Author's personal copy Wireless Netw (2016) 22:2275–2285 DOI 10.1007/s11276-015-1100-3

Capacity of large hybrid erasure networks with random node distribution Won-Yong Shin1 • Cheol Jeong2

Published online: 20 October 2015  Springer Science+Business Media New York 2015

Abstract The Gupta–Kumar’s nearest-neighbor multihop routing with/without infrastructure support achieves the optimal capacity scaling in a large erasure network in which n wireless nodes and m relay stations are regularly placed. In this paper, a capacity scaling law is completely characterized for an infrastructure-supported erasure network where n wireless nodes are randomly distributed, which is a more feasible scenario. We use two fundamental path-loss attenuation models (i.e., exponential and polynomial power-laws) to suitably model an erasure probability. To show our achievability result, the multihop routing via percolation highway is used and the corresponding lower bounds on the total capacity scaling are derived. Cut-set upper bounds on the capacity scaling are also derived. Our result indicates that, under the random erasure network model with infrastructure support, the achievable scheme based on the percolation highway routing is order-optimal within a polylogarithmic factor of n for all values of m. Keywords Achievability  Capacity scaling  Erasure network  Infrastructure  Percolation highway  Relay station  Upper bound

& Won-Yong Shin [email protected] Cheol Jeong [email protected] 1

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

2

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

1 Introduction Mobile data traffic has been explosively increasing as the number of mobile smart devices is increasing rapidly in recent years [1]. The next-generation wireless communication systems have intensively been studied in order to lead to a significant performance improvement over the conventional ones, thus enabling to support a huge amount of traffic demands [2]. Characterizing the sum-throughput of large-scale wireless networks (e.g., a wireless machineto-machine network [3] and a wireless sensor network) has been taken into account as one of the most challenging issues in evaluating the performance of next-generation wireless communication systems supporting multiple devices. In this paper, a capacity scaling law for large-scale hybrid erasure networks having m regularly-placed relay stations (RSs) and n randomly-distributed wireless nodes is completely analyzed using two fundamental path-loss attenuation models, i.e., exponential and polynomial power-laws. To show our achievability result, instead of the Gupta– Kumar’s nearest-neighbor multihop [4], the multihop routing via percolation highway [5] is utilized with/without the help of RSs, and the corresponding lower bounds on the total capacity scaling are derived. Our result indicates that, under random networks using both path-loss attenuation models, the achievable throughput scaling laws are the same as those shown under the regular network model, with no performance loss coming from the additional randomness. Furthermore, upper bounds on the total capacity scaling are derived using the cut-set bound. It turns out that, even under random erasure networks with RS support, the achievable throughput scaling based on the percolation

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highway routing protocol matches our upper bound on the capacity scaling within a polylogarithmic factor of n for any parameter m (i.e., for all the network conditions). To validate the derived analytical results for finite values of system parameters n and m, numerical evaluation is also shown via computer simulations. Our main contributions can be summarized as follows: – – – –

Design of the percolation highway multihop routing with/without infrastructure support in erasure networks Analysis of the achievable throughput scaling laws under two path-loss attenuation models Analysis of the upper bounds on the total capacity scaling under two path-loss attenuation models Numerical evaluation via simulations

The rest of this paper is organized as follows. The related work is summarized in Sect. 2. Sect. 3 describes our system and channel models. In Sect. 4, the achievable scheme and the corresponding lower bounds on the capacity scaling are shown. In Sect. 5, the cut-set upper bounds are derived. The numerical results are also shown in Sect. 6. Finally, Sect. 7 summarizes the paper with some concluding remarks.

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 by resources, one can maximize the end-to-end throughput by carefully selecting a routing path. In [6], the spatial reusability of wireless channels was exploited to improve the end-to-end throughput along with multihop routing protocols. Multicast routing protocols were designed to improve the performance in terms of energy efficiency, throughput, and fairness in lossy wireless networks [7, 8]. In [9], a topology control algorithm was introduced in mobile ad hoc networks so as to improve the quality of service (QoS) in terms of delay. Meanwhile, to improve the spectrum utilization, research in the field of cognitive radio networks has grown dramatically. It was addressed in [10] how to design good multihop routing metrics for cognitive radio networks. Resource allocation for secondary users based on their quality of experience (QoE) and priority was also studied in cognitive radio networks [11]. On the one hand, small cell networks have attracted great attention in increasing the network capacity with a low-cost and selforganized infrastructure. On the other hand, heterogeneous networks (HetNets) evolve and expand the traditional approach by complementing existing macrocells with a

