 Research
 Open Access
SINRbased scheduling in multipath multihop multiradio multichannel mmWave WPANs
 Yunfeng Liu^{1},
 Zhiyong Feng^{1}Email authorView ORCID ID profile,
 Zhiqing Wei^{1} and
 Zebing Feng^{2}
https://doi.org/10.1186/s1363801810749
© The Author(s) 2018
 Received: 2 May 2017
 Accepted: 7 March 2018
 Published: 20 March 2018
Abstract
Millimeter wave (mmWave) communications is a prospective candidate technology for multigigabit rates multimedia applications. To combat the severe propagation attenuation of mmWave, the high gain directional antenna is commonly employed at the nodes. Moreover, exploiting multiple radios over multiple channels is also a promising technology to improve the throughput and delay performance of mmWave communications. In this paper, considering the signal to interference plus noise ratio (SINR) constraint, we develop a multipath multihop multiradio multichannel (MPMHMRMC) concurrent transmission scheduling algorithm to fully exploit spatial reuse in mmWave wireless personal area networks (WPANs). The problem of MPMHMRMC scheduling is formulated as a mixedinteger linear programming (MILP) to clear all the flows within a minimum number of time slots, which is generally NPhard. We further propose a heuristic MPMHMRMC scheme with low computational complexity to solve the problem. Finally, through extensive simulations, we demonstrate that MPMHMRMC can significantly improve the network performance in terms of network throughput and transmission delay under various traffic patterns.
Keywords
 Multipath multihop (MPMH) scheduling
 Multiradio multichannel (MRMC) network
 Spatial reuse
 Wireless personal area networks (WPANs)
1 Introduction
Over the past decade, Millimeter wave (mmWave) communications in the 60 GHz band has gained considerable attention from academia, industry, and standards bodies. Due to the unprecedented 7 GHz unlicensed spectrum, mmWave communications is able to support multigigabit wireless multimedia services, such as uncompressed highdefinition TV, instantaneous music, and image data transmission. With the recent advances in mmWave transceivers design [1], considerable interests have shown in standardizing specifications for mmWave systems, including IEEE 802.15.3c and IEEE 802.11 VHT [2].
Since the free space propagation loss scales as the square of the carrier frequency, the mmWave communications in the 60 GHz band suffers severe attenuation. The free space propagation loss at 60 GHz is 28 dB more than that at 2.4 GHz [3]. Therefore, the mmWave communications range is limited, and it is mainly applied in shortrange indoor communications, such as wireless personal area networks (WPANs). To combat the severe propagation loss, the high gain directional antenna is commonly implemented at both the transmitter and receiver. As a result of directional listening and transmission, the deafness problem occurs when applying the carrier sense to avoid contentions within current transmissions. Besides, the multiuser interference (MUI) is relatively low, thus more concurrent transmissions can be supported to further increase the network capacity [4]. However, multiple communication links transmitting in the same time slot would lead to a high MUI, which might decrease the network throughput. Further, it is beneficial to employ multihop for transmission when the direct link is suffering low channel quality [1, 5]. Therefore, how to schedule appropriate concurrent transmissions to improve the network throughput is a significant and challenging issue.
The problem of concurrent transmission scheduling for mmWave communications has been investigated in the literatures. Most of the existing works focus on the singlepath singlehop singleradio singlechannel (SPSHSRSC) scenario [2–4, 6, 7]. A flipbased heuristic scheduling algorithm was proposed in [3] to maximize the number of flows scheduled in the network such that the quality of service (QoS) requirement of each flow was satisfied. In [4], Z. Yan et al. transformed the concurrent transmission maximization problem for one time slot into finding the feasible maximum clique of the conflict graph. In [8], L. Zhou et al. extended the scenario to multiradio multichannel (MRMC) network, which was mapped into a simple virtual SRSC network. A multipath multihop (MPMH) scheduling scheme was proposed in [1], where the traffic across links of low channel quality was transmitted through MPMH to take advantage of spatial reuse.
Links in the mmWave network can be regarded as pseudowired, i.e., interference is negligible [9]. Motivated by this fact, we propose a novel MPMHMRMC scheduling scheme to boost the spatial reuse in mmWave WPANs in this paper. By transmitting the traffic of flows with high traffic demand or low channel quality between the transmitters and receivers through MPMH in the MRMC situation, the potential of spatial reuse is released.
The main contributions of this paper are threefold. Firstly, we formulate MPMHMRMC scheduling problem into a mixedinteger linear programming (MILP) to accommodate the traffic of all flows within a minimum number of time slots. The signal to interference plus noise ratio (SINR) constraint is considered in the MILP. Secondly, a lowcomputation heuristic scheduling algorithm is proposed to solve the MILP, which is NPhard. The results of the algorithm are close to the optimal solution. Finally, extensive simulation results are provided to verify the superior performance in terms of delay and throughput compared with existing works.
