Physical-Layer Network Coding Based Throughput ... - IEEE Xplore

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 4, APRIL 2018

Physical-Layer Network Coding Based Throughput-Optimal Transmission for Bidirectional Traffic Wonjong Noh

, Member, IEEE, Sung Hoon Lim

Abstract—In this study, we develop a physical-layer network coding based throughput-optimal transmission scheduling policy, which we call network coding aware maximum differential backlog (NCMDB), and demonstrate its throughput optimality. As the proposed policy requires an NP-hard network-wide centralized algorithm to determine the optimal transmission schedule, we propose a distributed NCMDB based on link prioritization (D-NCMDB) as a lower bound approach. We further propose a bidirectionalityaware route establishment scheme for the best application of the NCMDB and D-NCMDB policies. Simulations confirm that compared to the signal-to-interference-based interference avoidance transmission approach and random maximal physical-layer network coding approach, the proposed D-NCMDB scheme offers 48% and 27% increased throughput, 78% and 63% reduced access delay, and 80% and 20% enlarged stability, respectively. The proposed distributed transmission scheduling scheme can be employed in future communication networks in which both the number of nodes and the amount of interference increase exponentially. Index Terms—Bidirectional path establishment, distributed scheduling, physical-layer network coding, throughput-optimal transmission control.

I. INTRODUCTION HE recent revolution in intelligent information systems has brought rapid changes in communication environments. For example, the number of devices has grown exponentially as new smart devices are added and the Internet of Things (IoT) becomes a reality. Further, communication patterns are diversifying from single-hop to multihop heterogeneous communications. These trends can result in significantly increased interference in networks as well as saturation of available wireless resources. In this situation, one of the

T

Manuscript received December 12, 2016; revised May 27, 2017, September 3, 2017, and November 9, 2017; accepted November 13, 2017. Date of publication November 23, 2017; date of current version April 16, 2018. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under Grant 2017R1D1A1B03036526. The review of this paper was coordinated by Prof. H.-F. Lu. W. Noh is with the Samsung Electronics Co., Ltd., Suwon 16677, South Korea (e-mail: [email protected]). S. H. Lim is with the Korea Institute of Ocean Science and Technology, Busan 49111, South Korea (e-mail: [email protected]). T. Kim is with the Department of Business Administration, Pai Chai University, Daejeon 35345, South Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2017.2777141

, Member, IEEE, and Tae-Suk Kim

major challenges in wireless multihop networks is to design a transmission scheduling scheme that can efficiently share the common spectrum among links within a given geographic area to avoid the problems of interference and resource saturation. To solve the problem, a centralized scheduling policy that achieves the maximum attainable throughput region was presented in a seminal paper by Tassiulas and Ephremides [1]. However, owing to the lack of central control in wireless multihop networks, many distributed scheduling algorithms were also proposed. [2], [3] and [4]–[6] transformed centralized algorithms into distributed algorithms with message passing by using approximation and randomization approaches, respectively. They are known to have polynomial computational complexity. Recently, a synchronous distributed maximal scheduler was also proposed in [7], [8]. However, these prior studies focused on interference avoidance strategies that may result in inefficient use of the wireless resource. The studies treated wireless broadcast signals as harmful interference. Building on this observation, some studies focused on transmission schemes based on interference exploitation and cancellation, which have been shown to improve spatial reuse opportunities and network throughput. For example, [10]–[14] proposed random-linear network coding (RLNC) [9]-based transmission control schemes. More specifically, [10], [11] proposed joint conflict-free transmission scheduling and practical network coding schemes using a time-dependent flow optimization formulation. They analyzed the impact of a realistic MAC protocol, transmission schedule, and packet combination strategy for network coding. [12], [13] proposed opportunistic transmission scheduling schemes. They opportunistically selected scheduled nodes, packets to be coded, and an employed modulation level according to timevarying channel conditions and packet length. [14] proposed a congestion-aware transmission scheduling scheme to prevent unexpected packet dropping. To determine the optimal schedule, this approach used the procedure of transmission-mode preassignment and cross-layer formulation that maximizes network utility. By contrast, [16]–[20] proposed physical-layer network coding (PLNC) [15]-based transmission control schemes. More specifically, [16] proposed a medium access control (MAC) protocol for amplify-and-forward (AF)-based PLNC in a large-scale wireless network. This approach considered PHYspecific conditions such as synchronization and channel state information (CSI) jointly with distributed scheduling at the

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NOH et al.: PHYSICAL-LAYER NETWORK CODING BASED THROUGHPUT-OPTIMAL TRANSMISSION FOR BIDIRECTIONAL TRAFFIC

MAC layer. [17] proposed a MAC protocol in which the relay node coordinates the PLNC transmission by making aware the queue status of its neighbor nodes. The protocol can work with any relaying PLNC scheme such as AF, compute and forward (CF), denoise and forward (DNF), or decode and forward (DF), but it does not use a network-coded queue model. [18] proposed a CF-based PLNC control scheme in a random access network. This scheme provided a generalization of both the PLNC and the multipacket reception strategy. [19], [20] proposed joint routing and scheduling schemes in small- to medium-size wireless networks with PLNC. They found the optimal configuration of routing and scheduling using a cross-layer optimization framework that maximizes the optimal max-min throughput of the flows. In this paper, we propose network-coded queue-based centralized and distributed transmission scheduling schemes with a new PLNC-enhancing routing process. The main contributions of this study, and its differences compared to previous studies, can be summarized as follows: 1) As a centralized scheduling approach, we develop a throughput-optimal transmission scheduling scheme that is based on a network coding-aware maximum differential backlog (NCMDB) policy. Unlike the schemes proposed in [16]–[18], this work develops a network-coded queue evolution model and suggests a network-coded queue backlog-based throughput-optimal transmission control. Furthermore, we analytically prove the throughput optimality. To the best of our knowledge, this has not been addressed to date. 2) As a distributed scheduling approach, we propose a prioritization-based distributed NCMDB transmission scheduling scheme, referred to as D-NCMDB. It is a lower-bound approach of the proposed NCMDB. Unlike the schemes proposed in [16]–[18], the proposed scheme is based on a conflict-free scheduling approach. We show that the proposed scheme has polynomial complexity and overhead so that it can be employed in practical networks. This scheme exploits the interference efficiently and reuses the wireless resource maximally. 3) We propose a bidirectionality-aware route establishment scheme for the best exploitation of the NCMDB and D-NCMDB policies. The proposed scheme provides a maximized bidirectional route setup. In support, we design tight and relaxed delay functions that guarantee bidirectional routes. Unlike the schemes proposed in [19], [20], the proposed scheme is a low-complexity routing setup scheme that can be used in conjunction with legacy routing setup protocols without modification. 4) We demonstrate that the proposed scheme improves network performance in terms of the throughput, delay, and stability region through system-level simulations. This work can be used with any DF-based PLNC scheme. However, if we use the queue as information storage instead of bits, then it can be applied to any other PLNC scheme. The remainder of the paper is organized as follows. Section II presents the system model. Section III presents the developed

Fig. 1.

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Bidirectional physical-layer network coding.