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layer of small cells using future wireless communications technologies. In [12], both femtocell downlink cellbreathing control and voting-based direct frameworks were proposed to characterize a tradeoff between cell coverage and throughput and a tradeoff between efficiency and fairness, respectively. Feasible schemes enabling small cell dynamic time division duplexing (TDD) transmissions were introduced in both homogeneous networks [13] and HetNets [14]. In [15], analysis on the signal-to-interference-plus-noise ratio of dynamic TDD transmissions was also performed in homogeneous small cell networks. Besides, in orthogonal frequency division multiple access networks, resource allocation was studied in terms of interference mitigation [16–18]. With regard to game-theoretic approaches, game dynamics and learning schemes were studied in heterogeneous networks [19]. There were other interesting topics including a software defined network [20] and a biology-inspired optimization in network design [21]. 2.2 Studies on the capacity scaling law In [4], Gupta and Kumar introduced and characterized the sum throughput scaling in a large wireless network with additive Gaussian noise. They showed that, for the network where n nodes are randomly located in a unit area, the total pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi throughput scales as Hð n= log nÞ.1 This throughput scaling is achieved in such a way that data is delivered from a node to another node in a multihop fashion. There have been further studies on multihop in the literature [5, 23–26], while the total throughput scales far less than HðnÞ. Besides, the almost linear throughput scaling law Hðn1 Þ for an arbitrarily small  [ 0 was derived using a hierarchical cooperation scheme [27] in the Gaussian network model of unit area. As an alternative approach to improving the total throughput up to a linear scaling, infrastructure nodes, or equivalently RSs, can be deployed to the wireless ad hoc network [28–30], where RSs are assumed to have high bandwidth connections to each other. It is strictly necessary for the number of RSs, m, to exceed a certain threshold in order to achieve a linear throughput scaling in m for the hybrid network. In [31], optimal capacity scaling was characterized for a more general hybrid network, where multiple antennas are equipped at each RS and the achievability is based on choosing one of RS-supported single-hop and multihop routings, pure multihop transmission [4], and hierarchical cooperation [27]. 1

We use the following notation: i) f ðxÞ ¼ OðgðxÞÞ means that there exist positive constants C0 and c0 such that f ðxÞ  C0 gðxÞ for all x [ c0 . ii) f ðxÞ ¼ XðgðxÞÞ if gðxÞ ¼ Oðf ðxÞÞ. iii) f ðxÞ ¼ HðgðxÞÞ if f ðxÞ ¼ OðgðxÞÞ and gðxÞ ¼ Oðf ðxÞÞ [22].

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Furthermore, as another fundamental class of networks, a wireless erasure network has been introduced in [32], where a precise characterization of capacity region was shown for some simple multicast problems. In the erasure network, signals are either successfully delivered or completely lost. Such erasure channel model is appropriate for systems with Automatic Repeat reQuest (ARQ)-like mechanisms, in which all information transmission is packetized. As the physical distance between nodes increases, the link quality of the nodes is degraded. In order to incorporate this phenomenon into the erasure channel model, either an exponential or polynomial power-law [33–37] can be invoked in computing the path-loss attenuation. In [33], the exponential decay model was considered in wireless erasure networks, where the probability of successful transmission decays exponentially with a distance between two nodes, whereas, in [34, 35], the capacity scaling was shown under the polynomial decay model. Moreover, a hybrid erasure network equipping multiple infrastructure nodes was characterized and its optimal capacity scaling law was derived, where the results were derived under the exponential decay model [36] and the polynomial decay model [37]. However, in [36, 37], a rather simple regular network was assumed, where wireless nodes are regularly placed over the network, which is not feasible in practice. It is thus not straightforward to intuitively see how the optimal capacity scales for hybrid erasure networks with random node distribution. It is also not obvious how to constructively design an achievable scheme guaranteeing the order optimality under the random erasure network model.

3 System and channel models Consider a two-dimensional extended network [27, 31, 38] of unit node density that consists of n wireless nodes uniformly and independently distributed on a square. We randomly pick a match of source–destination (S–D) pairs, so that each node is the destination of exactly one source. Suppose that the whole area is divided into m square cells, each of which is covered by one single-antenna RS at its center (see Fig. 1). It is assumed that n nodes are located except for the area covered by RSs. The minimum distance between each RS center and its nearest wireless nodes is assumed to be given by 0\dmin  1. For analytical convenience, let us state that parameters n and m are related according to m ¼ nb for b 2 ½0; 1Þ. Moreover, as in [28, 29, 31, 39], it is assumed that the RS-to-RS links have infinite bandwidth connections to each other and these RSs are neither sources nor destinations. Suppose that each node transmits at a rate T(n) / n, where T(n) denotes the total throughput of the network.

