The Multicast Capacity of Large Multihop Wireless ... - Semantic Scholar
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The Multicast Capacity of Large Multihop Wireless Networks† Srinivas Shakkottai,
Xin Liu, and
Abstract—We consider wireless ad hoc networks with a large number of users. Subsets of users might be interested in identical information, and so we have a regime in which several multicast sessions may coexist. We first calculate an upper-bound on the achievable transmission rate per multicast flow as a function of the number of multicast sources in such a network. We then propose a simple comb-based architecture for multicast routing which achieves the upper bound in an order sense under certain constraints. Compared to the approach of constructing a Steiner tree to decide multicast paths, our construction achieves the same order-optimal results while requiring little location information and no computational overhead.
I. I NTRODUCTION Wireless ad hoc networks are of use when there is a lack of fixed communication infrastructure. Such situations might arise in calamity environments, sensor network applications, and a variety of other civilian and military contexts. In many applications, multicast data transfer is more predominant than unicast data transfer. In military networks, it is often stated that multicast traffic dominates due to the need for group communications. In the civilian context, an emerging application that has already been tested is the use of wireless ad hoc networks to broadcast replays during football games. A situation like a football game would have a large number of spectators, each having a mobile device and a desire for a replay of an important moment in the game. There is almost no infrastructure available from which they could obtain such data, and there is a strong incentive to form an ad hoc network for this purpose. Some of the users might be close to data sources (perhaps if they were close to an Internet access point), and they would act as sources for the multicast traffic. Other nodes would act as relays and sinks for the data. The questions arise as to how many multicast sessions can be supported by such a network, what the total capacity of the network would be, and how to achieve the capacity in a simple and practical manner. Consider Figure 1. There is a finite area with a number of wireless nodes. There are 3 multicast flows in progress, with nodes receiving each multicast flow labeled 1, 2 and 3, respectively. The sources of these flows are labeled as † Research supported in part by DARPA CBMANET grant, NSF grants CNS-0519691, CNS-0904520, CNS-0963818, CNS-0520126, NSF CAREER Award CNS-0448613, DTRA grant HDTRA1-09-1-0051, and Qatar Telecom, Doha, Qatar. An earlier version of this paper [1] appeared in MobiHoc’07, Montreal, Canada, September 2007. S. Shakkottai ([email protected]) is with the Dept. of Electrical and Computer Engineering,Texas A&M University. X. Liu ([email protected]) is with the Dept. of Computer Science, University of California at Davis. R. Srikant ([email protected]) is with the Coordinated Science Laboratory and Dept. of ECE, University of Illinois at Urbana-Champaign.
R. Srikant
1
S2
2
S1 1
2 1,2
2
2
1
1
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1
S3 3
1
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1 1,3
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Fig. 1. Example of multicast flows in a wireless ad hoc network. Receivers are labeled with the flows that they are interested in. Some nodes act as pure relays, in which case they are not labeled. The system throughput is determined by the number of flows that pass through each node.
S1 , S2 , and S3 , respectively. The hops are labeled with the sessions that they carry. Some nodes may be neither sources nor destinations and merely act as relays (unlabeled nodes). Suppose each hop, if scheduled, could carry one bit per time slot. The first constraint on the system capacity is that we may not be able to schedule all the hops simultaneously due to interference. Even if all hops could be scheduled simultaneously, we see that the throughput of each source is at most 0.5 bits per time slot in this example, since all the multicast flows contain at least one node that is shared with one other multicast flow. The total throughput of the system would then be 1.5 bits per second. Thus, we see that there are two main sources of interference that limit the multicast capacity of the network: 1) The channel interference constraint: While a large transmission radius may allow multiple receivers to receive packets from a multicast flow simultaneously, a larger radius means a transmitting node interferes with more nodes, which limits the number of simultaneous transmissions. A good multicast structure should balance the tradeoff. 2) The flow interference constraint: if multiple flows pass through the same node, the rate that each flow obtains is a fraction of the node’s transmission capacity. In other words, system capacity is divided among the flows. Therefore, we consider the multicast capacity as a function of the number of multicast sources.
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In this paper, we will study how the throughput of multicast wireless ad hoc networks scales with the number of sources, the number of destinations per source, and the total number of nodes present in the system. We will concentrate on the case when the number of nodes in the system is large. We consider a unit square in which n nodes are dropped at random, ns of these nodes are randomly chosen as sources, and each of these sources is associated with nd random destinations, making a total of ns nd source-destination pairs in the system. We would like to understand the throughput scaling laws in the network as n, ns , nd → ∞. A number of studies consider such questions with regard to unicast flows, starting with the seminal paper by Gupta and Kumar [2], where achievable upper bounds are derived for wireless ad hoc networks in a finite region with a large number of nodes. The Gupta-Kumar result was re-derived in a much simpler manner in [3]. Related ideas can also be found in [4], [5]. A number of papers have also looked at delay-throughput tradeoffs in mobile models for such networks [5]–[9], as well as broadcast networks [10], [11] although we do not study these issues in our paper. Multicast in wireless networks has been studied in [12], [13], but these papers do not deal with scaling laws. Some examples of multicast protocols are presented in [14]–[16]. In comparison, our focus in this paper is not to develop a protocol, but to show that a simple routing structure can be capacity achieving in an order-optimal sense. To the best of our knowledge, the only prior works that deal with scaling laws in wireless networks with multicast flows are [17]–[22]. However, in [17] and [18] the authors use a Steiner tree approach to construct a multicast tree, and they assume that every node in the network is a multicast source in [17]. In [18], the authors build a Euclidean spanning tree with Manhattan routing. The authors also extend the results to hybrid networks where there are ad hoc nodes and base stations in [19], and to networks under Gaussian channel model [20]. Results in [21] apply to multicast with mobile nodes and the scaling of throughput subject to the nodes willing to tolerate delays. This is a different framework than our work in which we study randomly located but fixed nodes. The multicast results and achievability scheme presented in [22] are closely related to our paper. However, its assumption that every node in the network is a multicast source implies that is a special case of our results, and fits in the framework of Theorem 3 of our paper. We analyze a general case of different ratios of sources to destinations. The achievability result in [22] is developed through a comb architecture that is similar to our paper, a shorter version of which was originally published in a conference [1]. Our main contribution is to propose a simple architecture that achieves the same capacity in the order sense as the maximum possible throughput. For instance, in comparison to [17], [18], our scheme requires neither a central control to compute the multicast tree nor global location information of the multicast destinations. Furthermore, our proof of the upperbound is different from that in [17], [18]. Last, in contrast to [22], we present complete results with proofs of the scaling of the upperbound as well as the achievability scheme as a
function of the number of multicast sources and destinations. Main Results We express our results in the order sense. We say f (n) ∈ O(g(n)) with high probability (w.h.p) if given δ > 0, ∃ c, and m(δ) such that P{f (n) ≤ c g(n)} ≥ 1 − δ
∀n ≥ m(δ).
Similarly, we say f (n) ∈ Θ(g(n)) with high probability (w.h.p) if given δ > 0, ∃ c1 , c2 , and m(δ) such that P{c1 g(n) ≤ f (n) ≤ c2 g(n)} ≥ 1 − δ
∀n ≥ m(δ).
We first derive an upper bound on the multicast capacity of a wireless ad hoc network. The main technical challenge is to take into account the broadcast nature of wireless media where multiple nodes can receive simultaneously. We show that when nd = nǫ − 1, 0 < ǫ < 1, the total number of nodes in each multicast flow nm , nd + 1 = nǫ , and the number of multicast flows ns = O(n1−ǫ ), the upper bound on the sum of the source rates that the network can support is √ n w.h.p, O √ nm log n with a per flow throughput capacity of √ n √ w.h.p. O ns nm log n The achievability of the upper bound is the main contribution of this paper. To achieve the upper bound, we propose a simple routing architecture to transfer multicast data in the network. The architecture consists of a tree called the multicast comb, which is constructed independent of the receiver locations. The receivers then complete the multicast tree by attaching themselves to the comb using shortest path routing in a small vicinity. Using this simple architecture, we show that the achievable throughput matches the upper bound √ √ in an order sense if 1) ns nm = Ω( n log n); or 2) it is allowed to drop an arbitrarily small fraction of traffic; or 3) one can pose constraints on the locations of the source and destination nodes. In comparison, when there are ns nd unicast sourcedestination pairs, the per flow throughput capacity for unicast is √ n √ w.h.p. Θ ns nd log n following the Gupta-Kumar result. Therefore, by exploiting the properties of multicast, one can obtain multicast gain of √ Θ( nd ) for relatively large ns nd . An earlier version of this work is presented in [1]. The main differences in the current paper are: (i) the proof of the upper bound has been completely rewritten since our original proof did not accurately account for the broadcast nature of wireless networks, and (ii) Section V has been rewritten for clarity and to extend the results in [1] to a more general setting.
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A
Organization of the Paper We begin in Section II with a description of the system model that we consider. We derive the upper bound on multicast capacity and compare it to the unicast upper bound in Section III. We then proceed in Section IV to design a simple architecture that we call the multicast comb structure that achieves the upper bound in the order sense if the sources and destinations can be chosen suitably. However, since our goal is to study the case of randomly chosen sources and destinations, in Section V we find the lower bound on the achievable transmission rate. We conclude with possible extensions in Section VI. II. S YSTEM M ODEL We consider a system with n wireless nodes. Let the set of all node-indices be denoted N = [1, 2, · · · , n]. We select ns source nodes, and associate each such source with nd = nǫ −1 distinct destination nodes. The number of nodes associated with each multicast flow is then 1 + nd , nm . For simplicity let node 1 be the source node for multicast flow 1, with nodes [2, 3, · · · , nm ] being its destinations. Similarly, let nm + 1 be the source for multicast flow 2, with destinations being [nm + 2, · · · , 2nm ]. We repeat this process ns ≤ n1−ǫ times until we have ns multicast flows. We then drop all the n nodes uniformly and at random into a square of unit area. Thus, each node i ∈ N has a location Pi = [Pix , Piy ] where Pix and Piy are uniformly distributed in the interval [0, 1]. We use the protocol model to model interference between transmissions, as proposed in [2]. The model is illustrated in Figure 2. We assume that all nodes choose identical transmit radii r, which is large enough for the network to be connected. Suppose that node i transmits to node j. Node j receives the transmission successfully if every other node that transmits simultaneously is at a distance of at least (1 + ∆)r from j. This implies that circles of radius ∆r 2 around each receiver must be disjoint [2]. A node can transmit 1 bit of data per unit of time if its transmission is successful.
(1+∆ ) r R T
r
Fig. 2. Illustration of the protocol model for wireless transmission. The assumption is that a transmission would be successful as long as there is no other transmission in a circle of radius (1 + ∆)r from the intended receiver.
A
R
R
B
B C
a) A minimum Steiner tree
Fig. 3.
C b) A (wireless) multicast tree
A multicast tree in a wireless network.