NCMDB policy and its distributed version, D-NCMDB, in detail. Section IV describes the proposed bidirectionality-based route setup scheme in detail. Section V presents simulation results that demonstrate the performance of the proposed schemes. Finally, Section VI summarizes our findings and concludes the paper with the scope of future work. II. SYSTEM MODEL In this section, to clearly illustrate the system model, we use a simple example network in Fig. 1. Here, we assume that nodes 1 and 2 have their data to send to nodes 2 and 1, respectively. In addition, node n has some data to send to node 1 and node 2, respectively. Here, Xa→b and Xb→a denote the data that nodes a and b send to nodes b and a, respectively, and Xa→b ⊕ Xb→a denotes the physical-layer network-coded data of Xa→b and Xb→a . Using conventional protocols that support only unidirectional traffic flows, exchanging the data pair Xa→b and Xb→a takes four time slots. On the other hand, when PLNC is employed, only two time slots are needed to complete the communication, i.e., PLNC provides a twofold capacity increase in the network [15]. A. Network Model In larger wireless multihop networks, considerable opportunities exist for exploiting this PLNC gain. However, several challenges need to be overcome when applying PLNC to practical wireless networks [16]. The first challenge is to deal with the asynchrony of symbol-level time, frequency, and phase between the signals transmitted by multiple transmitters. Many previous works [15], [21], [22] reported that symbol misalignment and phase offset result in appreciable performance penalties. These earlier investigations led to a common belief that near-perfect symbol and carrier-phase synchronizations are important for good performance in PLNC. However, recent works [23]–[27] have shown that this is not exactly true, and that PLNC can have asynchrony reward instead of asynchrony penalty when appropriate methods such as channel coding are applied. In particular, both symbol misalignment and phase offset improve the bit-error rate (BER) performance when channel coding is used. In addition, the performance spread arising from all combinations of symbol and phase offsets is only approximately 1(dB). The second challenge is to estimate the channel state information (CSI) at the end nodes. For example, when node a decodes its data, CSI on the link between nodes n and b is assumed to be available in addition to the link between node a and n. The estimation of CSI on the link between nodes a and n by node a is relatively simple because it can be directly estimated. However, node a cannot directly estimate CSI on the distant link between nodes n and b. Therefore, a mechanism needs to be

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Fig. 2.


Queue model at node n.

developed for node a to obtain CSI on the link between nodes n and b. To acquire two-hop neighbor’s CSI information, many works [16], [17], [28] proposed distributed or cooperative CSI exchange protocols. In this work, based on previous works, we assume that symbols among all nodes are perfectly synchronized, and that the two-hop neighbor’s CSI information is also available. This work focuses on the transmission scheduling problem, especially conflict-free scheduling-based MAC. The network topology is described by a directed graph G = (N, E), where N is the set of network nodes and E is the set of communication links. It is assumed that all channel states remain constant in a slot consisting of multiple symbols, and that they vary according to a slow-fading model. We represent the topology state of the network as G(t). Here, the topology state of the network is identically and independently distributed over slots with probabilities πg = P r[G(t) = g]. The data are sent to node i ∈ N = {1, . . . , N } and are labeled as commodity ci .

Fig. 3.

Arrivals and departures at node n.

B. Queue Model: Data Arrivals and Departures Each node n has different buffer types: uncoded queue, network-coded queue, and virtual queue. An uncoded commod(c ) ity ci is buffered in its dedicated queue Qn i , and its queue (c ) length at the end of slot t is denoted as Qn i (t). A networkcoded commodity cj ⊕ ck is buffered in its dedicated queue (c ⊕c ) Qn j k , and its queue length at the end of slot t is denoted as (c ⊕c ) Qn j k (t). On the other hand, data that have already been sent but will be required for later network-coded packet decoding (v ) are buffered in the virtual queue Qn . For example, node n in (c 1 ) (c 2 ) (c ⊕c ) Fig. 1 keeps queues Qn and Qn for uncoded data, Qn 1 2 (v ) for network-coded data, and Qn for virtual data, as shown in Fig. 2. The queue length dynamics of the uncoded data at node n can be described as Qn(c i ) (t + 1) ⎧ ⎞ ⎫ ⎛ ⎪ ⎪ ⎪ N N ⎬ ⎨ P L ,(c ) ⎟ ⎪ ⎜ (c i ) i (c i ) ⎟,0 μ (t) + μ (t) = max Qn (t) − ⎜ n ,b n ,b ⎝ ⎪ ⎠ ⎪ ⎪ ⎪ b=1 b=1

⎭ ⎩ (B )

+

(C )

N

N μ ¯n(c,ai ) (t) + μ ¯nP ,aL ,(c i ) (t) + An(c i ) (t),

a=1 a=1 (E )

(F )

(A )

(1)

Fig. 4. Queue evolution model at node n. Here, dotted queues denote virtual queues.

where 1) (A) is the exogenous arrival rate of a commodity ci on slot (c ) t. Here, An i (t) is independent and identically distributed over the slots with a finite second moments and means, and satisfies the strong law of large numbers (SLLN), i.e., τ −1 1 (c i ) An (t) = λn(c i ) , τ →∞ τ

lim

with probability one,

t=0

(2) (n ) where λn = 0, i.e., there are no self-destined data arrivals. This is shown by (A) in Figs. 3 and 4. 2) (B) denotes the departure rate of a commodity ci for a neighboring node b on slot t. Here, ci will not be networkcoded at node b. This is shown by (B) in Figs. 3 and 4. 3) (C) also denotes the departure rate of a commodity ci for a neighboring node b on slot t. However, ci will be network-


Fig. 5.

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denotes a link connecting nodes n and a. A directional link from node n to a or node a to n is denoted by l(n, a) or l(a, n), respectively. On the other hand, a two-hop PLNC link connecting node n with its neighbor nodes a and b is denoted by l(n, (a, b)) or l((a, b), n). Likewise, l(n, (a, b)) and l((a, b), n) denote outgoing bidirectional PLNC links from node n and incoming bidirectional PLNC links to node n, respectively.

Link model.

coded with other commodities at node b. For the later physical-layer network decoding, ci is also temporarily saved at the virtual queue of node n. This is shown by (C) in Figs. 3 and 4. 4) (E) denotes the arrival rate of a commodity ci from a neighboring node a on slot t. Here, ci will not be networkcoded at node n. This is shown by (E) in Figs. 3 and 4. 5) (F) denotes the arrival rate of a network-coded commodity from a neighboring node a on slot t, which will be resolved to ci at node n. For example, the arrived network-coded commodity c1 ⊕ c2 is converted to c1 by the cancellation of c2 that was saved at the virtual queue of node n. This is shown by (F) in Figs. 3 and 4. The queue length dynamics of the network-coded data at node n can be described as ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ N N ⎨ ⎬ (c j ⊕c k ) (c j ⊕c k ) (c j ⊕c k ) Qn (t + 1) = max Qn (t) − μn ,(a,b) (t), 0 ⎪ ⎪ ⎪ ⎪ a=1 b=1

⎩ ⎭

D. Interference and Transmission Model We assume that nodes have a single transceiver. Therefore, they can only transmit or receive in a given time slot t. Furthermore, they use a fixed transmit power. When data transmission is carried out over a link, its interfering links are defined as the neighboring transmission links that affect the signal-tointerference ratio (SIR) of the transmission to a certain level or less. The transmission over a link is successful in a slot if and only if no interfering link tries to insert its data packets into the slot. E. Network Stability Region n (t) represent the total queue backlog at node n on Let Q slot t. ⎛ ⎞ N N N (c ⊕c ) n (t) := ⎝ Q Qn(c i ) (t) + Qn j k (t)⎠ , (6) j =1 k =1

i=1

(D )

+

N N a=1 b=1

(c j ⊕c k ) μ ¯n ,(a,b) (t),

(G )

(3)

where 1) (D) denotes the departure rate of a network-coded commodity cj ⊕ ck for neighbor nodes a and b. This is shown by (D) in Figs. 3 and 4. 2) (G) denotes the arrival rate of a network-coded commodity cj ⊕ ck . Here, the commodities cj and ck are received from neighbor nodes a and b, respectively. This is shown in (G) of Figs. 3 and 4. These two different queues in (1) and (3) are coupled through (C) and (G) and through (D) and (F). In addition, we assume a symmetric PLNC transmission considering a slow-fading model and minimum end-to-end cut-set bound: (c ⊕c )

P L ,(c j )

= μn ,b

(c ⊕c )

P L ,(c j )

=μ ¯n ,b

j k (t) = μn ,a μn ,(a,b) j k μ ¯n ,(a,b) (t) = μ ¯n ,a

P L ,(c k ) P L ,(c k )

,

(4)

.

