I Introduction
5G cellular communications are embracing mmwave frequencies between and , where the availability of large chunks of untapped bandwidth makes it possible to support stringent data rate requirements for future cellular systems [1]. Carrier frequencies up to with a bandwidth per single carrier up to has been standardized by 3GPP new radio (NR) [2]. Nevertheless, mmwave signals suffer from increased isotropic pathloss and can be severely vulnerable to blockage, which results in outages and intermittent channel quality [3].
The combination of high propagation attenuation and blockage phenomenon advocates for a highdensity deployment of infrastructure nodes [4]. In this regard, heterogeneous networks (HetNets), where a core macrocell seamlessly cooperates with small cells, have been treated as an available realization of network densification. Nevertheless, equipping all small cells with high performance fiberbased backhaul seems to be economically infeasible. As an attractive costefficient substitute to the wired backhaul, selfbackhauling, which has been investigated by 3GPP NR as a part of the integrated access and backhaul (IAB) study item [2], provides coverage extension and capacity expansion to fully exploit the heterogeneity of HetNets [5].
Another encouraging approach to cope with high isotropic pathloss and the sensitivity to blockage effect is the exploitation of beamforming techniques that form narrow beams with high antenna gain for data transmissions [6]. This is possible as the small wavelength, which is one of the distinctive features of mmwave bands, allows a large number of antenna arrays to be placed in a compact form factor. In general, directional antennas with beamforming reduce multiuser interference, where multiple links can be simultaneously transmitted to fully exploit spatial multiplexing gain.
Hereof, how to maximize the performance of HetNets with selfbackhauling and directional transmission becomes an interesting issue, particularly on the design of link scheduling, resource allocation, and path selection. A naive scheduling that lets the macrocell base station (BS) serve all users in a round robin fashion is neither practical nor efficient [7]. By contrast, the limited interference at mmwave bands makes it possible to schedule simultaneous transmissions, where the same radio resource can be allocated to multiple links to improve spatial reuse [8]. At the same time, when the backhaul link, which connects the associated small cell access point (AP) of a user to the macrocell BS, is weaker compared to the backhaul links to other nearby APs, a dynamic multihop routing scheme is much more favorable to improve overall performance.
Ia Related Works
Increasing cellular capacity through selfbackhauled small cells for IAB has become the primary motivation of many previous works. Some studies emphasized the placement of relay nodes inside cells to improve the signal quality at cell edges [9, 10]. More general approaches to enhance cellular capacity with multihop backhauling have been considered in [11, 12]. However, these works limit the number of links at each network node (BS, AP, user equipment (UE), etc.) to a single steerable beam, which does not take advantage of potential spatial reuse provided by highly directional mmwave antenna arrays.
When it comes to the exploitation of spatial reuse, the appropriate design of efficient scheduling policy has been suggested as a key challenge in realizing full benefits of multiplexing gain brought by simultaneous links [13, 14, 15, 16, 17]. Nevertheless, with the exception of some studies that cover limited models [16, 17], none of the aforementioned works to date has considered the degree of spatial link isolation. In other words, the interference in mmwave communications has a much weaker effect than in sub networks, but not negligible. We think that the question of whether the hypothesis of fully isolated pseudowired like links in certain scenarios of mmwave communications is realistic remains open, as also addressed in [11]. The available capacity on simultaneously transmitted link should not be always considered as the same.
Furthermore, recent research efforts, particularly for mmwave HetNets, have addressed versatile aspects of resource allocation, including frequency resource allocation [18], power allocation [19], joint power and time allocation [20], and joint scheduling and power allocation optimization [21]. Besides the consideration of resource allocation, authors in [7] proposed a polynomial time algorithm for joint scheduling and routing, unlike traditional NPhard solutions, by extending the work in [22]. Even though the aforementioned research studies cover a set of resource allocation and routing schemes that enhance system performance under different network configurations and constraint models, the authors focused on either one aspect of resource allocation (e.g., frequency allocation in [18] and power allocation in [19]), or limited combination (transmission duration and power allocation in [21], or scheduling and routing in [7]). The joint solution of scheduling, resource allocation, and routing, have not been considered in any of above works. By contrast, we formulate a joint optimization model with an accurate scheduledependent representation of data rate taking into account the resource allocation and routing to derive our main results.
IB Contributions
In this paper, we apply binary interference classification, i.e., an interference condition that prevents two links from being simultaneously active and is widely used in the literature ([11, 16, 17, 7, 22, 23]) for designing scheduling algorithms, to the joint scheduling, and resource allocation, and routing optimization. Nevertheless, on top of the binary classification, we formulate the data rate of scheduled links as a function that depends on actual signaltointerferenceplusnoiseratio (SINR), where links that experience interlink interference are not blocked completely. Specifically, multiple links are allowed to be simultaneously transmitted provided that mutual interferences among these links are below than a configurable threshold, which is usually considered in practical physical layer implementation. The main contributions of this paper are summarized as follows:

We formulate the optimization problem of joint scheduling and resource allocation (JSRA) for a typical HetNet with multihop IAB structure into a mixed integer nonlinear programing (MINLP) problem, in which the data rate, determined by scheduling, transmission duration, and power constraints, as well as by actual interference, is maximized by fully enabling simultaneous transmission to harvest spatial multiplexing gain.