When nd = 1, the model is the same as a unicast problem. When nd = n − 1, the source can broadcast its data in one √ hop to all the destinations with a transmission radius of 2. We would like to understand the capacity scaling law and its practical achievability between these two extremes. In particular, we study the case where nm = nd + 1 = nǫ , and 0 < ǫ < 1. In this paper, a multicast tree refers to a transmission structure that takes into account the broadcast nature of the wireless medium, which is different from a minimum spanning tree. The difference is shown in Figure II. In the figure, A is the multicast source with B and C as its receivers. The left figure shows a Steiner tree1 , where R is a relay node. In this case, A transmits once to R, and R transmits to B and C, respectively. The right figure shows a multicast tree in a wireless network where B and C can receive A’s transmission simultaneously (i.e., only one transmission is needed). The length of the wireless multicast tree is greater than or equal to that of the minimum Steiner tree while the number of transmissions may be smaller. Therefore, taking into account such an effect is critical to deriving an upperbound and to deciding the optimal transmission scheme for multicast in wireless networks. In a multicast tree, all leaf nodes are multicast receivers. III. U PPER B OUND Our first objective is to derive an upper bound on the throughput capacity of multicast wireless networks. We have the following theorem. Theorem 1: The throughput of each multicast source in a random wireless ad-hoc network is upper bounded by √ n O min 1, √ w.h.p. ns nm log n
as n → ∞, nm = nǫ , 0 < ǫ < 1. To prove this theorem, we still exploit the idea that communication consumes space as in [2], [17]. However, the focus is to take into account the properties of multicast. The questions that need to be answered to obtain a useful bound on the throughput capacity are: (i) what should the radius r be?, and (ii) how many transmissions are required to reach all destinations? When we have the answer to the above two related questions, we can derive the upperbound for capacity as follows. Let the transmission radius be r, since our square is of size 1, 1 A Steiner tree is a minimum length tree connecting a set of given points, where additional points may be added to minimize the length of the tree.
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Circle II
the maximum number of simultaneous transmissions can be bounded [2] as S≤
1 π
∆r 2 2
=
4 . π∆2 r2
Circle I
(1)
Suppose there are ns multicast flows and flow i requires at least Hi (r) transmissions, 1 ≤ i ≤ ns , the capacity of a multicast flow satisfies 4 P . (2) λ≤ π∆2 r2 i Hi (r)
Thus, λ is the largest uniformly achievable rate per source node. To prove Theorem 1, we first present two lemmas. The first lemma is on the degree of multicast trees. The second lemma is on the length of multicast trees. The objective is to show that even with the broadcast capability of the wireless medium, if the transmit radius is limited, the number of transmissions needed would be of the same order as that of the shortest multicast tree. Definition We say that a rooted, directed multicast tree, where the transmitter is the root and the receivers are the leaves of the tree, satisfies property P if each node in the tree has at most twelve children that are required to transmit to ensure that each packet reaches all destination. Thus, the degree of each transmitter is at most 12. Lemma 1: Given a multicast tree in a wireless network, either the multicast tree satisfies property P or one can construct another multicast tree with the same root and leaf nodes which satisfies property P such that the number of transmissions in the new tree is less than or equal to the number of transmissions in the original tree. Additional relay nodes may be necessary to construct the new multicast tree. Proof: Consider a multicast tree T . First, we remove all leaf nodes since the definition of property P only considers transmitting nodes. If all remaining nodes in the original tree have at most twelve children nodes, then the lemma trivially holds. If not, we construct an alternative tree that has a bounded degree of twelve and a smaller number of transmissions compared to the original tree T . The idea is shown in Figure 4 and discussed in the following. Consider a node A whose transmit children degree is greater or equal to 13 (i.e., node A has at least 13 children nodes that need to transmit). Then all receivers of its children are within Circle II (red). Now we arbitrarily add relay nodes to construct the desired new multicast tree. We add 6 square (blue) nodes on the boundary of circle I and 6 triangle (yellow) nodes within the boundary of Circle II as shown in in Figure 4. Replace Node A’s transmit children nodes by these new nodes. Node A plus these new nodes will cover all receivers in the original tree that are covered by node A and its children. To maintain the tree structure, we make A the parent of the 6 square nodes and all leaf nodes in Circle I in the original tree. Each square node is assigned as the parent of a unique triangular node; and all leaf nodes between Circle II and Circle I on the original tree can be connected to one of the square nodes. The new tree has a bounded transmit children degree of six at node A and a
2 1 A
Fig. 4. The small circles (eg. Circle III) represent the transmission ranges of the nodes at the center of these circles. Nodes may be placed in such a way that the number of transmit children of any node is finite.
smaller number of overall transmissions than the original tree T. We can follow the same procedure to eliminate all nodes of degree greater than 12 starting at the root of the tree. For example, suppose node A has a degree larger than 12 and its children may also have a degree larger than 12. We first replace children of node A following the above described procedure. The new children of node A may still have a degree larger than 13, say nodes 1 and 2, as shown in the figure. In this case, we apply the same procedure on nodes 1 and 2, respectively. Note that because of the regularity of the children node deployment, it is possible that a new child of node 1 will collocate with node 2. We consider them as two separate nodes for the ease of argument (although one of them and its corresponding transmission can be eliminated). We note that because of the tree structure, this procedure will not change the degree of a parent node, a sibling node, or nodes in other branches of the tree. Hence, we can use the same procedure to replace all nodes with degree larger than 12. Then the final tree has a maximum degree of 12. We now need the following results from [23], [24] for the length of the multicast tree in random networks. Result 1 Suppose we drop m nodes in a unit square, where the position of each node is chosen uniformly at random in the square. Then as m → ∞, the length of the minimum spanning tree L(m) satisfies √ 1) E(L(m)) ∼ C1 m (Lemma 3.3 from [24]), and 2) V ar(L(m)) ≤ C2 log m (Lemma 4.1 from [24]), where C1 and C2 are positive constants, and the notation ∼ is used to denote asymptotic equality. Also, the length of the optimal Euclidean Steiner tree connecting any m nodes √ satisfies S(m) ≥ 23 L(m) [23] . We next present an upper bound on the length of multicast trees. Let Mi (nm ) be the length (the sum of the euclidean distances of the hops in the tree) of a multicast tree associated with source i.
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Lemma 2: The average length of a multicast tree is √ Ω( nm ) w.h.p.; i.e., ) (n s X √ Mi (nm ) (3) > C nm → 1, P ns i=1 where C is a positive constant, when
Mi (nm ) be the (Euclidean) length of T ′ associated with source i. From Lemma 2, we have ) (n s X √ Mi (nm ) (5) P > C nm → 1. ns i=1 From Lemma 1 we know that Mi (nm ) 12rHi′ (r) > . ns ns
log nm lim = 0. n→∞ ns Proof: Let Li (nm ) be the length of the minimum spanning tree, and Si (nm ) be the length of its Steiner tree. We can lower bound the total length ns X i=1
Mi (nm ) ≥
ns X i=1
(a)
Si (nm ) ≥
√
3/2
ns X
Li (nm ),
i=1
where (a) follows from Result 1. Now, ) ( n s X Li (nm ) − E(Li (nm )) P ≥δ ns i=1
(b)
V ar(L1 (nm )) ns δ 2 (c) C log n 2 m ≤ ns δ 2 → 0 as n → ∞,
≤
where (b) follows from Chebyshev’s inequality and the i.i.d nature of the multicast trees (following the construction of multicast flows), and (c) follows from Result 1. Also we have used the hypothesis that log nm /ns → 0 as n → ∞. Thus, as n → ∞, ) ( n s X Li (nm ) − E(L1 (nm )) < δ → 1 P ns i=1 (n ) s X Li (nm ) ⇒P > E(L1 (nm )) − δ → 1. ns i=1
√ Now, from Result 1, ∃C ′ , such that E(Li (nm ))− δ ≥ C ′ nm for large nm . Hence, we have ) (n s X √ Mi (nm ) (4) P > C nm ns i=1 (n ) s X Li (nm ) ′√ ≥ P > C nm ns i=1 ) (n s X Li (nm ) > E(L1 (nm )) − δ → 1. ≥ P ns i=1 Now we are ready to prove Theorem 1. Proof: To use (2) to obtain an upper-bound onP throughput, we need the minimum number of transmissions i Hi (r) as a function of the radius r. Consider a multicast tree T for source i. Following Lemma 1, we can construct a new tree, T ′ , that has a maximum transmit degree 12. Let the number of transmissions required on T ′ be denoted Hi′ (r). Also, let
(6)
Thus, from (5) and (6) ) (n s X √ 12rHi′ (r) P > C nm → 1. ns i=1 Since Hi (r) ≥ Hi′ (r), the above implies that (n ) s X √ 12rHi (r) P > C nm → 1. ns i=1
(7)
(8)
Finally, using the relation (2), we have λ≤
48 √ , π∆2 Crns nm
w.h.p.
(9)
In addition, by the capacity limit on each node, we have λ ≤ 1. Thus, from (9) and the above √ n λ ≤ min 1, √ , ns nm log n
w.h.p.
(10)
In order to contrast the multicast case with the unicast case, we take the number of source destination pairs to be the same in both cases. This would enable us to characterize the gain that could potentially be achieved using multicast. Since the unicast regime consists of n source destination pairs, we take ns nm = n. In particular, to get Θ(n) s-d pairs, we take nd = nǫ − 1 and ns = n1−ǫ , where 0 ≤ ǫ ≤ 1. We have the following corollary: Corollary 2: The throughput of each multicast source in a random wireless ad hoc network is upper bounded by ! 1 w.h.p. Θ p n1−ǫ log n
Notice that as ǫ → 0, the multicast throughput capacity is essentially the same as the unicast capacity as it should be. Also notice that as ǫ → 1, the benefit of using multicast is lost since all nodes could be reached by one broadcast hop. As an example, consider the case where n = 10, 000 and ǫ = 1/2 (i.e., there are 100 sources, each with 100 destinations). The per-source throughput that can achieved by using multicast is 10 times that of unicast. We now move to the problem of designing a simple scheduling and routing algorithm that could be used to implement the multicast idea. We first show that in a network where we are allowed to choose the sources and destination, the multicast capacity can easily be attained. We then consider the case of randomly chosen sources and destinations.