(5)

On the other hand, the packets in the virtual queue do not affect the queue back-pressure because they have already been sent and are needed for the decoding of a future network-coded packet. Therefore, they are not included in the queue model of (1) and (3). C. Link Model As in Fig. 5, There are two link types: single-hop and twohop PLNC links. For example, a single-hop l(n, a) or l(a, n)

Then, we can define the following: Definition 1: A node is said to be stable if it has a bounded time average backlog lim sup τ →∞

τ −1 1 E Qn (t) < ∞, τ

(7)

t=0

i.e., a node is stable if the total incoming rate of traffic equals the total outgoing rate of traffic from the node. Definition 2: A network is stable if all nodes in the network are stable. (c ) (c ) Definition 3: Let λn = (λn 1 , . . . , λn N ) be the exogenous arrival rate vector of the commodities at node n. Then, the throughput region of a transmission policy is the set of all arrival rate vectors ρ = (λ1 , . . . , λN ) such that the network is stable for any arrival process of ρ under the transmission policy. Definition 4: Network capacity (stability) region Λ describes the set of all arrival rate vectors ρ for which the network is stable considering all possible transmission policies. For any ρ in the network capacity region, if a transmission policy can stabilize the network without prior knowledge of ρ, then the transmission policy is said to be throughput-optimal [29]. III. PLNC-AWARE THROUGHPUT-OPTIMAL TRANSMISSION A. Centralized Transmission Control: NCMDB We propose a new centralized transmission policy, NCMDB, in Theorem 1. Theorem 1: A throughput-optimal transmission scheduling policy for a PLNC-based cooperative network is given by the

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solution of the following: N N N (c ) (c ) max Qn(c i ) (t) − Qb i (t) μn ,bi (t)In ,b n =1

+

3) Finalizing routing variables: For each scheduled link, if the differential backlog for the optimal commodity is negative, (c o ) then no data are transferred over the link. Otherwise, μn ,bi (t),

N N N N

(c ⊕c k )

Qn j

(t)

a=1 b=1 j =1 k =1

(c ) (c i ⊕c j ) − Qa(c i ) (t) + Qb j (t) μn ,(a,b) (t)In ,(a,b) +

N N N N

(c j )

Qa

(c k )

(t) + Qb

(c ∗ ⊕c ∗ )

(c o ⊕c o )

j j k k μn ,(a,b) (t), or μ ¯n ,(a,b) (t) units of coi , coj ⊕ cok , or c∗j ⊕ c∗k is transferred, respectively. However, it is impractical to search for all transmission rate vectors in the set FG(t) because FG(t) may contain an infinite number of possible transmission rate vectors. Furthermore, it is extremely challenging to maintain and process the entire network information on a slot-by-slot basis. Therefore, the network-wide problem in Theorem 1 needs to be solved in a distributed manner.

b=1 i=1

(t)

a=1 b=1 j =1 k =1 (c ⊕c k )

− Qn j

(c j ⊕c k ) (t) μ ¯n ,(a,b) (t)I(a,b),n ,

B. Distributed Transmission Control: D-NCMDB (8)

subject to N

In ,b +

N N

In ,(a,b) +

a=1 b=1

b=1

N N

I(a,b),n ≤ 1, ∀n.

a=1 b=1

(9) Proof: See Appendix A. In Theorem 1, (8) denotes the proposed maximal scheduling. Here, In ,b , In ,(a,b) , and I(a,b),n are the indicator functions that have a value of 1 if links l(n, b), l(n, (a, b)), and l((a, b), n) are scheduled for transmission, respectively, and zero otherwise. Further, (9) denotes that each node can schedule only one link among its links. The proposed NCMDB policy differs from the conventional backpressure policy in particular owing to the coef(c ⊕c ) (c ) (c ) ficient terms ±(Qn j k (t) − Qa j (t) − Qb k (t)) that reflect the queue coupling effect induced by PLNC. To solve the problem in Theorem 1, the following steps are performed on every slot: 1) Choosing optimal commodities: For each link l(n, b), l(n, (a, b)) and l((a, b), n), the central controller finds optimal commodities coi , coj ⊕ cok , and c∗j ⊕ c∗k that have a maximal differential backlog. (c i )

coi = arg max Qn(c i ) (t) − Qb i∈N

(c ⊕c k )

coj ⊕ cok = arg max Qn j j ∈N,k ∈N

c∗j ⊕ c∗k = arg max

j ∈N,k ∈N

(c j )

Qa

(t),

(c ) (c ) (t) − Qa j (t) + Qb k (t) , (c k )

(t) + Qb

(c ⊕c ) (t) − Qn j k (t).

2) Choosing scheduling link and rates: Once the optimal commodities have been determined for each link l(n, b), l(n, (a, b)), and l((a, b), n), the central controller finds the set ¯n ,(a,b) (t) of transmission link rates μn ,b (t), μn ,(a,b) (t), and μ available under the topology state G(t), which is referred to as FG(t) using the genie-aided approach [3], [30]. Then, the node allocates the full link rate to the optimal commodity and the zero rate to all other commodities for each link. Then, the cen(c o ) tral controller chooses the optimal transmission control μn ,bi (t), (c o ⊕c o )

(c o ⊕c o )

j j k k μn ,(a,b) (t), and μ ¯n ,(a,b) (t) that maximizes (8) from the set FG(t) .

We propose a distributed transmission control algorithm called prioritization-based distributed NCMDB (D-NCMDB). 1) Node Priority Determination: Nodes determine their n (t), i.e., the total queue backnode priority L based on Q log on slot t. Here, the node with the longest backlog has the highest priority, L = 1. To this end, nodes use the well-known distributed ranking algorithm [31] that finds the node priority without knowing the information of all other nodes in the network. 2) Link Scheduling: Following the order of node priority L, each node n determines its transmission link using (10): (c ) (c ) max max Qn(c i ) (t) − Qb i (t) μn ,bi (t), b,i∈N

max

a,b,j,k ∈N

max

a,b,j,k ∈N

(c ⊕c k )

Qn j

(c j )

Qa

(c ) (c j ⊕c k ) (c ) (t) − Qa j (t) + Qb k (t) μn ,(a,b) (t), (c k )

(t) + Qb

(c ∗ ⊕c ∗ ) (c j ⊕c k ) (t) − Qn j k (t) μ ¯n ,(a,b) (t) . (10)

Case L = 1: First, the highest-priority node determines its transmission link. 1) The node first finds the optimal commodities coi , coj ⊕ cok , and c∗j ⊕ c∗k that maximize the differential backlog for all single-hop links l(n, b), outgoing PLNC links l(n, (a, b)), and incoming PLNC links l((a, b), n). 2) Next, the node finds its supportable link rates μn ,b (t), ¯n ,(a,b) (t). However, there is still no link μn ,(a,b) (t), and μ scheduled so that the node estimates these rates using the signal-to-noise ratio (SNR) with no interference. For example, if the candidate link is l(n, a), then its supportable link rate is calculated as log2 (SN Rn ,a ). SN Rn ,a =

Sn ,a , σa2

(11)

where Sn ,a denotes the signal strength, and σa2 is the average received noise power at node a. On the other hand, the supportable PLNC link rate is calculated using a minimum end-to-end cut-set bound. If the link is l(n, (a, b)), then its supportable rate is min{log2 (1 + SN Rn ,a ), log2 (1 + SN Rn ,b )},

(12)


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TABLE II SCHEDULING WEIGHT AT NODE n 0 commodity

Fig. 6. Queue-length-based transmission scheduling at node n 0 . The node n 0 has four neighbor nodes n 1 to n 4 , and six sessions, A, B, C, D, E, and F that flow through.