The constrained optimization problem is then demonstrated to be NPhard. In order to obtain feasible solution, we systematically decompose the problem into three subproblems, namely simultaneous transmission scheduling, transmission duration allocation, and transmission power allocation. Based on this, we propose a heuristic scheduling algorithm referred to as conflict graph maximum independent set (CGMIS) algorithm, a proportional fair transmission duration algorithm, and waterfilling power allocation algorithm to solve the optimization problem based on fixed routing decision.

The fixed routing decision, however, may degrade the system performance when the traffic from different users is congested at some network nodes. Therefore, we propose a dynamic routing (DR) algorithm, where the selection of path between BS and UE depends on realtime network statistics and the user traffic is routed along lightly loaded links, to investigate the ability of further improvement in the data rate achieved by the above scheduling and resource allocation algorithms.

Extensive simulations have been conducted under numerous system parameters to demonstrate that the proposed algorithms outperform benchmark schemes for multiplexing and interference mitigation, in terms of achievable data rate, and closely approach the theoretical optimum, yet with lower latency. The system performance of the proposed algorithms under different frame structures and duplex schemes are also analyzed. In particular, the potential of the proposed algorithms in boosting the system performance of pointtopoint (P2P) communications, has also been studied as an extension to the pointtomultipoint (P2M) scenario for the considered HetNets.
The remainder of this paper is organized as follows: Section II presents the system model and formulates the joint optimization problem of scheduling and resource allocation. The link scheduling algorithm, the transmission duration allocation algorithm, and the transmission power allocation algorithm are proposed in Section III. In Section IV, we introduce the routing algorithm. The performance of the proposed algorithms are evaluated by extensive simulations in Section V, followed by a summary concluding this paper in Section VI.
A conference version of this paper has appeared in [8]. The current paper extends the previous work with the design of routing algorithm and the application of the proposed joint scheduling, resource allocation, and routing algorithms to P2P communications with focus on data delivery in vehicle platoon. The current paper also includes all the derivations, discussion of the extensions, and more detailed simulations.
Ii System Overview and Problem Formulation
In this section, we first introduce the network model, the available connection between BS and UE, and the frame structure that incorporates the concept of spacedivision multiple access (SDMA) group, in which multiple links are simultaneously scheduled, in Section IIA. Then, we show the channel model that is applied to the calculation of achievable link capacity in Section IIB. Finally, the JSRA optimization problem is formulated in Section IIC, where the complexity of the optimization problem is demonstrated to be NPhard, such that we are motivated to develop heuristic algorithms to solve the problem efficiently. The important notations and system parameters defined in this section are summarized in Table I and will be used in the rest of this paper.
Notation  Description  Value 

Frame length  
Pathloss of a link at carrier frequency and distance  See (1)  
Carrier frequency  
Speed of light  
Pathloss exponent  LOS, NLOS:  
SF  Shadowing factor  LOS, NLOS: 
Probability of a link with distance to be LOS  See (2)  
Parameters in model  
Parameters in model  
Antenna gain of link (Antenna array vertical horizontal)  BS/AP, UE: ,  
SINR of link  See (3)  
Transmission power of link  See (18)  
Thermal noise power  
Schedule indicator of link in SDMA group  See (4)  
Channel capacity of link  See (5)  
Allocated bandwidth of link  See (17)  
Number of slots in SDMA group  See (14)  
Set of simultaneously transmitted links from node in SDMA group  See (11)  
Maximum transmission power of BS/AP/UE  for BS/AP, UE: ,  
Interlink interference threshold  
System bandwidth 
Iia Network, Connection, and Frame Structure
We consider a typical HetNet that consists of a macrocell BS, a set of small cell APs, and UEs that are associated either directly with the BS, or with the geographically closest AP and connected to the BS via multihop. Fig. 1 shows an example of the considered HetNet with one BS, two APs and four UEs. We represent the network as a directed graph , where indicates the set of nodes (BS, APs and UEs) and indicates the set of links. For the exampled network in Fig. 1, an abstracted graph model, which we refer to as link graph, is also illustrated in Fig. 1. Admissible connections are BSAP, BSUE, and APUE, with both downlink and uplink traffic flows along arbitrary routes.
