2.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010 ni ni forwarding based on their relay priorities – If the ﬁrst node L pii1 1 piiL in the set has received the packet successfully, it forwards the ni p ii 2 packet towards the destination while all other nodes suppress ni 2 themselves from duplicate forwarding. Otherwise, the second node in the set is arranged to forward the packet if it has Fig. 1. A transmitter ni is transmitting a packet, and its one-hop neighbors received the packet correctly. Otherwise the third node, the niq (1 ≤ q ≤ L) can correctly receive this packet with probability piiq . fourth node, etc. A forwarding candidate will forward the the unreliability of wireless links, there is a packet reception packet only when all the other candidates with higher priorities ratio (PRR) associated with each transmission link. In this failed to do so. Existing MAC protocols have been proposed to paper, we assume that there is no power control scheme, and ensure the relay priority among the candidates. For example, the link quality on each channel is independent and can be in [1], a batch map is used to indicate the packets known to obtained by the existing measurement schemes [11]. In order have been received by higher-priority candidates, thus prohibit to analyze the throughput bound, we assume that packet trans- the lower-priority candidates from relaying duplicate copies of mission/forwarding at an individual node and radio/channel the packets. Only when none of the forwarding candidates has allocation can be perfectly scheduled by an omniscient and successfully received the packet, the sender will retransmit the omnipotent central entity. Thus, we do not concern ourselves packet if retransmission is enabled. The forwarding reiterates with issues such as MAC contention or coordination overhead until the packet is delivered to the destination. that may be unavoidable in a distributed network. This is a very commonly used assumption for theoretical studies [7], III. P ROBLEM F ORMULATION [12], [13]. In this section we present our methodology to compute the throughput bound between two end nodes in a multi- A. Opportunistic Routing Primer radio multi-channel multi-hop wireless network. We ﬁrst study Different from TR, OR basically runs in such a way that which opportunistic modules can coexist at the same time for each local packet forwarding, a set of next-hop forwarding under the constraints of wireless interference and radio in- candidates are selected at the network layer and one of them terface limits. We then formulate the capacity of OR as an LP is chosen as the actual relay at the MAC layer according to problem which jointly solves the radio-channel assignment and their instantaneous availability and reachability at the time transmission scheduling. of transmission. We introduce the concept of opportunistic module for OR. As illustrated in Fig. 1, it consists of a A. Concurrent Transmission Sets transmitter (ni ), all of its one-hop neighbors {ni1 , ..., niL } In this subsection, we will discuss which opportunistic mod- (denoted as Ci ), and the corresponding wireless links from ules in the network can be activated at the same time. The set the transmitter to the candidates with each link (denoted as of opportunistic modules which can be activated at the same liiq ) associated with a PRR piiq (1 ≤ q ≤ L). A subset time is named as concurrent transmission set (CTS). The of the one-hop neighbors will be selected as forwarding motivation of building concurrent transmission set is similar to candidates. Only the forwarding candidates will help forward those of building independent set in [12] and concurrent trans- the packet. To avoid packet duplication, only one of the mission patterns in [10]. That is, taking the beneﬁt of time- forwarding candidates becomes the actual forwarder of each sharing scheduling of different concurrent transmission sets, packet. There is a forwarding priority among these forwarding we could achieve a collection of capacity graphs, associated candidates to decide who should forward the packet if multiple with capacity constraint on each link. OR can be performed forwarding candidates correctly receive the same packet. We on the underlying capacity graph to achieve the maximum use an ordered set Fi , the forwarding candidate sequence, throughput. However, the methodology of constructing CTS which is one permutation of the forwarding candidates, to for OR is quite different from those in [10], [12] for TR. Be- represent the forwarding priority. The order of the elements in cause for OR, any of the forwarding candidates can become the Fi corresponds to their priority in relaying a received packet. actual forwarder for each transmission, and the instantaneous For example, Fi = ni1 , ni2 , ..., nir indicates