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IV. A C OMB A RCHITECTURE FOR M ULTICAST T REES We develop a simple architecture, whereby we may achieve the upper bound found above (in the order sense), when we are allowed to select the source and destination nodes appropriately while still keeping the average distance from a multicast source to its destinations a constant. This location constraint is introduced here to simplify the discussion and highlight the comb structure. We will remove the assumption in Section V. As before, we study the system with ns multicast sources each with nd destinations, with nm = nd + 1 = nǫ . The main features of our construction are as follows. • We divide the region into squarelets in the manner of [3]. The squarelet size is large enough so that there is at least one node in each squarelet with high probability, and nodes in adjacent squarelets are capable of communicating with each other. Let the side-length of a squarelet be denoted sn . • The scheduling algorithm will be chosen such that the squarelets that are Ksn apart are scheduled simultaneously, where K is chosen such that the wireless interference constraint is satisfied. • We construct multicast comb structures using the constructed squarelets. There is one comb corresponding to each multicast flow. The combs are constructed so that at most 2 flows pass through a squarelet, in order to keep interference between flows minimal. In the comb structure, the width between two consecutive comb teeth is determined by nm . Each multicast source/destination node can reach any one tooth on the comb to send/receive multicast data. This can be done in the vicinity of the destination node using any routing algorithms, such as shortest path routing. To construct the comb, the location of the destination nodes is not required and no central control is needed. In addition, the cost of the comb structure is the same as an optimal Steiner tree in an order sense. We now present the details of the architecture and derive its throughput capacity. Recall that the length of a squarelet is sn . We have the following useful result from [4]: Result 2 For a squarelet size √ κ log n sn = √ , n the number of nodes mi in any squarelet i satisfies κ κ mi 2n− 8 P ≤ ≤ 4κ ∀i > 1 − , 2 log n κ log n
(11)
for sufficiently large n. Thus, for κ > 8, each squarelet has order log n nodes with high probability (and so each squarelet contains at least one node with high probability). We need to ensure that a node in one corner of a squarelet can transmit to a node in the opposite corner of an adjacent √ squarelet, i.e., the transmission radius is chosen as r = 5sn . We can guarantee successful reception at a receiving node, if no other transmission takes place within a distance of
√ 5sn (1 + ∆). Recall that Ksn is the distance such that squarelets that are this distance apart can be scheduled simultaneously. Then, as illustrated in Figure 5, we have √ √ (K − 2)sn ≥ (1 + ∆) 5sn ⇒ K = 2 + (1 + ∆) 5. We call the subset of squarelets a distance of K squarelets sn
Ks n
S 1
Fig. 5. Illustration of scheduling constraints with ∆ = 0.5. The source is in the squarelet labeled S, and the squarelet with the X is where the receiver is located. K is chosen so that all possible receiver nodes in this squarelet are guaranteed successful reception.
from each other, capable of simultaneous transmission as an equivalence class. So the number of such equivalence classes is K 2 . We then have a system in which the periodicity with which any squarelet is scheduled is K 2 time slots. Using the above idea, we construct the following comb structure for multicast traffic. Suppose that we are able to select the source and receiver nodes of the multicast. We first construct multiple combs, one for each multicast flow as shown in Figure 6. In the figure we have illustrated two multicast combs (one lightly shaded (cyan), and the other dark (magenta)) corresponding to two multicast flows. We define √ n 1 =√ S , (12) sn κ log n √ n 1 . (13) =√ N , √ sn nm κnm log n Here, S is the number of squarelets in any row/column and N is the number of squarelets between two consecutive teeth, which is also the number of distinct multicast flows the network can support without overlapping. Consider a multicast flow i ≤ N. The comb corresponding to this flow consists of (i) squarelets spq , where p indicates the row and q indicates the column of the squarelet. These squarelets form the teeth and the spine of the comb as follows: (i) Teeth: {spq : p ∈ [i, i + N, · · · , i + S/N ], q ∈ [1, 2, · · · , S]}. (i) Spine: {spq : p ∈ [1, 2, · · · , S], q = i}.
Note that the distance between the teeth of each comb is exactly √n1 m . For i > N we reuse the combs by letting multicast
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flow i use comb j, where j satisfies i = j + kN, k ∈ [1, 2, ...] and i ≤ n1−ǫ . The Euclidean length of the comb routing for a particular multicast flow is From Result 1, we have that the Euclidean length of the √ optimal Steiner tree is Θ( nm ), which is the same as our comb structure in an order sense. The construction of the Steiner tree requires global location information and a centralized controller. In comparison, the construction of the comb structure only needs information on the number of destination nodes and does not require central computation of the tree. Each comb is positioned one squarelet farther to the right and below to the previous one. Note that by using this structure, we have ensured that the maximum number of multicast flows that use any particular squarelet is 2. Since each squarelet is scheduled with a periodicity of K 2 slots and the channel has a capacity of 1 bit per time slot (if scheduled), if only 2 multicast flows share a squarelet, their individual throughput would be 1/2K 2 bits per time slot.
1 nm
1
1/2K 2 . This results in a total source rate of √ n √ , 2K 2 κnm log n
(15)
which is of the same order of magnitude as the upper bound calculated in the previous section. In summary, if ns ≤ √ √ n/ κn total√source rate of the flows is √ √ m log n, then the √ n/ κnm log n. If ns > n/ κnm log n, then the combs are reused over multiple sources and the total throughput remains the same. ǫ Consider √ √an example where nm = n , then the condition ns ≤ n/ κnm log n can be written as √ n1−ǫ ns ≤ √ . κ log n Thus, the network multicast throughput increases as ns increases up to the above bound but remains constant after that, as illustrated in Figure 7.
λ ns
n 1− κ log n
ns
Fig. 7. The upper bound for multicast using the comb method. We see that the bound increases linearly with the number of multicast sources as the comb gets filled. The throughput capacity is constant afterward.
Fig. 6. The comb idea for multicast data transfer. Each comb carries a data from a different source. The size of the squarelet places a fundamental limit on the number of possible coexistent combs.
Since we can choose the locations of the source and destination nodes as desired, we let the source and destinations of a particular multicast flow lie on its corresponding comb. The average distance of a source and its destinations is still Θ(1). For example, in Figure 6, the sources and destinations associated with the lightly shaded (cyan) comb would lie somewhere on the comb. As the distance between the teeth of a comb is √n1 m , the number of such combs that can be constructed is 1 √ . sn nm √ √ In other words, since sn = κ log n/ n, the proposed system can accommodate a total of √ n √ (14) κnm log n sources without using one comb for multiple flows. As explained above, since there are a maximum of 2 flows using each squarelet, the sources can each transmit at a rate of
We have proposed a simple multicast architecture, and shown that its throughput in ideal circumstances is identical (in the order sense) to the upper bound. However, we have yet to study its performance when we are not at liberty to place the sources and destinations. We proceed to study this case in the following section. V. ACHIEVABLE M ULTICAST C APACITY IN R ANDOM N ETWORKS We have just seen how the comb architecture is capacityachieving in the case where source-destination placements can be made as desired. We will now study the case where the sources and destinations are randomly chosen. In this case, both sources and destinations must reach the comb in a multihop manner. Once they reach the comb, they would have access to the multicast traffic on that comb. The limiting factor is that as the number of such access paths increases, the load of each squarelet does as well, leading to reduction of throughput for the flows passing through the squarelet. We will study this effect in this section. For a given multicast flow i, both the source and its destinations choose the shortest path to reach the closest tooth on the corresponding comb. As shown in Figure 8, these paths are simple to construct (they are either above or below the teeth). Note that the maximum length of the path is 2√1nm , which means that the increase in length of the multicast tree due to
8
1 nm
1 nm
1
Fig. 8. Using the comb method for randomly placed sources and destinations. A source would be assigned a comb and would connect to the closest possible tooth. Destinations would do likewise.
√ these vertical paths would be just nm × 2√1nm = Θ( nm ). Thus, comparing with the comb length of Θ(nm ), we see that Euclidean length of the tree is unaffected in the order sense by these vertical paths. However, it might so happen that a particular squarelet has a large number of such vertical paths passing through it, which might cause excessive load. Consider the load on a squarelet. There would be horizontal flows passing through corresponding to the comb, and this number can be determined exactly since we construct the comb deterministically. There would also be a number of vertical paths passing through this squarelet, taken by nodes in order to reach their desired comb tooth. We define the number of vertical branches through a node as the number of distinct (corresponding to different multicast flows) such paths that pass through it. We divide the square area into bins √ of length l = 1/ nm and width √ sn . The number of bins √ is denoted M , 1/lsn = nnm / κ log n. Each squarelet then belongs to a particular bin, as shown in Figure 9. If we consider a vertical branch as belonging to the bin in which it originates, then the number of such branches passing through any given squarelet is at most that of the branches belonging to two bins—the bin that the squarelet belongs to, and the bin immediately above or below it, as illustrated in Figure 9. Now, let us select an arbitrary bin j. For this bin, let χij be a random variable, with χij = 0 if our selected bin has no branch associatedPwith multicast stream i, and χij = 1 ns otherwise. Let Xj = i=1 χij , i.e., it is the number of distinct flows that have vertical branches that originate in our selected bin. Define the average load on any bin ρ , E[Xj ]. Then we have ρ = ns 1 − (1 − ls2n )nm √ n κ log n m = ns 1 − 1 − √ nnm √ √ nm κ log n (a) √ ∼ ns 1 − exp − 2 n √ √ ns nm κ log n (b) √ ∼ . (16) n
1
Fig. 9. Division of the area into bins. Two bins have been highlighted, and the vertical branches belonging to them are shown. The load on any squarelet is at most that of the branches belonging to two bins—the bin that the squarelet belongs to, and the bin immediately above or below it.
Above, (a) and (b) hold for large n, and ∼ indicates asymptotic equality. Finally, recall that the average load on each squarelet is at most twice the average number of vertical branches passing through the bin to which it belongs, plus that of the comb teeth passing through it. Thus, the average load on a squarelet is at most ρ′
=
ns
2ρ + sn
1 √ nm
√ √ √ √ ns nm κ log n ns nm κ log n √ √ ∼ 2 + . n n
(17)
Comparing with (10), we see that the average load on a squarelet is such that the comb architecture can achieve the capacity upper bound. However, the question arises as to whether the actual load in each bin is the same as this average in an order sense. We will consider two cases below. For a large value of ns and nm although the average load is large, the number of branches in bin is similar to the average in an order sense, much like the asymptotic result of Result 2. However, when ns and nm are small, although the average load is not to large, it is possible that some particular squarelets might have an above average number of vertical branches. We might need to drop a small fraction of the nodes to achieve maximum throughput for the remaining nodes in this case.