The rest of this paper is organized as follows. Related works are presented in Section 2. We describe the system overview including directional MAC structure, transmission model, interference level, and problem overview in Section 3. In Section 4, the optimal MPMHMRMC problem is formulated as an MILP. In Section 5, a lowcomputation heuristic scheduling algorithm is proposed to solve the MILP. In Section 6, we evaluate the performance of MPMHMRMC under various traffic patterns and simulation parameters. This paper is summarized in Section 7.
2 Related work
There has been considerable work on transmission scheduling for mmWave communications in the literatures. As time division multiple access (TDMA) is adopted in the standards for 60 GHz WPANs [1], i.e., ECMA387 and IEEE 802.15.3c, some schemes were proposed based on TDMA [3, 10–14]. Motivated by the fact that mmWave can attain high space utilization with highly directional pointtopoint communications, many concurrent transmission algorithms were presented. These algorithms can be classified into two categories, namely, conflict graphbased and SINRbased scheduling schemes.
In the conflict graphbased scheduling schemes, maximum independent setbased scheduling was proposed [5, 8, 15]. In these protocols, interference was modeled as a conflict graph, where the interference between the links in an independent set is zero. The interference model is constructed on the assumption that zero beamwidth [15] or ideal antenna gain, i.e., constant antenna gain within the beamwidth and zero gain outside the beamwidth [5, 8].
In the SINRbased scheduling schemes, considering the sidelobe antenna gain, the concept of exclusive region was introduced to enable concurrent transmissions [12, 16, 17]. When the exclusive region conditions are satisfied, multiple links transmitting simultaneously always outperforms TDMA scheme. Furthermore, J. Qiao et al. proposed a concurrent transmission scheduling algorithm for indoor IEEE 802.15.3c WPANs [3]. When MUI is below a specific threshold, multiple links are scheduled to transmit concurrently to maximize the number of flows scheduled in the network such that the QoS requirement of each flow is satisfied.
Since each node in WPANs can reach each other without blockage, considerable work has been put forward to settle the single hop concurrent scheduling problem [3, 4, 6, 8, 18, 19]. To address the link outage problem caused by blockage and combat severe attenuation to improve network throughput, multihop concurrent transmission (MHCT) scheme was proposed [5, 20]. The MHCT scheme concurrently transmits the traffic of flows through multiple hops without causing harmful interference. Y. Niu et al. proposed an MPMH scheduling scheme in [1], where the traffic across links of low channel quality is transmitted through MPMH to enhance spatial reuse. However, the potential of MPMH is not fully boosted in the SRSC situation. L. Zhou et al. exploited multiple radio interfaces over multiple channels in [8] to enhance the performance. Nevertheless, it is applied in the single hop scene employing conflict graph. To the best of our knowledge, the previous works in mmWave WPANs employed MPMH or MRMC for data transmission. Taking into account the unique features of mmWave WPANs (such as line of sight (LOS) transmission, high propagation loss, and directivity), it is likely that employing MPMH and MRMC jointly in mmWave WPANs can improve the network throughput and transmission delay.
3 System overview
3.1 Directional MAC structure
3.2 Transmission model
3.2.1 Antenna model
The directional antenna model proposed in [4] has been widely used, which has a main lobe of Gaussian form and constant level of side lobes. In this model, the directional antenna gain expressed in units of dB is defined as G(α,θ_{−3dB}), where α is the angle between the transmission direction (receiving direction) and center line of the transmitter’s (receiver’s) sector beam, θ_{− 3dB} is the antenna’s halfpower bandwidth (HPBW).
3.2.2 Channel model
3.3 Interference level
where l_{ j } denotes the link concurrent transmitted with link l_{ i } at the same frequency. We define I_{ i } as the cumulative interference level of link l_{ i }. With the definition of cumulative interference level, the SINR condition is converted to a summation form.
