IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XXXX 2007

1

Improved Unicast Capacity Bounds for General Multi-channel Multi-radio Wireless Networks Liangping Ma

Abstract—This letter extends and improves prior unicast transport capacity results for the general multi-channel, multi-radio, multi-hop wireless networks, where channels may have different data rates, and the number of interfaces per node may differ from the number of channels available in the network. Both the upper bound and the lower bound are rigorously derived for the Arbitrary Networks with interference modeled by the Protocol Model. Also presented is a minor but important improved use of the Protocol Model in the derivation of the transport capacity upper bound. Index Terms—network capacity, unicast, multi-radio, multichannel.

I. I NTRODUCTION AND R ELATED W ORK

T

HE unicast transport capacity of wireless networks was first studied by Gupta and Kumar [1]. By denoting the number of interfaces (radios) as m and the number of channels as c, the case investigated in [1] is m = c. Recently, the case m < c was investigated in [2] but in a less rigorous way since all results are in the asymptotic form. Also it is assumed in [2] that all channels have equal data rates, which may not always be valid in practice. In this letter, we investigate the unicast transport capacity [1] of a general multi-channel, multi-radio, multi-hop wireless network without any constraint on the parameters c, m, and the data rates. In particular, we derive the upper bound and the lower bound for the Arbitrary Networks [1] by taking a first-principle approach using the Protocol Model as the interference model. In addition, we present a minor but important improved use of the Protocol Model in the derivation of the transport capacity upper bound. The remainder of this letter is organized as follows. Section II presents unicast transport capacity bounds for the general multi-channel multi-radio multi-hop wireless networks under the Arbitrary Network model. Section III gives an improved use of the interference model (Protocol Model).

Chien-Chung Shen

Model [1] (the interference model), a transmission from node i (located at Xi ) to node j (located at Xj ) over some channel is successful if |Xk − Xj | ≥ (1 + ∆)|Xi − Xj | for every node k (located at Xk ) that simultaneously transmits over the same channel. A direct consequence [1] of the Protocol Model is that for any two simultaneous transmissions (e.g., i → j, k → l) to be mutually interference free, it is required that |Xl − Xj | ≥

∆ (|Xk − Xl | + |Xi − Xj |), 2

which can be interpreted as that each receiver “consumes” a disk of radius of ∆/2 times the length of the hop. The transport capacity [1], defined as the maximum sum of the products of bits and the distances over which the bits are carried, is used as the measure of the unicast network capacity. Additionally, the following assumptions are used in our analysis: (A1) n nodes are arbitrarily located in a disk of area A. (A2) Each node has m half-duplex interfaces (radios) which can be tuned to any of the c orthogonal channels. Each channel is associated with a fixed data rate wi (bits/sec), i = 1, ..., c. Without loss of generality, we assume w1 ≥ w2 ≥ ... ≥ wc . (A3) The network transports λnT bits over T seconds. (A4) The average distance between the source and the destination of a bit is L. Together with (A1), this implies that a bit-meters/second of λnL is achieved. (A5) Transmissions are slotted into synchronized slots of length τ seconds. (A6) Each node tunes at most one interface to a given channel. The above assumptions include assumptions made in [1] and [2] as special cases since (A2) imposes no constraint on the relationship between m and c, while [1] assumes m = c, and (A2) does not require w1 = ... = wc , while [2] does. Let the total number of interfaces tuned to channel i be ni , i = 1, · · · , c. Then it follows from (A6) that 0 ≤ ni ≤ n,

II. C APACITY B OUNDS A. Models and Assumptions In an Arbitrary Network [1], node locations, traffic patterns and the transmission powers are all arbitrary. In the Protocol Manuscript received April 22, 2007; revised June 19, 2007, and July 24, 2007; approved by IEEE COMMUNICATIONS LETTERS Associate Editor Dr. Alex Sprintson. Liangping Ma is with Argon ST, San Diego, CA 92121 ([email protected]), and his work was supported in part by NSF grant CNS-0721230. Chien-Chung Shen is with University of Delaware, Newark, DE 19716 ([email protected]), and his work was supported in part by NSF grants CNS-0347460 and CNS-0721361.

(1)

i = 1, · · · , c

(2)

which states that there are at most n interfaces tuned to a given channel. B. Upper Bound Lemma 1. Under assumption (A6), the following holds c X i=1

min(c,m)

ni wi ≤ n

X

wi ,

i=1

where w1 ≥ w2 ≥ ... ≥ wc according to assumption (A2).