ni1 has the throughput can take place on any link from the transmitter to highest forwarding priority, then ni2 , ..., then nir . We call the any forwarding candidate. So the CTS is constructed based candidate selection and prioritization a forwarding strategy. on opportunistic modules (involving multiple links sharing Denote a forwarding strategy as H = (Φ, P), where Φ is an the same transmitter) instead of individual links. Furthermore, indicator function on the one-hop neighbors deﬁned in Eq. (1), besides the co-channel interference, radio interface limits in and P is a permutation function of the one-hop neighbors. the multi-radio system also impose constraint on concurrent So Φ represents selection of forwarding candidates and P transmissions in the network. represents prioritization of forwarding candidates. We denote We introduce the concept of transceiver conﬁguration, P(ij ) < P(ik ) if nij has higher forwarding priority than k vi , which indicates node ni operating on the channel k nik . Thus, a speciﬁed H can uniquely decide a forwarding (1 ≤ k ≤ K). Each transceiver conﬁguration can be in either candidate sequence Fi . transmission or reception state, and we call it transmitter or 1, niq is a forwarding candidate; receiver, respectively. We say there is a wireless link lij (i = j) k Φ(iq ) := φiq = (1) k k k 0, otherwise. when vi is a transmitter and vj is a receiver and vj is in the k k k The opportunistic routing works by letting the source ns transmission range of vi . Link lij is usable when vj is not in forward the packet to the receivers in its forwarding candi- the interference range of any other transmitters; otherwise, it is date sequence Fs . One of the candidate nodes continues the unusable. When a link is usable, its transmitter and receiver
3.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010 are also usable. Let V = {vi |i = 1...N, k = 1...K}, and k Riiq = Ri · φiq · piiq (1 − φik · piik ) (8) E = {lij |i, j = 1...N, i = j, k = 1...K}. k P(ik )<P(iq ),nik ∈Ci A CTS Tα can be represented by an indicator vector on all where Ri is the data transmission rate at transmitter ni . wireless links, written as Tα = {ψij |lij ∈ E}. kα k The effective forwarding rate indicates that according to the relay priority, only when higher-priority forwarding candidates 1, lij is usable in CTS Tα ; k ψij = kα (2) do not receive the packet correctly, a lower-priority candidate 0, otherwise. may have a chance to relay the packet if it does. Denote the following indicator variable to represent the Then the effective forwarding rate from a transmitter ni to transceiver conﬁguration status in CTS Tα : its forwarding candidate sequence Fi is the summation of the effective forwarding rate to each forwarding candidate in the 1, vi is usable in CTS Tα ; k ηi = kα (3) sequence: 0, otherwise. An opportunistic module in a CTS Tα can be represented RiFi = Riiq = Ri · (1 − (1 − φiq piiq )) (9) as (vi , {vj |lij ∈ E, ψij == 1}). Note that according to the k k k kα niq ∈Fi niq ∈Ci k unique property of OR, when a transmitter vi is usable, its Note that, the effective forwarding rate from a transmitter multiple receivers can be usable at the same time. While a to a set of its forwarding candidates only depends on the usable receiver can only correspond to one transmitter. This transmission rate and the PRRs on the corresponding links, can be formally represented by: but does not depend on the priority among the forwarding candidates. ηi = min(1, kα ψij + kα ψji ), ∀ i = 1...N, k = 1...K kα lij ∈E k lji ∈E k C. Capacity Region of An Opportunistic Module (4) In this subsection, we study the capacity region of the Although any two active links operating on different chan- opportunistic module shown in Fig. 1. This capacity region nels do not interfere with each other, due to radio interface will serve as a bound of a rate vector corresponding to the constraint, the number of channels being used on one node links in the opportunistic module. cannot exceed the number of radios installed on this node. To By applying the proved result in [15], we have the capacity satisfy this constraint, we have region of the outgoing links from a transmitter ni to its one- K hop neighbors indicated in Inequality (10). ηi ≤ ti , ∀ i = 1...N kα (5) L L k=1 μq ·φiq ≤ Ri (1− (1−piiq φiq )), ∀ [φi1 , ..., φiL ] ∈ {0, 1}L If two wireless links are concurrently usable on the same q=1 q=1 channel, they should either share the same transmitter or do (10) not interfere with each other. This can be represented by where μq (1 ≤ q ≤ L) the rate from ni to niq . ψij + ψpq ≤ 1 + I(lij , lpq ), ∀ k = 1...K kα kα k k (6) The physical meaning of Inequality (10) is that any subset where summation of the rate vector − must be bounded by the effec- → μ 1, i == p, or lij and lpq do not interfere; k