For link l_{ i }, each radio interface of transmitter t_{ i } and receiver r_{ i } is equipped with a directional antenna that can be tuned to a specific direction and channel for data transmission. We denote the set of radios of node t_{ i } and r_{ i } as \(\mathbf {R}_{t_{i}}\) and \(\mathbf {R}_{r_{i}}\). We also define the set of channels as C. During the scheduling, we should specify the radio that node t_{ i } employs, the radio that node r_{ i } uses, and the channel occupied by link l_{ i }. The definition of tuple link is proposed for this purpose. The link l_{ i } between t_{ i } and r_{ i } can be further defined as a noderadiochannel tuple, i.e., tuple link. The set of tuple links of link l_{ i } is defined as L_{ i }. According to the method of permutation and combination, the number of tuple links of link l_{ i } is given by \(\left  {{\textbf {L}_{i}}} \right  = \left  {{\textbf {R}_{{t_{i}}}}} \right  \cdot \left  {{\textbf {R}_{{r_{i}}}}} \right  \cdot \left  \textbf {C} \right \). The xth tuple link of link l_{ i } is expressed as \(l_{i}^{x}\). For example, the link from node A to node B in Fig. 1 is denoted as l_{2}. Assuming that R_{ A }={R_{A,1},R_{A,2}}, R_{ B }={R_{B,1},R_{B,2}}, and C={C_{1},C_{2}}. The tuple link set of link l_{2} is given by \({\textbf {L}_{2}} = \left \{ {\left ({{R_{A,{z_{1}}}},{R_{B,{z_{2}}}},{C_{{z_{3}}}}} \right),{z_{1}},{z_{2}},{z_{3}} = 1,2} \right \}\), which has L_{2}=8 tuple links. Tuple link (R_{A,1},R_{B,2},C_{1}) means that there is a link from node A to B, node A employs radio 1, node B employs radio 2, and channel 1 is occupied by the link.

Case 1: \({l_{i}^{x}}\) does not exist in reality, i.e., t_{ i }=r_{ i }.

Case 2: \({l_{i}^{x}} = {l_{j}^{y}}\), which means that i=j,x=y. \(l_{i}^{x}\) and \(l_{j}^{y}\) are the same tuple link.

Case 3: \({l_{i}^{x}} \ne {l_{j}^{y}}\), which means that \(l_{i}^{x}\) and \(l_{j}^{y}\) are different tuple links. They operate on a diverse channel with disparate radios.

Case 4: \({l_{i}^{x}} \ne {l_{j}^{y}}\), they share the same radio.

Case 5: \({l_{i}^{x}} \ne {l_{j}^{y}}\), they have the common node(s) and operate on the same channel with disparate radios. The common node(s) transmit and receive simultaneously [15].

Case 6: Except for the above cases, we obtain the interference level W_{i,j} with (6).
3.4 Problem overview
To improve the throughput of the network, the flow with low transmission rate on the direct link will be transmitted through multiple paths to enhance the transmission efficiency. The object of the schedule is to clear all the flows within a minimum number of time slots.
4 Problem formulation
Notation in problem formulation
Symbol  Description 

K  The number of pairings in a schedule 
V  The number of flows in the network 
M _{ v }  The number of paths of the vth flow 
H _{ vp }  The number of hops of the pth path of the vth flow 
d _{ v }  The traffic demand of the vth flow 
d _{ vp }  The traffic demand distributed to the pth path 
l _{ vpi }  The ith link of the pth path of the vth flow 
c _{ vpi }  The transmission rate of link l_{ vpi } 
L _{ uqj }  The set of tuple links of link l_{ vpi } 
\({l_{vpi}^{x}}\)  The xth tuple link of link l_{ vpi } 
t _{ vpi }  The transmitter of link l_{ vpi } (tuple link \({l_{vpi}^{x}}\)) 
r _{ vpi }  The receiver of link l_{ vpi } (tuple link \({l_{vpi}^{x}}\)) 
\(a_{vpi}^{k}\)  A binary variable to indicate whether link l_{ vpi } is scheduled in the kth pairing 
\(a_{vpi}^{xk}\)  A binary variable to indicate whether tuple link \({l_{vpi}^{x}}\) is scheduled in the kth pairing 
\(\textbf {R}_{{t_{pi}}}\)  The set of radios of node t_{ vpi } 
C  The set of channels in the network 
\({W_{vpi,uqj}^{x,y}}\)  The interference level of tuple link \({l_{vpi}^{x}}\) caused by \({l_{uqj}^{y}}\) 
4.1 Problem formulation and analysis
We assume that there are V flows to be scheduled by the PNC. For flows with low transmission rates or high traffic demands, more time slots are allocated to handle their traffic demands. Since these extra time slots cannot be used in concurrent transmissions, the system performance will deteriorate. To improve throughput, they should be transmitted through multiple paths to enhance spatial reuse. The flow transmitted through multiple paths will be selected with the method in [1].