(3)

IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XXXX 2007

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Proof: Note that the total number of interfaces is mn. Case a: mn ≥ cn. It follows directly from (2) that c X

ni wi ≤ n

i=1

c X

wi ,

(4)

i=1

where the equality is attained if each node tunes c of its interfaces to the c channels respectively. Case b: mn ≤ cn. For all ni , i = 1, · · · , c such that 0 ≤ ni ≤ n, we have n

m X

wi

=

i=1

= ≥ = ≥ ≥ =

m X i=1 m X i=1 m X i=1 m X i=1 m X i=1 m X i=1 c X

nwi

(5)

m X ni wi + (n − ni )wi

ni wi +

which follows from the fact that the line segment has the shortest length among all curves passing two given points. Consider interfaces tuned to a given channel i. Since at most half of the interfaces can simultaneously transmit on channel i, we have λnT X h(b) X

wi τ ni 2 b=1 h=1 (14) where 1(·) is the indicator function, taking value 1 if the statement inside the parentheses is true, and value 0 otherwise. Summing over all slots s and channels i and noting that there are no more than T /τ slots in time interval T , we have 1(bit b’s hth hop is over channel i in slot s) ≤

(6)

i=1 m X

H :=

(n − ni )wm+1

λnT X

h(b) ≤

b=1

(7)

i=1

ni wi + (mn −

m X

≤ ni )wm+1

(8)

i=1

ni wi + ni wi +

c X i=m+1 c X

ni wm+1 ni wi

(9) (10)

i=m+1

ni wi

λnT X h(b) X

where (7) follows from Pc wi ≥ wm+1 , ∀i = 1, · · · , m, (9) and (10) from from the constraint i=1 ni ≤ mn,P Pcwm+1 ≥ m wi , ∀i = m + 1, · · · , c. That is, n i=1 wi ≥ i=1 ni wi . Note that there is a configuration ni = n, ∀i = 1, · · · , m and ni = 0, ∀i = m + 1, · · · , c that satisfies the equality. Summarizing both cases completes the proof. ¤ Remark: The upper bound in Lemma 1 is tight because the equality can be attained in both cases. This helps to tighten the upper bound of the unicast transport capacity. We now present an upper bound for the unicast transport capacity of the multi-channel, multi-radio, multi-hop wireless network. The proof follows the line of reasoning in [1] but with significant extension to account for the more general multichannel multi-radio scenario.

b=1 h=1

Theorem II.1. In the Protocol Model, the bit-meters/sec λnL (and hence the transport capacity) is bounded as follows: v r u min(c,mn/2) min(m,c) X X 8 1u tnA λnL ≤ wi wj . (12) π∆ i=1 j=1 Proof: Consider an arbitrary bit b, where 1 ≤ b ≤ λnT . Let the number of hops that bit b traverses during its trip from the source to the destination be h(b), and the distance of hop h be rbh . Noting assumption (A1), we have λnT X h(b) X b=1 h=1

nT 2

(15)

min(m,c)

X

wi

(16)

i=1

where (16) follows from Lemma 1. Recall that simultaneous interference-free transmissions “consume areas” (see (1) or [1]). Considering the data transmission in a tick t(i) (defined as the duration of transmitting one bit by one hop, which is shorter than a typical time slot and varies from channel to channel) and the edge effect, we have

(11)

i=1

c T X ni wi 2 i=1

1(the hth hop of bit b is over channel i

π∆2 h 2 (r ) ≤ A. 16 b Summing over all ticks in time slot s gives in tick t(i))

λnT X h(b) X

(17)

1(the hth hop of bit b is over channel i

b=1 h=1

π∆2 h 2 (r ) ≤ Awi τ. (18) 16 b Note that the maximum number of pairs of interfaces available to simultaneous communications is no more than mn/2, which limits the maximum number of channels that can be used for simultaneous communications. Also note that the total number of channels is c. Therefore, there are no more than min(c, mn/2) channels that can be selected for simultaneous communications. Summing over all such channels gives in time slot s)

X

λnT X h(b) X

1(the hth hop of bit b is over channel i

selected i b=1 h=1

in time slot s)

π∆2 h 2 (r ) ≤ Aτ 16 b

X

wi (19)

selected i

min(c,mn/2)

≤ Aτ

X

wi (20)

i=1

rbh ≥ λnT L,

(13)

where (20) follows from the above argument and assumption (A2).

IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XXXX 2007

3

s4

Summing over all time slots yields h(b) λnT XX b=1 h=1

π∆2 h 2 (r ) ≤ AT 16 b

wi .