A B C D E F A⊕B A⊕C D⊕E D⊕F A⊕B A⊕C D⊕E D⊕F

link

ΔQ

μ

ΔQ · μ

l(n 0 , n 4 ) l(n 0 , n 1 ) l(n 0 , n 1 ) l(n 0 , n 3 ) l(n 0 , n 2 ) l(n 0 , n 2 )

2 −3 2 1 1 −1 5 4 2 1 3 2 1 2

2 1 1 3 2 2 min{1, 2} min{1, 2} min{2, 3} min{2, 3} min{1, 2} min{1, 2} min{2, 3} min{2, 3}

4 −3 2 3 2 -2 5 4 4 2 3 2 2 4

l(n 0 , (n 1 , n 4 )) l(n 0 , (n 1 , n 4 )) l(n 0 , (n 2 , n 3 )) l(n 0 , (n 2 , n 3 )) l((n 1 , n 4 ), n 0 ) l((n 1 , n 4 ), n 0 ) l((n 2 , n 3 ), n 0 ) l((n 2 , n 3 ), n 0 )

TABLE I INFORMATION AT NODE n 0 ΔQ(s)

ΔQ o u t (s ⊕ bi(s))

ΔQ i n (s ⊕ bi(s))

A

2

B C D

−3 2 1

E F

1 −1

5, (bi(s) = B) 4, (bi(s) = C ) 5, (bi(s) = A) 4, (bi(s) = A) 2, (bi(s) = E) 1, (bi(s) = F ) 2, (bi(s) = D) 1, (bi(s) = D)

3, (bi(s) = B) 2, (bi(s) = C ) 3, (bi(s) = A) 2, (bi(s) = A) 1, (bi(s) = E) 2, (bi(s) = F ) 1, (bi(s) = D) 2, (bi(s) = D)

session s

and if the link is l((a, b), n), then its supportable rate is min{log2 (1 + SN Ra,n ), log2 (1 + SN Rb,n )}.

(13)

3) Then, the node allocates the full link rate to the optimal commodity and the zero rate to all other commodities for each link. 4) Finally, the node selects the best transmission control (c oj ⊕c ok ) (c ∗j ⊕c ∗k ) (c o ) (t), and μ ¯n ,(a,b) (t) in terms of among μn ,bi (t), μn ,(a,b) the maximization of (10). Fig. 6 shows an example in which node n0 maintains the following information: S, prev(s), next(s), bi(s), ΔQ(s), ΔQou t (s ⊕ bi(s)), and ΔQin (s ⊕ bi(s)). Here, S is the set of all sessions (or commodities) flowing through node n0 ; prev(s) and next(s) are neighboring incoming and outgoing nodes of session s, respectively; and bi(s) denotes the bidirectional session of session s. For example, bi(A) de(s) (s) notes session B or C, ΔQ(s) := Qn 0 − Qn ext(s) is the differential queue backlog of session s, ΔQou t (s ⊕ bi(s)) := (s⊕bi(s)) (s) (s) Qn 0 − (Qn ext(s) + Qn ext(bi(s)) ) is the differential queue backlog of the outgoing bidirectional session of s and bi(s), and (s) (s) (s⊕bi(s)) ΔQin (s ⊕ bi(s)) := (Qn ext(s) + Qn ext(bi(s)) ) − Qn 0 is the differential queue backlog of the incoming bidirectional sessions s and bi(s). Let us assume that the information at node n0 is given in Table I. We also assume that the supportable

link rates of l(n0 , n1 ), l(n1 , n0 ), l(n0 , n2 ), l(n2 , n0 ), l(n0 , n3 ), l(n3 , n0 ), l(n0 , n4 ), and l(n4 , n0 ) are 1, 1, 2, 2, 3, 3, 2, and 2, respectively. Then, node n0 can calculate the scheduling weights of its transmissions as in Table II, where ΔQ and μ denote the longest differential queue backlog and the transmission rate over each link, respectively. From Table II, the bidirectional outgoing transmission of sessions A and B has the highest scheduling weight, 5, i.e., node n0 schedules the bidirectional outgoing sessions A and B over the link l(n0 , (n1 , n4 )). Case (2 ≤ L ≤ |N |): Following the transmission link decision of the highest-priority node, all nodes determine their transmission scheduling in the order of node priority. For a node with priority L, if one of the node’s links has been already selected for transmission by the nodes with higher priority 1 to L − 1, then the node does not determine its transmission link. However, if none of the node’s links have been selected by the nodes with higher priority 1 to L − 1, then the node determines its transmission link. To this end, as in Case L = 1 above, the node with priority L first finds the optimal commodities coi , coj ⊕ cok , and c∗j ⊕ c∗k that maximize the differential backlog for single-hop links l(n, b), outgoing PLNC links l(n, (a, b)), and incoming PLNC links l((a, b), n). Following this, to determine the supportable link rate of each candidate link, the node performs as follows: 1) If transmission over a candidate link gives excessive interference, i.e., signal-to-interference and noise ratio (SINR) damage beyond a certain value, to the already scheduled transmissions by the nodes with higher priority 1 to L − 1, then its supportable link rate becomes 0, and transmission over the link will not be scheduled. This is called transmitter-yielding. 2) If a candidate link receives excessive interference from the already scheduled transmissions by the nodes with higher priority 1 to L − 1, its supportable link rate becomes 0, and transmission over the link will not be scheduled. This is called receiver-yielding.

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3) If transmission over a candidate link does not experience transmitter- or receiver-yielding, it can be selected for transmission scheduling. Currently, the supportable link rate is determined by the SINR value based on the interference by the scheduled transmissions of the nodes with higher priority 1 to L − 1. For example, if the candidate link is l(n, a), then its supportable link rate is calculated as log2 (SIN Rn ,a ), Sn ,a , SIN Rn ,a = L −1 2 k =1 Ik + σa

(14)

where Ik denotes the interference from the scheduled transmission by a node with priority k. On the other hand, if the link is l(n, (a, b)), then its supportable rate is

Fig. 7. Route setup (RS) packet forwarding. In this figure, RS 1 , RS 2 , and RS 3 packets are assumed to sequentially arrive from n 1 , n 2 , and n 3 at T 1 , T 2 , and T 3 , respectively. Then, the FIFO-based forwarding in (a) forwards the RS 1 packet that arrives first. However, the delayed forwarding in (b) forwards the RS 3 packet that has the earliest deadline after ω i delay for RS i packet, i = 1, 2, 3.

min{log2 (1 + SIN Rn ,a ), log2 (1 + SIN Rn ,b )}, (15) and if the link is l((a, b), n), then its supportable rate is min{log2 (1 + SIN Ra,n ), log2 (1 + SIN Rb,n )}. (16) Then, in Case L = 1 above, the node with priority L allocates the full link rate to the optimal commodity and the zero rate to all other commodities for each link. Finally, the node with pri(c o ) ority L selects the best transmission control among μn ,bi (t), (c oj ⊕c ok ) μn ,(a,b) (t),

(c ∗j ⊕c ∗k ) μ ¯n ,(a,b) (t)

and in terms of the maximization of (10). 3) Rate Scheduling: In the previous link scheduling phase, the nodes temporarily calculate the supportable link rates considering interference only from the higher-priority nodes. This can provide an inaccurate transmission rate, i.e., an inaccurate modulation and coding rate, in the data transmission phase. Therefore, after all nodes determine their scheduling links, they recalculate accurate transmission rates based on wideband pilots from all scheduled links. For example, if the scheduled link is l(n, a), then it recalculates its actual transmission rate as log2 (SIN Rno ,a ). Sn ,a . 2 k ∈N Ik + σa

SIN Rno ,a =

(17)

On the other hand, if the scheduled link is l(n, (a, b)), then its actual transmission rate is min{log2 (1 + SIN Rno ,a ), log2 (1 + SIN Rno ,b )},

(18)

and if the link is l((a, b), n), its actual transmission rate is o o min{log2 (1 + SIN Ra,n ), log2 (1 + SIN Rb,n )}.

(19)

The proposed distributed transmission scheduling scheme leads to efficient spatial packing and channel- and queue-aware maximal matching.

Fig. 8.

