Author's personal copy Wireless Netw (2016) 22:1133–1144 DOI 10.1007/s11276-015-1017-x

HierHybNET: Cut-set upper bound of ad hoc networks with cost-effective infrastructure Cheol Jeong1 • Won-Yong Shin2

Published online: 23 July 2015  Springer Science+Business Media New York 2015

Abstract This paper introduces an information-theoretic upper bound on the capacity scaling law for a hierarchical hybrid network (HierHybNET), consisting of both n wireless ad hoc nodes and m base stations (BSs) equipped with l multiple antennas per BS, where the communication takes place from wireless nodes to a remote central processor through BSs in a hierarchical way. We deal with a general scenario where m, l, and the backhaul link rate scale at arbitrary rates relative to n. Then, a generalized cut-set upper bound under the HierHybNET model is derived by cutting not only the wireless connections but also the wired connections. In addition, the corresponding infrastructurelimited regime is identified. Keywords Ad hoc network  Backhaul  Base station (BS)  Capacity scaling  Cost  Infrastructure  Remote central processor (RCP)

1 Introduction We introduce a large-scale hierarchical hybrid network (HierHybNET) assuming an arbitrary rate scaling of backhaul links, where the HierHybNET is composed of both n wireless ad hoc nodes and m base stations (BSs) & Won-Yong Shin [email protected] Cheol Jeong [email protected] 1

DMC R&D Center, Samsung Electronics, Suwon 443-742, Republic of Korea

2

Department of Computer Science and Engineering, Dankook University, Yongin 448-701, Republic of Korea

equipped with l multiple antennas per BS, where the communication takes place from wireless nodes to a remote central processor (RCP) through BSs in a hierarchical way. Provided that three scaling parameter (i.e., the number of BSs, the number of antennas at each BS, and the rate of each backhaul link) scale at arbitrary rates relative to the number of wireless nodes, n, a generalized upper bound on the capacity scaling is derived for our HierHybNET with finite-capacity infrastructure based on the cut-set theorem [4]. In order to obtain a tight upper bound on the aggregate capacity, we consider two different cuts under the network model. The first cut divides the network area into two halves by cutting the wireless connections. An interesting case is the use of a new cut (i.e., the second cut), which divides the network area into another halves by cutting not only the wireless connections but also the wired connections. In addition, using the above cut-set upper bound, we show the case where the HierHybNET is in the infrastructure-limited regime. To validate the derived upper bound for finite values of system parameters n, m, and l, numerical evaluation is also shown via computer simulations. The rest of this paper is organized as follows. The related work is summarized in Sect. 2. The system and channel models are described in Sect. 3. A generalized cutset upper bound on the capacity scaling is derived in Sect. 4. The corresponding infrastructure-limited regime is identified in Sect. 5. The numerical results are also shown in Sect. 6. Finally, Sect. 7 summarizes our paper with some concluding remarks. Throughout this paper, bold upper and lower case letters denote matrices and vectors, respectively. The superscripts T and y denote the transpose and conjugate transpose, respectively, of a matrix (or a vector). The matrix IN is an

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N  N identity matrix. Unless otherwise stated, all logarithms are assumed to be to the base 2.

2 Related work 2.1 Studies on a variety of wireless networks There have been a great number of studies on the performance of various types of wireless networks. Since the capacity of wireless channels is fundamentally limited, one can maximize the end-to-end throughput by carefully selecting a routing path. In [5], the spatial reusability of wireless channels was exploited to improve the end-to-end throughput along with multihop (MH) routing protocols. Multicast routing protocols were designed to improve the performance in terms of energy efficiency, throughput, and fairness in lossy wireless networks [6, 7]. In [8], a topology control algorithm was introduced in mobile ad hoc networks (MANETs) so as to improve the quality of service (QoS) in terms of delay. As one of MANET types, a vehicular ad hoc network (VANET) has been deployed to provide communications between vehicles as well as vehicles and infrastructure. In such VANET environments, a positioning-based routing algorithm was proposed in [9], and a ranging algorithm for positioning was investigated in [10]. Meanwhile, to improve the spectrum utilization, research in the field of cognitive radio networks has grown dramatically. In cognitive radio networks, resource allocation for secondary users based on their quality of experience (QoE) and priority was studied in [11]. Security challenges in cognitive radio networks were also discussed in [12]. Besides, an information centric network was introduced to solve inherent drawbacks of current Internet [13, 14]. In orthogonal frequency division multiple access networks, resource allocation was studied for interference mitigation [15, 16]. With regard to game-theoretic approaches, game dynamics and learning schemes were studied in heterogeneous networks [17] and a distributed channel assignment algorithm was introduced in wireless mesh networks [18]. There were other interesting topics including a routing in broadband networks [19], a software defined network [20], and a composite-radio infrastructure [21].