k tive forwarding rate from the transmitter to the corresponding I(lij , lpq ) = k k 0, otherwise. forwarding candidate set. Now are ready to formulate the end- (7) to-end throughput bound of OR in multi-radio multi-channel According to Eq. (4), (5), and (6), we can construct all systems by making use of the CTS and the capacity region of the CTS’s. One CTS represents one radio-channel assignment. the opportunistic module. Note that the number of all the CTS’s is exponential in the number of nodes, radios and channels. However, it may not be necessary to ﬁnd all of them to maximize an end-to-end D. Capacity of OR in Multi-radio Multi-channel Multi-hop throughput. Some heuristic algorithm similar to that in [14], Networks or column generation technique [10] can be applied to ﬁnd a Assume we have found all the CTS’s {T1 , T2 ...TM } in subset of all the CTS’s to approach the throughput bound. We the network. At any time, we activate all the transmitters will not go into detail of the technologies of ﬁnding CTS’s. in one CTS. Let λα denote the time fraction scheduled Next we discuss which link rate (or rate vector) is supportable to CTS Tα (α = 1...M ). Then the maximum throughput by OR in an opportunistic module. problem can be converted to an optimal scheduling problem that schedules the activation of the CTS’s to maximize the B. Effective Forwarding Rate end-to-end throughout. Therefore, considering communication A fundamental difference of OR from TR is that effective between a single source, ns , and a single destination, nd , throughput can take place from a transmitter to any of its with opportunistic routing, we formulate the throughput ca- forwarding candidates at any instant. To capture the unique pacity problem between the source and the destination as a property of OR, we apply the deﬁnition of effective forward- linear programming problem corresponding to a maximum- ing rate in [7] to represent the throughput on each link from ﬂow problem under additional constraints in Fig. 2. a transmitter to each of its forwarding candidate according to In Fig. 2, μkα and μkα denote the ﬂow rate on link lij ij iiq k a forwarding strategy. For a given transmitter ni and its one- and liiq in the CTS Tα , respectively. Recall that E is a k hop neighbor set Ci , under an OR strategy H = (Φ, P), the set of all the wireless links, and V is the set of all the effective forwarding rate on link liiq is deﬁned in Eq. (8): transceiver conﬁgurations. The maximization states that we
4.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010 K M M ax kα μsi (11) n3 n3 0.5 0.8 0.8 0.5 k=1 lk ∈E α=1 si n1 n4 n1 n4 s.t. 0.5 0.8 0.8 0.5 n2 n2 K M K M kα kα μij = μji , (a) PRRs from the source (b) PRRs from the source k=1 lk ∈E α=1 ij k=1 lk ∈E α=1 ji (n1 ) to relays (n2 and n3 ) (n1 ) to relays (n2 and n3 ) ∀ i = 1...N, i = s, i = d (12) are worse than that from are better than that from K M the relays to the destina- the relays to the destina- kα tion (n4 ) tion (n4 ) μis =0 (13) k=1 lk ∈E α=1 Fig. 3. Four-node networks under different channel conditions (link PRRs). is K M kα μdi = 0 (14) can be operated on two orthogonal channels. The PRR is k=1 lk ∈E α=1 indicated on each link. For simplicity, we assume the PRR is di kα μij ≥ 0, ∀ k = 1...K, lij ∈ E k (15) identical under different channels in each network. We assume M each node is in the interference range of each other. So there is λα ≤ 1 (16) only one transmitter can be active on the same channel at any α=1 instant in the network. By applying the methodology in Section λα ≥ 0, ∀ α = 1...M (17) kα k III, we solve the joint radio-channel assignment, routing, μiiq · φiq ≤ λα Ri (1 − (1 − piiq · φiq )), C C scheduling problem for maximizing the throughput from n1 k kα C = {niq |liiq ∈ E, ψiiq == 1}, to n4 . We summarize the results for Fig. 3(a) and Fig. 3(b) in k |C| Table I and II, respectively. The optimal throughput from n1 to ∀ vi ∈ V, α = 1...M, ∀ Φ(C) ∈ {0, 1} (18) n4 for each ﬁgure is 0.58 and 0.5, respectively. An interesting Fig. 2. LP formulations to optimize the end-to-end throughput of OR observation in Table II is that the opportunistic routing is not wish to maximize the sum of the ﬂow rates out of the source, used when n1 is transmitting packets. Since in Fig. 3(b), the which is the accumulated ﬂow rates on all outgoing links channel conditions from the source to the relays are better and all channels from the source in all CTS’s. The constraint than that from the relays to the destination, the maximum (12) represents ﬂow-conservation, i.e., at each node, except throughput is constrained by the bottleneck links from the the source and the destination, the accumulated incoming relays to the destination. So we should allow more concurrent ﬂow rate is equal to the accumulated outgoing ﬂow rate. The