A. Capacity-achieving Region Without Node-Dropping √ √ Theorem 3: When ns nm = Ω( n log n), the comb scheme is order optimal for each flow. In other words, the comb scheme enables per flow throughput of √ n √ w.h.p. Θ ns nm log n for all flows. Proof: Consider a particular bin j. The probability that any vertical branch corresponding to flow i originates in the
9
chosen bin is P(χij = 1) = 1 − (1 − lsn )nm . Recall that Xj
=
ns X
χij ,
i=1
where χij are i.i.d since the set of sources-destinations are independent for √ each multicast flow. By the hypothesis that √ ns nm = Ω( n log n), we have ρ = Ω(log n) > 1. Given some W ∈ R+ and using the Chernoff bound, we have E eXj . P(Xj ≥ W max(ρ, 1)) = P(Xj ≥ W ρ) ≤ eW ρ We calculate E eXj next as follows: ns X ns k Xj p (1 − p)ns −k ek = E e k k=0
=
(1 + (e − 1)p)ns ,
where p = 1 − (1 − lsn )nm and ρ = ns p. Therefore,
(1 + (e − 1)p)ns eW ρ ns (e−1)p e ≤ eW ρ e(e−1)ρ = eW ρ (e−1−W )ρ ≤ e , (18) √ √ By the hypothesis ns nm = Ω( n log n), we have √ √ ns nm ≥ k n log n, where k is a constant, for large enough n. In this case, ρ ≥ k ′ log n (where k ′ is constant). Let W = e − 1 + 1/k ′ . We have P(Xj ≥ W ρ)
≤
1 (19) P(Xj ≥ W max(ρ, 1)) = P(Xj ≥ W ρ) ≤ . n There are a total of M bins. Using a union bound and (19), we have 1 P(max Xj ≥ W ρ) ≤ M P(X k ≥ W ρ) ≤ √ , j log n
which has a limit of 0 as n → ∞. Therefore, the load of each bin (and from (17) every squarelet) is at most a constant factor of that of the average w.h.p. The throughput of each flow is thus √ 1 n √ . =Θ λ=Θ ρ ns nm log n Compared with Theorem 1, the comb scheme is order optimal for each flow. B. Capacity Achievement by Dropping a Fraction of Nodes Theorem 3 shows conditions on ns and nm under which the comb scheme is order optimal for each flow. √ We now consider √ the complementary case where ns nm = o( n log n). Under this regime, there might be certain squarelets that have a large number of interfering flows compared to the average, causing a loss of throughput. But how many squarelets would actually have such overcrowding? If the fraction of overcrowded squarelets were small, then we could simply drop the destinations belonging to these squarelets. Thus, we propose to use a majority rule, which can provide a high multicast rate
(on the same order of the capacity upper bound) to a majority of users if we are allowed to sacrifice an arbitrarily small percentage of users in overcrowded areas. We show below that a node can achieve order optimal throughput with a probability of at least 1−Pth , which is a constant, but can be set arbitrarily close to 1. Theorem 4: Given any threshold Pth > 0, with probability at least 1 − Pth , a node can receive the order optimal per flow throughput capacity using the comb structure by dropping a small percentage of traffic. Proof: As in the previous proof, we divide the network into disjoint bins. Each bin has l/sn squarelets. A vertical branch may span at most two bins. We say a bin is congested if the most congested squarelet in the bin has more than W times of the average load, where W is a constant that is to be determined later. The traffic to/from a node may be dropped if the node’s branch overlaps with a congested bin. A node can receive the order optimal throughput if its branch does not overlap with congested bins and neither does its source node. We will show that this probability is at least 1 − Pth . First, we calculate the probability that a bin is congested. Consider a bin. As before, let Xj be the number of distinct flows that have vertical branches that originate in bin j. Recall that χij = 1 is the event that bin j contains at least a node associated with multicast flow i. Also, let ρ be the average load of a squarelet as before. Now, following the proof of Theorem 3, we can show that P(Xj ≥ W max(ρ, 1)) ≤ e(e−1−W ) max(ρ,1) .
(20)
Let ′ 4 (1 − Pth ) = 1 − Pth ,
and let W = e − 1 + ln
1 ′ . Pth
We have P(Xj ≥ W max(ρ, 1))
≤ e(e−1−W ) max(ρ,1) ≤ e(e−1−W ) ′ = Pth .
′ In other words, with probability 1−Pth , a bin is not congested. Recall that the maximum number of vertical branches through a squarelet may at most come from two bins. Therefore, with ′ 4 probability at least (1 − Pth ) , the branches of a destination node and its source node do not correspond to a congested bin, and thus can receive order optimal throughput. We have thus shown that by dropping an arbitrarily small fraction of the traffic, we can achieve order optimal throughput for the rest of the nodes. Note that as Pth approaches 0, the per-source throughput also approaches zero. However, the rate of decrease is slow: it is proportional to the logarithm of 1/Pth . This kind of policy would be acceptable in real situations where it is important that most of the users obtain high rate transmission, rather than having to cut down the rate for all users so as to satisfy a small percentage of overcrowded areas. We also note that if a source node passes a congested area, it is intuitive to let the source node transmit first in the
10
area. However, there exists a probability (taken into account in Pth through the calculation of p) that multiple source nodes congest a bin. In this case, all receivers in these multicast groups suffer. Nonetheless, this probability is taken into account in Pth . Thus, the statement still holds. VI. C ONCLUSIONS In this paper we have developed an analytical framework for studying the multicast capacity of wireless ad hoc networks. We started with a comparison of the unicast case that has been studied in detail earlier, and showed how the multicast capacity is a function of the number of multicast sources and destinations. We developed a new and simple scheme that we called the comb architecture that would achieve this upper bound if we were at liberty to place the sources and destinations. We also studied the random network case and showed that the price paid in terms of throughput capacity for the simple and robust architecture is not high in the order sense. In the future we would like to study experimental wireless multicast networks.
[15] P. Sinha, R. Sivakumar, and V. Bharghavan, “MCEDAR: Multicast CoreExtraction Distributed Ad Hoc Routing,” in Proceedings of the Wireless Communications and Networking Conference (WCNC), New Orleans, LA, September 1999. [16] S.-J. Lee, M. Gerla, and C.-C. Chiang, “On-Demand Multicast Routing Protocol,” in Proceedings of the Wireless Communications and Networking Conference (WCNC), New Orleans, LA, September 1999. [17] P. Jacquet and G. Rodolakis, “Multicast scaling properties in massively dense ad hoc networks,” in ICPADS ’05: Proceedings of the 11th International Conference on Parallel and Distributed Systems - Workshops (ICPADS’05). Washington, DC, USA: IEEE Computer Society, 2005, pp. 93–99. [18] X. Li, S. Tang and O. Frieder,“Multicast Capacity of Large Scale Wireless Ad Hoc Networks” in ACM MobiCom 2007. [19] X. Mao, X-Y. Li, and ShaoJie Tang, “Multicast capacity for hybrid wireless networks” in ACM MobiHoc 2008. [20] S. Li and Y. Liu and X-Y. Li, “Capacity of large scale wireless networks under Gaussian channel model” in ACM MobiCom 2008. [21] S. Toumpis and A. Goldsmith, “Performance Bounds for Large Wireless Networks with Mobile Nodes and Multicast Traffic,” in IEEE International Workshop on Wireless Ad Hoc Networks, Vienna, Austria, May 2004. [22] S. Toumpis, “Asymptotic Capacity Bounds for Wireless Networks with Non-Uniform Traffic Patterns,” IEEE Transactions on Wireless Communications, vol. 7, no. 6, pp. 2231–2242, June 2008. [23] D. Z. Du and F. K. Hwang, “A proof of the Gilbert-Pollak’s conjucture on the Steiner ratio,” Algorithmica, no. 45, pp. 121–135, 1992. [24] M. Steele, “Growth rates of Euclidean minimal spanning trees with power weighted edges,” The Annals of Probability, vol. 2, no. 16, pp. 1767–1787, 1988.
R EFERENCES [1] S. Shakkottai, X. Liu and R. Srikant,“The Multicast Capacity of Large Multihop Wireless Networks” in Proceedings of ACM MobiHoc 2007, Montreal, Canada, September 2007. [2] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. IT-46, no. 2, pp. 388–404, March 2000. [3] S. R. Kulkarni and P. Viswanath, “A deterministic approach to throughput scaling in wireless networks,” IEEE Transactions on Information Theory, vol. 50, no. 6, pp. 1041–1049, 2004. [4] S. Toumpis and A. J. Goldsmith, “Large Wireless Networks under Fading, Mobility, and Delay Constraints,” in Proceedings of IEEE INFOCOM 2004, Hong Kong, March 2004. [5] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “ThroughputDelay Trade-off in Wireless Networks,” in Proceedings of IEEE INFOCOM 2004, Hong Kong, March 2004. [6] M. Grossglauser and D. Tse, “Mobility Increases the Capacity of Adhoc Wireless Networks,” IEEE/ACM Transactions on Networking, vol. 10, no. 4, pp. 477–486, August 2002. [7] X. Lin and N. B. Shroff, “Towards Achieving the Maximum Capacity in Large Mobile Wireless Networks,” Journal of Communications and Networks, Special Issue on Mobile Ad Hoc Wireless Networks, vol. 6, no. 4, December 2004. [8] M. J. Neely and E. Modiano, “Capacity and Delay Tradeoffs for Ad-Hoc Mobile Networks,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 1917–1937, June 2005. [9] G. Sharma, R. R. Mazumdar, and N. B. Shroff, “Delay and Capacity Trade-offs in Mobile Ad Hoc Networks: A Global Perspective,” in Proceedings of IEEE INFOCOM 2006, Barcelona, Spain, April 2006. [10] A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi,“Broadcast capacity in multihop wireless networks” in Proceedings of ACM MobiCom’06, Los Angeles, CA, September 2006. [11] B. Tavli, “Broadcast capacity of wireless networks,” IEEE Communication Letters, vol. 10, no. 2, pp. 68–69, June 2005. [12] J. E. Wieselthier, G. D. Nguyen, and A. Ephremides, “On the Construction of Energy-Efficient Broadcast and Multicast Trees in Wireless Networks,” in Proceedings of IEEE INFOCOM 2000, Tel-Aviv, Israel, March 2000. [13] P.Chaporkar and S. Sarkar, “Wireless Multicast: Theory and Approaches,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 1954–1972, June 2005. [14] E. M. Royer and C. E. Perkins, “Multicast Operation of the Ad Hoc On-Demand Distance Vector Routing Protocol,” in Proceedings of the 5th annual ACM/IEEE International Conference on Mobile Computing and Networking (Mobicom), Seattle, WA, August 1999.
Srinivas Shakkottai (S ’00-M ’08) received his Bachelor of Engineering degree in electronics and communication engineering from the Bangalore University, India, in 2001 and his M.S. and Ph.D degrees from the University of Illinois at Urbana-Champaign in 2003 and 2007, respectively, both in electrical engineering. He was postdoctoral scholar at Stanford University until December 2007, and is currently an Assistant Professor at the Dept. of ECE, Texas A&M University. He has received the Defense Threat reduction Agency Young Investigator Award (2009) as well as research awards from Cisco (2008) and Google (2010). His research interests include peer-to-peer systems, pricing approaches to network resource allocation, game theory, congestion control, and the measurement and analysis of Internet data.
Xin Liu (S ’99-M ’03) received her Ph.D. degree in electrical engineering from Purdue University in 2002. She is currently an associate professor in the Computer Science Department at the University of California, Davis. Before joining UC Davis, she was a postdoctoral research associate in the Coordinated Science Laboratory at UIUC. She received the Best Paper of year award of the Computer Networks Journal in 2003 for her work on opportunistic scheduling. She received the NSF CAREER award in 2005 for her research on “Smart-Radio-Technology-Enabled Opportunistic Spectrum Utilization”. She received the Outstanding Engineering Junior Faculty Award from theCollege of Engineering, University of California, Davis in 2005. Her research is on wireless communication networks, with a focus on resource allocation and dynamic spectrum management.