The number of paths of the vth flow is denoted as M_{ v }. We indicate the number of hops of the pth path of the vth flow as H_{ vp }. For the vth flow, we denote its traffic demand and the traffic demand distributed to the pth path as d_{ v } and d_{ vp }, respectively. The links of a path are sorted by the links’ hop number, i.e., the link of the first hop of the pth path is denoted as the first link of the pth path. The ith link of the pth path of the vth flow is defined as l_{ vpi }, and its transmission rate is c_{ vpi }. According to the previous definition, there exists L_{ vpi } tuple links between transmitter t_{ vpi } and receiver r_{ vpi }. Their transmission rates are also c_{ vpi }. We assume that there are K pairings in the schedule to handle all the traffics, and the kth pairing consists one time slot. We also use indicator function \({a_{vpi}^{k}}\), \(a_{vpi}^{xk}\) to express whether link l_{ vpi }, tuple link \({l_{vpi}^{x}}\), i.e., the xth tuple link of link l_{ vpi }, is scheduled in the kth pairing, respectively. \({a_{vpi}^{k}}\)/\(a_{vpi}^{xk}\) is assigned 1 when l_{ vpi }/\({l_{vpi}^{x}}\) is scheduled. Besides, the interference level caused by tuple link \(l_{uqj}^{y}\) to \(l_{vpi}^{x}\) is defined as \({W_{vpi,uqj}^{x,y}}\).
For example, we assume the flow from node A to B in Fig. 1 is the first flow, then we have M_{1}=3,H_{11}=1,H_{12}=3, and H_{13}=3. Besides, the traffic demand of the first flow and the traffic demand distributed to the three paths are given by d_{1}=18,d_{11}=3,d_{12}=9, and d_{13}=6. The link from A to B is the first link of the first path of the first flow. It is denoted as l_{111}, and the rate of link l_{111} is c_{111}. Similarly, the link from D to F is given by l_{132}. Then, the transmitter t_{111} and receiver r_{111} represent node A and B, respectively. The number of tuple links between A and B is L_{111}. Besides, \(a_{111}^{2}\) and \(a_{111}^{32}\) are indicators to express whether link l_{111} and tuple link \({l_{111}^{3}}\), i.e., the third tuple link of link l_{111} are scheduled in the second pairing. The interference level caused by tuple link \(l_{132}^{2}\) to \(l_{111}^{3}\) is defined as \({W_{111,132}^{3,2}}\).

Constraint (10) indicates regular flow restriction.

Constraints (11) and (12) indicate that link l_{ vpi } should be scheduled at most ⌈d_{ vp }/c_{ vpi }⌉ times in the K pairings.

Constraints (13) and (14) indicate that at most \(\min \left \{ \left  {{\textbf {R}_{{t_{vpi}}}}} \right ,\left  {{\textbf {R}_{{r_{vpi}}}}} \right , \left \textbf {C}\right \right \}\) tuple links between two nodes t_{ vpi },r_{ vpi } could be scheduled in one pairing.

Constraint (15) indicates that the traffic of link l_{ vpi } should be accommodated in the schedule.

Constraints (16)–(18) represent scheduling restrictions. Constraint (16) indicates the inherent order of transmission in each path, the (i+1)th link of the pth path of the vth flow should be scheduled after the ith link. Constraints (17)–(18) indicate SINR restriction for concurrent transmissions.
4.2 Problem reformulation
Problem P1 has a nonlinear constraint (18). We apply the RLT to obtain a linear relaxation [21]. Problem P1 will become a standard MILP, which can be solved by some existing sophisticated algorithms, such as the branchandbound method.
4.3 Example
As an example, a 6node WPAN is illustrated in Fig. 1. Without loss of generality, the interference level between tuple link \({l_{vpi}^{x}}\) and \({l_{uqj}^{y}}\), i.e., \(W_{vpi,uqj}^{x,y}\), of case 6 is assumed to be 0.1 in Fig. 1. As mentioned before, the schedule contains five time slots in the 2radio 2channel situation and the scheduling process is illustrated in Fig. 2. However, the transmission phase of the schedule consists of 18 time slots when using the SPSHSRSC method. It consumes nine time slots when applying the MPMH method [1]. MPMHMRMC reduces the time slots for transmission by 72 and 44% compared with the SPSHSRSC and MPMH methods. However, the reformulated MILP P2 is NPhard. Applying an opensource MILP solver, lp solver ver 5.5.2.0 [2] to obtain the optimal solution will take significantly long computation time, e.g., several minutes. It is not practical for mmWave WPANs where the duration of a time slot is only several μs. In the next section, we propose a lowcomputation heuristic algorithm.
