducing energy from 20 Joules to 16.6 Joules. This simulation was performed by using Greedy-WMMSE, when $K=10$ nodes, $N=6$ antennas for both transmit and receive and $S=3$ subchannels.