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Infrastructure node (or RS)

Wireless node

Fig. 1 Illustration of the random extended network with infrastructure support

Now, let us describe the channel between any two nodes, which is modeled as a memoryless erasure channel with erasure events over all channels being independent. The channel model in the uplink is first shown as follows. The erasure probability ki for transmission between node i 2 f1; . . .; ng and RS k 2 f1; . . .; mg is modeled as an increasing function of distance. In this work, we take into account two fundamentally different types of path-loss attenuation models: exponential and polynomial decay models. When we use the exponential decay model [33, 36] such that the probability of successful transmission decays exponentially with distance dki between node i and RS k, it then follows that ki ¼ 1  ddki ;

ð1Þ

where 0\d\1. When we consider the polynomial decay model [34, 35, 37], the erasure probability is given by ki ¼ 1 

1 ; dkia

ð2Þ

where a [ 0. In addition, to further incorporate the broadcast feature of wireless networks, we consider the case of finite-field additive interference. When, in each time slot, each node i 2 f1; . . .; ng chooses a single symbol xi from the finite-field alphabet Fq , the received symbol yk at RS k is given by X yk ¼ cki xi ; i2I

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where I is the set of simultaneously transmitting nodes and cki is the binary random variable that takes the value 0 with probability ki . Note that the output is the sum of all unerased symbol. The above channel models also hold in downlink and wireless node-to-node communication environments in a similar manner.

Horizontal highway

Source Vertical highway

4 Achievability results In this section, routing protocols are first described, and then the corresponding lower bounds on the capacity scaling law are derived under the two path-loss attenuation models.

Destination

(a)

4.1 Routing protocols In this subsection, routing protocols with/without infrastructure support are described using percolation highway [5]. For the two routing protocols, as in [4, 27, 31], a time division multiple access (TDMA) operation is used to avoid causing any huge interference.

Vertical highway

4.1.1 Delivery routing without infrastructure The basic procedure of the percolation highway delivery [5] follows three steps: draining, highway, and delivery phases (note that this routing strategy was used in random ad hoc networks with additive Gaussian noise [5], but can also be applied to random erasure networks with a slight modification). Let us first explain how to construct a backbone network. We divide the area into equal square grids of edge length c for a constant c [ 0, independent of n. Next, we divide the network area into equal horizontal pffiffi pffiffiffi pffiffiffi rectangle of size n  2cp log l, where l ¼ pffiffin . Here, the 2c

parameters c and p are determined to generate Hðlog lÞ horizontal disjoint open paths that cross each rectangle from left to right. Each of the rectangles thus has l  log l grids in the percolation model. The area can also be divided into l= log l equal vertical rectangles to generate vertical disjoint paths from bottom to top. As illustrated in Fig. 2(a), the overall procedure is summarized as follows. –





Draining phase: A source in each horizontal rectangle sends its packets directly via single-hop to a node on a horizontal path of the backbone network. Highway phase: The packets are transported along the horizontal path using multihop routing and then reach a vertical path. Delivery phase: A node in the vertical path sends the packets directly via single-hop to the corresponding destination.

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RS Horizontal highway

Source

(b) Fig. 2 Illustration of the percolation delivery routing. a Percolation delivery routing without the help of infrastructure in the network. b Percolation delivery routing with the help of infrastructure in a cell

We refer to [5] for the detailed description. Note that the pffiffiffi average number of active S–D pairs is given by Hð nÞ pffiffiffi with high probability since there exist Hð nÞ horizontal and vertical paths simultaneously, with all the rectangles. 4.1.2 Infrastructure-supported delivery routing When the number of RSs, m, is higher than a certain level, the throughput scaling of RS-supported networks can be improved. In other words, the RS-supported protocol may achieve a better throughput scaling than that of the pure ad hoc transmission with no help of the RSs when m is sufficiently large. In an extended network, as depicted in

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Fig. 2(b), the RS-supported percolation highway multihop routing is described as follows:

where 0\d\1, given in (1). Let us assume that we choose a value k [ d lnln d2 so that ð1  dkd Þ2 [ 14, which follows that



the probability PI is upper-bounded by PI \32dðk1Þdc . We have PI \1 by choosing

• •



Divide the network into equal square cells of area n / m, each having one RS at the center of each cell. In each cell, horizontal and vertical highways are formed, similarly as in the delivery routing without infrastructure. For the access routing, one source in each cell transmits its packets to the corresponding RS via percolation highway. The RS that completes decoding its packets transmits them to another RS closest to the corresponding destination by wired RS-to-RS links. For the exit routing, the percolation delivery routing from an RS to the corresponding destination is performed, similarly as in the access routing case.