Examples of link bidirectionality of a link l(n i , n j ).

setup by FIFO-based RS packet forwarding does not consider the bidirectionality of traffic flows, it cannot effectively support the proposed PLNC-based transmission scheme. Therefore, we propose a new RS packet forwarding scheme that can enhance the bidirectionality between established routes. This is referred to as a delayed forwarding scheme [see Fig. 7(b)]. Before elaborating on this proposed RS packet forwarding scheme, we define two new metrics: link bidirectionality and route bidirectionality. Definition 5: The link bidirectionality of a link l(ni , nj ) is defined as the number of bidirectional traffic sessions over the link. This is denoted as K(ni , nj ) := min{T (l(ni , nj )), T (l(nj , ni ))},

(20)

where T (l(ni , nj )) is the number of traffic flows from node ni to node nj . Link bidirectionality has a symmetric property, i.e., K(ni , nj ) = K(nj , ni ). Fig. 8 shows examples of link bidirectionality. Definition 6: The route bidirectionality of a route from a source to a destination is defined as Γ=

min

l(n i ,n j )∈{R(n s ,n d )}

K(ni , nj ),

(21)

where {R(ns , nd )} denotes the set of links that belong to a route R(ns , nd ) from a source ns to a destination nd . The proposed forwarding scheme uses a delay function to set up a source-to-destination route that has more bidirectional traffic flows. Here, the delay function f (·) should satisfy the following conditions:

IV. BIDIRECTIONAL PATH ESTABLISHMENT

(C1) f (K) ≥ f (K + n), for 0 ≤ K ≤ |N| − 1 and n > 0,

In wireless multihop networks, conventional routing schemes such as dynamic source routing (DSR) [32] and ad-hoc on-demand distance vector (AODV) [33] find a source-todestination route using first-in first-out (FIFO)-based route setup (RS) packet forwarding, as shown in Fig. 7(a). However, as route

(C2) f (K) ≥ (|N| − 1) · f (K + 1), for 0 ≤ K ≤ |N| − 1, (C3) (|N| − 1) · f (0) ≤ Dlim it , where K represents link bidirectionality, and |N| denotes the number of nodes in the network. (C1) means that the forwarding


delay should decrease as the link bidirectionality increases. (C2) means that the RS packet passing along a route with larger bidirectionality should always arrive earlier. For example, let us assume that there are two different routes, R1 (ns , nd ) and R2 (ns , nd ). The routes have Γ1 and Γ2 bidirectionality, respectively, where Γ1 < Γ2 . Then, the RS packet passing along R2 (ns , nd ) can experience at most a (|N| − 1) · f (Γ2 ) end-toend delay, while the RS packet passing along R1 (ns , nd ) can experience at least a f (Γ1 ) end-to-end delay. Thus, if the delay f (Γ1 ) is larger than (|N| − 1) · f (Γ2 ), it can be strictly guaranteed that the RS packet passing along R2 (ns , nd ) always arrives earlier than the RS packet passing along R1 (ns , nd ). Finally, (C3) means that the largest end-to-end delay, (|N| − 1) · f (0), should be smaller than a route setup delay constraint Dlim it . Based on these conditions, we can design the delay function in Property 1. Property 1: The delay function f (K) is f (K) =

1 · Dlim it . (|N| − 1)K+1

(22)

Proof: See Appendix B. On the other hand, some problems might be encountered when employing the delay function in (22). First, if K is large, f (K) becomes very small. That is, when K is large, very fast and accurate delay control is demanded at the nodes. However, owing to software and hardware processing limitations, the forwarding delays that the nodes can handle may not be less than a certain time. Second, as K becomes larger, routes with greater bidirectionality can be established. However, this can drive burst traffic on some links, adversely affecting network load balancing and resource utilization. Third, the delay function can induce longer paths while providing paths with greater bidirectionality. ¯ To address the first and second problems, we introduced K, which is the maximum bidirectionality K that the delay function is applied to. If the link bidirectionality K becomes greater than ¯ the forwarding delay for the link is set to f (0) to minimize K, additional traffic passing through the link. Moreover, to address the last problem, we introduced α, which scales the delay function as f (K) = α · (|N| − 1) · f (K + 1) in (C2), where 1 |N|−1 ≤ α ≤ 1. This means that the RS packet passing along the path with route bidirectionality Γ = K + 1 can arrive earlier than the RS packet passing along the 1-hop direct path with route bidirectionality Γ = K only if the route length of the path with Γ = K + 1 is shorter than α · (|N| − 1). Considering these relaxations along with the smallest for¯ K) can warding delay γ, a new relaxed delay function f (α, K, be built. ¯ K) is Property 2: The relaxed delay function f (α, K, 1 1 ¯ · · Dlim it , K ≤ K, α K (|N|−1) K+ 1 ¯ f (α, K, K) = ¯ f (0), K > K, where

1 1 ∗ ¯ K ≤ max m ∈ Z : m · Dlim it ≥ γ . α (|N| − 1)m +1 Proof: See Appendix C.

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Algorithm 1: Proposed Route Setup Packet Forwarding. 1: Any intermediate node in the network, say nj , performs the following. 2: while (1) do 3: T ← current system time 4: if there is an arrival of RSi from a neighbor ni then 5: Ti ← T 6: Calculate the link bidirectionality K(ni , nj ) 7: Calculate corresponding delay f (K(ni , nj )) 8: Insert the RSi to the arrived RS set, Φ 9: end if 10: Check RSk , k = arg min{Ti + f (K(ni , nj ))} i∈Φ

11: if Tk + f (K(nk , nj )) ≤ T then 12: Forward the RSk packet 13: Discard the other RS packets 14: Break 15: end if 16: end while 1 ¯ = 0, all We can observe two extreme cases: If α = |N|−1 or K nodes use the same forwarding delay regardless of the link bidirectionality. Then, the delayed forwarding is reduced to legacy ¯ = |N| − 1, FIFO-based forwarding. Otherwise, if α = 1 and K all nodes perfectly satisfy the forwarding delay conditions (C1) to (C3), thus corresponding to strict delayed forwarding. There¯ any relaxed RS packet forwarding fore, by controlling α and K, between two extreme forwarding schemes can be configured. Algorithm 1 summarizes the proposed RS packet forwarding scheme. Remark 1: The delay function includes no link propagation delay because the link propagation delay is negligible. If necessary, however, the average link propagation delay can be incorporated into the delay function.

V. PERFORMANCE EVALUATION A. Complexity and Overhead Analysis The distributed transmission schemes can be analyzed in terms of computational complexity, communication overhead, and storage overhead. To perform an analysis, we decompose the transmission schemes into route determination, node priority decision, link scheduling, and rate scheduling. First, the proposed scheme has the following computational complexity: 1) Route determination: Each node calculates RS packets’ forwarding delay using a closed-form formula, which has O(1) complexity. 2) Priority decision: Each node finds its priority in a distributed fashion by employing the distributed ranking algorithm in [31], which has O(N 4 ) complexity. 3) Link scheduling: Each node finds its optimal commodity for each possible scheduling link, supportable link rates, and the best transmission using (8), which has O(N 2 ), O(N ), and O(N ) complexity, respectively.

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4) Rate scheduling: Each node determines the final transmission rate of its scheduled link, which has O(1) complexity. Second, the proposed scheme has the following communication overhead. In a distributed environment, the success of the proposed scheduling control relies on the message exchange in the control channel between neighboring nodes. 1) Route determination: Nodes do not exchange additional information with neighbor nodes. 2) Priority decision: Nodes exchange O(N 2 ) messages with neighbors [31]. 3) Link scheduling: Nodes exchange their queue and channel information with neighbor nodes, demanding O(N 2 ) messages, respectively. In addition, nodes exchange SINR information for yielding decisions. This demands O(N ) messages. 4) Rate scheduling: Nodes exchange SINR information for the actual transmission rate determination. This demands O(N ) messages. Third, the proposed scheme has the following storage overhead: 1) Route determination: Each node saves its bidirectionality information, which needs O(N ) storage. 2) Priority decision: Each node saves its priority information, which needs O(N ) storage. 3) Link scheduling: Each node saves its own and neighbors’ queue information, which needs O(N 2 ) storage. Because each node maintains network-coded queues, it has more complicated storage maintenance overhead. 4) Rate scheduling: Each node does not save any information. The complexity and overhead of the proposed scheme is mainly owing to the management of bidirectionality, node priority, differential backlog, and implicit channel information for the proposed scheme. However, the complexity and overhead are polynomial and hence can be efficiently managed in practical networks. B. Throughput, Delay, and Stability We evaluated the performance of the proposed transmission scheduling scheme. Nodes were randomly placed in a 1,000 (m) × 1,000 (m) network, with two-thirds of the nodes selected as source and destination nodes. The location of the node was fixed, but the wireless channel continued to change according to a slow-fading model. At this time, link rescheduling and rate adaptation were applied depending on the channel variation. The average transmit power of each node was set to 23 (dBm). All results were obtained by taking the average of 20 independent runs, with each run executed for 200 time slots of duration 1(ms) apiece. Using this setup, the following schemes were compared under the system-level simulation assumptions listed in Table III: 1) An SNR-based scheme is a signal-to-noise-ratio-based transmission scheduling approach such as Wi-Fi [34]. Any transmitter that senses high signal energy from neighboring transmitters or gives high signal energy to its neighboring receivers is not permitted to transmit. The energy threshold is set to −76 (dBm). Here, the end-to-end


