Joint Mode Selection and Resource Allocation for Mobile Relay-Aided Device-to-Device Communication

Abstract

Device-to-Device (D2D) communication underlaying cellular networks is a promising add-on component for future radio communication systems. It provides more access opportunities for local device pairs and enhances system throughput (ST), especially when mobile relays (MR) are further enabled to facilitate D2D links when the channel condition of their desired links is unfavorable. However, mutual interference is inevitable due to spectral reuse, and moreover, selecting a suitable transmission mode to benefit the correlated resource allocation (RA) is another difficult problem. We aim to optimize ST of the hybrid system via joint consideration of mode selection (MS) and RA, which includes admission control (AC), power control (PC), channel assignment (CA) and relay selection (RS). However, the original problem is generally NP-hard; therefore, we decompose it into two parts where a hierarchical structure exists: (i) PC is mode-dependent, but its optimality can be perfectly addressed for any given mode with additional AC design to achieve individual quality-of-service requirements. (ii) Based on that optimality, the joint design of MS, CA and RS can be viewed from the graph perspective and transferred into the maximum weighted independent set problem, which is then approximated by our greedy algorithm in polynomial-time. Thanks to the numerical results, we elucidate the efficacy of our mechanism and observe a resulting gain in MR-aided D2D communication.

1. Introduction

Device-to-Device (D2D)1 communication is viewed as a promising add-on component in future radio communication systems [1-3]. It is a novel communication paradigm where an instance of user equipment (UE) transmits data directly to another UE in the vicinity without requiring a base station (BS) as a relay. D2D has the potential to enhance spectral efficiency and improves quality-of-service (QoS). Moreover, it offloads the relay burden from the BS, shortens transmission latency and enables emerging context-aware applications [4]. For these reasons, mainstream telecommunications organizations and companies, such as the 3rd Generation Partnership Project [4-5], the Nokia Research Center [1], the European Union’s Research and Innovation Funding Programme [2] and IEEE 802.15 (Personal Area Networks) [3], are addressing the standardization of D2D communication over licensed bands.

Table 1.The abbreviations used most often in this paper

Unfortunately, intra-cell interference is inevitable as a result of spectral sharing; therefore, mitigating severe cross-tier interference while exploring the proximate gain residing in adjacent communicating UEs is a key challenge when integrating D2D communication into cellular networks. Resource allocation (RA) has proven to be efficient for addressing the cross-tier interference issue and enhancing both system-wide and individual performance. In addition, apart from conventional RA techniques, e.g., admission control (AC), power control (PC) and channel assignment (CA), mode selection (MS) is introduced to each applying D2D link (DL), i.e., a D2D transmitter (DT) and its corresponding D2D receiver (DR), as a novel resource. Three transmission modes are available as explained below.

MS design ushers in new diversity and requires careful cooperation with conventional RA techniques, and thus has motivated many studies in the recent literature. [6-11] focused on the scenario involving one CL and one DL. Specifically, [6-7] maximized system throughput (ST) of two such links with QoS protection only for the prioritized CL via joint MS and PC. Those results were extended [8-11] by introducing further fixed relays to facilitate the performance of the CL. Considering Mode 2 and Mode 3 under fixed power settings, [8] found that Mode 3 was preferable when the received strength of the desired signal at DR exceeded a threshold, but the study neglected the interference imposed from DL to CL. Compared to [8], PC was complemented in [9] for ST maximization under the same simplified interference model. The preceding drawback was resolved in [10], where only MS was considered using a fixed power setting. The QoS protection of the DL was first raised in [11], in which MS and PC were jointly designed to optimize both ST and energy conservation (EC) under a given data rate constraint, but the imposed interference from DL to CL was also ignored as in [8-9]. The investigation was expanded in [12-20] to a multiuser scenario that further explored the gain when the allocations of multidimensional resources were in cooperation. Specifically, [12-13] considered a code-division-multiple-access based system and optimized EC [12] and energy efficiency (EE) [13], respectively, by focusing on MS and PC, but as the problems were NP-hard, they derived the optimality via an exhaustive brute-force search and did not fully emphasize the feasibility check or AC. A lightly loaded system was considered in [14-15], but that study accounted for the selection only between Mode 1 and Mode 2, i.e., each channel could be only assigned to at most one link. An attempt to maximize EC under individual QoS requirements by joint MS, CA and PC was made in [14], which proposed a heuristic algorithm to solve the originally NP-hard problem. In contrast, [15] modeled the joint MS and CA problem into a stationary stochastic process under time-varying channel conditions, which was optimally solved by an opportunistic sub-channel scheduling algorithm. Compared to [12-15], [16-20] dealt with all three modes. In particular, [16-17] came up with a two-step mechanism to determine the MS based on partial path-gain information in step one and to adjust the RA of each link individually using a specified priority order in step two. The joint MS, CA and PC problem for ST was decomposed in [18] into two parts: PC optimization as in [21], and the joint MS and CA, which was transformed into an integer linear programming (ILP) task. Again, this was generally NP-hard; consequently, two heuristic algorithms were devised to approximate the optimality according to system load level. The study by [19] can be viewed as an enhanced version of [16-17] and intended to maximize EC, while MS and RA of each link was performed progressively based on the superiority of channel conditions and the gap between the currently achieved data rate and the individual QoS requirement. Another study [20] also accounted for modulation and coding schemes compared to [16-19]. Their solution was made up of three components with the different variables optimized sequentially. Furthermore, the precoding, PC and CA of underlaid DLs were jointly studied in [22] from the game perspective in the uplink multiple inputs and multiple outputs (MIMO) cellular network. In contrast to the scheme in [22], the above variables were merged into one single variable, i.e., PORAVdm, in [23] for a multi-cell MIMOcellular system, and the new variable maintained the group sparse structure, which could be exploited to derive a stationary solution using advanced information theory. But, the greedy optimizations of sum data rate or ST in [22-23] will cause the serious fairness problem due to the absence of QoS protection for the underlaid DLs.