11
R. Srikant (S ’90-M ’91-SM ’01-F ’06) received his B.Tech. from the Indian Institute of Technology, Madras in 1985, his M.S. and Ph.D. from the University of Illinois at Urbana-Champaign in 1988 and 1991, respectively, all in electrical engineering. He was a Member of Technical Staff at AT&T Bell Laboratories from 1991 to 1995. He is currently with the University of Illinois at Urbana-Champaign, where he is Fredric G. and Elizabeth H. Nearing Endowed Professor in the Department of Electrical and Computer Engineering, and a Research Professor at the Coordinated Science Laboratory. He has served as an associate editor of Automatica, the IEEE/ACM Transactions on Networking and IEEE Transactions on Automatic Control. He has also served on the editorial boards of special issues of the IEEE Journal on Selected Areas in Communications and IEEE Transactions on Information Theory. He was the chair of the 2002 IEEE Computer Communications Workshop in Santa Fe, NM, program co-chair of IEEE INFOCOM, 2007, and program co-chair of GameNets 2007. His research interests include communication networks, stochastic processes, queueing theory, information theory, and game theory.
The Multicast Capacity of Large Multihop Wireless Networks† Srinivas Shakkottai,
Xin Liu, and
Abstract—We consider wireless ad hoc networks with a large number of users. Subsets of users might be interested in identical information, and so we have a regime in which several multicast sessions may coexist. We first calculate an upper-bound on the achievable transmission rate per multicast flow as a function of the number of multicast sources in such a network. We then propose a simple comb-based architecture for multicast routing which achieves the upper bound in an order sense under certain constraints. Compared to the approach of constructing a Steiner tree to decide multicast paths, our construction achieves the same order-optimal results while requiring little location information and no computational overhead.
I. I NTRODUCTION Wireless ad hoc networks are of use when there is a lack of fixed communication infrastructure. Such situations might arise in calamity environments, sensor network applications, and a variety of other civilian and military contexts. In many applications, multicast data transfer is more predominant than unicast data transfer. In military networks, it is often stated that multicast traffic dominates due to the need for group communications. In the civilian context, an emerging application that has already been tested is the use of wireless ad hoc networks to broadcast replays during football games. A situation like a football game would have a large number of spectators, each having a mobile device and a desire for a replay of an important moment in the game. There is almost no infrastructure available from which they could obtain such data, and there is a strong incentive to form an ad hoc network for this purpose. Some of the users might be close to data sources (perhaps if they were close to an Internet access point), and they would act as sources for the multicast traffic. Other nodes would act as relays and sinks for the data. The questions arise as to how many multicast sessions can be supported by such a network, what the total capacity of the network would be, and how to achieve the capacity in a simple and practical manner. Consider Figure 1. There is a finite area with a number of wireless nodes. There are 3 multicast flows in progress, with nodes receiving each multicast flow labeled 1, 2 and 3, respectively. The sources of these flows are labeled as † Research supported in part by DARPA CBMANET grant, NSF grants CNS-0519691, CNS-0904520, CNS-0963818, CNS-0520126, NSF CAREER Award CNS-0448613, DTRA grant HDTRA1-09-1-0051, and Qatar Telecom, Doha, Qatar. An earlier version of this paper [1] appeared in MobiHoc’07, Montreal, Canada, September 2007. S. Shakkottai ([email protected]) is with the Dept. of Electrical and Computer Engineering,Texas A&M University. X. Liu ([email protected]) is with the Dept. of Computer Science, University of California at Davis. R. Srikant ([email protected]) is with the Coordinated Science Laboratory and Dept. of ECE, University of Illinois at Urbana-Champaign.
R. Srikant
1
S2
2
S1 1
2 1,2
2
2
1
1
1
1
S3 3
1
3
3
1 1,3
3
3
Fig. 1. Example of multicast flows in a wireless ad hoc network. Receivers are labeled with the flows that they are interested in. Some nodes act as pure relays, in which case they are not labeled. The system throughput is determined by the number of flows that pass through each node.
S1 , S2 , and S3 , respectively. The hops are labeled with the sessions that they carry. Some nodes may be neither sources nor destinations and merely act as relays (unlabeled nodes). Suppose each hop, if scheduled, could carry one bit per time slot. The first constraint on the system capacity is that we may not be able to schedule all the hops simultaneously due to interference. Even if all hops could be scheduled simultaneously, we see that the throughput of each source is at most 0.5 bits per time slot in this example, since all the multicast flows contain at least one node that is shared with one other multicast flow. The total throughput of the system would then be 1.5 bits per second. Thus, we see that there are two main sources of interference that limit the multicast capacity of the network: 1) The channel interference constraint: While a large transmission radius may allow multiple receivers to receive packets from a multicast flow simultaneously, a larger radius means a transmitting node interferes with more nodes, which limits the number of simultaneous transmissions. A good multicast structure should balance the tradeoff. 2) The flow interference constraint: if multiple flows pass through the same node, the rate that each flow obtains is a fraction of the node’s transmission capacity. In other words, system capacity is divided among the flows. Therefore, we consider the multicast capacity as a function of the number of multicast sources.
2
In this paper, we will study how the throughput of multicast wireless ad hoc networks scales with the number of sources, the number of destinations per source, and the total number of nodes present in the system. We will concentrate on the case when the number of nodes in the system is large. We consider a unit square in which n nodes are dropped at random, ns of these nodes are randomly chosen as sources, and each of these sources is associated with nd random destinations, making a total of ns nd source-destination pairs in the system. We would like to understand the throughput scaling laws in the network as n, ns , nd → ∞. A number of studies consider such questions with regard to unicast flows, starting with the seminal paper by Gupta and Kumar [2], where achievable upper bounds are derived for wireless ad hoc networks in a finite region with a large number of nodes. The Gupta-Kumar result was re-derived in a much simpler manner in [3]. Related ideas can also be found in [4], [5]. A number of papers have also looked at delay-throughput tradeoffs in mobile models for such networks [5]–[9], as well as broadcast networks [10], [11] although we do not study these issues in our paper. Multicast in wireless networks has been studied in [12], [13], but these papers do not deal with scaling laws. Some examples of multicast protocols are presented in [14]–[16]. In comparison, our focus in this paper is not to develop a protocol, but to show that a simple routing structure can be capacity achieving in an order-optimal sense. To the best of our knowledge, the only prior works that deal with scaling laws in wireless networks with multicast flows are [17]–[22]. However, in [17] and [18] the authors use a Steiner tree approach to construct a multicast tree, and they assume that every node in the network is a multicast source in [17]. In [18], the authors build a Euclidean spanning tree with Manhattan routing. The authors also extend the results to hybrid networks where there are ad hoc nodes and base stations in [19], and to networks under Gaussian channel model [20]. Results in [21] apply to multicast with mobile nodes and the scaling of throughput subject to the nodes willing to tolerate delays. This is a different framework than our work in which we study randomly located but fixed nodes. The multicast results and achievability scheme presented in [22] are closely related to our paper. However, its assumption that every node in the network is a multicast source implies that is a special case of our results, and fits in the framework of Theorem 3 of our paper. We analyze a general case of different ratios of sources to destinations. The achievability result in [22] is developed through a comb architecture that is similar to our paper, a shorter version of which was originally published in a conference [1]. Our main contribution is to propose a simple architecture that achieves the same capacity in the order sense as the maximum possible throughput. For instance, in comparison to [17], [18], our scheme requires neither a central control to compute the multicast tree nor global location information of the multicast destinations. Furthermore, our proof of the upperbound is different from that in [17], [18]. Last, in contrast to [22], we present complete results with proofs of the scaling of the upperbound as well as the achievability scheme as a
function of the number of multicast sources and destinations. Main Results We express our results in the order sense. We say f (n) ∈ O(g(n)) with high probability (w.h.p) if given δ > 0, ∃ c, and m(δ) such that P{f (n) ≤ c g(n)} ≥ 1 − δ
∀n ≥ m(δ).
Similarly, we say f (n) ∈ Θ(g(n)) with high probability (w.h.p) if given δ > 0, ∃ c1 , c2 , and m(δ) such that P{c1 g(n) ≤ f (n) ≤ c2 g(n)} ≥ 1 − δ
∀n ≥ m(δ).
We first derive an upper bound on the multicast capacity of a wireless ad hoc network. The main technical challenge is to take into account the broadcast nature of wireless media where multiple nodes can receive simultaneously. We show that when nd = nǫ − 1, 0 < ǫ < 1, the total number of nodes in each multicast flow nm , nd + 1 = nǫ , and the number of multicast flows ns = O(n1−ǫ ), the upper bound on the sum of the source rates that the network can support is √ n w.h.p, O √ nm log n with a per flow throughput capacity of √ n √ w.h.p. O ns nm log n The achievability of the upper bound is the main contribution of this paper. To achieve the upper bound, we propose a simple routing architecture to transfer multicast data in the network. The architecture consists of a tree called the multicast comb, which is constructed independent of the receiver locations. The receivers then complete the multicast tree by attaching themselves to the comb using shortest path routing in a small vicinity. Using this simple architecture, we show that the achievable throughput matches the upper bound √ √ in an order sense if 1) ns nm = Ω( n log n); or 2) it is allowed to drop an arbitrarily small fraction of traffic; or 3) one can pose constraints on the locations of the source and destination nodes. In comparison, when there are ns nd unicast sourcedestination pairs, the per flow throughput capacity for unicast is √ n √ w.h.p. Θ ns nd log n following the Gupta-Kumar result. Therefore, by exploiting the properties of multicast, one can obtain multicast gain of √ Θ( nd ) for relatively large ns nd . An earlier version of this work is presented in [1]. The main differences in the current paper are: (i) the proof of the upper bound has been completely rewritten since our original proof did not accurately account for the broadcast nature of wireless networks, and (ii) Section V has been rewritten for clarity and to extend the results in [1] to a more general setting.
3
A
Organization of the Paper We begin in Section II with a description of the system model that we consider. We derive the upper bound on multicast capacity and compare it to the unicast upper bound in Section III. We then proceed in Section IV to design a simple architecture that we call the multicast comb structure that achieves the upper bound in the order sense if the sources and destinations can be chosen suitably. However, since our goal is to study the case of randomly chosen sources and destinations, in Section V we find the lower bound on the achievable transmission rate. We conclude with possible extensions in Section VI. II. S YSTEM M ODEL We consider a system with n wireless nodes. Let the set of all node-indices be denoted N = [1, 2, · · · , n]. We select ns source nodes, and associate each such source with nd = nǫ −1 distinct destination nodes. The number of nodes associated with each multicast flow is then 1 + nd , nm . For simplicity let node 1 be the source node for multicast flow 1, with nodes [2, 3, · · · , nm ] being its destinations. Similarly, let nm + 1 be the source for multicast flow 2, with destinations being [nm + 2, · · · , 2nm ]. We repeat this process ns ≤ n1−ǫ times until we have ns multicast flows. We then drop all the n nodes uniformly and at random into a square of unit area. Thus, each node i ∈ N has a location Pi = [Pix , Piy ] where Pix and Piy are uniformly distributed in the interval [0, 1]. We use the protocol model to model interference between transmissions, as proposed in [2]. The model is illustrated in Figure 2. We assume that all nodes choose identical transmit radii r, which is large enough for the network to be connected. Suppose that node i transmits to node j. Node j receives the transmission successfully if every other node that transmits simultaneously is at a distance of at least (1 + ∆)r from j. This implies that circles of radius ∆r 2 around each receiver must be disjoint [2]. A node can transmit 1 bit of data per unit of time if its transmission is successful.