2.2 Studies on the capacity scaling law It was shown in [1] that the aggregate throughput achieved by the nearest-neighbor MH routing scales as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hð n= log nÞ for a wireless network having n nodes randomly distributed in a unit area.1 MH schemes were further developed and analyzed in [23–25]. The aggregate

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pffiffiffi throughput scaling was improved to Hð nÞ by using percolation theory [23]. The effect of multipath fading channels on the throughput scaling was studied in [24]. In [25], the tradeoff between power and delay was examined in terms of scaling laws. Together with the studies on the MH, it was shown that a hierarchical cooperation (HC) strategy [3, 26] achieves an almost linear throughput scaling, i.e., Hðn1 Þ for an arbitrarily small  [ 0, in dense networks of unit area. There have been alternative approaches to achieving a linear scaling in dense networks by using networks with node mobility [27], interference alignment [28, 29], directional antennas [30–32], and infrastructure support [33]. Due to the long delay and high cost of channel estimation in ad hoc networks with only wireless connectivity, hybrid networks consisting of both wireless ad hoc nodes an infrastructure nodes, or equivalently BSs, have been introduced and analyzed in [2, 33–36]. In [2], two new routing protocols, termed infrastructure-supported singlehop (ISH) and infrastructure-supported MH (IMH) protocols, were introduced in hybrid networks where the number of antennas at each BS is assumed to scale at an arbitrary rate relative to the number of wireless nodes. In the ISH protocol, either a single-hop multiple-access or a singlehop broadcast is used for communications between all wireless source nodes and its belonging BS. In the IMH protocol, the nearest-neighbor MH routing is used for communications between source nodes in each cell and its belonging BS. Using the two new routing protocols, the optimal capacity scaling was characterized by deriving the corresponding cut-set upper bound. In hybrid networks with ideal infrastructure [2, 33–36], it is assumed that BSs are interconnected by infinite-capacity wired links. In practice, since the backhaul link rate is one of factors that determine a cost of operators, it is of importance to understand what are the fundamental capabilities of hybrid networks with cost-effective rate-limited backhaul links in supporting multiple nodes that wish to communicate concurrently with each other. The throughput scaling was studied in [37, 38] for a simplified practical hybrid network, where BSs are connected only to their neighboring BSs via a finite-rate backhaul link. Recently, a general hybrid network deploying multi-antenna BSs was studied in [39] when each BS-to-BS link has a finite rate. More practically, packets arrived at a certain BS in a radio access network (RAN) are delivered to a core network (CN) in a hierarchical way, and then are transmitted from 1

We use the following notation: (1) f ðxÞ ¼ OðgðxÞÞ means that there exist constants C and c such that f ðxÞ  CgðxÞ for all x [ c, (2) f ðxÞ ¼ f ðxÞ ¼ 0, (3) f ðxÞ ¼ XðgðxÞÞ if oðgðxÞÞ means that limx!1 gðxÞ gðxÞ ¼ Oðf ðxÞÞ, (4) f ðxÞ ¼ wðgðxÞÞ if gðxÞ ¼ oðf ðxÞÞ, and (5) f ðxÞ ¼ HðgðxÞÞ if f ðxÞ ¼ OðgðxÞÞ and gðxÞ ¼ Oðf ðxÞÞ [22].

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pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi If l ¼ wð n=mÞ and l ¼ Oðn=mÞ, then n=m antennas are regularly placed on the BS boundary and the remaining antennas are uniformly placed inside the boundary. pffiffiffiffiffiffiffiffiffi If l ¼ Oð n=mÞ, then l antennas are regularly placed on the BS boundary.

the CN to other BSs in the RAN. In this paper, the HierHybNET that is well suited to this realistic scenario [40– 44] is taken into account in analyzing the informationtheoretic upper bound on the aggregate capacity scaling law.

1.

3 System and channel models

Such an antenna deployment guarantees both the nearest neighbor transmission around the BS boundary and the enough spacing between the antennas of each BS. If antennas are uniformly placed inside the BS boundary, then the transmission rate may be reduced due to a relatively long hop distance between an antenna and the nearest neighbor node. We thus need to place the BS antennas first on the BS boundary. This antenna configuration will be widely used for future massive MIMO systems. For analytical convenience, we assume that the parameters n, m, and l are related according to