Inspired by [6-23], it is beneficial to incorporate multidimensional resource variables into the optimization model, but any gain will decrease when the distance between DLs rises, as shown in [18]; thus, Mode 1 and Mode 2 become more preferable (as in [14-15]) even under high system load levels, which results not only in poor spectral efficiency but also in a small number of admitted links. As the number of UEs has increased exponentially during recent years [1-3], it may be wise to extend the range of D2D communication via mobile relays (MR) [24-25] to allow more admissions and spectral sharing opportunities for DLs. Consequently, the design of MR-aided DLs has inspired another line of research as a means of cooperative communication [26], and introduced two additional transmission modes.

When MRs are enabled, the MS mechanisms proposed in [6-20] are no longer efficient, and new questions are raised, e.g., determining when MR-aided modes are preferable and which MR should be chosen. In addition, when Mode 4 or Mode 5 is selected, there are choices to be made about which channel and how much power should be assigned to the two-hop links described above under a given optimization target with additional individual QoS requirement. In [27], a distributed relay selection (RS) scheme was proposed to maximize EC together with PC and CA, but that study considered only one DL and only Mode 3 and Mode 5. In [28], a RS scheme was proposed to reduce the averaged transmission time delays from a cross-layer perspective in a multiuser scenario, but the QoS protection was not fully stressed due to the absence of PC, and only two modes were involved, as in [27]. Another study [29] analyzed the outage probability of Mode 5 in a two-way amplify-and-forward (AF) relay protocol [26] with fixed power settings. The achievable transmission capacity of DLs under a QoS requirement was minimized in [30] with respect to the power ratio between the CLs and DLs and a relay selection range based on stochastic geometry. However, only two modes were involved, just as in [27-28] with the decode-and-forward (DF) relay protocol [26]. In addition, the above performance was derived under an aloha-like multiple access protocol, which was not aligned with cellular system standardization, e.g., long term evolution (LTE) applies a channelized multiple access protocol for better interference coordination with a central controller (BS) for multiuser scheduling. The performance of two-hop D2D communications was evaluated in [31], which devised a heuristic MS along with a distributed PC to balance the optimization of ST and EC, but the design of both RS and CA was omitted.

Motivated by [6-23,27-31], our goal is to present a comprehensive view for the joint optimization of MS and RA by including AC, PC, CA and RS in our hybrid cellular system when MR is further enabled to facilitate DLs (under either the AF or DF relay protocol) in case the channel condition of their desired link is unfavorable. Our contributions are threefold:

Fig. 1.An illustration of the scenario and content in this paper

The remainder of this paper is organized as follows: In Section 2, we briefly introduce the system model and performance evaluation in all five available transmission modes, followed by the formulation of our joint MS and RA problem. In Section 3, we decompose the original problem into two parts and address the optimality of each part in sequence. Section 4 discusses the extensive simulations we conducted to demonstrate the efficacy of our joint optimization mechanism. Finally, we draw the conclusions in Section 5.

 

2. Model Framework

2.1 System Model

We consider an isolated cell by assuming that the inter-cell interference is well-controlled via cell planning [33] and that only uplink channels are available to be reused by DLs. We denote the number of channels as N, where the first M(M ≤ N) have been pre-assigned to CLs using scheduling that is out of the scope of this paper. Therefore, the system load level is defined as η = M / N, and there are L= N − M remaining vacant channels. There are J DLs trying to get admitted into the system using one of the five available transmission modes as explained in Section 1. Moreover, there are K vacant MRs with the potential to provide relay service for DLs. Compared to Layer 3 fixed relays [34], MRs are both energy- and function-limited, so it is reasonable to assume that each MR can provide the relay service for at most one flow at a time, and channel reassignment is forbidden, i.e., the same channel must be used during the two-hop transmission. In addition, to avoid the negative impact of dense spectral sharing and various signaling measurements among multiple UEs [35-36], we add the restrictions that one channel can be reused by at most one DL and that one DL can reuse at most one channel. Finally, the entire set of channels and vacant channels can be defined as H = {1,...,N} and V, respectively, so if M < N, V = {M + 1,...,N}; otherwise, V = ∅. We also assume that the label of a CL is the same as that of the channel it is assigned to, so the set of CLs is C = {1,...,M}, while the set of DLs and MRs are D = {1,...,J} and R = {1,...,K}, respectively. Finally, Fig. 1 shows an illustration of our system model where three local pairs (DL1-3) are requesting for admission, and five MRs (MR1-5) are available to provide two-hop transmission for adjacent DLs, and three CLs (CL1-3) are already in service and pre-assigned to channels 1-3, respectively, and channels 4-5 are unused. The major symbols used in this paper are listed in Table 2.