transmissions to saturate the bottleneck links instead of making constraint (13) states that the incoming accumulated ﬂow rate use of OR to push more ﬂows out of the sender. Differently, to the source node is 0. The constraint (14) indicates that the when the bottleneck links are between the sender and relays outgoing accumulated ﬂow rate from the destination node is (Fig. 3(a)), OR is used to push more ﬂows out the sender. This 0. The constraint (15) restricts the amount of ﬂow rate on observation is expected to provide a guideline on designing each link to be non-negative. The constraint (16) represents distributed radio-channel assignment for OR in multi-radio that at any time, at most one CTS will be scheduled to be multi-channel multi-hop wireless networks. active. The constraint (17) indicates that the scheduled time fraction should be non-negative. In the constraint (18), Φ(C) is B. Simulation of Random Networks a vector of φj ’s with length |C|. The constraint (18) states that the ﬂow rates out of a transmitter in an opportunistic module In this subsection, we investigate the throughput bound of within a CTS must be in the capacity region discussed in OR and TR in multi-radio multi-channel multi-hop networks Section III-C. That is, in any CTS, any sub-summation of the and compare the results with that in single-radio single- ﬂow rates from a transmitter to its usable receivers is bounded channel systems. In the simulation, we randomly deploy 12 by the effective forwarding rate from the transmitter to the nodes in a rectangle area of 200 units × 300 units. We select corresponding receivers. node n1 at the left corner of the network as the destination, The solution of the objective function (11) is the upper then calculate the throughput bound from other nodes to the bound of the throughput between two nodes for OR. The destination using the LP formulations in Fig. 2. Therefore, byproduct of the LP in Fig. 2 is the radio-channel assign- there are 11 different source-destination pairs considered in ment (CTS’s {Tα |α = 1...M }) and transmission scheduling the evaluation. In all the simulations, we assume the packet ({λα |α = 1...M }). We also get the ﬂow rate μkα on each link ij reception ratio is inversely proportional to the distance with k lij in each CTS Tα from the LP. gasussian random variation, which simulates the log-normal fading and two-ray path loss model. The interference range RI = 2RT . The transmission range RT is set as 100 units. The IV. P ERFORMANCE E VALUATION performance metric is the normalized end-to-end throughput In this section, we show the results of joint radio-channel bound (by assuming the transmission rate is unit one). assignment, routing, and scheduling for optimizing an end- Fig. 4 shows the simulation result. In the legend, “TR” to-end throughput solved by our methodology for two simple represents traditional routing, “OR” represents opportunist scenarios, and simulation results for more general networks. routing, “xRyC-z” represents x radios and y channels, with All the simulations are implemented in Matlab. z maximal number of forwarding candidates. We can see that with the number of radios and channels increasing, the throughput of TR and OR are both increased. Generally A. Two Scenarios with Different Link Qualities OR achieves higher throughput than TR, and the multi- Consider two four-node network scenarios in Fig. 3 with radio/channel capability has greater impact on the throughput different link qualities. Suppose each node has one radio which of TR than OR. When the source is farther away from
5.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010 CTS 1 1 1 {(v1 , v2 , v3 )} 1 1 1 {(v1 , v3 , v2 )} 1 1 2 2 {(v1 , v2 ), (v3 , v4 )} 1 1 2 2 {(v1 , v3 ), (v2 , v4 )} Time fractions 0.14 0.14 0.36 0.36 TABLE I C HANNEL ASSIGNMENT, ROUTING , AND SCHEDULING OF OPPORTUNISTIC FORWARDING STRATEGIES FOR F IG . 3( A ). CTS 1 1 2 2 {(v1 , v3 ), (v2 , v4 )} 1 1 2 2 {(v1 , v2 ), (v3 , v4 )} 1 1 {(v2 , v4 )} 1 1 {(v3 , v4 )} Time fractions 0.354 0.354 0.146 0.146 TABLE II C HANNEL ASSIGNMENT, ROUTING , AND SCHEDULING OF OPPORTUNISTIC FORWARDING STRATEGIES FOR F IG . 3( B ). that 1) OR can achieve better performance than TR under Normalized end−to−end throughput bound 1.8 TR−1R1C TR−1R2C different radio/channel conﬁgurations. However, in particular 1.6 TR−2R2C scenarios (e.g. bottleneck links exist between the sender to OR−1R1C−2 OR−1R2C−2 relays), TR can be more preferable than OR; 2) OR can OR−2R2C−2 1.4 OR−1R1C−3 achieve comparable or even better performance than TR by OR−1R2C−3 OR−2R2C−3 using less radio resource; 3) for OR, the throughput gained 1.2 OR−1R1C−4 OR−1R2C−4 from increasing the number of potential forwarding candidates 1 OR−2R2C−4 becomes marginal. Just involving a few “good” forwarding candidates is enough to approach optimal throughput. As for 0.8 the future work, we are interested in designing practical joint radio-channel assignment and opportunistic routing protocols 0.6 in multi-radio multi-channel systems based on our theoretical study and observations in this paper. 