In an extended HierHybNET of unit node density, n nodes are uniformly and independently distributed on a square of area n, except for the area where BSs are placed. We randomly pick S–D pairings, so that each node acts as a source and has exactly one corresponding destination node. Assume that the BSs are neither sources nor destinations. As illustrated in Fig. 1, the network is divided into m square cells of equal area, where a BS with l antennas is located at the center of each cell. The total number of BS antennas in the network is assumed to scale at most linearly with n, i.e., ml ¼ OðnÞ. It is assumed that the radius of each pffiffiffiffiffiffiffiffiffi BS scales as 0 n=m, where 0 [ 0 is an arbitrarily small constant independent of n, m, and l. This radius scaling would ensure enough separation among the antennas since per-antenna distance scales (at least) as the average pernode distance Xð1Þ for any parameters n, m, and l. If the radius scaling scales slower than Hð1Þ, then per-antenna distance may become vanishingly small, which is undesirable under our infrastructure-supported routing protocols. The antenna configuration basically follows that of [2, 39, 45]. According to the radius scaling of each BS, the antennas of each BS are placed as follows:

2.

n ¼ m1=b ¼ l1=c ; where b; c 2 ½0; 1Þ with a constraint b þ c  1. This constraint is reasonable since the total number of antennas of all BSs deployed over the network does not need to be greater than the number of nodes in the network. As depicted in Fig. 1, it is assumed that all the BSs are fully interconnected by wired links through one RCP. Without loss of generality, it is assumed that the RCP is located at the center of the network. The packets transmitted from BSs are received at the RCP and are then conveyed to the corresponding BSs. In the previous studies [2, 33–36], the rate of backhaul links has been assumed to

RCP

RBS

RBS

RBS

BS

BS

RBS

BS

BS RBS

RBS

BS

BS

Fig. 1 The HierHybNET with limited backhaul link rate RBS between a BS and an RCP

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(a) Wireless node

L1

S

Infrastructure node

(b) BS-to-RCP link

L2

RCP

D

D

S

D

Fig. 2 The cuts L1 and L2 in the HierHybNET. The BS-to-RCP or RCP-to-BS links are not shown in (a) since they are not in effect under L1 . a The cut L1 . b The cut L2

be unlimited so that the links are not a bottleneck when packets are transmitted from one cell to another. In practice, however, it is natural for each backhaul link to have a finite capacity that may limit the transmission rate of infrastructure-supported routing protocols. In this paper, we assume that each BS is connected to one RCP through an errorless wired link with finite rate RBS ¼ ng for g 2 ð1; 1Þ. It is also assumed that the BS-to-RCP or RCP-to-BS link is not affected by interference. The uplink channel vector between node i and BS b is denoted by " ðuÞ hbi

¼

ðuÞ

ðuÞ

ejhbi;1

ejhbi;2

rbi;1

rbi;2

; ðuÞa=2

; . . .; ðuÞa=2

ðuÞ

ejhbi;l

#T

ðuÞa=2

ðuÞ

ðuÞ

rbi;t denotes the distance between node i and the tth antenna of BS b, and a [ 2 denotes the path-loss exponent. The downlink channel vector between BS b and node i is  ðdÞ  ðdÞ ðdÞ jh jh jh ðdÞ e ib;1 e ib;2 e ib;l similarly denoted by hib ¼ ðdÞa=2 ; ðdÞa=2 ; . . .; ðdÞa=2 . The rib;2

rib;l

channel between nodes i and k is given by ejhki a=2

rki

ð2Þ

For the uplink-downlink balance, it is assumed that each BS satisfies an average transmit power constraint nP / m, while each node satisfies an average transmit power constraint P. Then, the total transmit power of all BSs is the same as the total transmit power consumed by all wireless

123

n!1

log Tn ða; b; c; gÞ : log n

4 Cut-set upper bound

where hbi;t represents the random phases uniformly distributed over ½0; 2pÞ based on a far-field assumption, which is valid if the wavelength is sufficiently small [2, 23]. Here,

hki ¼

eða; b; c; gÞ ¼ lim

ð1Þ

;

rbi;l

rib;1

nodes. This assumption is based on the same argument as duality connection between multiple access channel (MAC) and broadcast channel (BC) in [46]. Suppose that each source transmits with the same average transmission rate Rn . The total throughput of the network is then defined as Tn ða; b; c; gÞ ¼ nRn and its scaling exponent is given by2

In this section, to see the fundamental limit of the HierHybNET with rate-limited BS-to-RCP (or RCP-to-BS) links, a generalized cut-set upper bound on the aggregate capacity scaling based on the information-theoretic approach is derived. As illustrated in Fig. 2, in order to provide a tight upper bound, two cuts L1 and L2 are taken into account. Similarly as in [2], the cut L1 divides the network area into two halves by cutting the wireless connections between wireless source nodes on the left of the network and the other nodes, including all BS antennas and one RCP. In addition, to fully utilize the main characteristics of the network with finitecapacity infrastructure, we consider another cut L2 , which divides the network area into another two halves by cutting the wired connections between BSs and the RCP as well as the wireless connections between all nodes (including BS

2

To simplify notations, Tn ða; b; c; gÞ will be written as Tn if dropping a, b, c, and g does not cause any confusion.