As in [3], BS and AP support mmwave bands for backhaul and access transmission and reception (inband backhauling). The transmission requests and corresponding time/frequency synchronization information is assumed to be collected by sub communications. For the data transmission of each UE associated with APs, a predefined route is selected, where we are able to focus on the JSRA optimization. Nevertheless, the design of DR is introduced in Section IV as an extension to further improve the network performance.
We further consider a timedivision duplex mmwave frame structure as shown in Fig. 2, where the system time is divided into consecutive frames with period . Each cycle begins with a beacon phase, followed by a data transmission phase modeled as a slottedbased time period, in which transmissions between any valid pair of nodes can be scheduled.
Timedivision multiple access (TDMA) is widely adopted for mmwave channel access in 5G networks [13, 17, 16], where within the period of each frame, it is assumed that network topology and channel condition remain unchanged. In TDMA scheme, each slot is exclusively occupied by a single link. By enabling the possibility of spatial multiplexing, multiple simultaneous transmissions can be scheduled in each slot. Hence, we can allocate more transmission duration to each link, such that the achievable data rate of each link is improved without other sophisticated techniques. An example of the comparison of TDMA and simultaneous transmission in slot allocation for six transmission links in a frame with a tenslot data phase is illustrated in Fig. 3, where the corresponding number of allocated slots are written below the links.
IiB Channel, Traffic, and Link SINR
We define a SDMA group as a transmission interval that consists of consecutive slots allocated to a link when simultaneous transmission is enabled. For simplicity, in the rest of the paper we use the term “group” to represent the SDMA group. In the example depicted in Fig. 3, the data phase consisting of ten slots is separated in three groups, in which different links are simultaneously scheduled. However, each link can be scheduled only once in a frame.
For an accurate generation of link capacity, we compute the isotropic pathloss for link transmitted from node towards node at distance in meters, denoted as , given by [6] as
(1) 
where link is determined to be LOS or NLOS with probability according to [6, 24] as
(2) 
Here, indicates the carrier frequency in , and represents the pathloss exponent. is the speed of light. The impact of objects such as trees, cars, etc. is modeled separately using the shadowing factor (SF). Further, an additive white Gaussian noise (AWGN) channel is assumed within all links, and the channel knowledge is assumed to be available at the BS.
We further apply similar modeling of antenna gain as in [24] and denote the antenna gain of link as . Unlike the recent works [13, 14, 17], which have assumed a pseudowired behavior for mmwave links, we do not assume that the interference is negligible but rather compute real mutual interference between simultaneous links within each frame. Then, the instantaneous SINR of link , denoted as , is provided by
(3) 
where , , , and represent the transmission power, the antenna gain, the pathloss, and the thermal noise power of link , respectively. The experienced interference of link is model by which summarizes the receive power of all interfered link at link .
IiC Problem Formulation
We consider transmission links to be scheduled in a given frame that consists of slots. For each link , we defined a logic indicator that controls the schedule policy of whether link is scheduled in group . Specifically, the policy is modeled as
(4) 
Then, the actual capacity of link , denoted as , can be represented according to Shannon channel capacity equation as
(5) 
where indicates the available bandwidth at link in . We further assume that the slots of the given frame are allocated into groups, and the number of slots allocated in each group is denoted as .
Define a scheduling policy , a slot allocation policy , and a power allocation policy
as the set of binary vector
, the set of slot allocation , and the set of power allocation , and , respectively. Then, we can define the achievable data rate for all the links in the considered frame with groups, given the scheduling policy , the slot allocation policy , and the power allocation policy , as(6) 
As the total number of slots in each frame is fixed, then the objective function, which is the achievable data rate defined in (6) and to be maximized, can be further simplified as
(7) 
Now, we analyze the system constraints of the optimization problem. First of all, the scheduling policy is ruled by the following two constraints:
(8) 
and
(9) 
Here, (8) indicates that each link can be scheduled only once in each frame (in one of the groups), as demonstrated in Section IIB. However, each link scheduled in group can be allocated with multiple slots, namely . Further, due to halfduplex constraint, the sequential links (e.g. backhaul and access links as edge and in Fig. 1) cannot be scheduled in the same group, which is governed by the constraint demonstrated in (9).
Next, the constraint on the slot allocation policy , where the total number of allocated slots in all groups should not be larger than , is represented as
(10) 
Lastly, the summarized allocated power of the links transmitted from the same node, say node , in group , should not exceed the total available transmission power of the transmitter. Denoting the set of simultaneously transmitted links from node in group as , the constraint on the power allocation policy is given by
(11) 
where indicates the maximum transmission power of BS/AP/UE.
Finally, we can formulate our JSRA optimization problem to maximize the data rate , under the constraints of scheduling policy , slot allocation policy , and power allocation policy , as follows:
Problem 1.