Table 2.The major symbols used in this paper

We assume all the involved links experience independent block fading, and the overall path gain from the transmitter of the m-th CL to the BS is defined as is the large-scale distance-dependent path gain, and includes antenna gains at both transmitter and receiver, and is the small-scale channel fading component. In a like manner, the overall path gain from the transmitter of the m-th CL to the j-th DR and to the k-th MR can be defined as respectively. Moreover, the path gain from the j-th DT to its desired DR, to the BS and to the k-th MR, and those from the k-th MR to the j-th DR and to the BS, as well as that from the BS to the j-th DR, are denoted as respectively.

We assume that the BS is able to obtain full channel state information, which can be done by decoding the sounding reference signal in LTE based cellular networks. Consequently, we are dedicated to designing centralized joint MS and RA mechanisms for better performance and to setting an upper bound benchmark for distributed ones.

2.2 Performance Evaluation of All Five Available Transmission Modes

In this section, we evaluate the contribution of a DL to ST in any transmission mode. To avoid loss of generality, we consider only the j-th DL (j∈D). A transmission frame is divided into two slots. If Mode 2 and Mode 3 are chosen, the direct link occupies both slots; otherwise, the DT transmits to the relay node (the BS or MR) in the first slot, while the corresponding relay node transmits to the DR in the second slot.

A. Mode 1: Cellular mode. The j-th DL transmits to the BS on the l-th vacant uplink channel (l∈V) in the first slot, and the BS decodes and forwards the data to the corresponding DR on a downlink channel in the second slot. We assume that when the j-th DL is in cellular mode, its channel will not be reused by other DLs. Therefore, the signal-to-interference-plus-noise ratio (SINR) of the first hop is then reduced to the signal-to-noise ratio (SNR), denoted as and defined in (1-a) below,

where is the transmit power of the j-th DT, and σ2 is the addictive white Gaussian noise (AWGN) at each receiver. Similarly, the SINR of the second hop can also be reduced to the SNR as defined in (1-b), due to the orthogonality in downlink cellular scheduling, where is the transmit power of the BS. To guarantee a successful transmission, a SINR threshold, i.e., should be satisfied for each hop and achieved by analyzing the QoS class identifier in the user bearer. Finally, if then the data rate of the j-th DL under Mode 1 can be denoted as in (2); otherwise, indicating infeasibility.

In general, the BS is equipped with a much larger power budget than UEs, i.e., 46 dBm at the BS compared to 23 dBm at a UE in the LTE system. Thus we assume the downlink rate can always satisfy can be simplified into (3) as in [12-20].

B. Mode 2: Overlaid direct mode. The j-th DT transmits to its corresponding DR using both slots on the l-th vacant channel (l∈V). The SNR of a single-hop transmission can be denoted as in (4). If then the data rate under Mode 2 can be defined in (5) as otherwise, indicating infeasibility.

C. Mode 3: Underlaid direct mode. The j-th DT transmits to its corresponding DR using both slots by reusing the same channel with the m-th CL (m∈C). In this way, mutual interference is imposed between the two links, and their SINRs can be defined in (6) as respectively, where is the transmit power of the m-th CL. Apart from is introduced to guarantee the QoS requirement for the prioritized CL, and if the data rate under Mode 3 is defined in (7) as otherwise, indicating infeasibility.

D. Mode 4: Overlaid MR-aided mode. The j-th DT transmits to the k-th MR (k∈R) in the first slot, while the k-th MR relays the data to the j-th DR in the second hop, and both hops occupy the same l-th vacant channel (l∈V). Thus the SNR of each hop can be defined in (8) as respectively,

where is the transmit power of the k-th MR in the second hop. Then, the data rate of the two-hop transmission can be classified into one of the following two cases based on the relay protocol implemented at the MRs:

(i) Case 1: under the AF relay protocol. If the data rate under Mode 4 can be defined in (9) as [26]; otherwise, indicating infeasibility.

(ii) Case 2: under the DF relay protocol. If the data rate under Mode 4 can be defined in (10) as [26]; otherwise, indicating infeasibility.

E. Mode 5: Underlaid MR-aided mode. The j-th DT transmits to the k-th MR (k∈R) in the first slot, while the k-th MR relays the data to the j-th DR in the second hop, and both hops occupy the same channel with the m-th CL (m∈C). To create more opportunity for successful transmission, we assume that the CL occurs in only one of the two slots, thus the performance of this mode could be divided into one of the following four cases depending on the hop occupied by the m-th CL and the relay protocol implemented at the MRs.

When the m-th CL occupies the first hop, the SINR of the first hop in the j-th DL and the m-th CL can be defined in (11) as respectively, while the SNR of the second hop in the j-th DL is similar to that in (8-b) and defined in (12) as

(i) Case 1: under AF relay protocol. If the data rate under Mode 5 is defined in (13) as otherwise, indicating infeasibility.

(ii) Case 2: under the DF relay protocol. If the data rate under Mode 5 is defined in (14) as otherwise, indicating infeasibility.

When the m-th CL occupies the second hop, the SINR of the second hop in the j-th DL and the m-th CL can be defined in (15) as respectively, while the SNR of the first hop in the j-th DL is similar to that in (8-a) and defined in (16) as

(iii) Case 3: under the AF relay protocol. If the data rate under Mode 5 is defined in (17) as otherwise, indicating infeasibility.