5 The MPMHMRMC scheme
Notation in MPMHMRMC
Symbol  Description 

R _{0}  The number of radios of a node in the network 
N  The number of nodes 
P _{ s }  Set of selected paths of all flows 
H  Set of links in P_{ s } 
l _{ pi }  The ith link of the pth path 
\({l_{pi}^{x}}\)  The xth tuple link of link l_{ pi } 
w _{ pi }  The initial weight of link l_{ pi } 
\({w_{pi}^{t}}\)  The remaining weight of link l_{ pi } in the tth pairing 
t _{ pi }  The transmitter of link l_{ pi } (tuple link \({l_{pi}^{x}}\)) 
r _{ pi }  The receiver of link l_{ pi } (tuple link \({l_{pi}^{x}}\)) 
\(\textbf {R}_{{t_{pi}}}^{t}\)  Set of unused radios of transmitter t_{ pi } in the tth pairing 
\(\textbf {R}_{{r_{pi}}}^{t}\)  Set of unused radios of receiver r_{ pi } in the tth pairing 
F _{ p }  The hop number of the first unscheduled link of the pth path 
\({\textbf {P}_{muh}^{t}}\)  Set of unvisited paths with the largest number of unscheduled links 
\(\textbf {P}_{u}^{t}\)  Set of unvisited paths in the tth pairing 
H ^{ t }  Set of links in the tth pairing 
\(\textbf {H}_{0}^{t}\)  Set of tuple links in the tth pairing 
δ ^{ t }  Number of time slots of the tth pairing 
\({I_{pi}^{x}}\)  The cumulative interference level of tuple link \({l_{pi}^{x}}\) 
5.1 Symbols in MPMHMRMC scheme
Assuming that the number of radios of each node in the network is the same, which is denoted as R_{0}. The number of channels C is no less than R_{0}. In a pairing, the SINR of each tuple link should be larger than its threshold. It means that the maximum number of tuple links in the same pairing is R_{0}·⌊N/2⌋. According to the traffic of each path, the number of time slots for a link to accommodate its traffic is set as the weight of this link.
In the transmission scheduling, the set of selected paths of all flows is defined as P_{ s }. The set of links in P_{ s } is denoted as H. As mentioned in Section 4.2, the links of a path are sorted by the links’ hop number. We denote the ith link of the pth path of P_{ s } as l_{ pi }. The xth tuple link of link l_{ pi } is given by \({l_{pi}^{x}}\). The initial weight of each link l_{ pi } is defined as w_{ pi }. The remaining weight of link l_{ pi } in the tth pairing is defined as \({w_{pi}^{t}}\). The tth pairing consists of at least one time slot. The transmitter and receiver of link l_{ pi }/tuple link \({l_{pi}^{x}}\) are denoted as t_{ pi } and r_{ pi }, respectively. The sets of unoccupied radios of nodes t_{ pi } and r_{ pi } in the tth pairing are expressed as \({\textbf {R}_{{t_{pi}}}^{t}}\) and \({\textbf {R}_{{r_{pi}}}^{t}}\). Let F_{ p } denote the hop number of first unscheduled link of the pth path. The set of unvisited paths with the largest number of unscheduled hops in the tth pairing is given by \(\textbf {P}_{muh}^{t}\). The set of paths that are not visited in the tth pairing is denoted as \(\textbf {P}_{u}^{t}\). The sets of links and tuple links of the tth pairing are defined as H^{ t } and \({\textbf {H}_{0}^{t}}\), respectively. Besides, we denote the number of time slots of the tth pairing as δ^{ t }.
For example, the network illustrated in Fig. 1 has one flow from A to B. Then, P_{ s } contains three paths: A→B (first path, p=1), A→C→E→B (second path, p=2), and A→D→F→B (third path, p=3). Thus, H consists of seven links. The first link of the third path is A→D, which is denoted by l_{31}. The initial weight of link l_{31} is w_{31}=1. The remaining weight of link l_{31} in the tth pairing is defined as \({w_{31}^{t}}\). The transmitter t_{31} and receiver t_{31} of link l_{31} represent nodes A and B, respectively. Initially, we have F_{ p }=1,p=1,2,3, which means that the first link of the three paths has not been scheduled. Since path 2 and path 3 both contain 3 links before the schedule, \(\textbf {P}_{muh}^{1}\) contains 2 paths: the second and third path. We have \(\textbf {P}_{muh}^{1} = \left \{ {p\left  {p = 2,3} \right.} \right \}.\)
5.2 Working mechanism of MPMHMRMC scheme
In the MPMHMRMC scheme, we define two mechanisms, namely, the link selection mechanism and the tuple link schedule mechanism.
If I_{0}≤1, tuple link \(l_{pi}^{x_{0}}\) will be added to \({\textbf {H}_{0}^{t}}\) and scheduled. Link l_{ pi } will be added to H^{ t } accordingly.