(1+∆ ) r R T
r
Fig. 2. Illustration of the protocol model for wireless transmission. The assumption is that a transmission would be successful as long as there is no other transmission in a circle of radius (1 + ∆)r from the intended receiver.
A
R
R
B
B C
a) A minimum Steiner tree
Fig. 3.
C b) A (wireless) multicast tree
A multicast tree in a wireless network.
When nd = 1, the model is the same as a unicast problem. When nd = n − 1, the source can broadcast its data in one √ hop to all the destinations with a transmission radius of 2. We would like to understand the capacity scaling law and its practical achievability between these two extremes. In particular, we study the case where nm = nd + 1 = nǫ , and 0 < ǫ < 1. In this paper, a multicast tree refers to a transmission structure that takes into account the broadcast nature of the wireless medium, which is different from a minimum spanning tree. The difference is shown in Figure II. In the figure, A is the multicast source with B and C as its receivers. The left figure shows a Steiner tree1 , where R is a relay node. In this case, A transmits once to R, and R transmits to B and C, respectively. The right figure shows a multicast tree in a wireless network where B and C can receive A’s transmission simultaneously (i.e., only one transmission is needed). The length of the wireless multicast tree is greater than or equal to that of the minimum Steiner tree while the number of transmissions may be smaller. Therefore, taking into account such an effect is critical to deriving an upperbound and to deciding the optimal transmission scheme for multicast in wireless networks. In a multicast tree, all leaf nodes are multicast receivers. III. U PPER B OUND Our first objective is to derive an upper bound on the throughput capacity of multicast wireless networks. We have the following theorem. Theorem 1: The throughput of each multicast source in a random wireless ad-hoc network is upper bounded by √ n O min 1, √ w.h.p. ns nm log n
as n → ∞, nm = nǫ , 0 < ǫ < 1. To prove this theorem, we still exploit the idea that communication consumes space as in [2], [17]. However, the focus is to take into account the properties of multicast. The questions that need to be answered to obtain a useful bound on the throughput capacity are: (i) what should the radius r be?, and (ii) how many transmissions are required to reach all destinations? When we have the answer to the above two related questions, we can derive the upperbound for capacity as follows. Let the transmission radius be r, since our square is of size 1, 1 A Steiner tree is a minimum length tree connecting a set of given points, where additional points may be added to minimize the length of the tree.
4
Circle II
the maximum number of simultaneous transmissions can be bounded [2] as S≤
1 π
∆r 2 2
=
4 . π∆2 r2
Circle I
(1)
Suppose there are ns multicast flows and flow i requires at least Hi (r) transmissions, 1 ≤ i ≤ ns , the capacity of a multicast flow satisfies 4 P . (2) λ≤ π∆2 r2 i Hi (r)
Thus, λ is the largest uniformly achievable rate per source node. To prove Theorem 1, we first present two lemmas. The first lemma is on the degree of multicast trees. The second lemma is on the length of multicast trees. The objective is to show that even with the broadcast capability of the wireless medium, if the transmit radius is limited, the number of transmissions needed would be of the same order as that of the shortest multicast tree. Definition We say that a rooted, directed multicast tree, where the transmitter is the root and the receivers are the leaves of the tree, satisfies property P if each node in the tree has at most twelve children that are required to transmit to ensure that each packet reaches all destination. Thus, the degree of each transmitter is at most 12. Lemma 1: Given a multicast tree in a wireless network, either the multicast tree satisfies property P or one can construct another multicast tree with the same root and leaf nodes which satisfies property P such that the number of transmissions in the new tree is less than or equal to the number of transmissions in the original tree. Additional relay nodes may be necessary to construct the new multicast tree. Proof: Consider a multicast tree T . First, we remove all leaf nodes since the definition of property P only considers transmitting nodes. If all remaining nodes in the original tree have at most twelve children nodes, then the lemma trivially holds. If not, we construct an alternative tree that has a bounded degree of twelve and a smaller number of transmissions compared to the original tree T . The idea is shown in Figure 4 and discussed in the following. Consider a node A whose transmit children degree is greater or equal to 13 (i.e., node A has at least 13 children nodes that need to transmit). Then all receivers of its children are within Circle II (red). Now we arbitrarily add relay nodes to construct the desired new multicast tree. We add 6 square (blue) nodes on the boundary of circle I and 6 triangle (yellow) nodes within the boundary of Circle II as shown in in Figure 4. Replace Node A’s transmit children nodes by these new nodes. Node A plus these new nodes will cover all receivers in the original tree that are covered by node A and its children. To maintain the tree structure, we make A the parent of the 6 square nodes and all leaf nodes in Circle I in the original tree. Each square node is assigned as the parent of a unique triangular node; and all leaf nodes between Circle II and Circle I on the original tree can be connected to one of the square nodes. The new tree has a bounded transmit children degree of six at node A and a
2 1 A
Fig. 4. The small circles (eg. Circle III) represent the transmission ranges of the nodes at the center of these circles. Nodes may be placed in such a way that the number of transmit children of any node is finite.
smaller number of overall transmissions than the original tree T. We can follow the same procedure to eliminate all nodes of degree greater than 12 starting at the root of the tree. For example, suppose node A has a degree larger than 12 and its children may also have a degree larger than 12. We first replace children of node A following the above described procedure. The new children of node A may still have a degree larger than 13, say nodes 1 and 2, as shown in the figure. In this case, we apply the same procedure on nodes 1 and 2, respectively. Note that because of the regularity of the children node deployment, it is possible that a new child of node 1 will collocate with node 2. We consider them as two separate nodes for the ease of argument (although one of them and its corresponding transmission can be eliminated). We note that because of the tree structure, this procedure will not change the degree of a parent node, a sibling node, or nodes in other branches of the tree. Hence, we can use the same procedure to replace all nodes with degree larger than 12. Then the final tree has a maximum degree of 12. We now need the following results from [23], [24] for the length of the multicast tree in random networks. Result 1 Suppose we drop m nodes in a unit square, where the position of each node is chosen uniformly at random in the square. Then as m → ∞, the length of the minimum spanning tree L(m) satisfies √ 1) E(L(m)) ∼ C1 m (Lemma 3.3 from [24]), and 2) V ar(L(m)) ≤ C2 log m (Lemma 4.1 from [24]), where C1 and C2 are positive constants, and the notation ∼ is used to denote asymptotic equality. Also, the length of the optimal Euclidean Steiner tree connecting any m nodes √ satisfies S(m) ≥ 23 L(m) [23] . We next present an upper bound on the length of multicast trees. Let Mi (nm ) be the length (the sum of the euclidean distances of the hops in the tree) of a multicast tree associated with source i.
5
Lemma 2: The average length of a multicast tree is √ Ω( nm ) w.h.p.; i.e., ) (n s X √ Mi (nm ) (3) > C nm → 1, P ns i=1 where C is a positive constant, when
Mi (nm ) be the (Euclidean) length of T ′ associated with source i. From Lemma 2, we have ) (n s X √ Mi (nm ) (5) P > C nm → 1. ns i=1 From Lemma 1 we know that Mi (nm ) 12rHi′ (r) > . ns ns
log nm lim = 0. n→∞ ns Proof: Let Li (nm ) be the length of the minimum spanning tree, and Si (nm ) be the length of its Steiner tree. We can lower bound the total length ns X i=1
Mi (nm ) ≥
ns X i=1
(a)
Si (nm ) ≥
√
3/2
ns X
Li (nm ),
i=1
where (a) follows from Result 1. Now, ) ( n s X Li (nm ) − E(Li (nm )) P ≥δ ns i=1
(b)
V ar(L1 (nm )) ns δ 2 (c) C log n 2 m ≤ ns δ 2 → 0 as n → ∞,
≤
where (b) follows from Chebyshev’s inequality and the i.i.d nature of the multicast trees (following the construction of multicast flows), and (c) follows from Result 1. Also we have used the hypothesis that log nm /ns → 0 as n → ∞. Thus, as n → ∞, ) ( n s X Li (nm ) − E(L1 (nm )) < δ → 1 P ns i=1 (n ) s X Li (nm ) ⇒P > E(L1 (nm )) − δ → 1. ns i=1
√ Now, from Result 1, ∃C ′ , such that E(Li (nm ))− δ ≥ C ′ nm for large nm . Hence, we have ) (n s X √ Mi (nm ) (4) P > C nm ns i=1 (n ) s X Li (nm ) ′√ ≥ P > C nm ns i=1 ) (n s X Li (nm ) > E(L1 (nm )) − δ → 1. ≥ P ns i=1 Now we are ready to prove Theorem 1. Proof: To use (2) to obtain an upper-bound onP throughput, we need the minimum number of transmissions i Hi (r) as a function of the radius r. Consider a multicast tree T for source i. Following Lemma 1, we can construct a new tree, T ′ , that has a maximum transmit degree 12. Let the number of transmissions required on T ′ be denoted Hi′ (r). Also, let
(6)
Thus, from (5) and (6) ) (n s X √ 12rHi′ (r) P > C nm → 1. ns i=1 Since Hi (r) ≥ Hi′ (r), the above implies that (n ) s X √ 12rHi (r) P > C nm → 1. ns i=1
(7)
(8)
Finally, using the relation (2), we have λ≤
48 √ , π∆2 Crns nm
w.h.p.
(9)
In addition, by the capacity limit on each node, we have λ ≤ 1. Thus, from (9) and the above √ n λ ≤ min 1, √ , ns nm log n
w.h.p.
(10)
In order to contrast the multicast case with the unicast case, we take the number of source destination pairs to be the same in both cases. This would enable us to characterize the gain that could potentially be achieved using multicast. Since the unicast regime consists of n source destination pairs, we take ns nm = n. In particular, to get Θ(n) s-d pairs, we take nd = nǫ − 1 and ns = n1−ǫ , where 0 ≤ ǫ ≤ 1. We have the following corollary: Corollary 2: The throughput of each multicast source in a random wireless ad hoc network is upper bounded by ! 1 w.h.p. Θ p n1−ǫ log n
Notice that as ǫ → 0, the multicast throughput capacity is essentially the same as the unicast capacity as it should be. Also notice that as ǫ → 1, the benefit of using multicast is lost since all nodes could be reached by one broadcast hop. As an example, consider the case where n = 10, 000 and ǫ = 1/2 (i.e., there are 100 sources, each with 100 destinations). The per-source throughput that can achieved by using multicast is 10 times that of unicast. We now move to the problem of designing a simple scheduling and routing algorithm that could be used to implement the multicast idea. We first show that in a network where we are allowed to choose the sources and destination, the multicast capacity can easily be attained. We then consider the case of randomly chosen sources and destinations.