(iv) Case 4: under DF relay protocol. If the data rate under Mode 5 is defined in (18) as otherwise, indicating infeasibility.

2.3 Problem Formulation

To give a simplified form of the original problem, we first introduce binary variables xj,a,n with respect to the triple (j,a,n) ∈ D × A × H, where A = {1,...,L,L+1,....,L,L+J,L+J+1,...,L+J+K} is used to denote the set of relay nodes, and two direct modes, i.e., Mode 2 and Mode 3, can be viewed as two special cases when DR coincides with the relay node. We assume that if a ∈ A1 = {1,...,L}, it represents the case when BS is the relay node, i.e., in Mode 1, and if a ∈ A2 = {L+1,...,L+J}, it represents the case when the (a-L)-th DR is the relay node, i.e., in direct modes (Mode 2 and Mode 3); otherwise, i.e., if a ∈ A3 = {L+J+1,...,L+J+K}, it represents the case when the (a-L-J)-th MR is the relay node, i.e., in the MR-aided relay modes (Mode 4 and Mode 5). Finally, if the j-th DL communicates on the n-th channel via the a-th relay node, then xj,a,n = 1; otherwise, xj,a,n = 0. Moreover, Rj,a,n is introduced to denote the data rate related to the triple (j,a,n) ∈ D × A × H. In this way, the joint MS and RA for ST maximization can be formulated as Problem 1 (P1) in (19),

where (19-a) conveys that we aim to maximize ST with respect to xj,a,n and and the designs of MS, CA and RS have been included in xj,a,n. It’s noteworthy that the individual QoS requirement has been implied in the value of Rj,a,n, and specifically, if feasibility is violated, Rj,a,n = −∞ as described in Section 2.2. (19-b) shows the binary feature of xj,a,n, and the constraints from (19-c) to (19-e) limit a DL to occupying at most one flow at a time, each relay node can serve at most one flow at a time, and dense spectral sharing is forbidden, respectively. Finally, (19-f) represents the local power budgets at the DTs, CLs and MRs. Apparently, P1 requires mixed-integer non-linear programming, which is in general NP-hard, thus efficient mechanisms are preferable for achieving a tradeoff between performance and complexity.

 

3. Joint MS and RA Mechanism

Because the constraints in (19) are separable between xj,a,n and we can switch the order of the two maximizations in (19-a) and derive the compound optimality in sequence, i.e., Therefore, P1 can be decomposed into two sub-problems: The one only pertinent to as Problem 2 (P2) in (20), and the one only pertinent to xj,a,n as Problem 3 (P3) in (21). Specifically, P2 is intended to achieve the optimal Rj,a,n for any given triple (j,a,n) ∈ D × A × H, which is then used as a weight value to variable xj,a,n in P3, i.e., in (21-a). Thus, the above two sub-problems maintain a hierarchical structure, and P3 optimizes xj,a,n based upon the optimality of P2. In what follows, we are going to elaborate on the design for each sub-problem sequentially as follows: (i) investigating how to derive all feasible by PC with AC, and (ii) using all , investigating how to effectively solve the constrained ILP for the joint MS, CA and RS.

3.1 The Optimization of PC with Additional AC

Under any given triple (j,a,n) ∈ D × A × H, the optimality of P2 is mode-dependent and highly coupled with both the spectral sharing condition and the selected relay node as well as with its adopted relay protocol. So, we address the optimality of PC and AC separately for each mode.

3.1.1 The Modes on Dedicated Channels (Mode 1, Mode 2 and Mode 4)

Because mutual interference is canceled in this case, the optimal data rate is achieved when the DT and/or the MR emits at its maximum power on the condition that feasibility is guaranteed.

For the triple (j,a,n) ∈ D × A × H, if a ∈ A1 = {1,...,L}, n ∈ V, this means that the j-th DL is in Mode 1, and with (1-a) and (3) if otherwise. Moreover, if a = L+j ∈ A2 = {L+1,...,L+J}, n ∈ V, then the j-th DL is in Mode 2, and with (4) and (5) if otherwise. Finally, if a = L+J+k ∈ A3 = {L+J+1,...,L+J+K}, n ∈ V, it means that the j-th DL is in Mode 4 with data relayed via the k-th MR. When AF is applied, with (8) and (9) if where otherwise, and when DF is applied, with (8) and (10) if otherwise.

3.1.2 The Modes on Reused Channels (Mode 3 and Mode 5)

A. Mode 3: For the triple (j,a,n) ∈ D × A × H, if a = L+j ∈ A2, n ∈ C, then the j-th DL chooses underlaid direct mode, and P2 is specified into Problem 4 (P4). We first check its feasibility, i.e., whether the constraint set is non-empty or the SINR requirement in both (22-d) and (22-e) can be achieved within (22-b) and (22-c), which can be summarized in Proposition 1. Then, we proceed to derive its optimality upon above feasibility as concluded in Proposition 2.