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L2

L1

Fig. 3 The partition of destinations under the cuts L1 and L2 in the extended HierHybNET. To simplify the figure, one and two BSs are shown in (a) and (b) with the RCP, respectively. a The cut L1 . b The cut L2

BS

BS RCP

D

(1) L1

RCP BS

D

(2) L1

(3 a) L1

D

Wired link

DL 2

(3b) L1

D

antennas) located on the left of the network and all nodes (including BS antennas and the RCP) on the right. Upper bounds obtained under the cuts L1 and L2 are ð1Þ Tn

h  i y Tnð1Þ  max E log det In2þml þ HL1 QL1 HL1 QL1  0

denoted by and respectively. By the cut-set theorem, the total capacity is upper-bounded by n o Tn  min Tnð1Þ ; Tnð2Þ : ð3Þ The following two lemmas will be used to derive an upper bound on the capacity in the remainder of this section. Lemma 1 In our two-dimensional extended network, where n nodes are uniformly distributed and there are m BSs with l regularly spaced antennas, the minimum distance between any two nodes or between a node and an antenna on the BS boundary is greater than 1=n1=2þ1 whp for an arbitrarily small 1 [ 0. The Lemma 1 can be easily proved by essentially following the same line as [3, Sect. V-B] and [2, Lemma 6]. Lemma 2 Assume a two-dimensional extended network. When the network area with the exclusion of BS area is divided into n squares of unit area, there are less than log n nodes in each square whp. The proof of Lemma 2 is given by slightly modifying the proof of in [23, Lemma 1]. Let us first focus on the cut L1 . Let SL1 and DL1 denote the sets of sources and destinations, respectively, for L1 in the network. Then, all wireless ad hoc nodes on the left half of the network are SL1 , while all ad hoc nodes on the right half and all BS antennas in the network are destinations DL1 (see Fig. 2a). Note that the wired BS-toRCP (or RCP-to-BS) links do not need to be considered under L1 since all BSs in the network act as destinations DL1 .   In this case, the n2  n2 þ ml MIMO channel between the two sets of nodes and BSs separated by the cut L1 is formed.

ð4Þ

y ¼ max E½log detðIHðnÞ þ HL1 QL1 HL1 Þ

ð2Þ Tn ,

QL1  0

where the equality comes from the fact that n ¼ XðmlÞ.3 Here, the channel matrix HL1 consists of the uplink channel ðuÞ

vectors hbi in (1) for i 2 SL1 , b 2 B, and hki in (2) for i 2 SL1 , k 2 Dr , where B and Dr denotes the set of BSs in the network and the set of wireless nodes on the right half, respectively. The matrix QL1 is the positive semidefinite input covariance matrix whose kth diagonal element satisfies ½QL1 kk  P for k 2 SL1 . In order to obtain a tight upper bound, it is necessary to narrow down the class of S– D pairs according to their Euclidean distances. As illustrated in Fig. 3a, the set DL1 is partitioned into the folð1Þ

ð2Þ

lowing four groups according to their locations: DL1 , DL1 , ð3aÞ

ð3bÞ

ð3aÞ

ð3bÞ

DL1 , and DL1 . The set DL1 [ DL1 ð1Þ

ð3Þ

is denoted by DL1 .

ð2Þ

The sets DL1 and DL1 represent the sets of destinations located on the rectangular slab with width one immediately to the right of the centerline (cut) L1 including the RCP and on the ring with width one immediately inside each BS ð3Þ

boundary (cut) on the left half, respectively. The set DL1 is ð1Þ

ð2Þ

given by DL1 n ðDL1 [ DL1 Þ. By generalized Hadamard’s inequality [47] as in [3, 48], we then have h  i ð1Þy ð1Þ Tnð1Þ  max E log det Ipffiffin log nþ1 þ HL1 QL1 HL1 QL1  0

h  i ð2Þy ð2Þ ffi þ max E log det IOðpffiffiffiffi mnÞ þ HL1 QL1 HL1 QL1  0

h  i ð3Þy ð3Þ þ max E log det IHðnÞ þ HL1 QL1 HL1 : QL1  0

ð5Þ

ð1Þ

The total throughput Tn for sources on the left half is bounded by the capacity of the MIMO channel between SL1 and DL1 , and thus is given by

3

Here and in the sequel, the noise variance is assumed to be one to simplify the notation.