TABLE III SYSTEM-LEVEL SIMULATION PARAMETERS Parameter Network Size Node Deployment Minimum Node Distance Bandwidth Node Transmit Power Path-Loss Shadowing Fading Node Mobility

Assumption 1,000 (m) × 1,000 (m), Wrap-around Random 50 (m) CF: 2 GHz, BW: 10 MHz 23 (dBm) 140.7 + 36.7 log 10 (d(km))(dB) 8 (dB) Slow Rayleigh fading Static

paths are established using the FIFO-based RS forwarding scheme. This scheme is an interference-avoidance-based transmission scheduling approach. 2) An SIR-based scheme is a signal-to-interference-ratiobased transmission scheduling approach proposed in [8]. It employs nonprioritized random maximal scheduling, where in each time slot the scheduler chooses the scheduling links in a random order. A transmission can be scheduled if it does not cause excessive SIR damage to already scheduled transmissions and does not receive excessive SIR damage from the scheduled transmissions. The SIR threshold is set to 9 (dB). Here, end-to-end paths are established using the FIFO-based RS forwarding scheme. This scheme is also an interference-avoidance-based transmission scheduling approach. 3) An opportunistic-XOR-based scheme opportunistically applies exclusive-OR (XOR) coding proposed in [35], [36]. It has no queue-based control or priority-based link scheduling control. Here, end-to-end paths are established using the FIFO-based RS forwarding scheme. This scheme is an interference-exploitation-based transmission scheduling approach. 4) A random-PLNC-based scheme applies the random maximal PLNC control proposed in [37], [38]. It has no queue-based control or priority-based link scheduling control. Here, end-to-end paths are established using the FIFO-based RS forwarding scheme. This scheme is an interference-exploitation-based transmission scheduling approach. 5) Our proposed scheme is the PLNC-based distributed scheduling scheme (D-NCMDB) with an SIR threshold of 9 (dB). Here, end-to-end paths are established by the proposed delayed RS forwarding scheme. In the evaluation, ¯ to 2 because it provides more bidirectional seswe set K sions with a balanced network-resource utilization. This scheme is an interference exploitation-based transmission scheduling approach. Fig. 9 compares the achievable system throughput as the number of nodes increases to 100. 1) When the proposed scheme is employed with FIFO routing, it has up to 10%, 22%, and 28% enhanced throughput as compared to random-PLNC, opportunisticXOR, and SIR-based schemes, respectively. This performance gain comes purely from the proposed transmission


Fig. 9.

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System throughput according to the number of nodes in the networks. Fig. 11. Stability with 50 nodes under the Poisson arrival rates into the networks.

Fig. 10.

Access delay according to the number of nodes in the networks.

scheduling, i.e., without the proposed bidirectional route establishment. 2) If the proposed transmission scheduling scheme is em¯ = 2, α = 0.5), the throughput gains inployed with (K crease up to 21%, 35%, and 41%, respectively. 3) If the proposed transmission scheduling scheme is em¯ = 2, α = 1.0), the throughput gains ployed with (K increase up to 27%, 41%, and 48%, respectively. The performance gain was more enhanced as the proposed scheme provides more bidirectional sessions. Fig. 10 compares the access delay, which is defined as the time the nodes wait to send data. 1) When the proposed scheme is employed with FIFO routing, it offers 2.3(ms) access delay under approximately a 100-node deployment. This represents an 8%, 30%, and 39% reduction in access delay as compared to randomPLNC, opportunistic-XOR, and SIR-based schemes, respectively. 2) If the proposed transmission scheduling scheme is em¯ = 2, α = 0.5), the access delay of the ployed with (K

proposed scheme is reduced by 32%, 48%, and 55%, respectively. 3) If the proposed transmission scheduling scheme is em¯ = 2, α = 1.0), the access delay of the ployed with (K proposed scheme is reduced by 63%, 75%, and 78%, respectively. Fig. 11 shows the network stability result with respect to the uniform arrival rate. The network stability is indirectly compared using a critical point (the arrival rate at which the maximum queue length in the network begins to increase), as in [39]–[41]. As the critical point becomes larger, the network becomes more stable. 1) When the proposed scheme is employed with FIFO rout¯ = 2, α = 0.5), the critical points are 0.42 ing and (K and 0.5 (bps/Hz). This has an enhanced stability region as compared to random-PLNC, opportunistic-XOR, and SIR-based schemes, respectively. ¯ = 2, α = 2) When the proposed scheme is employed with (K 1), its critical point is approximately 0.55 (bps/Hz). With respect to the critical point, the proposed scheme provides an approximately 20%, 60%, and 80% larger stability than random-PLNC, opportunistic-XOR, and SIR-based schemes, respectively. ¯ = 2. In fact, finding the optiIn this evaluation, we used K ¯ mal K is beyond the scope of this study. It is chosen through numerical simulation instead of any mathematical analysis or ¯ in optimization. We will determine and apply the optimal K future work. Moreover, if the proposed network-coded queuebased throughput-optimal scheduling and bidirectional route establishment are applied with other advanced algorithms in [16]-[20], then a better performance gain can be achieved. VI. CONCLUSION In this work, we developed a physical-layer network codingbased throughput-optimal transmission scheduling scheme called NCMDB policy. We analytically derived the policy and proved its throughput optimality. As the scheme requires

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searching for all rate vectors and maintaining information on the entire network, for the sake of practicality, we further proposed a prioritization-based distributed transmission scheduling scheme for the NCMDB policy called D-NCMDB. We also proposed a new route establishment scheme that exploits network bidirectionality to enhance the effectiveness of NCMDB and D-NCMDB. Simulations confirmed that the proposed distributed transmission scheduling scheme enhances the system performance in terms of throughput, access delay, and stability in a random network topology. Compared to an SIR-based interference avoidance transmission scheduling approach and a random maximal physical-layer network coding approach, the proposed D-NCMDB scheme offers 48% and 27% increased throughput, 78% and 63% reduced access delay, and 80% and 20% enlarged stability. The proposed distributed transmission scheduling scheme can be efficiently employed in future communication systems such as multihop heterogeneous networks, relay-assisted cellular networks, and IoT networks. In future work, the tightness of the capacity region of the proposed scheme will be demonstrated, and further advanced distributed transmission scheduling schemes with enhanced maximal matching and optimal bidirectionality control will be investigated. Furthermore, in the bigger picture, the PLNC scheme can be thought as one example of the non-orthogonal multiple access (NOMA) technology [42], [43]. Therefore, we will also extend this work to more general schemes including NOMA with successive interference cancellation (SIC) and power control. APPENDIX A PROOF OF THEOREM 1

(c )

(c )

= max Qn(c i ) (t) −

N b=1

a=1

(28)

Then, (26) can be simply expressed as Qn(c i ) (t + 1) = max Qn(c i ) (t) −

μ ¯n(c,ai ) (t) +

N a=1

(c )

μn ,bi (t) +

N

N

! (c ) υn ,bi (t), 0

b=1

+

N

υ¯n(c,ai ) (t) + An(c i ) (t),

(29)

a=1

On the other hand, the queue length dynamics of the networkcoded data at node n can be described as ! N N (c j ⊕c k ) (c j ⊕c k ) (c j ⊕c k ) Qn (t + 1) = max Qn (t) − μn ,(a,b) (t), 0 a=1 b=1

+

N N

(c ⊕c )

j k μ ¯n ,(a,b) (t).