Proposition 1. P4 is feasible if where and

Proof. As one’s transmit power is proportional to its SINR, we can derive the Pareto-optimal power for EC by forcing (22-d) and (22-e) to take equality as in (23). From (23-b), we have and to make it achievable within for any given b1>0 and By substituting in (23-a), we have and to satisfy thus requires

Proposition 2. When feasibility is guaranteed as claimed in Proposition 1, the optimal power pair, i.e., to P4 lies within P,X,Q,Y,I, which is both feasible and reaps the largest in (22-a), i.e., with and

Proof. Based on [37], for any feasible power pair is strictly increasing in α > 1 when is still feasible with the help of (24), thus the optimal is obtained when at least one of the upper bound constraints in (22-b) and (22-c) is activated. Therefore, in what follows, we proceed to analyze the properties of and By the monotonicity of log2(·) function, is reduced to so we derive the first- and second-order derivatives of ψ in (25), if and ψ is strictly increasing in otherwise, and ψ is thus convex in . Due to the symmetry, is also either strictly increasing or convex in In conclusion, can lie only at corner points of the feasible zone when and the structure of the optimality can be categorized into the three cases illustrated in Fig. 2, where the shadow region is the feasible zone constructed by (23) and Specifically, when as shown in Fig. 2(a), where at point Y, the CL tends to impose a strong interference to the DL, i.e., a small lies within P,Q. In contrast, when as shown Fig. 2(b), where at point X, the above mutual interference is weak, i.e., both are small, and lies within P,Y,I. Finally, when as shown in Fig. 2(c), the DL tends to impose a strong interference to the CL, i.e., a small and a large and lies within X,Y. ■

Fig. 2.Structure of optimality in P4 for Mode 3

B. Mode 5: For the triple (j,a,n) ∈ D × A × H, if a = L + J + k ∈ A3, n ∈ C, then the j-th DL chooses the underlaid MR-aided mode with data relayed via the k-th MR. When the AF relay protocol is adopted at the MRs and the n-th CL occupies the first slot, P2 is specified into Problem 5 (P5), and the feasible condition and the optimality upon feasibility can be jointly summarized below, in Proposition 3.

Proposition 3: P5 is feasible if and where with Upon feasibility, the optimal power pair to P5 lies within P',X',Q',Y',I' with which is both feasible and reaps the largest

Proof. We note that the objective function is strictly increasing in thus when optimal. If we define P5 can be reduced to (27).

Compared to P4, (27) has an additional constraint (27-e), but it is independent to thus Proposition 1 can be readily extended here for the feasibility check only by switching respectively, in b1,b2,b3, and further limiting that (27-e) is maintained. Moreover, compared to P4, (27) has a different objective function, but it can be readily verified that (27-a) has the same properties as (22-a), elaborated in Proposition 2 as follows: (i) For any feasible power pair is still strictly increasing in α > 1 when is still feasible with the help of (24) and ∂log2(1+βx/(x+β+1))/∂x>0, and (ii) for we can derive the first-order derivative as in (28), where and if (29) is valid, otherwise, we can readily verify that and we omit the math for brevity. In contrast to Proposition 2, the symmetry is violated, so we proceed to analyze the property of the first-order derivative of which is shown in (30), where The minimum value of is obtained when so if (31) is valid, otherwise, we can readily verify that and we omit the math for brevity. Therefore, both are either monotone or convex, thus the optimality to P5 adopts the identical structure of that in P4 only by switching the channel coefficients at the potential optimal points P',X',Q',Y',I'. Similar to Fig. 2, we first filter out the feasible points from P',X',Q',Y',I', and then find out the one with the maximum objective function in (27-a). The details are omitted for brevity.■

Remark 1. When the n-th CL occupies the second slot, P2 can be specified similarly to P5. Compared to (27), the channel coefficients between the two interfering links are changed, but the structure of the optimality stays the same, so conclusions on the feasibility check and the optimality when the n-th CL occupies the second slot have the same form as Proposition 3 only by changing into respectively, in c1,c2,c3 and P',X',Q',Y',I' after In sum, if we denote the optimal power pairs in the above two cases as respectively, then the optimal data rate of Mode 5 under the AF relay protocol is

On the other hand, when the DF relay protocol is adopted at the MRs and the n-th CL occupies the first slot, P2 is specified into Problem 6 (P6), and the feasible condition and the optimality upon feasibility can be jointly summarized below, in Proposition 4.

Proposition 4: P6 is feasible if and where and Upon feasibility, the optimal power pair for P6 lies within P",X",Q",Y",S",Z",I" with which is both feasible and reaps the largest in (33-a), i.e.,

Proof. Due to the characteristics of DF-based two-hop transmissions (i.e., the maximum data rate of the second hop is achieved when and the performance of the two-hop transmission is determined by the hop with the worst data rate), P6 can be reformulated below,

Compared to P4, an additional upper bound constraint is imposed on SINR i.e., in (33-d), but it is always valid after so the conclusion of the feasibility check in P4 can be directly extended here only by switching respectively, on the condition that (33-e) is further satisfied. Moreover, the objective function in (33-a) is in the same form as (22-a) in P4, so similar to Proposition 2, both are either monotonically increasing or convex in respectively. Finally, the optimal power pair can lie only at corner points of the feasible zone when lies within the points P",X",Q",Y",S",Z",I", which is both feasible and reaps the largest in (33-a), i.e., Specifically, if where are the at points P",S", constraint (33-e) is redundant, and the structure of is exactly the same as that in P4, as illustrated in Fig. 3(a) through Fig. 3(c) by switching P,X,Q,Y,I into P",X",Q",Y",I", where the shadow region is the feasible zone constructed by (34) and otherwise, the structure of can be depicted as in Fig. 3(d) through Fig. 3(f). In Fig. 3(d), lies within S",Q". In Fig. 3(e), lies within S",Y",I". In Fig. 3(f), lies within Y",Z".■