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h i ðtÞ ðtÞ where HL1 is the matrix with entries HL1 for i 2 SL1 , ki

ðtÞ

k 2 DL1 , and t ¼ 1; 2; 3. By analyzing either the sum of the capacities of the multiple-input single-output (MISO) channel or the amount of power transferred across the network, an upper bound for each term in (5) can be derived. We now establish the following lemma, which shows an upper bound under the cut L1 . Under the cut L1 in Fig. 3a, an upper bound

Lemma 3

ð1Þ

on the aggregate capacity, Tn , of the HierHybNET with rate-limited infrastructure is given by    rffiffiffiffi

m a=21 n pffiffiffi 2a=2 Tnð1Þ ¼ O n max ml ; m min l; ; n; n ; n m

( ð1Þ Tn;3

¼

Oðn2a=2þ Þ if 2\a\3

ð9Þ Oðn1=2þ Þ if a  3: pffiffiffiffiffiffiffiffiffi If l ¼ Xð n=mÞ, using an upper bound for the power ð3aÞ

ð1Þ

transfer from the set SL1 to the set DL1 , the term Tn;3 is upper-bounded by [2] 8 ma=2

>  2a=2 > > c4 n max n ; nl if 2\a\3 > < n ð1Þ ( )  Tn;3  > pffiffiffi n ml a=2 >  > p ffi c n max n ; if a  3 > : 4 l n ð10Þ

ð6Þ

where c4 [ 0 is some constant independent of n. Using (7)– (10) finally yields (6), which completes the proof of the lemma. h

Proof Let Tn;1 , Tn;2 , and Tn;3 denote the first to third terms in (5), respectively. By further applying generalized Hadamard’s

The upper bound Tn matches the achievable throughput scaling within a factor of n in the network with infinite-capacity infrastructure (g ! 1), which

where  [ 0 is an arbitrarily small constant. ð1Þ

ð1Þ

ð1Þ

ð1Þ

inequality [47], the term Tn;1 is upper-bounded by 0 1 X X ð1Þ 2 Tn;1  log@1 þ P jhki j A ð1Þ

k2DL

1

i2SL1

ð1Þ

indicates that Tn does not rely on the parameter g. Note that the first to fourth terms in the max operation of (6) represent the amount of information transferred to the ð7Þ

pffiffiffi  c1 ð n log n þ 1Þ log n pffiffiffi  c2 n n

where c1 and c2 are some positive constants, independent of n, and  [ 0 is an arbitrarily small constant. The second inequality follows from the fact that the minimum distance between any source and destination (including the RCP) is greater than 1=n1=2þ1 whp for an arbitrarily small 1 [ 0 by Lemma 1 and pffiffiffi ð1Þ there exist no more than n log n þ 1 nodes in DL1 whp by Lemma 2. Since the remaining terms (i.e., the second and third terms) in (5) can be computed regardless of the presence of the RCP, they are derived by basically following the same approach as that in [2]. From the antenna configuration in our network, ð1Þ

the second term in (5), Tn;2 , is bounded by [2] ( pffiffiffiffiffiffiffiffiffi c3 ml log n if l ¼ oð n=mÞ ð1Þ Tn;2  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi c3 mn log n if l ¼ Xð n=mÞ rffiffiffiffi

n  c3 m min l; log n m

ð8Þ

where c3 [ 0 is some constant independent of n. An upper ð1Þ

bound for the third term in (5), Tn;3 , is derived now. If pffiffiffiffiffiffiffiffiffi ð3aÞ l ¼ oð n=mÞ, there is no destination in DL1 (see Fig. 3a) ð3Þ

and thus the information transfer to the set DL1 is the same as that in wireless network with no infrastructure (that is, the ð3bÞ

information transfer to the set DL1 ). Hence, it follows that [3]

123

ð1Þ

ð3aÞ

ð2Þ

ð1Þ

ð3bÞ

destination sets DL1 , DL1 , DL1 , and DL1 , respectively. The third and fourth terms characterize the cut-set upper bound of wireless networks with no infrastructure. By employing infrastructure nodes, it is possible to get an additional information transfer for a given cut L1 , corresponding to the first and second terms in the max operation of (6). In addition to the cut L1 , we now turn to the cut L2 in Fig. 2b to obtain a tight cut-set upper bound in the network with rate-limited infrastructure. Let SL2 and DL2 denote the sets of sources and destinations, respectively, for L2 in the network. More precisely, under L2 , all the wireless and infrastructure nodes on the left half are SL2 , while all the nodes on the right half including the RCP are included in DL2 (see Fig. 2b). Unlike the case of L1 , we take into account information flows over the wired connections as well as the wireless connections. In consequence, an upper bound on the aggregate capacity is established based on using the min-cut of our network, and is presented in the following theorem. Theorem 1 In the HierHybNET with the backhaul link rate RBS , the aggregate throughput Tn is upper-bounded by  ma=21 Tn ¼ O max min max n ml ; n rffiffiffiffi



n  1=2þ 2a=2þ n m min l; ; mRBS ; n ;n ; m ð11Þ

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Table 1 The operating regime for an extended HierHybNET with finite-capacity infrastructure Regime