(JSRA optimization)
s.t.  (12) 
The maximization problem indicated in (12) with constraints (8)–(11) is a MINLP problem [25], where the variables are categorized as

Integer variables: ,

Continuous variables: ,
In general, one among the simplest MINLP problem is the 0–1 Knapsack problem, which is a class of problems and proven to be typically NPhard [25]. Nevertheless, the multiplication of the above variables in (12) further complicates the proposed optimization problem 1 and makes it even more complex than the 0–1 Knapsack problem. Specifically, there exists the thirdorder term in (12), in which the coupling exists among the constraints where slot and power allocation policies rely on the results of link scheduling. Therefore, the considered optimization problem 1 is NPhard. In the next section, we propose a heuristic JSRA algorithm to decouple the correlated variables and solve problem 1 efficiently with low complexity.
Iii Scheduling and Resource Allocation Algorithm
As mentioned in Section II, the scheduling policy , the slot allocation policy , and the power allocation policy are three key and correlated terms for solving the optimization problem 1. However, the slot and power allocation policy cannot be determined until the scheduling decision is made. Specifically, the slot allocation depends on the demand of links that are scheduled in each SDMA group, and the number of scheduled links affects the power allocation policy of the corresponding group. Therefore, in this section we propose a heuristic JSRA algorithm to solve problem 1, where simultaneous transmission is fully exploited to increase data rate.
Iiia CGMIS Scheduling Algorithm
To better demonstrate the relationship among different links presented by constraints (8) and (9), we introduce an appropriate concept named conflict graph (CG). Different from the link graph depicted in Fig. 1, in a conflict graph, each node represents one link in the network, and there is an edge connecting two nodes if “conflict” exists. Specifically, the conflicts can be derived from sequential links (e.g., backhaul and access links) as described in (9), due to halfduplex constraint, or from links that interfere with each other severely. For the latter case, we define an interference threshold as the criterion that judges whether an edge exists between two nonsequential links (nodes)^{1}^{1}1The interference information is assumed to be acquired from e.g. interference sensing procedure. This issue, as well as other signaling designs, have been addressed by the other works of us and are beyond the scope of this paper.: An edge exists if the SINR of any of the two corresponding links is larger than . An example of CG construction is illustrated in Fig. 4.
Based on the CG, we propose a maximum independent set (MIS) based scheduling algorithm, namely the CGMIS algorithm, to distribute links into different groups. In graph theory, an independent set is a subset of nodes in a graph, in which no pair of nodes are adjacent. A maximal independent set is either an independent set such that adding any other node to the set forces the set to contain an edge or the set of all nodes of an empty graph. It is demonstrated in [26] that the computational complexity of finding the MIS of a general graph is NPhard, and there are no very efficient algorithms that are capable to find optimal solution in polynomial time. Therefore, we utilize the minimumdegree greedy algorithm to solve the problem of finding the MIS of a CG [27].
We denote the CG as , where and represent the set of nodes and the set of edges in the CG, respectively. For any node , we define its neighbors as , which consists of all adjacent nodes of . The degree of any node is denoted by . Then, the CGMIS scheduling algorithm is summarized in Algorithm 1. In the algorithm, links are iteratively scheduled into each group until all links have been traversed, as indicated in line 1. At the beginning of each iteration, the set for recording the scheduled links in the corresponding group (say group ), denoted as , is initialized as empty. Line 1–1 describe the minimumdegree greedy scheduling algorithm of group . In line 1, the traversed links are removed from the set , which is later used for updating the nontraversed set for next group in line 1.
The computational complexity of the algorithm is , compared to of naive brute force scheme, where indicates the cardinality of the node set , namely the number of nodes in the set. The performance analysis of the algorithm is presented as below.
Lemma 1.
Minimumdegree greedy algorithm outputs an independent set such that where is the maximum degree of any node in the graph.
Proof.
Denote as the complement of set for set , namely . Then, we upper bound the number of nodes in , i.e. , as follows: A node is in because it is removed as a neighbor of some node when greedily added to . Associate to . A node can be associated at most times as it has at most neighbors. Hence, we have . Then, as every node is either in or , we have and therefore , which implies that . ∎
Corollary 1.
Minimumdegree greedy algorithm gives a approximation for MIS in graphs of degree at most .
Proof.
A straightforward result of Lemma 1. ∎
As the maximum degree of a graph varies with the graph topology, we are interested in the performance comparison between greedy algorithm and brute force scheme in general case, where the performance ratio can be characterized by the average degree of the graph, denoted as . This is presented in the following theorem:
Theorem 1.
Minimumdegree greedy algorithm achieves a performance ratio for MIS in graphs of average degree .
Proof.