6
IV. A C OMB A RCHITECTURE FOR M ULTICAST T REES We develop a simple architecture, whereby we may achieve the upper bound found above (in the order sense), when we are allowed to select the source and destination nodes appropriately while still keeping the average distance from a multicast source to its destinations a constant. This location constraint is introduced here to simplify the discussion and highlight the comb structure. We will remove the assumption in Section V. As before, we study the system with ns multicast sources each with nd destinations, with nm = nd + 1 = nǫ . The main features of our construction are as follows. • We divide the region into squarelets in the manner of [3]. The squarelet size is large enough so that there is at least one node in each squarelet with high probability, and nodes in adjacent squarelets are capable of communicating with each other. Let the side-length of a squarelet be denoted sn . • The scheduling algorithm will be chosen such that the squarelets that are Ksn apart are scheduled simultaneously, where K is chosen such that the wireless interference constraint is satisfied. • We construct multicast comb structures using the constructed squarelets. There is one comb corresponding to each multicast flow. The combs are constructed so that at most 2 flows pass through a squarelet, in order to keep interference between flows minimal. In the comb structure, the width between two consecutive comb teeth is determined by nm . Each multicast source/destination node can reach any one tooth on the comb to send/receive multicast data. This can be done in the vicinity of the destination node using any routing algorithms, such as shortest path routing. To construct the comb, the location of the destination nodes is not required and no central control is needed. In addition, the cost of the comb structure is the same as an optimal Steiner tree in an order sense. We now present the details of the architecture and derive its throughput capacity. Recall that the length of a squarelet is sn . We have the following useful result from [4]: Result 2 For a squarelet size √ κ log n sn = √ , n the number of nodes mi in any squarelet i satisfies κ κ mi 2n− 8 P ≤ ≤ 4κ ∀i > 1 − , 2 log n κ log n
(11)
for sufficiently large n. Thus, for κ > 8, each squarelet has order log n nodes with high probability (and so each squarelet contains at least one node with high probability). We need to ensure that a node in one corner of a squarelet can transmit to a node in the opposite corner of an adjacent √ squarelet, i.e., the transmission radius is chosen as r = 5sn . We can guarantee successful reception at a receiving node, if no other transmission takes place within a distance of
√ 5sn (1 + ∆). Recall that Ksn is the distance such that squarelets that are this distance apart can be scheduled simultaneously. Then, as illustrated in Figure 5, we have √ √ (K − 2)sn ≥ (1 + ∆) 5sn ⇒ K = 2 + (1 + ∆) 5. We call the subset of squarelets a distance of K squarelets sn
Ks n
S 1
Fig. 5. Illustration of scheduling constraints with ∆ = 0.5. The source is in the squarelet labeled S, and the squarelet with the X is where the receiver is located. K is chosen so that all possible receiver nodes in this squarelet are guaranteed successful reception.
from each other, capable of simultaneous transmission as an equivalence class. So the number of such equivalence classes is K 2 . We then have a system in which the periodicity with which any squarelet is scheduled is K 2 time slots. Using the above idea, we construct the following comb structure for multicast traffic. Suppose that we are able to select the source and receiver nodes of the multicast. We first construct multiple combs, one for each multicast flow as shown in Figure 6. In the figure we have illustrated two multicast combs (one lightly shaded (cyan), and the other dark (magenta)) corresponding to two multicast flows. We define √ n 1 =√ S , (12) sn κ log n √ n 1 . (13) =√ N , √ sn nm κnm log n Here, S is the number of squarelets in any row/column and N is the number of squarelets between two consecutive teeth, which is also the number of distinct multicast flows the network can support without overlapping. Consider a multicast flow i ≤ N. The comb corresponding to this flow consists of (i) squarelets spq , where p indicates the row and q indicates the column of the squarelet. These squarelets form the teeth and the spine of the comb as follows: (i) Teeth: {spq : p ∈ [i, i + N, · · · , i + S/N ], q ∈ [1, 2, · · · , S]}. (i) Spine: {spq : p ∈ [1, 2, · · · , S], q = i}.
Note that the distance between the teeth of each comb is exactly √n1 m . For i > N we reuse the combs by letting multicast
7
flow i use comb j, where j satisfies i = j + kN, k ∈ [1, 2, ...] and i ≤ n1−ǫ . The Euclidean length of the comb routing for a particular multicast flow is From Result 1, we have that the Euclidean length of the √ optimal Steiner tree is Θ( nm ), which is the same as our comb structure in an order sense. The construction of the Steiner tree requires global location information and a centralized controller. In comparison, the construction of the comb structure only needs information on the number of destination nodes and does not require central computation of the tree. Each comb is positioned one squarelet farther to the right and below to the previous one. Note that by using this structure, we have ensured that the maximum number of multicast flows that use any particular squarelet is 2. Since each squarelet is scheduled with a periodicity of K 2 slots and the channel has a capacity of 1 bit per time slot (if scheduled), if only 2 multicast flows share a squarelet, their individual throughput would be 1/2K 2 bits per time slot.
1 nm
1
1/2K 2 . This results in a total source rate of √ n √ , 2K 2 κnm log n
(15)
which is of the same order of magnitude as the upper bound calculated in the previous section. In summary, if ns ≤ √ √ n/ κn total√source rate of the flows is √ √ m log n, then the √ n/ κnm log n. If ns > n/ κnm log n, then the combs are reused over multiple sources and the total throughput remains the same. ǫ Consider √ √an example where nm = n , then the condition ns ≤ n/ κnm log n can be written as √ n1−ǫ ns ≤ √ . κ log n Thus, the network multicast throughput increases as ns increases up to the above bound but remains constant after that, as illustrated in Figure 7.
λ ns
n 1− κ log n
ns
Fig. 7. The upper bound for multicast using the comb method. We see that the bound increases linearly with the number of multicast sources as the comb gets filled. The throughput capacity is constant afterward.
Fig. 6. The comb idea for multicast data transfer. Each comb carries a data from a different source. The size of the squarelet places a fundamental limit on the number of possible coexistent combs.
Since we can choose the locations of the source and destination nodes as desired, we let the source and destinations of a particular multicast flow lie on its corresponding comb. The average distance of a source and its destinations is still Θ(1). For example, in Figure 6, the sources and destinations associated with the lightly shaded (cyan) comb would lie somewhere on the comb. As the distance between the teeth of a comb is √n1 m , the number of such combs that can be constructed is 1 √ . sn nm √ √ In other words, since sn = κ log n/ n, the proposed system can accommodate a total of √ n √ (14) κnm log n sources without using one comb for multiple flows. As explained above, since there are a maximum of 2 flows using each squarelet, the sources can each transmit at a rate of
We have proposed a simple multicast architecture, and shown that its throughput in ideal circumstances is identical (in the order sense) to the upper bound. However, we have yet to study its performance when we are not at liberty to place the sources and destinations. We proceed to study this case in the following section. V. ACHIEVABLE M ULTICAST C APACITY IN R ANDOM N ETWORKS We have just seen how the comb architecture is capacityachieving in the case where source-destination placements can be made as desired. We will now study the case where the sources and destinations are randomly chosen. In this case, both sources and destinations must reach the comb in a multihop manner. Once they reach the comb, they would have access to the multicast traffic on that comb. The limiting factor is that as the number of such access paths increases, the load of each squarelet does as well, leading to reduction of throughput for the flows passing through the squarelet. We will study this effect in this section. For a given multicast flow i, both the source and its destinations choose the shortest path to reach the closest tooth on the corresponding comb. As shown in Figure 8, these paths are simple to construct (they are either above or below the teeth). Note that the maximum length of the path is 2√1nm , which means that the increase in length of the multicast tree due to
8
1 nm
1 nm
1
Fig. 8. Using the comb method for randomly placed sources and destinations. A source would be assigned a comb and would connect to the closest possible tooth. Destinations would do likewise.
√ these vertical paths would be just nm × 2√1nm = Θ( nm ). Thus, comparing with the comb length of Θ(nm ), we see that Euclidean length of the tree is unaffected in the order sense by these vertical paths. However, it might so happen that a particular squarelet has a large number of such vertical paths passing through it, which might cause excessive load. Consider the load on a squarelet. There would be horizontal flows passing through corresponding to the comb, and this number can be determined exactly since we construct the comb deterministically. There would also be a number of vertical paths passing through this squarelet, taken by nodes in order to reach their desired comb tooth. We define the number of vertical branches through a node as the number of distinct (corresponding to different multicast flows) such paths that pass through it. We divide the square area into bins √ of length l = 1/ nm and width √ sn . The number of bins √ is denoted M , 1/lsn = nnm / κ log n. Each squarelet then belongs to a particular bin, as shown in Figure 9. If we consider a vertical branch as belonging to the bin in which it originates, then the number of such branches passing through any given squarelet is at most that of the branches belonging to two bins—the bin that the squarelet belongs to, and the bin immediately above or below it, as illustrated in Figure 9. Now, let us select an arbitrary bin j. For this bin, let χij be a random variable, with χij = 0 if our selected bin has no branch associatedPwith multicast stream i, and χij = 1 ns otherwise. Let Xj = i=1 χij , i.e., it is the number of distinct flows that have vertical branches that originate in our selected bin. Define the average load on any bin ρ , E[Xj ]. Then we have ρ = ns 1 − (1 − ls2n )nm √ n κ log n m = ns 1 − 1 − √ nnm √ √ nm κ log n (a) √ ∼ ns 1 − exp − 2 n √ √ ns nm κ log n (b) √ ∼ . (16) n
1
Fig. 9. Division of the area into bins. Two bins have been highlighted, and the vertical branches belonging to them are shown. The load on any squarelet is at most that of the branches belonging to two bins—the bin that the squarelet belongs to, and the bin immediately above or below it.
Above, (a) and (b) hold for large n, and ∼ indicates asymptotic equality. Finally, recall that the average load on each squarelet is at most twice the average number of vertical branches passing through the bin to which it belongs, plus that of the comb teeth passing through it. Thus, the average load on a squarelet is at most ρ′
=
ns
2ρ + sn
1 √ nm
√ √ √ √ ns nm κ log n ns nm κ log n √ √ ∼ 2 + . n n
(17)
Comparing with (10), we see that the average load on a squarelet is such that the comb architecture can achieve the capacity upper bound. However, the question arises as to whether the actual load in each bin is the same as this average in an order sense. We will consider two cases below. For a large value of ns and nm although the average load is large, the number of branches in bin is similar to the average in an order sense, much like the asymptotic result of Result 2. However, when ns and nm are small, although the average load is not to large, it is possible that some particular squarelets might have an above average number of vertical branches. We might need to drop a small fraction of the nodes to achieve maximum throughput for the remaining nodes in this case.