Condition

eða; b; c; gÞ

B~

2\a\4  2b  2g

2  a2

a  4  2b  2g

bþg

2\a\4  2b  2g

2  a2

4  2b  2g  a\2 þ 2ðcgÞ 1b

bþg

2c 2 þ 2ðcgÞ 1b  a\1 þ 1b

1 þ c  að1bÞ 2

2c a  1 þ 1b

1þb 2

~ D

Proof By the min-cut of the network, the aggregate capacity is upper-bounded by (3). The upper bound on the ð1Þ

capacity, Tn , under the cut L1 directly follows from Lemma 3. ð2Þ

An upper bound on the capacity, Tn , under the cut L2 is derived below. Since the cut-set argument under L1 does not utilize the main characteristics of rate-limited backhaul links, we now deal with a new cut L2 , which divides the network into two equal halves. A wired link between two BSs that lie on the opposite side of each other from the cut is illustrated in   nþml  Fig. 3b. In this case, we get the nþml  2 þ 1 MIMO 2 wireless channel and the m2  1 MISO wired channel between the two sets SL2 and DL2 separated by L2 . We now derive the amount of information transferred by each channel. Let ð2Þ

Tn;wireless and Tn;wired denote the amount of information transferred through the wireless and wired channels under L2 , respectively. The aggregate capacity obtained under L2 is ð2Þ

ð2Þ

ð2Þ

then upper-bounded by Tn  Tn;wireless þ Tn;wired . ð2Þ

Let us first focus on deriving Tn;wireless , which is bounded by the capacity of the MIMO channel between SL2 and DL2 . Using the same power transfer argument as [3], an upper bound on ð2Þ Tn;wireless ,  

ð2Þ Tn;wireless

under the cut L2 in Fig. 2b, is given by ¼

pffiffiffi 2a=2  O n max n; n , where  [ 0 is an arbitrarily small constant. Next, we turn to analyzing the capacity of the MISO wired channel. There also exist wired links between the two sets separated by L2 (i.e., BSs on the left half of the network and the RCP placed at the center of the network). From the fact that the m2 BS-to-RCP (or RCP-to-BS) links ð2Þ

can be created under L2 , we have Tn;wired ¼ OðmRBS Þ since the capacity of each backhaul link is assumed to be RBS . In consequence, under the cut L2 , it follows that

4

ð2Þ

ð12Þ

Using (6) and (12), the total throughput Tn is finally upperbounded by Tn  minfTnð1Þ ; Tnð2Þ g (   m a=21 ¼ O min n max ml ; n rffiffiffiffi

n pffiffiffi 2a=2 ; n; n m min l; g; m )! maxfmRBS ; n1=2þ ; n2a=2þ g

where  [ 0 is an arbitrarily small constant.4

ð2Þ

ð2Þ

Tnð2Þ  Tn;wireless þ Tn;wired  n o ¼ O max mRBS ; n1=2þ ; n2a=2þ :

To simplify notations, the terms including  are omitted if dropping them does not cause any confusion.

(

ma=21 ¼ O max min max n ml ; n )! rffiffiffiffi

n  1=2þ 2a=2þ m min l; g; mRBS g; n ;n ; m

where the last equality follows from minfmaxfa; xg; maxfb; xgg ¼ maxfminfa; bg; xg. This completes the proof of Theorem 1. h Now we would like to examine in detail the amount of information transfer by each separated destination set. Remark 2 As mentioned earlier, each term in (11) is associated with one of the destination sets for a given cut. To be concrete, the first and the second terms in the max operation of (11) represent the amount of information ð3aÞ

ð2Þ

transferred to DL1 and DL1 over the wireless connections, respectively. The third term represents the information flows over the wired connections from the BSs on the left half to the BSs on the right half. The fourth and fifth terms represent the amount of information transferred ð1Þ

ð3bÞ

to DL1 and DL1 respectively.

over the wireless connections,

5 Infrastructure-limited regime We show the case where our HierHybNET is in the infrastructure-limited regime using the cut-set argument drawn under L1 and L2 . Using the identified infrastructurelimited regime, one can see whether or not the throughput scaling in a certain regime can be improved by increasing ~ and D ~ the backhaul link rate. Two operating regimes B causing an infrastructure limitation for some a are intro~ and D ~ become duced in Table 1. Note that Regimes B infrastructure-limited when a  4  2b  2g and 4  2b  2g  a\2 þ 2ðcgÞ 1b , respectively.