Note that m S–D pairs can be activated simultaneously with the above infrastructure-supported delivery routing. 4.2 Achievable throughput scaling under the exponential decay model Under the exponential decay model, a lower bound on the capacity scaling in the RS-supported random erasure network is first derived using the routing protocols with and without infrastructure support in Sect. 4.1. Theorem 1 Suppose a random erasure network with RSs under the exponential decay model, where the probability of successful transmission decays exponentially as in (1). Then, the total throughput is lower-bounded by pffiffiffi ð3Þ TðnÞ ¼ Xðmaxfm; ngÞ for all m ¼ nb satisfying b 2 ½0; 1Þ. Proof The proof technique essentially follows that of [33, Theorem 2] with a slight modification. We first derive the transmission rate achieved using the routing protocol without infrastructure in the network which is divided into squares of side length c [ 0. Let us denote the distance between a source node and its destination by d. Then, it is assumed that the routing is performed based on the t-TDMA scheme with t ¼ ðkd=cÞ2 (parameter k will be specified later). Under the t-TDMA scheme, the nearest interfering transmitter is at least distance ðk  1Þd  c away from the intended receiver. The probability PI that the symbol from at least one of the simultaneously interfering nodes is not erased is given by 1 X PI  ð8iÞdðik1Þdc i¼1

¼

8dðk1Þdc ð1  dkd Þ2

;

k[1 

5 ln 2 c þ ; d ln d d

which satisfies the inequality k [ d lnln d2. Then, we can achieve the following transmission rate to any destination within a distance d: dd ð1  PI Þ=t; where & 2 ’ d 5 ln 2 t¼  þ1 ¼ Hðd2 Þ: c c ln d In the draining or delivery phase, the distance between a node and the point on the highway is not greater than pffiffi pffiffiffi 2cp log pffiffi2nc. Note that there are Oðlog nÞ nodes in each square. The transmission rate in these phases is thus given by   X dd ð1  PI Þd 2 ðlog nÞ1  pffiffi  pffi 2cp logpffin 3 2c ðlog nÞ ¼X d  pffiffi  pffi  2cp logpffi n  3 2cd ðlog nÞ ¼X e  pffi2cp  ¼ X n 2d ðlog nÞ3 ; 

where dd ¼ ed=d and d is the critical distance determined according to d. If we choose pffiffiffi 2cp 1 \ ; 2d  2 then per-node transmission rate is limited only by the highway phase. From [5, Lemma 4], it is shown that pernode transmission rate for the nodes along the highways is   pffiffiffi X p1ffiffin . Hence, the total throughput Xð nÞ is achievable using the routing protocol without RSs. In a similar fashion, it is shown that the total throughput XðmÞ can be achieved using the routing protocol with RSs since per-cell transmission rate Xð1Þ can be achieved by the infrastructure-supported delivery routing, whose detailed derivation is omitted but is basically the same as the case of the percolation highway routing without infrastructure. Therefore, the achievable throughput scaling for the RS-supported random erasure network under the exponential decay model is given by (3), which completes the proof of the theorem.

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Based on Theorem 1 and [33, Theorem 2], it is easy to see that both regular and random erasure networks with infrastructure support achieve the same throughput scaling law. Thus, randomness in node positions does not occur any performance loss in scaling law. 4.3 Achievable throughput scaling under the polynomial decay model In this subsection, under the polynomial decay model, a lower bound on the capacity scaling in the RS-supported random erasure network is derived using the routing protocols with and without infrastructure support. Theorem 2 Suppose a random erasure network with RSs under the polynomial decay model, where the probability of successful transmission decays polynomially as in (2). Then, when a [ 2, the total throughput is lower-bounded by pffiffiffi TðnÞ ¼ Xðmaxfm; ngÞ ð4Þ

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Since per-node transmission rate is limited only by the   highway phase and scales as X p1ffiffin along the highways, the pffiffiffi total throughput scaling without RSs is given by Xð nÞ. Similarly, under the infrastructure-supported routing, it is shown that the total throughput scaling XðmÞ can be achieved. Therefore, the achievable throughput scaling for the RS-supported random erasure network under the polynomial decay model is given by (4), which completes the proof of the theorem. This result shows scaling behaviors consistent with the regular network case [37], which is rather obvious. More specifically, the lower bound under the polynomial decay model holds for a [ 2, unlike the case of the exponential decay model whose bound does not depend on the parameter d (refer to Theorem 1). In addition, from Theorems 1 and 2, it is seen that both path-loss attenuation models provide the same achievable throughput scaling under our network assumption.

is achievable for all m ¼ nb satisfying b 2 ½0; 1Þ. Proof As in the exponential model scenario, it is assumed that the routing is performed based on the t-TDMA scheme with t ¼ ðkd=cÞ2 , where d is the distance between a source node and its destination and c is the side length of each square in the network. We remark that the nearest interfering transmitter is at least distance ðk  1Þd  c away from the intended receiver. The probability PI that the symbol from at least one of the simultaneously interfering nodes is not erased is given by 1 X 8i PI  : ððik  1Þd  cÞa i¼1 For a [ 2, choosing an appropriate value k will lead to the probability PI that is less than 1. Then, we can achieve the following transmission rate to any destination within a distance d: ð1=d a Þð1  PI Þ=t;