The proof is provided in [27] in combination with a fractional relaxation technique and not addressed here due to space limitation. ∎
IiiB Slot Allocation Algorithm
With the simultaneous transmission scheduling policy, we propose a proportional fair time resource allocation scheme to determine the transmission duration (slot) for each group. This algorithm gets the required slot of each link by TDMA scheme (exampled in Fig. 3) in a group, calculates the maximal number of all the required slots, and proportionally allocate all slots in a frame to the links in each group. As more time slots are allocated to each link, the achievable data rate of these links are increased.
We denote the number of slots required by link as , then the maximal number of required slots among all links in group , denoted as , can be obtained by
(13) 
Based on this, the total slots in a frame are allocated to each group proportionally to its maximum number of required slots . Hence, the number of slots allocated to all links in group , denoted as , can be calculated as
(14) 
where represents the floor function. Pseudo code of the time resource allocation algorithm is summarized in Algorithm 2. In the algorithm, firstly, the required number of slots of all links scheduled in each group, , are collected, as indicated in line 2–2. Then, for each group , the maximal number of required slots, , are calculated as in line 2. Based on this, the actual allocatable number of slots are determined for all groups, as indicated in line 2–2. As each link can only be scheduled into one group, the computational complexity of the algorithm is .
IiiC Power Allocation Algorithm
By enabling spatial multiplexing, at some nodes (BS, AP), there will be multiple links to be simultaneously transmitted. Hence, power control is required at these nodes to fulfill the power splitting in P2M transmission situation. Specifically, after link scheduling and slot allocation, constraints (8)–(10) are satisfied and consequently the optimization problem 1 in (12) can be reformed as
Problem 2.
(Power allocation optimization)
s.t.  (15) 
Here, in (15), the objective function first summarizes the achievable channel capacity, instead of data rates, as the slot allocation policy has been determined, of all links scheduled in group (i.e., ) and further summarizes all groups of a frame (i.e., ). As the power allocation of links in one group is independent of that in the others, and the total transmission power of BS/AP remains unchanged for each group, the above maximization problem 2 provided by (15) can be further relaxed as
Problem 3.
(Relaxed power allocation optimization)
s.t.  (16) 
Here, we assume that the total system bandwidth of node is equally averaged to all links in , i.e.,
(17) 
Hence, the term is eliminated in (16) from (3). Moreover, the term that describes the experienced interference power of link is also eliminated in (16), as all the other links scheduled in group are isolated from link , namely these links lead to interference power that is less than the threshold , and are allowed to be simultaneously transmitted with link under tolerable interference level. Nevertheless, we still calculate the actual SINR of link for performance evaluation in Section V as addressed in Section IB.
Denoting as , which we refer to as channel quality, the relaxed optimization problem 3 provided by (16) can be solved by the following theorem (known as waterfilling solution).
Theorem 2.
The optimal solution of problem 3 is given by
(18) 
where is the optimal transmission power allocated to link , and the optimal Lagrangian multiplier is given by
(19) 
Proof.
The proof is provided in Appendix A. ∎
The pseudo code of the power allocation algorithm is presented in Algorithm 3. In the algorithm, firstly, the channel quality is sorted in a descending order, as indicated in line 3. The reason behind this falls into the fact that according to (18), once the optimal Lagrangian multiplier is determined, all links with will be allocated to zero power and do not contribute in the calculation of the optimal Lagrangian multiplier in (19). With the descending , the corresponding Lagrangian multiplier of each link is derived as indicated in line 3–3. Then, the index of the optimal Lagrangian multiplier is acquired by finding the largest Lagrangian multiplier that is greater than the reciprocal of channel quality the corresponding link, as indicated in line 3–3. Ultimately, the optimal Lagrangian multiplier is determined in line 3 and the transmission power is allocated to each link as indicated in line 3–3. Similar to the slot allocation algorithm, the computational complexity of the power allocation algorithm is also , as each link can only be scheduled into one group.
Iv Multihop Routing Algorithm Design for Path Selection
In the previous section, we assume that the multihop path connecting BS and UE is predefined, where the JSRA algorithm is designed based on fixed network topology. This assumption does not fully exploit the freedom given by a reconfigurable mmwave backhaul, which can be flexibly selected as an intermediate hop for the path connecting BS and UE, and the selection depends on realtime network load, i.e., achievable data rate that varies for different paths consisting of different intermediate hops. In this section, we propose a DR algorithm for path selection, where the optimal path is greedily generated for each UE in terms of achievable data rate taking into account the data rates achieved by applying the JSRA algorithm to the existing UEs in the network.
Iva Routing Algorithm Design
We introduce our DR algorithm as follows. The network is firstly established as link graph^{2}^{2}2As mentioned in Section IIA, UEs are associated either directly with the BS or with the geographically closest AP and connected to the BS via multihop. Depending on different network layouts specified by various use cases, this assumption can be further modified as UEs are associated with the network node (BS/AP) from which the strongest signal power is detected. illustrated in Fig. 1. With the network topology, the achievable data rate of each link is calculated by the JSRA algorithm. Then, for all UEs in the network that requires multihop path connecting to the BS, the optimal path is greedily selected considering the existed UE(s) in the network. When a path has been determined for a UE, the network will be updated where the achievable data rate of each link is recalculated. More details of the path selection algorithm and network update are elaborated in Section IVB and Section IVC, respectively. The routing algorithm terminates when the path of all UEs are generated. A flow chart of the proposed DR algorithm is illustrated in Fig 4(a).