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˜ B

˜ ˜

Fig. 4 The operating regimes with respect to b and c, where  12  g\0The operating regimes with respect to b and c, where 1 2  g\1

Fig. 5 The operating regimes with respect to b and c, where 0  g\ 12

The infrastructure-limited regime is identified using the upper bound derived under the two cuts. The network is shown to be fundamentally infrastructure-limited if the following two conditions are satisfied:   pffiffiffi iÞ mng ¼ X n maxf n; n2a=2 g ð13aÞ    rffiffiffiffi

m a=21 n g  iiÞ mn ¼ o n max ml ; m min l; ; n m ð13bÞ where the first and second conditions come from (12) and (6), respectively, corresponding to the aggregate throughput under the cuts L2 and L1 each. In order to satisfy the first condition (13a), the value g must be greater than or equal to  12. The infrastructure-limited regime is thus identified for the following three cases depending on g (   12). •

•

 12  g\0: The first condition (13a) holds if b  12  g and a  4  2b  2g, while the second condition (13b) always holds in this case since  n pffiffiffio g n mn ¼ o m min l; m . Hence, the network is infrastructure-limited if b  12  g and ~ a  4  2b  2g, which is depicted in Regime B of Fig. 4. 0  g\ 12: As in the previous range of g, b  12  g and a  4  2b  2g need to be satisfied so that the first condition (13a) holds. Let us now examine the case where the second condition (13b) follows according to the two-dimensional operating regimes with respect to b and c, which is non-trivial. –

b þ 2c\1: In this case, from the fact that the righthand side (RHS) of (13b) is simply given by ml, (13b) is satisfied if g\c. The regime under these

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˜

Fig. 6 The operating regimes with respect to b and c, where 12  g\1

–

~ below conditions corresponds to a part of Regime B the line b þ 2c ¼ 1 in Fig. 5. b þ 2c  1: Let us first consider the case where the pffiffiffiffiffiffi RHS of (13b) is given by mn. Then, (13b) is satisfied if b\1  2g, which corresponds to the ~ in Fig. 5. Let us now remaining part of Regime B turn to the case where the RHS of (13b) is given by  a=21 ml mn . In this case, (13b) is satisfied if a\2 þ 2ðcgÞ 1b . Our network is thus infrastructurelimited when 4  2b  2g\2 þ 2ðcgÞ 1b (equivalent to c [ b2 þ ðg  2Þb þ 1), which is depicted in ~ of Fig. 5. Note that the network is Regime D infrastructure-limited at the medium path-loss attenuation regime, corresponding to 4  2b  2g  a\2 þ 2ðcgÞ 1b .

Author's personal copy Wireless Netw (2016) 22:1133–1144 1 2

 g\1: We show the case where the two conditions in (13) follow according to the operating regimes with respect to b and c.

80

b þ 2c\1: In this case, the RHS of (13b) is given by ml. There is no infrastructure limitation since g [ c. b þ 2c  1: The conditions in (13) are satisfied if

60

–

–

2ðcgÞ 1b .

a  4  2b  2g and a\2 þ As a consequence, the network is infrastructure-limited 4  2b  2g\2 þ 2ðcgÞ 1b

when

R

=0

R

=1

R

=10

BS

70

BS BS

Upper bound (bps/Hz)

•

1141

50 40 30 20

(equivalent

to 2 ~ c [ b þ ðg  2Þb þ 1), which is depicted in Regime D of Fig. 6.

10 0

100

200

300

400

500

600

700

800

900

1000

The number of nodes, n

Fig. 7 The upper bound on the aggregate throughput Tn versus the number of nodes, n

6 Numerical evaluation In this section, to validate the performance of the upper bound in Sect. 4, we perform extensive computer simulations according to finite values of the system parameters n, m, l, and RBS . Using (3), the upper bound on the aggregate ð1Þ

More specifically, the term Tn is numerically computed as   the capacity of the n2  n2 þ ml MIMO wireless channel between SL1 and DL1 , which is expressed as (4). From the ð2Þ

ð2Þ

ð2Þ

ð2Þ

fact that Tn  Tn;wireless þ Tn;wired , the terms Tn;wireless and ð2Þ

Tn;wired are numerically computed as the capacity of the nþml  nþml m 2  2 þ 1 MIMO wireless channel and the 2  1 MISO wired channel, respectively, between SL2 and DL2 , which eventually leads to (12). We slightly modify our system model so that the model is suitable for numerical evaluation—to remove any randomness of node location, a regular network is assumed, where nodes are equally spaced over a square network such that the distance between nearest-neighbor nodes is 1 (m). The path-loss exponent a is assumed to be 4. In our Monte-Carlo simulations, each channel is randomly generated 1  103 times for given system parameters. The upper bound on the aggregate throughput Tn versus the number of nodes, n, is first evaluated in Fig. 7, where m ¼ 4, l ¼ 2, and RBS ¼ 0; 1; 10 (bps/Hz). In Fig. 8, the upper bound on the aggregate throughput Tn is evaluated according to the number of BSs, m, where n ¼ 256, l ¼ 2, and RBS ¼ 0; 1; 10. Figure 9 shows the upper bound on the aggregate throughput Tn according to the number of antennas per BS, l, where n ¼ 256, m ¼ 4, and RBS ¼ 0; 1; 10. In Figs. 8 and 9, a sufficient number of nodes (n ¼ 256) are deployed so that a large-scale network is suitably modelled in practice. Moreover, in Figs. 7, 8, 9 and 10, it is observed that

=0

R

=1

BS

600

RBS=10

ð2Þ

throughput is numerically computed as minfTn ; Tn g.