Fig. 3.Structure of optimality in P6 for Mode 5 when the DF relay protocol is adopted at MRs

Remark 2. When the n-th CL occupies the second slot, P2 can be specified similarly to P6. Compared to (33), the channel coefficients between the two interfering links are changed, but the structure of the optimality stays the same, so conclusions on the feasibility check and the optimality when the n-th CL occupies the second slot have the same form as Proposition 4 only by switching respectively, in d1,d2,d3 and P",X",Q",Y",S",Z",I" after In sum, if we denote the optimal power pairs in the above two cases as respectively, then the optimal data rate of Mode 5 under the DF relay protocol is

In summary, our original joint MS and RA problem, i.e., P1 in (19), can be perfectly decomposed into two sub-problems, i.e., P2 in (20) and P3 in (21), due to the separability of PC variables and joint MS, CA and RS variables. In addition, the above two sub-problems are highly related and maintain a hierarchical structure, i.e., P3 optimizes xj,a,n based on of P2, so we analyze P2 in front of P3. From (20), we learn that the design of PC and AC in P2 is mode-dependent, thus we address its optimality separately for each of the five available modes as described in Section 2.2. For the modes on dedicated channels, i.e., Modes 1, 2 and 4, the maximum data rate is achieved when all involved transmitters emit at the largest power as articulated in Section 3.1.1. While for the modes on reused channels, i.e., Modes 3 and 5, the optimal power pair for the maximum data rate lies within a finite set of points. In particular, when a DL is in Mode 3, P2 is specified into P4, and the optimality of P4 is concluded in Proposition 2 upon the feasible conditions given in Proposition 1. When a DL is in Mode 5 and the AF relay protocol is applied at its MR, P2 is specified into P5, and the optimality of P5 is summarized with the help of Proposition 3 and Remark 1. Besides, when a DL is in Mode 5 and the DF relay protocol is adopted at its MR, P2 is specified into P6, and the optimality of P6 is outlined with the help of Proposition 4 and Remark 2. Based on the above optimality of P2 under all five available modes, i.e., P1 is reduced to P3 in (21), which is the joint design of MS, CA and RS and will be studied in the following subsection.

3.2 Joint Design of MS, CA and RS

With all feasible hand, P3 is intended to maximize the sum data rate or the ST of all admitted DLs by joint MS, CA and RS under the restrictions that (i) each DL chooses at most one mode and one flow, i.e., constraint (21-c), (ii) each MR serves at most one flow at a time, i.e., constraint (21-d), and (iii) each channel is reused by at most one DL (no dense spectral sharing), i.e., constraint (21-e). In what follows, we view P3 in (21) from the perspective of graph theory, and an undirected weighted graph G = (V,E,W) is formulated as specified below:

It is worth noting that dense spectral sharing is disabled as assumed in Section 2.1 or (21-e), so there is no co-channel interference among admitted DLs, whereas the co-channel cross-tier interference between DLs and CLs does exist and has been amply addressed in Section 3.1 to derive the weights of vertices in our graph model. Then, we give definitions of independent set (IS) and maximum weighted independent set (MWIS) below before our final conclusion on P3.

Definition 1. Given an undirected weighted graph model G = (V,E,W), a subset of vertices, i.e., T ⊆ V, is an IS if there is no edge between any two nodes in T, i.e., (v1,v2) ∉ E, ∀v1 ∈ T, v2 ∈ T-{v1}, and if the sum weight is the maximum among all ISs, T is then called the MWIS in G.

Proposition 5. To derive the optimal solution to P3 is equivalent to searching for the MWIS of our undirected weighted graph as formulated above.

Proof. P3 is in respect to the binary variables xv = xj,a,n, and if xv = 1, vertex v = (j,a,n) is selected; otherwise, vertex v = (j,a,n) is left unselected. In this sense, P3 can be thought of as choosing a subset from V, e.g., T⊆V, and the triples represented by every two different vertices in T do not overlap, as interpreted from the constraints from (21-c) to (21-e); therefore, T falls into an IS of G by the definition of E. To maximize its sum weight as in (21-a), P3 searches for the MWIS of our undirected weighted graph by Definition 1. ■

Unfortunately, MWIS is a well-known NP-complete problem [38], so it is preferable to find a heuristic algorithm to approximate to the optimality with polynomial-time complexity. In this sense, we adopt the greedy scheme proposed in [38] to tackle the above issue, i.e., Algorithm 1 as shown in Table 3, which outputs an IS with a sum weight of at least where NB+(vi) = {vi}∪{vk∈V|(vi,vk)∈E} [38, Theorem 3.11]. In Algorithm 1, the vertices in the final MWIS are selected in an iterative manner. In each iteration, the vertex selection criteria is performed by evaluating a metric, i.e., as specified in Step (i), where the numerator W(v) indicates the weight or the contribution to the overall ST of vertex v, and the denominator indicates the sum weight of its neighbor vertices or the potential loss imposed on the overall ST by the admission of vertex v. Therefore, to achieve a higher ST, it’s preferable to admit the vertex with the largest metric, i.e., v* = argmaxv∈Vmetric(v), as specified in Step (ii). Due to the resource-sharing constraint shown in (21-c) through (21-e), when a vertex is selected, neighboring vertices will have no chance of being selected and are thus eliminated from V as specified in Step (iii). The iteration continues until no vertices are left in the graph, i.e., V = ∅.