0.4 0.2 R EFERENCES [1] S. Biswas and R. Morris, “Exor: Opportunistic multi-hop routing for 0 wireless networks,” in SIGCOMM’05, Philadelphia, Pennsylvania, Aug. 2 3 4 5 6 7 8 9 10 11 12 2005. Node ID (in order of distance to the destination) [2] H. Dubois-Ferriere, M. Grossglauser, and M. Vetterli, “Least-cost op- portunistic routing,” School of Computer and Communication Sciences, Fig. 4. Normalized end-to-end throughput bound under different number of EPFL, Technical Report LCAV-REPORT-2007-001, 2007. radios, channels and potential forwarding candidates in rectangle topology. [3] K. Zeng, W. Lou, J. Yang, and D. R. Brown, “On geographic collabo- rative forwarding in wireless ad hoc and sensor networks,” in WASA’07, the destination, the OR presents more advantage than TR. Chicago, IL, August 2007. The opportunistic forwarding by using multiple forwarding [4] ——, “On throughput efﬁciency of geographic opportunistic rout- candidates do help increase the throughput. An interesting ing in multihop wireless networks,” in QShine’07, Vancouver, British Columbia, Canada, August 2007. result is that, for nodes 7 to 12, the throughput of 1R2C [5] K. Zeng, W. Lou, and Y. Zhang, “Multi-rate geographic opportunistic case for OR is comparable with or even greater than that of routing in wireless ad hoc networks,” in Milcom’07, Orlando, FL, Oct. 2R2C case for TR. This result indicates that OR can achieve 2007. [6] S. Chachulski, M. Jennings, S. Katti, and D. Katabi, “Trading structure comparable or even better performance as TR by using less for randomness in wireless opportunistic routing,” in ACM SIGCOMM, radio resource. Kyoto, Japan, 2007. Another interesting observation is that the throughput gained [7] K. Zeng, W. Lou, and H. Zhai, “On end-to-end throughput of oppor- tunistic routing in multirate and multihop wireless networks,” in IEEE decreases as the number of forwarding candidates increases. Infocom’08, Phoenix, AZ, April 15-17 2008. This result is consistent with that found in [3], [4]. So it [8] M. Kodialam and T. Nandagopal, “Characterizing the capacity region is not necessary to involve all the usable receivers of the in multi-radio multi-channel wireless mesh networks,” in MobiCom ’05. New York, NY: ACM, 2005, pp. 73–87. transmitter into the opportunistic forwarding, and selecting [9] M. Alicherry, R. Bhatia, and L. E. Li, “Joint channel assignment a few “good” forwarding candidates is enough to approach and routing for throughput optimization in multi-radio wireless mesh optimal throughput. This theoretical observation may help us networks,” in MobiCom ’05. New York, NY: ACM, 2005, pp. 58–72. [10] J. Zhang, H. Wu, Q. Zhang, and B. Li, “Joint routing and scheduling design practical protocols. in multi-radio multi-channel multi-hop wireless networks,” in IEEE Broadnets, 2005. V. C ONCLUSIONS AND F UTURE W ORK [11] K.-H. Kim and K. G. Shin, “On accurate measurement of link quality in multi-hop wireless mesh networks,” in MobiCom ’06: Proceedings In this paper, we proposed a uniﬁed framework to compute of the 12th annual international conference on Mobile computing and the throughput bound of opportunistic routing between two networking. New York, NY, USA: ACM Press, 2006, pp. 38–49. [12] K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, “Impact of end nodes in multi-radio multi-channel multi-hop wireless interference on multi-hop wireless network performance,” in MobiCom networks. Our model accurately captures the unique property ’03. New York, NY, USA: ACM Press, 2003, pp. 66–80. of OR that throughput can take place from a transmitter [13] H. Zhai and Y. Fang, “Impact of routing metrics on path capacity in multirate and multihop wireless ad hoc networks,” in IEEE, ICNP, 2006. to any one of its forwarding candidates at any instant. Our [14] J. Tang, G. Xue, and W. Zhang, “Cross-layer design for end-to-end methodology can be used to calculate the end-to-end through- throughput and fairness enhancement in multi-channel wireless mesh put bound of OR and TR in multi-radio multi-channel multi- networks,” IEEE Transactions on Wireless Communications, vol. 6, no. 10, 2007. hop wireless networks, as well as to study the OR behaviors [15] L. Tassiulas and A. Ephremides, “Dynamic server allocation to parallel (such as candidate selection and prioritization). Leveraging our queues with randomly varying connectivity,” IEEE Transactions on analytical model, we gained the following insights into OR Information Theory, vol. 39, no. 2, pp. 466–478, 1993.
Be the first to comment