A. Capacity-achieving Region Without Node-Dropping √ √ Theorem 3: When ns nm = Ω( n log n), the comb scheme is order optimal for each flow. In other words, the comb scheme enables per flow throughput of √ n √ w.h.p. Θ ns nm log n for all flows. Proof: Consider a particular bin j. The probability that any vertical branch corresponding to flow i originates in the
9
chosen bin is P(χij = 1) = 1 − (1 − lsn )nm . Recall that Xj
=
ns X
χij ,
i=1
where χij are i.i.d since the set of sources-destinations are independent for √ each multicast flow. By the hypothesis that √ ns nm = Ω( n log n), we have ρ = Ω(log n) > 1. Given some W ∈ R+ and using the Chernoff bound, we have E eXj . P(Xj ≥ W max(ρ, 1)) = P(Xj ≥ W ρ) ≤ eW ρ We calculate E eXj next as follows: ns X ns k Xj p (1 − p)ns −k ek = E e k k=0
=
(1 + (e − 1)p)ns ,
where p = 1 − (1 − lsn )nm and ρ = ns p. Therefore,
(1 + (e − 1)p)ns eW ρ ns (e−1)p e ≤ eW ρ e(e−1)ρ = eW ρ (e−1−W )ρ ≤ e , (18) √ √ By the hypothesis ns nm = Ω( n log n), we have √ √ ns nm ≥ k n log n, where k is a constant, for large enough n. In this case, ρ ≥ k ′ log n (where k ′ is constant). Let W = e − 1 + 1/k ′ . We have P(Xj ≥ W ρ)
≤
1 (19) P(Xj ≥ W max(ρ, 1)) = P(Xj ≥ W ρ) ≤ . n There are a total of M bins. Using a union bound and (19), we have 1 P(max Xj ≥ W ρ) ≤ M P(X k ≥ W ρ) ≤ √ , j log n
which has a limit of 0 as n → ∞. Therefore, the load of each bin (and from (17) every squarelet) is at most a constant factor of that of the average w.h.p. The throughput of each flow is thus √ 1 n √ . =Θ λ=Θ ρ ns nm log n Compared with Theorem 1, the comb scheme is order optimal for each flow. B. Capacity Achievement by Dropping a Fraction of Nodes Theorem 3 shows conditions on ns and nm under which the comb scheme is order optimal for each flow. √ We now consider √ the complementary case where ns nm = o( n log n). Under this regime, there might be certain squarelets that have a large number of interfering flows compared to the average, causing a loss of throughput. But how many squarelets would actually have such overcrowding? If the fraction of overcrowded squarelets were small, then we could simply drop the destinations belonging to these squarelets. Thus, we propose to use a majority rule, which can provide a high multicast rate
(on the same order of the capacity upper bound) to a majority of users if we are allowed to sacrifice an arbitrarily small percentage of users in overcrowded areas. We show below that a node can achieve order optimal throughput with a probability of at least 1−Pth , which is a constant, but can be set arbitrarily close to 1. Theorem 4: Given any threshold Pth > 0, with probability at least 1 − Pth , a node can receive the order optimal per flow throughput capacity using the comb structure by dropping a small percentage of traffic. Proof: As in the previous proof, we divide the network into disjoint bins. Each bin has l/sn squarelets. A vertical branch may span at most two bins. We say a bin is congested if the most congested squarelet in the bin has more than W times of the average load, where W is a constant that is to be determined later. The traffic to/from a node may be dropped if the node’s branch overlaps with a congested bin. A node can receive the order optimal throughput if its branch does not overlap with congested bins and neither does its source node. We will show that this probability is at least 1 − Pth . First, we calculate the probability that a bin is congested. Consider a bin. As before, let Xj be the number of distinct flows that have vertical branches that originate in bin j. Recall that χij = 1 is the event that bin j contains at least a node associated with multicast flow i. Also, let ρ be the average load of a squarelet as before. Now, following the proof of Theorem 3, we can show that P(Xj ≥ W max(ρ, 1)) ≤ e(e−1−W ) max(ρ,1) .
(20)
Let ′ 4 (1 − Pth ) = 1 − Pth ,
and let W = e − 1 + ln
1 ′ . Pth
We have P(Xj ≥ W max(ρ, 1))
≤ e(e−1−W ) max(ρ,1) ≤ e(e−1−W ) ′ = Pth .
′ In other words, with probability 1−Pth , a bin is not congested. Recall that the maximum number of vertical branches through a squarelet may at most come from two bins. Therefore, with ′ 4 probability at least (1 − Pth ) , the branches of a destination node and its source node do not correspond to a congested bin, and thus can receive order optimal throughput. We have thus shown that by dropping an arbitrarily small fraction of the traffic, we can achieve order optimal throughput for the rest of the nodes. Note that as Pth approaches 0, the per-source throughput also approaches zero. However, the rate of decrease is slow: it is proportional to the logarithm of 1/Pth . This kind of policy would be acceptable in real situations where it is important that most of the users obtain high rate transmission, rather than having to cut down the rate for all users so as to satisfy a small percentage of overcrowded areas. We also note that if a source node passes a congested area, it is intuitive to let the source node transmit first in the
10
area. However, there exists a probability (taken into account in Pth through the calculation of p) that multiple source nodes congest a bin. In this case, all receivers in these multicast groups suffer. Nonetheless, this probability is taken into account in Pth . Thus, the statement still holds. VI. C ONCLUSIONS In this paper we have developed an analytical framework for studying the multicast capacity of wireless ad hoc networks. We started with a comparison of the unicast case that has been studied in detail earlier, and showed how the multicast capacity is a function of the number of multicast sources and destinations. We developed a new and simple scheme that we called the comb architecture that would achieve this upper bound if we were at liberty to place the sources and destinations. We also studied the random network case and showed that the price paid in terms of throughput capacity for the simple and robust architecture is not high in the order sense. In the future we would like to study experimental wireless multicast networks.
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R EFERENCES [1] S. Shakkottai, X. Liu and R. Srikant,“The Multicast Capacity of Large Multihop Wireless Networks” in Proceedings of ACM MobiHoc 2007, Montreal, Canada, September 2007. [2] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. IT-46, no. 2, pp. 388–404, March 2000. [3] S. R. Kulkarni and P. Viswanath, “A deterministic approach to throughput scaling in wireless networks,” IEEE Transactions on Information Theory, vol. 50, no. 6, pp. 1041–1049, 2004. [4] S. Toumpis and A. J. Goldsmith, “Large Wireless Networks under Fading, Mobility, and Delay Constraints,” in Proceedings of IEEE INFOCOM 2004, Hong Kong, March 2004. [5] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “ThroughputDelay Trade-off in Wireless Networks,” in Proceedings of IEEE INFOCOM 2004, Hong Kong, March 2004. [6] M. Grossglauser and D. Tse, “Mobility Increases the Capacity of Adhoc Wireless Networks,” IEEE/ACM Transactions on Networking, vol. 10, no. 4, pp. 477–486, August 2002. [7] X. Lin and N. B. Shroff, “Towards Achieving the Maximum Capacity in Large Mobile Wireless Networks,” Journal of Communications and Networks, Special Issue on Mobile Ad Hoc Wireless Networks, vol. 6, no. 4, December 2004. [8] M. J. Neely and E. Modiano, “Capacity and Delay Tradeoffs for Ad-Hoc Mobile Networks,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 1917–1937, June 2005. [9] G. Sharma, R. R. Mazumdar, and N. B. Shroff, “Delay and Capacity Trade-offs in Mobile Ad Hoc Networks: A Global Perspective,” in Proceedings of IEEE INFOCOM 2006, Barcelona, Spain, April 2006. [10] A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi,“Broadcast capacity in multihop wireless networks” in Proceedings of ACM MobiCom’06, Los Angeles, CA, September 2006. [11] B. Tavli, “Broadcast capacity of wireless networks,” IEEE Communication Letters, vol. 10, no. 2, pp. 68–69, June 2005. [12] J. E. Wieselthier, G. D. Nguyen, and A. Ephremides, “On the Construction of Energy-Efficient Broadcast and Multicast Trees in Wireless Networks,” in Proceedings of IEEE INFOCOM 2000, Tel-Aviv, Israel, March 2000. [13] P.Chaporkar and S. Sarkar, “Wireless Multicast: Theory and Approaches,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 1954–1972, June 2005. [14] E. M. Royer and C. E. Perkins, “Multicast Operation of the Ad Hoc On-Demand Distance Vector Routing Protocol,” in Proceedings of the 5th annual ACM/IEEE International Conference on Mobile Computing and Networking (Mobicom), Seattle, WA, August 1999.
Srinivas Shakkottai (S ’00-M ’08) received his Bachelor of Engineering degree in electronics and communication engineering from the Bangalore University, India, in 2001 and his M.S. and Ph.D degrees from the University of Illinois at Urbana-Champaign in 2003 and 2007, respectively, both in electrical engineering. He was postdoctoral scholar at Stanford University until December 2007, and is currently an Assistant Professor at the Dept. of ECE, Texas A&M University. He has received the Defense Threat reduction Agency Young Investigator Award (2009) as well as research awards from Cisco (2008) and Google (2010). His research interests include peer-to-peer systems, pricing approaches to network resource allocation, game theory, congestion control, and the measurement and analysis of Internet data.
Xin Liu (S ’99-M ’03) received her Ph.D. degree in electrical engineering from Purdue University in 2002. She is currently an associate professor in the Computer Science Department at the University of California, Davis. Before joining UC Davis, she was a postdoctoral research associate in the Coordinated Science Laboratory at UIUC. She received the Best Paper of year award of the Computer Networks Journal in 2003 for her work on opportunistic scheduling. She received the NSF CAREER award in 2005 for her research on “Smart-Radio-Technology-Enabled Opportunistic Spectrum Utilization”. She received the Outstanding Engineering Junior Faculty Award from theCollege of Engineering, University of California, Davis in 2005. Her research is on wireless communication networks, with a focus on resource allocation and dynamic spectrum management.
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R. Srikant (S ’90-M ’91-SM ’01-F ’06) received his B.Tech. from the Indian Institute of Technology, Madras in 1985, his M.S. and Ph.D. from the University of Illinois at Urbana-Champaign in 1988 and 1991, respectively, all in electrical engineering. He was a Member of Technical Staff at AT&T Bell Laboratories from 1991 to 1995. He is currently with the University of Illinois at Urbana-Champaign, where he is Fredric G. and Elizabeth H. Nearing Endowed Professor in the Department of Electrical and Computer Engineering, and a Research Professor at the Coordinated Science Laboratory. He has served as an associate editor of Automatica, the IEEE/ACM Transactions on Networking and IEEE Transactions on Automatic Control. He has also served on the editorial boards of special issues of the IEEE Journal on Selected Areas in Communications and IEEE Transactions on Information Theory. He was the chair of the 2002 IEEE Computer Communications Workshop in Santa Fe, NM, program co-chair of IEEE INFOCOM, 2007, and program co-chair of GameNets 2007. His research interests include communication networks, stochastic processes, queueing theory, information theory, and game theory.