Table 3.A greedy algorithm that approximates an answer to our MWIS problem (Algorithm 1)

There are three components that comprise Algorithm 1: (i) The vertices of G are constructed during initiation. The maximum number of vertices is JL+JN+JKN, where JL denotes the vertices for Mode 1, JN denotes the vertices for Modes 2 and 3, and JKN denotes the vertices for Modes 4 and 5. Moreover, with the help of Section 3.1, the optimality of the PC part occurs when either the DT or MR transmits at the maximum power as given in Section 3.1.1 for the modes on dedicated channels, i.e., Modes 1, 2 and 4, or when the transmit power pair of all involved links is within a finite set of points as shown in Section 3.1.2 for the modes on reused channels, i.e., Modes 3 and 5. Therefore, the complexity of the PC part can simply be denoted as O(1), and the sum complexity of component (i) is thus O((JL+JN+JKN)·1) = O(JL+JN+JKN). (ii) The neighbor table of G is constructed during initiation. The complexity of this part is related to the maximum number of nodes from graph theory and on the order of O((JL+JN+JKN)2). (iii) Iterative vertex selection is performed. The main computations at each iteration lie primarily in finding the vertex with the maximum weight, as in Step (ii). In the q-th iteration, the number of vertices is at most (L-q)(L+J+K-q)(N-q), thus the complexity of component (iii) is on the order of where △1 = min{J,N} and △2 = max{L+J+K,N}. Consequently, our algorithm to P3, i.e., Algorithm 1, is solvable in polynomial-time and thus greatly outweighs the exhaustive brute-force search scheme, whose complexity is where derives all possible matchings from D to A, and max{J,N}3 derives the optimal matching from each matching in D × A to N by the Hungary algorithm.

Remark 3. It is noteworthy that our joint resource optimization mechanism can be extended to optimize other targets only by switching (i) the weights in the graph formulation or (ii) the optimization model of P2. Specifically, for any vertex v = (j,a,n), (i) when W(v) is switched into a binary value: If W(v) = 1, it means the j-th DL can get admitted into the system on the n-th channel with data relayed via the a-th relay node; otherwise, it indicates infeasibility. Our mechanism is then changed to maximize the number of admitted DLs; (ii) when we switch the target in P2, i.e., Rj,a,n in (20-a), into and also change the meaning of weights in P3 accordingly, our mechanism is then converted to optimize EC or EE. We leave out the details due to space limitations and simply focus on the optimization of ST.

 

4. Numerical Results

Consider an isolated circular cell with a radius of 300 m, where the BS is located at cell center, and all transmitters are uniformly and independently distributed in the cell area. In addition, each DR is uniformly distributed in the disk area of its corresponding DT with a maximum radius of r. We assume that the BS and each UE is equipped with a single and omni-directional antenna with a processing gain of 14/0 dBi for BS/UEs and that the path loss model is the same as [18]. The bandwidth of each channel is 180 kHz, and the power spectral density of AWGN is -174 dBm/Hz, with a noise figure of 5/9 dB for BS/UEs.

A detailed description of the algorithms compared in this part is given below: (i) Alg1: Forced cellular mode, which is only available on vacant channels. To maximize ST, we need only arrange the DLs in a descending order by for any n ∈ V, and then admit the first ʟ DLs. The complexity of the above process is O(J). (ii) Alg2: Forced D2D mode. The problem for ST maximization can be reduced into a bipartite matching where is the utility for the (j-n) DL-channel pair, and can be optimality solved by the Hungary algorithm with the complexity being O(max(J,N)3). (iii) Alg3: Forced MR-aided mode under the DF relay protocol. (iv) Alg4: Forced MR-aided mode under the AF relay protocol. The problems in the above forced MR-aided modes can be formulated similarly to P1 in (19) only by switching A into A3, and thus can be effectively solved by our mechanism. (v) Alg5: When only cellular mode and D2D mode are enabled, the problem is still NP-hard, but it can be well approximated by the two algorithms in [18] based on the system load level. (vi) Our1: When all five transmission modes are optional and the DF relay protocol is adopted at the MRs, we apply Algorithm 1 to solve P3. (vii) Our2: When all five transmission modes are optional and the AF relay protocol is adopted at the MRs, we apply Algorithm 1 to solve P3. (viii) Opt1/Opt2/Opt3/Opt4: We solve the same problem as in Alg3/Alg4/Our1/Our2 using Mosek, which is a commercial optimization toolbox for complex problems, including the ILP. Finally, we observe the variations in ST and the number of admitted DLs using the change in system parameters as shown from Fig. 4 to Fig. 7. In Fig. 4, we change η from 0 to 1 using a step size of 0.1 by fixing In Fig. 5, we change K from 50 to 150 using a step size of 10 by fixing η = 0.8, r = 200m and In Fig. 6, we change r from 150 m to 250 m using a step size of 10 m by fixing In Fig. 7, we change from 5 mW to 55mWusing a step size of 5 mW by fixing η = 0.8, K = 100, r = 200m. In addition, we set and and all results are averaged over 1000 samples.