(30)

a=1 b=1

In (30), there is no exogenous arrival of the network-coded data because the network-coded data can be generated using the uncoded data that have already arrived. Using the following (max[x − y, 0] + z)2 ≤ x2 + y 2 + z 2 + 2x(z − y),

1

(31)

Qn(c i ) (t

2 2 + 1) ≤ Qn(c i ) (t) +

N

! P L ,(c i )

μn ,b

(t) , 0

2 (c ) υn ,bi (t)

b=1

+

N

2

υ¯n(c,ai ) (t) + An(c i ) (t)

a=1

+

i) 2Q(c n (t)

An(c i ) (t) +

N

υ¯n(c,ai ) (t) −

N

(c )

υn ,bi (t) ,

b=1

(32) and 2 2 (c ⊕c ) (c ⊕c ) Qn j k (t + 1) ≤ Qn j k (t)

(25)

The queue length dynamics of the uncoded data at node n can be described as

N

υ¯n ,bi (t) := μ ¯n(c,ai ) (t) + μ ¯nP ,aL ,(c i ) (t).

a=1

ΔL(t) = E [L(Q(t + 1)) − L(Q(t))|Q(t)] .

+

(27)

along with (29) and (30), we can observe

(24) We derive the proposed NCMDB policy using a Lyapunov drift representing the expected change in the Lyapunov function from one slot to the next slot:

+ 1)

(t),

j =1 k =1

i=1

P L ,(c i )

(c )

and set a Lyapunov function as below, ⎛ ⎞ N N N N 2 2 (c ⊕c ) ⎝ Qn(c i ) (t) + Qn j k (t) ⎠. L(Q(t)) =

Qn(c i ) (t

(c )

υn ,bi (t) := μn ,bi (t) + μn ,b

Proof: We define Q(t) as the entire queue status in the network on slot t, (c ⊕c ) Q(t) = Qn(c i ) (t), Qn j k (t) |i, j, k, n ∈ N , (23)

n =1

(c )

Here, we define υn ,bi (t) and υ¯n ,bi (t) as

+

N N

2 (c j ⊕c k ) μn ,(a,b) (t)

a=1 b=1 (c ⊕c k )

+ 2Qn j

(t)

+

N N

2 (c j ⊕c k ) μ ¯n ,(a,b) (t)

a=1 b=1 N N a=1 b=1

(c ⊕c )

j k μ ¯n ,(a,b) (t) −

N N

(c ⊕c )

j k μn ,(a,b) (t) .

a=1 b=1

(33)

b=1

μ ¯nP ,aL ,(c i ) (t) + An(c i ) (t).

Then, the Lyapunov drift in (25) can be expressed as (26)

ΔL(t) ≤ B(t) + W (t),

(34)


where B(t) = ⎡ ⎛ ⎛ ⎞ 2 N 2 N N N (c ) E⎣ ⎝ ⎝ υn ,bi (t) + υ¯n(c,ai ) (t) + An(c i ) (t) ⎠ n =1

+

i=1

⎛

N N

⎝

j =1 k =1

this amounts to minimizing the following (39): N 'N N N (c ) (c i ) (c i ) Qn (t) υ¯n ,a (t) − υn ,bi (t) min E n =1

N N

i=1

(35)

+

(c ⊕c k )

2Qn j

(t)

j =1 k =1

−

a=1 N N

(c j ⊕c k ) μn ,(a,b) (t)

a=1 b=1

+

i=1

b=1

N N

(c ⊕c k )

Qn j

(t)

j =1 k =1 N N

−

b=1

a=1 N N

(c ⊕c )

j k μn ,(a,b) (t)

a=1 b=1

$ ( $ $ $ Q(t) . $

(c j ⊕c k ) μ ¯n ,(a,b) (t)

a=1 b=1

(40)

(c ⊕c )

j k μ ¯n ,(a,b) (t)

a=1 b=1

N N

(39)

which is again equivalent to maximizing the following (40): 'N N N N (c ) (c i ) max E Qn (t) υn ,bi (t) − υ¯n(c,ai ) (t)

W (t) = 'N N N N (c ) (c i ) E 2Qn (t) An(c i ) (t) + υ¯n(c,ai ) (t) − υn ,bi (t)

(c ⊕c )

j k μ ¯n ,(a,b) (t)

$ ( $ $ $ Q(t) , $

(c j ⊕c k ) μn ,(a,b) (t)

a=1 b=1

n =1

i=1

(t)

b=1 N N a=1 b=1

N N

−

a=1 b=1

and

N N

(c ⊕c k )

Qn j

j =1 k =1

2 (c j ⊕c k ) μn ,(a,b) (t)

⎤ ⎞⎞$ N N 2 $ (c ⊕c ) $ j k + μ ¯n ,(a,b) (t) ⎠⎠$$ Q(t)⎦ , $ a=1 b=1

n =1

a=1

N N

+

a=1

b=1

3141

$ ( $ $ $ Q(t) . $

(36)

The expectation in (40) is then maximized by maximizing the function inside of it: N N N N (c ) max Qn(c i ) (t) υn ,bi (t) − υ¯n(c,ai ) (t) n =1

In (35), it is assumed that the second moment of arrival rate and maximum possible second moment of transmission rate under any algorithm for choosing the transmission rates are ¯ Further, in (36), finite. Then, B(t) is upper bounded by B. (c i ) (c i ) since E[An (t)] = λn ,

+

i=1

N N

(c ⊕c k )

Qn j

(t)

j =1 k =1

−

N N

a=1

b=1

N N

(c ⊕c )

j k μn ,(a,b) (t)

a=1 b=1 (c ⊕c )

j k μ ¯n ,(a,b) (t)

.

(41)

a=1 b=1

W (t) = 'N N N N (c ) (c i ) E 2Qn (t) λn(c i ) (t) + υ¯n(c,ai ) (t) − υn ,bi (t) n =1

+

i=1 N N

(c ⊕c ) 2Qn j k (t)

j =1 k =1

−

N N a=1 b=1

a=1 N N

P L ,(c j )

= μn ,b

(c ⊕c )

P L ,(c j )

=μ ¯n ,b

j k μ ¯n ,(a,b) (t) = μ ¯n ,a

(c j ⊕c k ) μ ¯n ,(a,b) (t)

$ ( $ $ $ Q(t) . $

(c ⊕c )

j k μn ,(a,b) (t) = μn ,a

b=1

a=1 b=1

(c j ⊕c k ) μn ,(a,b) (t)

Using (27), (28), and the following

(37)

P L ,(c k )

,

(42)

,

(43)

the maximization in (41) can be reorganized as shown below: N N N (c ) (c ) max Qn(c i ) (t) − Qb i (t) μn ,bi (t) n =1

+

Then, the Lyapunov drift can be again described as

P L ,(c k )

b=1 i=1

N N N N

(c ⊕c k )

Qn j

(t)

a=1 b=1 j =1 k =1

¯ + W (t). ΔL(t) ≤ B

(38)

System stability can be achieved by taking control actions that make the Lyapunov drift in the negative direction approach zero, i.e., as the Lyapunov drift becomes smaller, the system has better stability. Therefore, in (38), the right-hand side of the ¯ and λn(c i ) is constant, Lyapunov drift should be minimized. As B

(c ) (c j ⊕c k ) (c ) − Qa j (t) + Qb k (t) μn ,(a,b) (t) +

N N N N

(c j )

Qa

(c k )

(t) + Qb

(t)

a=1 b=1 j =1 k =1 (c ⊕c k )

− Qn j

(c j ⊕c k ) (t) μ ¯n ,(a,b) (t) .