5 Cut-set upper bounds In this section, to verify the order optimality of our achievability in Sect. 4, information-theoretic cut-set upper bounds [40] are derived for an RS-supported random erasure network. Let SL and DL denote the sets of sources and destinations, respectively, for a given cut L in the network (see Fig. 3). Following the same steps as those in [31, 36, 37], we consider the cut L dividing the network area into two halves. More precisely, under L, (wireless) source nodes SL are on the left, whereas all nodes on the right and all RSs are destinations DL . Here, as illustrated in Fig. 3, the set of destinations is assumed to consist of the two ð1Þ

ð2Þ

subsets DL and DL , which denote the set of destinations on the left half and the set of wireless and infrastructure nodes on the right half, respectively. We start from the following lemma, in which the cut-set bound for erasure

L

where t ¼ Hðd2 Þ. In the draining or delivery phase, the distance between a node and the point on the highway is pffiffi pffiffiffi not greater than 2cp log pffiffi2nc. Note that there are Oðlog nÞ nodes in each square. The transmission rate in these phases is thus given by   X ð1=d a Þð1  PI Þd2 ðlog nÞ1 0 1 1 B C ¼ X@pffiffiffi A pffiffi 2þa n 2cp log pffiffi2c log n   ¼ X ðlog nÞ3a :

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DL(1)

SL

DL(2)

Fig. 3 The sources and the partition of the destinations with cut L. To simplify the figure, one RS is shown in the left-half network

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networks is characterized assuming no interference, leading to an upper bound on the performance. Lemma 1 ([32]) For an erasure network divided into two sets SL and DL , the cut-set bound on the total throughput T(n) is given by ! X Y TðnÞ  1 ki ; ð5Þ i2SL

k2DL

where ki is the erasure probability between source i 2 SL and destination k 2 DL . Using the characteristics of random node distribution establishes the following lemma. Lemma 2 Assume a two-dimensional extended network. When the network area is divided into n squares of unit area, there are less than log n nodes in each square with high probability. This result can be obtained by applying our RS-based network and slightly modifying the proof of [5, Lemma 1]. By using Lemma 2, we would like to consider the network transformation resulting in a regular network with at most log n nodes at each square vertex, similarly as in [27, 31]. In the following two subsections, we derive upper bounds on the capacity according to the two path-loss attenuation models. Our proof technique essentially follows that of [36, 37] with a modification, but we provide simple proofs of all theorems for completeness. 5.1 Upper bound under the exponential decay model Under the exponential decay model, an upper bound on the capacity scaling in the RS-supported random erasure network is first derived in the following theorem. Theorem 3 Suppose a random erasure network with RSs under the exponential decay model, where the probability of successful transmission decays exponentially as in (1). Then, the total throughput is upper-bounded by  pffiffiffi  TðnÞ ¼ O ðlog nÞ2 maxfm; ng ð6Þ

where Q the secondPinequality follows from the fact that 1  i ð1  xi Þ  i xi for 0\xi \1. Let us first consider a regular network in which two neighboring nodes are regularly 1 unit of distant apart from each other. From Lemma 2, there are at most 8i log n nodes whose distance ð1Þ

from one RS in the set DL is given by dmin þ i  1 in the ith layer. Let us now focus on computing the first term in (7), whose upper bound is given by 1 X X log n ð8iÞddmin þi1 ð1Þ

k2DL

i¼1

1 X X

 8 log n

ð1Þ k2DL

¼ 8 log n

X

iddmin i ð8Þ

i¼1

d

dmin

dmin 2 Þ ð1Þ ð1  d

k2DL

 c1 m log n; where c1 [ 0 is some constant, independent of n. Here, the first inequality follows from the fact that dmin i  dmin þ i  1, and the second inequality holds ð1Þ

since there exist m / 2 RSs in DL . Moreover, the term log n in (8) comes from the network transformation to a regular network. More precisely, as illustrated in Fig. 4(a), the nodes in the set of sources, SL , are moved in a sense of decreasing the Euclidean distance between node i 2 SL and the corresponding RS. From Lemma 2, such a network transformation results in the regular network with at most log n nodes at each vertex of unitarea squares in SL . Next, let us turn to the second term in (7). Since the second term corresponds to the cut-set upper bound for erasure networks with no RS support, it can be upperbounded by that under the regular network with at most log n nodes at each vertex of unit-area squares, using the displacement as shown in Fig. 4(b). From such a node displacement, it follows that XX pffiffiffi ddki  ðlog nÞ2 n: ð9Þ ð2Þ i2S L

i2DL

b

for all m ¼ n satisfying b 2 ½0; 1Þ.

Hence, using (8) and (9), the total throughput scaling is finally upper-bounded by (6), which completes the proof of the theorem.