Fig. 4.Performance variation with change in system load level

Fig. 5.Performance variation with change in number of MRs

Fig. 6.Performance variation with change in maximum distance of DLs

Fig. 7.Performance variation with change in maximum power of DLs

As depicted in Fig. 4, as η rises, the number of vacant channels decreases; because DLs in forced cellular mode can use only dedicated channels, the number of admitted DLs and the ST in Alg1 drops sharply. Specifically, when η = 1, no DLs can be admitted. As η rises, two changes are introduced that influence the ST for all algorithms except Alg1. First, spectral reuse is enabled, and the data rate of CLs will be accounted in the aggregate throughput as defined in (7), (13), (14), (17) and (18), thus enlarging ST. Second, with the decrease in L, some DLs in unfavorable channel conditions that were originally on dedicated channels are removed from the system, which lowers the ST. Finally, the ST of all algorithms except Alg1 first increases and then decreases according to the degree of the above two influences. Compared to Alg2, Alg3 and Alg4 suffer an inferior ST due to the two-hop feature, but reap a higher number of admitted DLs with the help of the MRs. Compared to Alg4, Alg3 reaps a higher ST under the DF relay protocol at the MRs by comparing (13), (17) and (14), (18), but may suffer from security issues because a UE can decode others’ data. The same phenomenon can also be observed by comparing Our1 and Our2. When η ≤ 0.3, Alg5 achieves almost the same ST as our two schemes because dedicated cellular and direct mode favor contributing a higher ST compared to the MR-aided relay mode. As η rises, spectral sharing is inevitable, and some DLs cannot get admitted under Mode 1 due to channel shortages or under Mode 3 due to severe mutual interference, but they can still be accessed with the help of the MRs, as in our two schemes, i.e., the gain of the MRs. Specifically, when η = 1, Alg5 obtains the same performance as Alg2 because Mode 1 is forbidden, and Our1 or Our2 outwits Alg5 by 24.70% or 17.86% in ST, respectively, and by 63.37% or 50.05% in the number of admitted DLs, respectively. Moreover, Alg3 or Alg4 harvests almost the same number of admitted DLs as Our1 or Our2, but suffers a loss of 45.25% or 53.10% in ST, respectively, due to the two-hop feature. Algorithm 1 is effective in solving the ILP in P3, while Alg3/Alg4/Our1/Our2 suffer averaged losses of only 3.22/3.22/2.25/2.26% in ST, respectively, and 1.71/1.62/2.12/2.37% in the number of admitted DLs, respectively, compared to Opt1/Opt2/Opt3/Opt4.

As shown in Fig. 5, the performances of Alg1, Alg2 and Alg5 are not affected by the change in the number of MRs because MRs are not involved in their calculations. As K increases, it becomes more likely that two distant UEs could be admitted with the relay of an MR, thus contributing to an enlarged ST value in algorithms where the MRs are involved. As delineated in Fig. 6, as the maximum distance of DLs increases, each link suffers a raised path loss, and may fail to achieve its SINR threshold especially in the case of spectral sharing, which results in a reduced number of admitted DLs and an inferior ST using all algorithms except Alg1. As illustrated in Fig. 7, as the maximum power budget of DLs increases, the system tends to maintain a larger feasible zone under Mode 3 and Mode 5; thus all algorithms except Alg1 harvest a rise in the number of admitted DLs, thereby enhancing ST. In sum, from Fig. 5 to Fig. 7, the STs and the number of admitted DLs in Our1 and Our2 vary with different settings of system parameters, but they always outperform the rest by further exploring the benefits of MRs, especially under high system load levels. Moreover, they achieve polynomial-time complexity with only minor performance loss and are applicable when only MR-aided modes are optional, as in Alg3 and Alg4. Specifically, Our1 suffers averaged performance losses of only 2.71/2.55/2.62% in ST and of 4.70/3.86/3.08% in the number of admitted DLs with changes in when compared to Opt3, and verifies the effectiveness of our Algorithm 1 in solving the original ILP in P3, which can also be observed via the comparison between Alg3/Alg4/Our2 and Opt1/Opt2/Opt4. The details are omitted for brevity. Moreover, the performance gap between Alg3 (Alg4) and Opt1 (Opt2) is relatively smaller than that between Our1 (Our2) and Opt3 (Opt4) because the number of vertices in our graph model (searching space) is reduced when A is changed to A3.

 

5. Conclusions

We have investigated the joint design of MS and RA for ST maximization in hybrid cellular networks with MR-aided D2D communication. Due to the high complexity of the problem, we decompose it into two parts and address each part sequentially. It is noteworthy that the above two parts are highly correlated, and the optimality of the PC part helps to construct the graph formulation in the second part. With the help of simulations, we illustrate that our scheme can approximate the optimality in polynomial-time, and observe the gain when MR is enabled to facilitate DLs and multiple resource variables are in elaborate cooperation, which provides more admission and spectral sharing opportunities, especially under high system load levels. In the future, we plan to develop distributed mechanisms and to consider scenarios when dense spectral reuse [35-36] and multi-hop transmission are enabled, which will play an important role in future user-oriented radio communication systems (5G) in improving individual QoS experiences as well as resource utilization, e.g., in the promising D2D-based heterogeneous network architecture for mobile cloud computing [32]. Moreover, we will include (massive) MIMO, i.e., the spatial-dimension resource, into the design of RA, which is another crucial technique addressed in future 5G radio communication systems [2-3,22-23] and complements the D2D-based heterogeneous network architecture [32].