IvB Path Selection Algorithm
For a multihop transmission between BS and UE, a source and destination node pair, , is selected as either (BS,UE) or (UE,BS) for downlink or uplink transmission, respectively. Further, the achievable data rate is used as edge weight. Then, we provide the pseudo code of the proposed path selection algorithm in Algorithm 4.
In the beginning, both the set of traversed node and the set of next node are initialized as empty. The set of current node contains only source node , and for each node except , the weight of path starting from and ending at it, referred to as and defined as the minimum edge weight along the path (the achievable rate of the path), is initialized as , which means that in the beginning all the nodes are not connected. Then, the algorithm picks every in the set of current node (starting from ), and finds all its neighboring node in the set of nontraversed node , as described in line 4–4. Afterwards, there are three criteria for judging whether can be added to the path from to as the next node of :

Adding the node to the path will increase the weight of path . In this case, the optimal path from to in terms of maximizing achievable data rate goes exclusively through , as by adding the edge , the minimal weight of the path is increased. Therefore, the other paths from to through other parent nodes of in the set should be eliminated and is the only parent node of , as described in line 4–4.

Otherwise, adding to the path may decrease the weight of path , which is not the desired result for maximizing the weight of the path and will not be considered.
When the sets of currents nodes has been traversed, it will be updated as the set of next nodes, which are generated by selecting the proper neighboring nodes of nodes in by the above criteria, as described in line 4–4. The algorithm terminates until all nodes in the network have been assigned a path from .
IvC Update Network
After the optimal path between BS and one UE is generated, the achievable data rate of each link in the network needs to be updated. This procedure utilizes the proposed JSRA algorithm, where every time the path for one UE has been determined, the achievable data rates of all links in the network are obtained by (6), according to the scheduling policy , the slot allocation policy , and the power allocation policy , which are acquired by running the JSRA algorithm for all the existing UEs in the network. In this way, the objective of DR is reached, where for each user the optimal path is selected taking into account realtime network statistics (achievable data rate) and correspondingly the system performance is improved compared to the scenario of predefined routing.
V Numerical Evaluation and Discussion
In this section, we evaluate the performance of the proposed JSRA algorithm with and without the DR algorithm for the mmwave HetNets with both downlink and uplink traffics. The systemlevel evaluation setup is described in Section VA. For the evaluation, we first compare the proposed algorithms with some benchmark schemes for multiplexing and interference mitigation and also with the approach that achieves theoretical optimum, in terms of data rate and latency in Section VB and Section VC, respectively. Then, the impacts of frame structure and duplex mode on data rate are addressed in Section VD and Section VE, respectively. Finally, the capability of the proposed CGMIS scheduling algorithm in improving system performance of P2P communications, for which we focus on data delivery of a vehicular platoon, is demonstrated in Section VF.
Va Simulation Setup
We consider a HetNet deployed under a single Manhattan Grid [6, 8, 24], where square blocks are surrounded by streets that are 200 meters long and 30 meters wide. As illustrated in Fig. 4(b), one BS and nine APs, which are marked as a black circle with black cross and black triangles, respectively, are located at the crossroads. 100 UEs are uniformly dropped in the streets marked as small blue crosses. In addition, green arrows illustrate mmwave access links while red arrow depicts mmwave backhaul link, respectively. The adopted propagation and antenna model are explained in Section IIB. It is worth noting that UEs are assumed to be almost stationary so the pathloss and shadowing values are fixed during the simulation. The type of user traffic is set as full buffer, and the default duplex mode is assumed to be halfduplex. However, for different case studies addressed in Section VD and Section VE, the default traffic type and duplex mode can be modified to investigate the efficiency of applying the proposed algorithms to ubiquitous system configurations. Simulation samples are averaged over 1000 independent snapshots. The default system parameter values are summarized in Table I.