Proof

Substituting (1) into (5), we have ! X Y  dki TðnÞ  1 1d i2SL



XX

k2DL

ddki

i2SL i2DL



XX i2SL i2Dð1Þ L

ddki þ

ð7Þ XX i2SL i2Dð2Þ L

ddki ;

From Theorems 1 and 3, it is shown that a combination of the percolation highway routing protocols with and without infrastructure support is order-optimal within a polylogarithmic factor of n in our random hybrid network. We also remark that, unlike the case of Gaussian network models [31], the use of the hierarchical cooperation [27] or

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! Y 1 1 1 a TðnÞ  dki i2SL k2DL XX 1  da i2SL i2DL ki XX 1 XX 1  þ : a d da ð1Þ ki ð2Þ ki i2S i2S X

L

L

i2DL

ð11Þ

i2DL

Following the network transformation argument similar to the proof of Theorem 3, an upper bound on the first term in (11) is given by 1 X X 1 log n ð8iÞ ðd þ i  1Þa min ð1Þ i¼1 k2DL

 8 log n

1 X X ð1Þ

k2DL

¼

i¼1

i ðdmin iÞa

8 log n X 1 a dmin ia1 ð1Þ k2DL

 c2 m log n; where c2 [ 0 is some constant, independent of n. Here, the ð1Þ

Fig. 4 The displacement of nodes, indicated by arrows. a The displacement of the nodes in SL . b The displacement of the nodes in ð2Þ SL and DL

last inequality holds since there exist m / 2 RSs in DL and the sum converges when a [ 2. Since the second term in (11) corresponds to the cut-set upper bound for erasure networks with no RSs, from [35], we have XX 1 pffiffiffi  ðlog nÞ2 n; a d ð2Þ ki i2S L

any sophisticated multiuser detection scheme is not needed to improve the throughput scaling even in the random network setup.

5.2 Upper bound under the polynomial decay model In this subsection, under the polynomial decay model, an upper bound on the capacity scaling in the RS-supported random erasure network is given as in the following theorem. Theorem 4 Suppose a random erasure network with RSs under the polynomial decay model, where the probability of successful transmission decays polynomially as in (2). Then, when a [ 3, the total throughput is upper-bounded by  pffiffiffi  TðnÞ ¼ O ðlog nÞ2 maxfm; ng ð10Þ for all m ¼ nb satisfying b 2 ½0; 1Þ. Proof

Substituting (2) into (5), we have

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i2DL

where the inequality holds when a [ 3. Therefore, the total throughput is finally given by (10), which completes the proof of the theorem. As in the exponential decay model, from Theorems 2 and 4, it is shown that our achievable scheme is order-optimal within a polylogarithmic factor of n. It is also seen that the upper bound derived under the polynomial decay model holds for a [ 3, while that under the exponential decay model always holds regardless of the parameter d (refer to Theorem 3). In addition, it is worth noting that the derived upper bounds for both path-loss attenuation models are the same for a [ 3, even under the random hybrid network. In other words, performance for the two channel models shows the same trends as far as capacity scaling is concerned.

6 Numerical evaluation In this section, to validate the performance of the upper bounds in Sect. 5, we perform extensive computer simulations according to finite values of the system parameters

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1200 Exponential (β = 0.5, δ = 0.5) Polynomial (β = 0.5, α = 3.1)

Upper bound on T(n)

1000

800

600

400

200

0

0

2000

4000

6000

8000

10000

Number of nodes, n

Fig. 5 The upper bound on the total throughput T(n) versus the number of nodes, n

Exponential (β = 0.5, δ = 0.5) Polynomial (β = 0.5, α = 3.1)

Upper bound on T(n)

800

600

400

200

0

0

20

40

60

80

100

Number of RSs, m

Fig. 6 The upper bound on the total throughput T(n) versus the number of RSs, m

n and m.2 Using (5), the upper bounds on the total capacity [bps/Hz] is numerically computed as  P  Q i2SL 1  k2DL ki , where ki is given by (1) and (2) under the exponential and polynomial decay models, respectively. It is assumed that nodes are uniformly and independently distributed over a square network whose size pffiffiffi pffiffiffi is n  n [m], while RSs are regularly spaced over the network such that the distance between nearest-neighbor pffiffiffiffiffiffiffiffiffi RSs is n=m [m]. The parameter b is assumed to be 0.5, pffiffiffi i.e., m ¼ n is assumed. The decay parameters d and a are 2

7 Concluding remarks For the random erasure network with infrastructure support under the two fundamental path-loss decay models, the capacity scaling law was fully characterized by deriving both upper and lower bounds as a function of n and m. It was shown that our routing scheme using the multihop via percolation highway indeed achieves the same throughput scaling as the regular hybrid network case using the Gupta– Kumar’s nearest-neighbor multihop. We also proved that the routing scheme is order-optimal within a polylogarithmic factor of n for any values of m. As a result, it turned out that no performance loss coming from the additional randomness occurs compared to the regular network scenario.