(44)

3142


On the other hand, each node can schedule only one link among its links. Then, (44) can be expressed as N N N (c ) (c ) Qn(c i ) (t) − Qb i (t) μn ,bi (t)In ,b max n =1

+

b=1 i=1

N N N N

(c ⊕c k )

Qn j

a=1 b=1 j =1 k =1

− +

(c ) Qa j (t)

+

(c ) Qb k (t)

N N N N

(c j )

Qa

a=1 b=1

−

(49) Plugging (48) and (49) into W (t) in (37) yields: ⎛ ⎞ N N N N N (c ⊕c ) ¯ − 2δ ⎝ ΔL(t) ≤ B Qn(c i) (t) + Qn j k (t)⎠.

(c j ⊕c k ) μn ,(a,b) (t)In ,(a,b) (c k )

(t) + Qb

(c j ⊕c k ) μ ¯n ,(a,b) (t)I(a,b),n

n =1 i=1

(t)

,

(45)

subject to N b=1

In ,b +

a=1 b=1

(t)

a=1 b=1 j =1 k =1 (c ⊕c ) Qn j k (t)

following for all n = i: $ ( ' N N $ $ (c ) (48) υ¯n(c,ai ) (t) − υn ,bi (t)$ Q(t) < −δ, E λn(c i ) + $ a=1 b=1 $ ' N N ( N N $ (c ⊕c ) $ (c j ⊕c k ) j k E μ ¯n ,(a,b) (t) − μn ,(a,b) (t)$ Q(t) < −δ. $

n =1 j =1 k =1

as the total queue backlog in the network on If we define Q(t) slot t, ⎞ ⎛ N N N N N (c ⊕c ) j k := ⎝ Qn(c i ) (t) + Qn (t)⎠ , Q(t) n =1 i=1

N N a=1 b=1

In ,(a,b) +

N N

n =1 j =1 k =1

then, I(a,b),n ≤ 1, ∀n.

a=1 b=1

Here, In ,b , In ,(a,b) , and I(a,b),n are indicator functions, i.e., they have 1 if link l(n, b), l(n, (a, b)), and l((a, b), n) are scheduled for transmission, respectively, and zero otherwise. Then, (c j ⊕c k ) (c j ⊕c k ) (c ) (t) and μ ¯n ,(a,b) (t) to maximize one chooses μn ,bi (t), μn ,(a,b) (45) with the condition (46), which is the NCMDB policy in Theorem 1. Furthermore, the throughput-optimality of the proposed NCMDB policy can be shown as follows: For any arrival rate (c ) (c ) vector ρ = (λn 1 , . . . , λn N ) in the capacity region Λ, there is a capacity-achieving policy that chooses its transmission rates while observing only channel states in every slot t. The determined transmission rates are independent of current queue backlogs. This implies that for the determined transmission rates, the conditional expectation given Q(t) in (37) is the same as the unconditional expectation. By plugging the determined transmission rates into W (t) in (37), we assume that we can observe W ∗ (t). On the other hand, the proposed NCMDB policy minimizes W (t), which is referred to as W N C M D B (t). Then, we can say that W N C M D B (t) ≤ W ∗ (t).

¯ − 2δ Q(t). ΔL(t) ≤ B

(46)

(47)

That is, the proposed NCMDB policy can also satisfy the Lyapunov condition from the Lyapunov theory [44], and it can support all rates in the capacity region. This means that the proposed NCMDB policy is throughput-optimal. (c ) Moreover, for any arrival rate λn i in the interior of the network capacity region Λ, there exists a vector δ > 0 such (c ) (c ) (c ) that (λn i + δIn i ) ∈ Λ, where In i is 1 if n = i, and zero otherwise. Then, the proposed transmission scheme yields the

(50)

For any > 0, we can find the compact region as (c ⊕c ) ≤ B+ , Qn(c i ) (t), Qn j k (t) : Q(t) Ψ= 2δ

(51)

such that ΔL(t) < − . That is, if an arrival rate is in the capacity region Λ, the Lyapunov drift in (25) is less than − whenever Q(t) ∈ / Ψ, and a steady-state distribution on the queue status Q(t) exists. Further, summing (50) up for t ∈ {0, 1, . . . , T − 1}, ¯ − 2δ ΔL(T ) − ΔL(0) ≤ BT

T −1

Q(t).

(52)

t=0

Then, the time-averaged network queue size becomes lim sup T →∞

T −1 ¯ B 1 Q(t) = T 2δ

(53)

t=0

APPENDIX B PROOF OF PROPERTY 1 Proof: In (C1) and (C2), the tightest delay function f (·) should satisfy f (K) = (|N| − 1|) · f (K + 1), 0 ≤ K ≤ |N| − 1.

(54)

This implies that f (0) = (|N| − 1) · f (1) = · · · = (|N| − 1)|N|−1 · f (|N| − 1). (55) On the other hand, (|N| − 1) · f (0) ≤ Dlim it in (C3). Under the assumption that the route-setup delay time is fully used to the maximum Dlim it , it can be said that f (0) =

1 · Dlim it (|N| − 1)

(56)


Then, we can find the delay function f (K) using (55) and (56) as f (K) =

1 · Dlim it . (|N| − 1|)K+1

(57)

APPENDIX C PROOF OF PROPERTY 2 Proof: Along the same lines as Property 1, from (C1) and (C2), ¯ 0) = α · (|N| − 1|) · f (α, K, ¯ 1) f (α, K, = ··· ¯ K) = αK · (|N| − 1|)K · f (α, K, = ··· ¯ ¯ ¯ K), ¯ = αK · (|N| − 1|)K · f (α, K,

(58)

1 where |N|−1 ≤ α ≤ 1 since 1 ≤ α · (|N| − 1|) ≤ |N| − 1|. Furthermore, from (C3),

¯ 0) = f (α, K,

1 · Dlim it . (|N| − 1)

(59)

¯ K) Then, we can then find the relaxed delay function f (α, K, using (58) and (59) as 1 ¯ ¯ K) = 1 · · Dlim it , where K ≤ K. f (α, K, K α (|N| − 1)K+1 (60) On the other hand, the forwarding delay cannot be shorter than ¯ γ owing to the system constraint. Therefore, the maximum K should be limited to 1 ¯ ≤ max m ∈ Z∗ : 1 K · D ≥ γ , lim it αm (|N| − 1)m +1 where Z∗ is the set of nonnegative integers.

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Wonjong Noh (S’02–M’08) received the B.S., M.S., and Ph.D. degrees from the Department of Electronics Engineering, Korea University, Seoul, South Korea, in 1998, 2000, and 2005, respectively. From 2005 to 2008, he conducted his postdoctoral research at Purdue University, IN, USA and the University of California at Irvine, Irvine, CA, USA. Since 2008, he has been with the Samsung Electronics, Suwon, South Korea, as a Principal Engineer. His current research interest focuses on wireless communication networks. He is especially interested in studying issues related to performance modeling and analysis, network capacity, cloud computing, and machine-learning-based intelligent IoT systems control in 5G/6G wireless communication and networks. He was a recipient of the Samsung Best Paper Award in 2010.


Sung Hoon Lim (S’08–M’12) received the B.S. degree (with Hons.) in electrical and computer engineering from Korea University, Seoul, South Korea, in 2005, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, South Korea, in 2007 and 2011, respectively. From March 2012 to May 2014, he was with the Samsung Electronics and, from June 2014 to July 2016, he was a Postdoctoral Associate with the School of Computer and Com´ munication Sciences, Ecole Polytechnique F´ed´erale, Lausanne, Switzerland. He is currently with the Korea Institute of Ocean Science and Technology, Busan, South Korea. His research interests include information theory, communication systems, data compression, and coding theory.

Tae-Suk Kim received the M.S. degree and Ph.D. degree under the Industrial Engineering and Telecommunication Engineering Interdisciplinary Program from the Korea Advanced Institute of Science and Technology, Daejeon, South Korea, in 2000 and 2005, respectively. From 2005 to 2009, he was a Postdoctoral Research Associate at the University of Illinois at Urbana Champaign and the University of California, Riverside. He was a research staff at the Samsung Advanced Institute of Technology. He is currently a Professor with the Business Administration Department, Pai Chai University, Daejeon, South Korea. His research interests include resource management, cross-layer optimization, and machine-learning application of wireless networks.