R

BS

Upper bound (bps/Hz)

ð1Þ

700

500 400 300 200 100 0

20

40

60

80

100

120

140

The number of BSs, m

Fig. 8 The upper bound on the aggregate throughput Tn versus the number of BSs, m

the upper bound on Tn is monotonically increasing with respect to parameters n, m, and l, which shows trends consistent with our analytical behaviors in Theorem 1. It is also seen that the performance gets greatly improved when the backhaul link rate RBS increases up to 10. Finally, Fig. 10 illustrates the upper bound on Tn versus RBS , where n ¼ 256, m ¼ 4, and l ¼ 2. From Fig. 10, the following interesting observations are made under our simulation environments: • •

The upper bound on Tn is monotonically increasing as RBS \4. For RBS  4, the performance is not improved with increasing RBS .

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upper bound result, we concretely studied the case where our network is fundamentally infrastructure-limited. Our analysis was also validated using extensive computer simulations. Our comprehensive analysis sheds more light on the backhaul design policies and on operational issues for ad hoc networks with rate-limited infrastructure.

80 75

R

=0

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BS

70

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65

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BS

60 55

Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2054577) and by the research fund of Dankook University(BK21 Plus) in 2014. This paper was presented in part at the 2014 IEEE International Symposium on Information Theory, Honolulu, HI, June/July 2014.

50 45 40 35 30

5

10

15

20

25

30

References

The number of antennas per BS, l

Fig. 9 The upper bound on the aggregate throughput Tn versus the number of antennas per BS, l

46

Upper bound (bps/Hz)

44 42 40 38 36 34 32

0

6

8

The backhaul link rate, R

2

4

(bps/Hz)

BS

10

Fig. 10 The upper bound on the aggregate throughput Tn versus the backhaul link rate RBS

•

Hence, the minimum backhaul link rate RBS required to achieve the same capacity scaling law as in the infinitecapacity backhaul link case (i.e., RBS ! 1) is given by 4 (bps/Hz).

7 Conclusion A generalized cut-set upper bound was derived using two cuts under the HierHybNET model. In the HierHybNET with rate-limited infrastructure, it was shown that the upper bound matches the achievable throughput scaling for all the three-dimensional operating regimes with respect to m, l, and backhaul link rate RBS . Furthermore, based on the

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Author's personal copy 1144 Cheol Jeong received the B.S. degree in electrical and electronics engineering from Yonsei University, Seoul, Korea, in 2003, and the Ph.D. degree in electrical engineering from KAIST, Daejeon, Korea, in 2010. From August 2010 to July 2011, he was with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Canada, as a Postdoctoral Fellow. In September 2011, he joined the Samsung Electronics, where he is currently a senior engineer. His research interests include MIMO relay communications, physical layer security, ad hoc networks, and millimeter-wave communications. Won-Yong Shin received the B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, in 2002. He received the M.S. and the Ph.D. degrees in electrical engineering and computer science from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004 and 2008, respectively. From February 2008 to April 2008, he was a Visiting Scholar in the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA. From September 2008 to April 2009, he

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Wireless Netw (2016) 22:1133–1144 was with the Brain Korea Institute and CHiPS at KAIST as a Postdoctoral Fellow. From August 2008 to April 2009, he was with the Lumicomm, Inc., Daejeon, Korea, as a Visiting Researcher. In May 2009, he joined Harvard University as a Postdoctoral Fellow and was promoted to a Research Associate in October 2011. Since March 2012, he has been with the Division of Mobile Systems Engineering, College of International Studies and the Department of Computer Science and Engineering, Dankook University, Yongin, Korea, where he is currently an Assistant Professor. His research interests are in the areas of information theory, communications, signal processing, mobile computing, big data analytics, and their applications to multiuser networking issues. Dr. Shin has served as an Associate Editor for the IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS, COMMUNICATIONS, COMPUTER SCIENCES, for the IEIE TRANSACTIONS ON SMART PROCESSING & COMPUTING, and for the JOURNAL OF KOREA INFORMATION AND COMMUNICATIONS SOCIETY. He has also served as an Organizing Committee for the 2015 IEEE Information Theory Workshop.