Step 1: Select the tuple links to schedule in a pairing. Select link l_{ pi } according to the link selection mechanism until \(\textbf {P}_{u}^{t} = \phi \) or H^{ t }>R_{0}·⌊N/2⌋. The corresponding tuple link \({l_{pi}^{x}}\) is scheduled according to the tuple link schedule mechanism;
Step 2: Schedule the unoccupied radios of selected link l_{ pi } in H^{ t }. The corresponding tuple link \({l_{pi}^{x}}\) is scheduled according to the tuple link schedule mechanism;
Step 3: Determine the number of time slots of the pairing. Repeat step 2 until no more tuple links can be added to \({\textbf {H}_{0}^{t}}\). The minimum time slots needed for a link l_{ pi } in H^{ t } to accommodate its traffic is \(\delta _{pi}^{t}\). The number of time slots of the tth pairing is \({\delta ^{t}} = \mathop {\min }\limits _{{l_{pi}} \in {\textbf {H}^{t}}} \left \{ \delta _{pi}^{t}\right \}\);
Step 4: Update H, F_{ p }, and the weight of scheduled links. Go to step 1 if H>0, end the algorithm when H≤0.
The pseudo code of the heuristic traffic scheduling algorithm is presented in Algorithm 1. Line 4 obtains the suitable link l_{ pi }. Firstly, it gets the set of unvisited paths with the largest number of unscheduled hops \(\textbf {P}_{muh}^{t}\) in the tth pairing. Then, it gets the link of path \({p \in \textbf {P}_{muh}^{t}}\) with the minimum weight \(w_{p{F_{p}}}^{t}\), which is denoted as l_{ pi }. Line 5 selects the tuple link \({l_{pi}^{x}}\) that brings minimum interference to the tuple links in \(\textbf {H}_{0}^{t}\). Lines 8–12 utilize the idle radios of the transmitters and receivers of scheduled links in H^{ t } when there are remained data to transmit. Line 13 obtains the duration of tth pairing δ^{ t }. δ^{ t } is the minimum time needed for a link in H^{ t } to accommodate its traffic. Lines 14–17 update the weight of each scheduled link and delete the link l_{ pi } when all its traffic has been dealt with. Meanwhile, F_{ p } is renewed.
The pseudo code of the TLS algorithm is presented in Algorithm 2. It is used for selecting the tuple link \({l_{pi}^{x}}\) that brings minimum interference to the tuple links in \(\textbf {H}_{0}^{t}\). Line 1 examines the initial conditions for each selection. The algorithm starts when the transmitter and receiver own idle radios. Line 2 selects the tuple link \({l_{pi}^{x}}\) with minimum interference to the tuple links in \(\textbf {H}_{0}^{t}\) according to the cumulative interference level \({I_{qj}^{y}}\). With different \({l_{pi}^{x} \notin \textbf {H}_{0}^{t}}\), it calculates the cumulative interference level \({I_{qj}^{y}}\) for each tuple link \({l_{qj}^{y}}\) in \({\textbf {H}_{0}^{t} \cup \left \{ {l_{pi}^{x}} \right \}}\) and obtains the maximum value. Thus, it gets a maximum cumulative interference level set. The tuple link \(l_{pi}^{{x_{0}}}\) corresponding to the smallest value in the maximum cumulative interference level set will be selected. Line 3 gets the maximum cumulative interference level of links in \({\textbf {H}_{0}^{t} \cup \left \{ {l_{pi}^{{x_{0}}}} \right \}}\). Line 4 is a SINR condition to guarantee concurrent transmissions. Lines 6–8 renew \({\textbf {H}^{t}},\textbf {H}_{0}^{t},w_{pi}^{t},\textbf {R}_{{t_{pi}}}^{t}\), and \(\textbf {R}_{{r_{pi}}}^{t}\) once the SINR condition is met.
5.3 Example
Step 1: Initially, paths 2 and 3 both have three hops unscheduled, we have F_{ p }=1,p=1,2,3, and \(\textbf {P}_{muh}^{1} = \left \{ {p\left  {p = 2,3} \right.} \right \}\). According to the number of packets and rate of each link, we have \(w_{2,1}^{1} = 2\) and \(w_{3,1}^{1} = 1\). As δ^{1}=0, link l_{31} will be selected based on the criterion of line 4 in Algorithm 1. Since \(\textbf {H}_{0}^{1} = \phi \), each tuple link of link l_{31} is suitable to be added to \(\textbf {H}_{0}^{1}\) according to Algorithm 2. We choose the one with minimum subscript of radios and channel, which is denoted as (R_{A,1},R_{D,1},C_{1}). At the same time, link l_{31} will be added to H^{1}. Besides, \(w_{3,1}^{1}\) will be updated and is given by \(w_{3,1}^{1} = 0\). Then we have \(\textbf {P}_{muh}^{1} = \left \{ {p\left  {p = 2} \right.} \right \}\). Similarly, link l_{21} and tuple link (R_{A,2},R_{C,1},C_{2}) will be added to H^{1} and \(\textbf {H}_{0}^{1}\), respectively. \(w_{2,1}^{1}\) will be updated and is given by \(w_{2,1}^{1} = 1\). Meanwhile, \(\textbf {P}_{muh}^{1}\) will also be updated and is given by \(\textbf {P}_{muh}^{1} = \left \{ {p\left  {p = 1} \right.} \right \}\). However, according to algorithm 2, link l_{11} will not be added to H^{1} since the radios of transmitter t_{11} (node A) are all occupied. The while loop between lines 3 and 7 will stop as \(\left \textbf {P}_{u}^{t}\right  = 0\).