VB Case Studies: Data Rate
In Fig. 5(a)
, we plot the simulation results of the data rate for different schemes in the considered HetNet, versus the type of data rate. Here, the edge data rate is defined as the 5th percentile point of the cumulative distribution function (CDF) of data rates. In particular, TDMA and eICIC
[28] schemes are selected as benchmark schemes for the performance comparison with the proposed algorithms. As shown in Fig. 5(a), the proposed JSRA algorithm provides considerable improvement in both edge and average data rate, thanks to the spatial multiplexing gain achieved by enabling simultaneous transmission, compared to the benchmark schemes. Moreover, applying the DR algorithm to the JSRA algorithm taking into account realtime network statistics further improves the data rate compared to the JSRA algorithm with predefined routing. In particular, the proposed JSRA algorithm with the DR algorithm closely approximates to the optimal data rate, which is achieved by a general dynamic policy approach (DPP) for the network utility maximization problem [29], where at each time slot a maximum weighted sum rate problem with respect to singlehop instantaneous rate is solved, and weights are recursively updated in terms of the flow queues at each node.In Fig. 5(b), we plot the simulation results of the data rate for different schemes in the considered HetNet, versus the number of users in the network. On the one hand, as expected, increasing the number of users reduces the average data rate due to limited resource. However, the proposed JSRA algorithm (without and with the DR algorithm) still provides significant improvement compared to the benchmark schemes for different number of users. On the other hand, with the increased user density, the gap between the optimum (DPP) and the proposed algorithms shrinks. The reason behind this falls into the fact that when the number of users grows, the allocatable resource to each link in all schemes is limited and becomes the dominant factor in determining the data rate.
VC Case Studies: Latency
While the DPP scheme is (asymptotically) optimal for maximizing the data rate and outperforms the proposed JSRA algorithm (without and with the DR algorithm), due to the gain from slotwise scheduling and resource allocation compared to the framewise approach of the JSRA algorithm, it is not optimal when we wish to guarantee other system performance, e.g. latency. In other words, DPP scheme achieves high data rate by sacrificing some flow, which might never be scheduled (always stay in flow queue), to maintain the stability of each transmission node in terms of the balance between influx and outflow bits.
In Fig. 6(a), we plot the simulation results of the latency for different schemes in the considered HetNet, versus the number of users in the network. As stated above, DPP scheme experiences higher latency compared to the JSRA algorithm (without and with the DR algorithm). According to the path selection criteria of the DR algorithm, a link that achieves higher data rate is more likely to be picked as an intermediate hop for a path, which may lead to the situation that the optimal path in terms of maximizing achievable data rate consists of a large number of hops which eventually cause higher latency compared to fixed routing with limited number of hops.
VD Case Studies: Frame Structure
As the actual user demand, which is represented by the required transmission slot, varies across different frames, the results of the slot allocation policy for different frames may also be diverse. This brings an additional advantage for frame structure design, where the “switch point” for different node (BS/AP/UE), defined as the percentage of the number of downlink slots in each frame, can be flexibly adjusted according to actual allocated slots to downlink and uplink transmissions.
In Fig. 8, we plot the simulation results of switch points for four APs in the considered HetNet, versus the index of successive frames. The trend of switch point curves for the other BS/AP are expected to be similar to the depicted ones. As anticipated, the switch points of the selected APs fluctuate in the vicinity of the baseline (, which means the number of downlink and uplink slots are the same) for both algorithms, which shows the efficiency the proposed algorithm in allocating slots to fulfill actual user demands.
In Fig. 6(b), we plot the simulation results of the data rate for JSRA algorithm with frame structure of flexible switch point and with frame structure of fixed switch point () in the considered HetNet, versus the number of users in the network. Here, the scheduling policy acquired by the CGMIS algorithm for the above two cases are the same, but the slot allocation in the fixed switch point case is restricted to allocating total number of slots in each frame equally to all downlink and uplink links (namely – ). In the figure, we notice that the proposed algorithm, owing to the capability of the flexible adjustment of downlink/uplink slots, yields both higher edge and average data rates compared to that of fixed switch point.
VE Case Studies: Duplex Mode
In addition to the default halfduplex mode, in Fig. 8(a) and Fig. 8(b), we plot the simulation results of the data rate for different duplex modes in the considered HetNet, versus the type of data rate and the number of users, respectively. Here, the perfect fullduplex refers to the case that a reception link is isolated from the simultaneously scheduled transmission link of the same node without interference, while the interfered (abbreviated as interf. in the figures) case incorporates a
interference at the reception link caused by the simultaneously scheduled transmission link at the same node. The fullduplex modes are further classified into only enabling fullduplex mode at AP (green and red bars in Fig.
8(a) and Fig. 8(b)) and at both AP and BS (yellow and blue bars). Results show that by relaxing the scheduling criterion for simultaneous transmission (from halfduplex to fullduplex, from fullduplex at only AP to at both AP and BS, etc.), the data rate is gradually improved.VF Case Studies: P2P Communications
In this subsection, we focus on the capability of the proposed JSRA algorithm in P2P communication scenario. Specifically, we consider the data delivery in a platoon of vehicles, which is currently intensively studied by standardization association and research activities [30]. It is obvious that for vehicles in a platoon, P2P communications are more likely to be established, as vehicles move in a line and data is forwarded by the vehicles one after another. When the number of vehicles in a platoon increases, the delivery latency of data becomes a crucial target of performance optimization, for which we may concentrate on how to schedule the data streaming in a platoon. Moreover, as bumper antennas installed in vehicles are geographically separated and noncollocated, referring to the method of enabling simultaneous transmission and reception as fullduplex is no longer appropriate. Hence, the method in vehicular society has been proposed as bidirectional transmission (BT), which is designed to be differentiated from conventional fullduplex.