1200

1000

set to 0.5 and 3.1, respectively. In our Monte–Carlo simulations, the network topology (i.e., the node distribution) is generated 1  103 times for given system parameters. The upper bound on the total throughput T(n) versus the number of nodes, n, is first evaluated in Fig. 5 under the two fundamental path-loss decay models (i.e., the exponential and polynomial decay models). It is shown that T(n) increases almost with the square root of n, which is consistent with Theorems 3 and 4. Next, Fig. 6 shows the upper bound on T(n) versus the number of RSs, m, under both exponential and polynomial decay models. From Fig. 6, it is shown that T(n) increases linearly with m, which also matches our analytical results.

The lower bounds on the capacity in Sect. 4 can also be validated via simulations in a similar fashion, which is omitted in this paper.

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 (2014R1A1A205 4577).

References 1. CISCO. (2013). Cisco visual networking index: Global mobile data traffic forecast update, 2012–2017. White Paper. 2. ITU-R. (2013). IMT vision–Framework and overall objectives of the future development of IMT for 2020 and beyond. ITU-R WP 5D working document. 3. Lo, A., Law, Y. W., & Jacobsson, M. (2013). A cellular-centric service architecture for machine-to-machine (M2M) communications. IEEE Wireless Communications, 20, 143–151. 4. Gupta, P., & Kumar, P. R. (2000). The capacity of wireless networks. IEEE Transactions on Information Theory, 46, 388–404. 5. Franceschetti, M., Dousse, O., Tse, D. N. C., & Thiran, P. (2007). Closing the gap in the capacity of wireless networks via percolation theory. IEEE Transactions on Information Theory, 53, 1009–1018.

123

Author's personal copy 2284 6. Meng, T. Wu, F., Yang, Z., Chen, G., & Vasilakos, A. V. (2015). Spatial reusability-aware routing in multi-hop wireless networks. IEEE Transactions on Computers (to appear). 7. Li, P., Guo, S., Yu, S., & Vasilakos, A. V. (2012). CodePipe: An opportunistic feeding and routing protocol for reliable multicast with pipelined network coding. In Proceedings of IEEE INFOCOM, pp. 100–108. 8. Li, P., Guo, S., Yu, S., & Vasilakos, A. V. (2014). Reliable multicast with pipelined network coding using opportunistic feeding and routing. IEEE Transactions on Parallel and Distributed Systems, 25, 3264–3273. 9. Zhang, X. M., Zhang, Y., Yan, F., & Vasilakos, A. V. (2015). Interference-based topology control algorithm for delay-constrained mobile ad hoc networks. IEEE Transactions on Mobile Computing, 14, 742–754. 10. Youssef, M., Ibrahim, M., Abdelatif, M., Chen, L., & Vasilakos, A. V. (2014). Routing metrics of cognitive radio networks: A survey. IEEE Communications Surveys & Tutorials, 16, 92–109. 11. Jiang, T., Wang, H., & Vasilakos, A. V. (2012). QoE-driven channel allocation schemes for multimedia transmission of priority-based secondary users over cognitive radio networks. IEEE Journal on Selected Areas in Communications, 30, 1215–1224. 12. Wang, C.-Y., Ko, C.-H., Wei, H.-Y., & Vasilakos, A. V. (2015). A voting-based femtocell downlink cell-breathing control mechanism. IEEE/ACM Transactions on Networking (to appear). 13. Ding, M., Lopez-Perez, D., Vasilakos, A. V., & Chen, W. (2014). Dynamic TDD transmissions in homogeneous small cell networks. In Proceedings of IEEE International Conference on Communications (ICC) Workshop on Small Cell and 5G Networks, pp. 616–621. 14. Ding, M., Lopez-Perez, D., Xue, R., Vasilakos, A. V., & Chen, W. (2014). Small cell dynamic TDD transmissions in heterogeneous networks. In Proceedings of IEEE International Conference on Communications (ICC), pp. 4881–4887. 15. Ding, M., Lopez-Perez, D., Vasilakos, A. V., & Chen, W. (2014). Analysis on the SINR performance of dynamic TDD in homogeneous small cell networks. In Proceedings of IEEE Global Communications Conference (GLOBECOM), pp. 1552–1558. 16. Lopez-Perez, D., Vasilakos, A. V., & Claussen, H. (2013). Minimising cell transmit power: Towards self-organized resource allocation in OFDMA femtocells. In Proceedings of ACM SIGCOMM, pp. 410–411. 17. Lopez-Perez, D., Chu, X., Vasilakos, A. V., & Claussen, H. (2013). On distributed and coordinated resource allocation for interference mitigation in self-organizing LTE networks. IEEE/ ACM Transactions on Networking, 21, 1145–1158. 18. Lopez-Perez, D., Chu, X., Vasilakos, A. V., & Claussen, H. (2014). Power minimization based resource allocation for interference mitigation in OFDMA femtocell networks. IEEE Journal on Selected Areas in Communications, 32, 333–344. 19. Khan, M. A., Tembine, H., & Vasilakos, A. V. (2012). Game dynamics and cost of learning in heterogeneous 4G networks. IEEE Journal on Selected Areas in Communications, 30, 198–213. 20. Yang, M., Li, Y., Jin, D., Zeng, L., Wu, X., & Vasilakos, A. V. (2015). Software-defined and virtualized future mobile and wireless networks: A survey. Mobile Networks and Applications, 20, 4–18. 21. Liu, L., Song, Y., Zhang, H., Ma, H., & Vasilakos, A. V. (2015). Physarum optimization: A biology-inspired algorithm for the Steiner tree problem in networks. IEEE Transactions on Computers, 64, 818–832.