Step 2: No tuple links will be added to \(\textbf {H}_{0}^{1}\) during the while loop between lines 8 and 11 since node A has no free radios.
Step 3: We obtain the renewed δ^{1},w_{2,1}, and w_{3,1}, which are give by δ^{1}=1,w_{2,1}=1, and w_{3,1}=0.
Step 4: l_{3,1} will be removed from H. F_{3} will increase by 1 since all the traffic of l_{3,1} has been cleared, which is given by F_{3}=2. The process of the second to fifth pairings are similar to the first pairing.
The scheduling results of our algorithm in SRSC situation are as follows: in the first pairing, link A→D transmits for one time slot; in the second pairing, links A→C and D→F transmit for two time slots; in the third pairing, links D→F, C→E, and A→B transmit for one time slot; in the fourth pairing, links C→E and A→B transmit for two time slots; in the fifth pairing, link E→B transmits for two time slots; in the sixth pairing, link F→B transmits for one time slot. However, it consumes 10 time slots with the algorithm in [1] in the SRSC situation. As analyzed in Section 4.3, the optimal solution takes nine and five time slots. Therefore, our heuristic scheme obtains results close to the optimal solution.
The path selection algorithm has the computational complexity of \(O\left ({N^{{H_{\max }}}}\right)\), where H_{max} is the maximum number of hops for each selected path [1]. The computational complexity of the traffic distribution algorithm is negligible [1]. The transmission scheduling algorithm has the complexity of O(N^{2}). Thus, the total complexity of the scheme is \(O\left ({N^{{H_{\max }}}} + {N^{2}}\right)\), which is pseudopolynomial. Thus, our scheme is practical for WPANs.
With the introduction of MPMHMRMC, the control overhead will be increased since more scheduling information needs to be broadcasted to the nodes in the WPAN. Based on the directional MAC structure in Section 3.1, network synchronization and control messages for the (m−1)th transmission requests are broadcasted in the mth BP from the PNC. During the mth CAP period, the PNC receives several transmission requests [3]. As mentioned in [1, 2], the BP or CAP can be completed by the PNC in one time slot. The PNC makes the path selection and scheduling decision for the (m+1)th CTAP during the mth CTAP. One CTAP contains nearly 1000 time slots. Generally, only a few time slots are needed for the PNC to complete path selection and schedule computation [1]. Considering that one frame consists of 1000 time slots [1, 3] and the Gbps transmission rates of the mmWave communications, the increase of the control overhead can be neglected.
6 Simulation results
In this section, we evaluate the performance of the proposed algorithm with respect to average delay and network throughput under two different traffic patterns. The algorithm applied in SRSC and MRMC situation are named MPMHSRSC and MPMHMRMC, respectively. We compare MPMHSRSC and MPMHMRMC with MPMH [1], MPMH2, and TDMA schemes. MPMH2 scheme applies the MPMH scheme [1] in MRMC situation by utilizing the TLS algorithm proposed in our paper. TDMA scheme supports only one active transmission in each time slot. In the MRMC situation, each node contains four radios, and there are four orthogonal channels in the network.
6.1 Simulation setup
Simulation parameters
Symbol  Description  Value 

W  Channel bandwidth  1200 MHz 
P _{ t }  Transmission power  0.1 mW 
N _{0}  Background noise  −114 dBm/MHz 
ρ  Path loss exponent  2 
A _{LOS}  LOS path loss constant  32.5 dB 
θ _{−3 dB}  The antennas HPBW  60° 
T _{slot}  Slot duration  5 μs 
where L is the size of data packets in bit, V is the number of flows, and R is set to be 2 Gbps.
In the simulation, two traffic patterns are adopted.
1) Uniform traffic: All flows have the same arrival rate λ, i.e., λ_{ i }=λ, which is adopted in [1].
2) Random traffic: The arrival rate of flow i, λ_{ i }, is randomly generated.
The system performance is evaluated by the following two metrics:
1) Average delay: The average delay of received packets of all flows, and the unit is time slot. The delay caused by BP and CAP is not calculated.