Based on the above observations, in this subsection we study how the system performance of data delivery in a platoon benefits from the proposed CGMIS scheduling algorithm. We focus on a platoon of ten vehicles in the considered HetNet, where each vehicle has data to transmit to other vehicles in the platoon or to the BS, or vice versa. The vehicle speed is assumed to be , and the distance between two adjacent vehicles equals to vehicle speed (in ).
In Fig. 9(a) and Fig. 9(b), we plot the simulation results of the latency and the throughput for different scheduling schemes in the platoon, versus the number of BTenabled vehicles, respectively. On the one hand, the results suggest that the more BTenabled vehicles, the lower the latency and the higher the throughput are. Here, for BTenabled vehicles, transmission and reception links are simultaneous scheduled, which allows data to “flow” through these vehicles and eventually decreases the latency. On the other hand, the proposed CGMIS scheduling algorithm outperforms other classical scheduling algorithms, namely round robin and proportional fair algorithm, in both latency and throughput. The reason behind this lies in the fact that the CGMIS scheduling algorithm groups links that are suitable to transmit simultaneously according to achievable channel capacity, which is better than the round robin algorithm that always transmits the first data flow in a queue, and the proportional fair algorithm that delays links with low achievable channel capacity due to interference, respectively.
Vi Summary
In this paper, we have addressed the problem of maximizing the achievable data rate of mmwave multihop HetNets considering both downlink and uplink transmissions on backhaul and access links with IAB structure. To solve the maximization problem in an efficient way, we proposed a joint scheduling and resource allocation algorithm, where the optimization problem was decomposed into subproblems including link scheduling, transmission slot allocation, and transmission power allocation. The subproblems were then solved by the proposed maximum independent set based scheduling algorithm, the proportional fair slot allocation algorithm, and the waterfilling power allocation algorithm, respectively. Based on this, a dynamic routing algorithm, which incorporates a path selection algorithm to greedily search the optimal path connecting BS and UE by considering realtime network load and traffic, was investigated to further improve the data rate. By evaluating the proposed algorithms via extensive simulations, we concluded that the proposed joint scheduling and resource allocation algorithm, together with the dynamic routing algorithm, outperforms benchmark schemes in terms of achievable data rate, and closely approach the theoretical optimum, yet with lower latency. Besides, the proposed algorithms achieve higher data rate, by enabling the flexible adjustment of downlink and uplink slot allocation according to spontaneous user demand, and support both half and fullduplex modes with considerable performance enhancement. In particular, the proposed algorithms can deliver significant flexibility in terms of fulfilling various performance requirements for both pointtopoint and pointtomultipoint communications.
Future work can consider a joint optimization of link scheduling, resource allocation scheme, and also routing strategy to achieve even better overall system performance compared to the current solutions, while a potential increase in computational complexity should be also investigated. It would also be interesting to leverage the proposed solution to multiconnectivity scenarios where users are allowed to connect with multiple BSs, and/or to other interference models.
Appendix A Proof of Theorem 2
As the transmission power of a link is nonnegative, the relaxed optimization problem 3 provided by (16) needs to be modified as
s.t.  (20) 
where . For simplicity, we assume . Observe that all inequality constraints functions in (20) are affine, and at least a set of exists such that , and , which implies the KarushKuhnTucker (KKT) conditions are necessary. Moreover, the objective function in (20) is the sum of convex functions and consequently convex as well. Therefore, the KKT conditions are also sufficient, in which we can use standard KKT form to solve the problem. They are concluded as follows:

Primal Feasibility (PF):
(21)

Dual Feasibility (DF):
(22)

Complementary Slackness (CS):
(23)
As the channel capacity is increasing with , the optimal power allocation for link , denoted as , should satisfy . As a result, we can always increase some without violating the first PF condition to increase the sum rate, in case , which means the first CS condition is always satisfied and there is no further restriction on the Lagrangian multiplier besides the second DF condition. Actually, (22) also indicates that . Otherwise, assume , then the other two DF conditions require , and , which results in contradiction.
Now, let link gets positive transmission power, i.e., . Then, according to (22) and (23), we have and , which means . Note that this can hold only if , or equivalently, . Otherwise, to satisfy the first DF condition, we must have , and .
In conclusion, the optimal solution of problem 3 is given by
(24) 
where the optimal Lagrangian multiplier can be derived from
(25) 
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