123

Wireless Netw (2016) 22:2275–2285 22. Knuth, D. E. (1976). Big Omicron and big Omega and big Theta. ACM SIGACT News, 8, 18–24. 23. Gupta, P., & Kumar, P. R. (2003). Towards an information theory of large networks: An achievable rate region. IEEE Transactions on Information Theory, 49, 1877–1894. 24. Xue, F., Xie, L.-L., & Kumar, P. R. (2005). The transport capacity of wireless networks over fading channels. IEEE Transactions on Information Theory, 51, 834–847. 25. Wang, C., Jiang, C., Li, X.-Y., Tang, S., He, Y., Mao, X., et al. (2012). Scaling laws of multicast capacity for power-constrained wireless networks under Gaussian channel model. IEEE Transactions on Computers, 61, 713–725. 26. Shin, W.-Y., Chung, S.-Y., & Lee, Y. H. (2013). Parallel opportunistic routing in wireless networks. IEEE Transactions on Information Theory, 59, 6290–6300. ¨ zgu¨r, A., Le´veˆque, O., & Tse, D. N. C. (2007). Hierarchical 27. O cooperation achieves optimal capacity scaling in ad hoc networks. IEEE Transactions on Information Theory, 53, 3549–3572. 28. Liu, B., Liu, Z., & Towsley, D. (2003). On the capacity of hybrid wireless networks. In Proceedings of IEEE INFOCOM, pp. 1543–1552. 29. Zemlianov, A., & de Veciana, G. (2005). Capacity of ad hoc wireless networks with infrastructure support. IEEE Journal of Selected Areas on Communications, 23, 657–667. 30. Shin, W.-Y. (2011). Refined routing algorithm in hybrid networks with different transmission rates. IEEE Transactions on Communications, 59, 1242–1246. 31. Shin, W.-Y., Jeon, S.-W., Devroye, N., Vu, M. H., Chung, S.-Y., Lee, Y. H., et al. (2011). Improved capacity scaling in wireless networks with infrastructure. IEEE Transactions on Information Theory, 57, 5088–5102. 32. Dana, A., Gowaikar, R., & Hassibi, B. (2006). Capacity of wireless erasure networks. IEEE Transactions on Information Theory, 32, 789–804. 33. Smith, B., Gupta, P., & Vishwanath, S. (2007). Routing is orderoptimal in broadcast erasure networks with interference. In Proceedings of IEEE International Symposium on Information Theory (ISIT), pp. 141–145. 34. Smith, B. & Vishwanath, S. (2006). Asymptotic transport capacity of wireless erasure networks. In Proceedings of the 44th Allerton Conference on Communications, Control, and Computing, pp. 27–29. 35. Smith, B. Gupta, P., & Vishwanath, S. (2007). Routing versus network coding in erasure networks with broadcast and interference constraints. In Proceedings of IEEE Military Communications Conference (MILCOM), pp. 1–5. 36. Shin, W.-Y., & Kim, A. (2011). Capacity scaling of infrastructure-supported erasure networks. IEEE Communications Letters, 15, 485–487. 37. Jeong, C., & Shin, W.-Y. (2013). Capacity scaling of hybrid erasure networks based on polynomial power-law. IEEE Communications Letters, 17, 1024–1027. 38. Xie, L.-L., & Kumar, P. R. (2004). A network information theory for wireless communication: Scaling laws and optimal operation. IEEE Transactions on Information Theory, 50, 748–767. 39. Liu, B., Thiran, P., & Towsley, D. (2007). Capacity of a wireless ad hoc network with infrastructure. In Proceedings of ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc). 40. Cover, T. M., & Thomas, J. A. (1991). Elements of information theory. New York: Wiley.

Author's personal copy Wireless Netw (2016) 22:2275–2285 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, USA. From September 2008 to April 2009, he 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,

2285 mobile computing, big data analytics, and online social networks analysis. 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 and Computing, and for the Journal of Korea Information and Communications Society. He also served as an Organizing Committee for the 2015 IEEE Information Theory Workshop. 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.

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