(21)

i=1

d1 d2 d3

Since function f (x) = x2 is convex, we have h(b) λnT XX

(

b=1 h=1

or equivalently, λnT X h(b) X b=1 h=1

λnT h(b) 1 h 2 XX 1 h 2 rb ) ≤ (r ) , H H b

(22)

b=1 h=1

v uλnT h(b) XX 1 1 h u rb ≤ t (rh )2 , H H b

(23)

b=1 h=1

which together with (13), (21) and (16) yields (12). ¤ Remarks: Note that a special care is taken in (20) to derive a tight upper bound. We now discuss the relationship between Theorem II.1 and prior work. Let m = c, then the network model considered in Theorem II.1 reduces to the Arbitrary Network in [1]. Moreover, (12) reduces to r c 8 1 X √ λnL ≤ wi nA, (24) π ∆ i=1 Pc which agrees to [1] if W ≡ i=1 wi . Let wi = w, ∀i = 1, · · · , c. Then the network model in Theorem II.1 reduces to the Arbitrary Network in [2]. Moreover, (12) reduces to  q √ 8 1  wc nA if c < m  π ∆  q √ 8 1 λnL ≤ (25) if m ≤ c ≤ mn/2 π ∆ w nmcA  √   √2 1 if c > mn/2 wmn A π∆ which, in contrast to the result in [2], specifies the exact values for all the turning points of c in relation to m, n and the exact values for the upper bound, and also includes the case c < m. The result in [2] characterizes the network capacity in the asymptotic form, as a function of c, which is also in the c asymptotic form of m and n. For example, when m is Ω(n), the network capacity is upper bounded by O(W nm c ). C. Lower Bound Theorem II.2. (Quoted from [1].) There is a placement of nodes and an assignment of traffic patterns such that the √ W √n √ A network can achieve 1+2∆ bit-meters per second n+ 8π under the Protocol Model, for n being a multiple of 4. Pc Again, in the above result, W corresponds to i=1 wi in our derivation. See [1] for a construction leading to the above lower bound. Based on that construction, a lower bound for the multi-channel multi-radio case is obtained as follows. First, each node tunes one of its interfaces to channel 1. By applying the same node placement and traffic pattern to the interfaces on channel 1 as those used to derive Theorem √ w1 A √n √ is achieved. Next, each II.2, a bit-meters/sec of 1+2∆ n+ 8π node tunes one of its interfaces to channel 2. By keeping the same node placement and applying the same traffic pattern to the interfaces tuned to channel 2, a bit-meters/sec of

s s2 s1

min(c,mn/2)

X

s3

d4 (a)

d

(b)

Fig. 1. (a) An example network; (b) less than 1/4 of the area of the dotted circle is within the solid disk, which denotes the range of the network, and s and d are on either end of a diameter of the solid circle. √ w2 A √n √ 1+2∆ n+ 8π

is achieved. Continuing this procedure in the order of decreasing data rates until it terminates at channel min(m, c) leads to the following lower bound. Theorem II.3. There is a placement of nodes and an assignment of traffic patterns such that the network can achieve Pmin(c,m) √ wi A n i=1 √ √ bit-meters per second under the Protocol 1+2∆ n+ 8π Model, for n being a multiple of 4. III. I MPROVED U SE OF THE P ROTOCOL M ODEL We first quote a result from [1], which is for the case m = c. Theorem III.1. In the Protocol Model, the transport capacity for Arbitrary Networks is bounded as follows r 8 1 √ λnL ≤ (26) W nA. π∆ We now explain why an improved use of the Protocol Model is needed. As indicated in [1], the result (26) is not asymptotic, meaning that it holds for any n. However, (26) may not be true if ∆ is not taken good care of. For example, see Fig. 1(a), where node si (di ) chooses node di (si ) as its destination, i = 1, ..., 4.√Regardless of the value of ∆, this √ A bit-meters/sec (a lower bound of network can achieve 2W n π the network capacity) if each node uses an exclusive time slot to transmit and equally shares time with others. On the other √ hand, for ∆ = 2 2n3/2 , which may occur for small n or receivers extremely susceptible to interference, (26) gives√an √ √ A which is less than the lower bound 2W√ A upper bound W n π n π derived above, causing a contradiction. The reason of this problem is that the “consumed” disjoint area is overestimated, which is shown in Fig. 1(b). When ∆ is large enough such that the disjoint disk (the dashed circle) centered at receiver d touches the sender s, exactly 1/4 of the disk covers the network (solid circle of area A = πr2 ). If ∆ increases further (the dotted circle), the fraction will be less than 1/4. But 1/4 is considered the minimum fraction in [1], which holds only if ∆ 2 2r ≤ 2r, i.e., ∆ ≤ 2. If ∆ > 2, we may replace ∆ with 2 in (17) of this letter, or in (8) of [1]. In summary, we substitute min(∆, 2) for ∆ in (26) and (12). R EFERENCES [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Information Theory, vol. IT-46, pp. 388–404, March 2000. [2] P. Kyasanur and N. H. Vaidya, “Capacity of multi-channel wireless networks: impact of number of channels and interfaces,” in Proceedings of the 11th annual international conference on mobile computing and networking (MobiCom), pp. 43–57, Aug. 2005.

Improved Unicast Capacity Bounds for General Multi ...

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