2) Network throughput: The number of successfully transmitted packets of all flows during the entire simulation, and the unit is packet. A packet will be counted as a successfully transmitted packet when its delay is no more than the threshold.
6.2 Network throughput with different traffic load ratio
The increment of the random traffic is larger than that of the uniform traffic. The reasons are as follows. Under uniform traffic, the transmission durations of links in a pairing are diverse in MPMH [1]. The diversities bring about the waste of time slots in a pairing. The differences among the transmission durations of links in a pairing under random traffic are worse, which result in further degradations in the throughput performance. However, the transmission durations of links in a pairing are the same under both traffic patterns in MPMHSRSC. Thus, it outperforms the MPMH under uniform traffic and has greater performance increment under random traffic.
With the introduction of MRMC, MPMHMRMC can better utilize MPMH than the MPMHSRSC, and the throughput of MPMHMRMC increases linearly even under heavy traffic load. With the traffic load ratio from 10 to 20, MPMHMRMC surpasses MPMHSRSC by about 82.34% under uniform traffic and 108.29% under random traffic, respectively. Compared with MPMH2, MPMHMRMC increases the network throughput by about 14.09% under uniform traffic and 25.8% under random traffic on average with the traffic load ratio from 12 to 20. The reason is the same with the one in SRSC situation. When the traffic load ratio is over 3, the number of discarded packets in TDMA exceeds MPMH and MPMHSRSC, which results in smaller network throughput in TDMA. Since the delay threshold is larger than the frame length, the packets that cannot be scheduled in current frame will be transmitted in the next frame. With the increase of the traffic load ratio, the number of packets successful transmitted in the system becomes stable and the network throughput curves become flat.
6.3 Network throughput with different number of flows with multipath
Considering that the paths of all flows construct a directed graph, the degree of one node will increase when introducing the MPMH. With dense flows, MPMHMRMC has sufficiently utilized MRMC in the SPSH situation. When the flows are transmitted through multipath, the degree of a node is beyond the ability of the MRMC, which results in the degradation of the network throughput. However, with sparse flows in the network, the MRMC of a node is underutilized until more flows are transmitted through MPMH. MPMHSRSC and MPMH are employed in the SRSC situation, one node has fully exploited its radio even when the SPSH is applied. As a result, the performance of MPMHSRSC and MPMH decrease rapidly with the increase of number of flows transmitted through MPMH in both situations. For the similar reason, the performance of MPMH2 decreases when traffic load ratio is 25 with 10 flows and increases about 7.58% when traffic load ratio is 10 with 4 flows.
6.4 Average delay with different traffic load ratio
As mentioned in Section 6.2, MPMHMRMC can better utilize MPMH than the MPMHSRSC, which results in the decrease of delay. Compared with MPMHSRSC, the average transmission delay of MPMHMRMC decreases by about 74.20% under uniform traffic and 70.91% under random traffic with the traffic load ratio from 10 to 20. With the traffic load ratio from 10 to 20, MPMHMRMC outperforms MPMH2 by about 29.21% under uniform traffic and 49.04% under random traffic on average, respectively. The reason is identical with the one in network throughput. When the traffic load ratio is over 3, the number of discarded packets in TDMA exceeds MPMH and MPMHSRSC, which leads to lower average delay in TDMA.
6.5 Average delay with different number of flows with multipath
Compared with MPMH, MPMHSRSC decreases the delay by about 21.86% when the traffic load ratio is 25 with 10 flows and 25.01% when traffic load ratio is 10 with 4 flows, respectively. MPMHMRMC surpasses MPMHSRSC by about 49.96 and 66.72% correspondingly. Besides, MPMHMRMC outperforms MPMH2 by about 39.33 and 49.04%. The results are consistent with the ones of the network throughput.
7 Conclusions
In this paper, we have proposed a scheduling algorithm for mmWave WPANs, which boosts the potential of spatial reuse by concurrent transmissions scheduling through MPMHMRMC. It can sufficiently improve the performance of network throughput and average delay. The proposed heuristic algorithm can achieve the results close to the optimal solution with the complexity of \(O\left ({N^{{H_{\max }}}} + {N^{2}}\right)\). Compared with MPMH, MPMHSRSC enhances the performance of network throughput and delay by about 22.09 and 24.30% on average, respectively. MPMHMRMC outperforms MPMHSRSC in terms of the network throughput and delay by about 95.32 and 72.56% on average. Performance under different number of flows transmitted through MPMH indicates the number of flows transmitted through MPMH should be selected according to the flow density to optimize network performance.
Declarations
Funding
This work was sponsored by the National Natural Science Foundation of China (61631003).
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contribute to the concept, the design and developments of the theory analysis and heuristic algorithm, and the simulation results in this manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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