Maximum Throughput and Fair Bandwidth Allocation in Multi ...

specifies the amount of flow routed through each wirelesslink) to carry out the desired transmissions. We denote theaggregated flow on each link e by f e . What we really want tohave is a bandwidth allocation for every non-gateway nodeand a corresponding flow routing solution (which specifiesthe route and the amount of flow allocated to each link onthe route for traffic between each non-gateway node and thewired network), not just the aggregated flow allocation vector.However, once we have the aggregated flow allocation vector,we can easily compute a corresponding flow routing solution,which will be explained in more detail in Section V. Wewish to emphasize that the bandwidth allocation is a noderelated concept, while the flow allocation is a link relatedconcept. In order to differentiate them, we will call the laterone aggregated link flow allocation and the correspondingvector aggregated link flow allocation vector in the following.Definition 1 (Feasible Bandwidth Allocation Vector): Anaggregated link flow allocation vector f = [f 1 ,f 2 ,...,f m ]corresponding to a given bandwidth allocation vectorb = [b 1 ,b 2 ,...,b N ] assigns an aggregated flow f e ≥ 0for each link e ∈ E such that at each non-gateway nodes i , the total flow out of 1 node s i minus the total flowinto node s i is equal to b i . f is said to be a feasibleaggregated link flow allocation vector (corresponding tobandwidth allocation vector b) if for every link e ∈ E, wehave A(e) = (C e − ∑ e ′ ∈IE ef e ′) ≥ 0, where C e is thecapacity of link e. We call A(e) the residual capacity onlink e. A bandwidth allocation vector is said to be a feasiblebandwidth allocation vector if there exists a correspondingfeasible aggregated link flow allocation.We note that the above method for residual capacity computationis a worst-case computation. Suppose that links e 1 ande 2 are the links interfering with link e, but do not interferewith each other. Then traffic on e 1 and e 2 may be transmittedsimultaneously according to 802.11 DCF. In this case, morethan (C e − f e1 − f e2 ) traffic can be transmitted along linke. Therefore, A(e) is essentially a lower bound for the actualresidual capacity. However, this bound is tight because it isachievable in the worst case. We overestimate the interferenceimpact and corresponding bandwidth usage a little bit becausewe try to provide bandwidth guarantee for users in the WMN,which is very important for those multimedia applications.We try to propose a network layer (routing) solution anddepend upon the 802.11 DCF for transmission scheduling.If we allocate bandwidth and routing flows based on sucha computation, it is most likely that the bandwidth allocatedto each node in S is achievable even though the wirelesschannel is not reliable and 802.11 DCF is a random accessprotocol. Similar estimation methods have also been used byseveral previous QoS routing papers such as [29], [31]. Sinceneighboring nodes need to periodically exchange maintenancemessages, such as HELLO messages, a small amount ofbandwidth should be reserved for those routine traffic toprevent them from being destroyed by strong interference.As a result, the link capacity C e in Definition 1 should beslightly smaller than the physical capacity of link e. Nowwe1 note our discussion in the second paragraph of this section.are ready to define the optimization problems to be studied.Since our ultimate goal is to maximize network throughput,we first formulate a problem which seeks bandwidth allocationand its corresponding aggregated link flow allocation withmaximum throughput. Let the network topology G(V,E) andnon-gateway node set S be given.Definition 2 (MBA Problem): The Maximum throughputBandwidth Allocation (MBA) problem seeks a feasiblebandwidth allocation vector for all nodes in S along with acorresponding feasible aggregated link flow allocation vectorsuch that the throughput of this bandwidth allocation vectoris maximum among all feasible bandwidth allocation vectors.As discussed before, simply maximizing the throughputmay starve some wireless mesh nodes. Therefore, in orderto achieve a good bandwidth allocation with regards to bothfairness and throughput, we formulate two fair bandwidthallocation problems.Definition 3 (MMBA Problem): A feasible bandwidth allocationvector b is said to be a feasible max-min guaranteedbandwidth allocation vector if for any other feasiblebandwidth allocation vector ˆb, we have min{b i |i ∈{1, 2,...,N} ≥ min{ˆb i |i ∈ {1, 2,...,N}. The Max-minguaranteed Maximum throughput Bandwidth Allocation(MMBA) problem seeks a feasible max-min guaranteed bandwidthallocation vector for all nodes in S along with acorresponding feasible aggregated link flow allocation vectorsuch that the throughput of this bandwidth allocation vector ismaximum among all feasible max-min guaranteed bandwidthallocation vectors.For a bandwidth allocation vector b =[b 1 ,b 2 ,...,b N ],wewill use r =[r 1 ,r 2 ,...,r N ] to denote the sorted version ofb such that r 1 ≤ r 2 ≤ ... ≤ r N . Similarly, for a bandwidthallocation vectors ˆb, we will use ˆr to denote its sorted version.Definition 4 (LMMBA Problem): A feasible bandwidth allocationvector b is said to be a feasible lexicographicalmax-min bandwidth allocation vector if for any otherfeasible bandwidth allocation vector ˆb, either r i = ˆr i fori =1, 2,...,N or there exists an integer j ∈{1, 2,...,N}such that r i =ˆr i for i ˆr j .TheLexicographicalMax-Min Bandwidth Allocation (LMMBA) problem seeksa feasible lexicographical max-min bandwidth allocation vectorfor all nodes in S along with a corresponding feasibleaggregated link flow allocation vector.The fairness model behind the MMBA problem is a simplemax-min model. The bandwidth allocation vector obtainedby solving the MMBA problem is guaranteed to have themaximum minimum bandwidth value, but is not necessarilya feasible LMM bandwidth allocation vector. The LMM is awell used fairness model ([12]) because it is believed to beable to provide a very good tradeoff between throughput andfairness.V. THE PROPOSED BANDWIDTH ALLOCATION SCHEMESIn this section, we present LP formulations for the MBA andMMBA problem, and an LP-based polynomial time optimal algorithmfor the LMMBA problem. First, we need to constructan auxiliary directed graph G ′ (V ′ ,E ′ ) according to a givenThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the Proceedings IEEE Infocom.

3network topology G(V,E). For each node v ∈ V , V ′ containsQ v nodes v λ1(v) ,v λ2(v) ,...,v λQv (v) , where λ 1 (v)

subject to∑f vw − ∑w≠v u≠v∑f uv =0;f vw − ∑w≠v u≠v∑f xy ≤ C uv ;(x,y)∈IE uvf uv = b v ; ∀v ∈ S ′∀v ∈ V ′ \{S ′ ⋃ {t}}∀(u, v) ∈ E ′ Of uv ≥ 0; ∀(u, v) ∈ E ′b v ≥ α; ∀v ∈ S ′By solving LP 2, we can obtain a maximum minimumbandwidth value α, i.e., we can ensure that for any feasiblebandwidth allocation vector ˆb, we have min{ˆb i |i =1, 2,...,N}≤α = min{b i |i =1, 2,...,N}.Compared with LP 1, the objective of LP 3(α) is to maximizethe network throughput while making sure that eachnon-gateway node v has a bandwidth allocation of at least α.Therefore, solving LP 2 to obtain α and then solving LP 3(α)can provide a max-min guaranteed maximum throughputbandwidth allocation.It is more challenging to compute a feasible LMM bandwidthallocation vector and its corresponding feasible aggregatedflow allocation vector. We will present a polynomial timealgorithm for this problem by solving a sequence of LPs.Before proceeding with the presentation of the algorithm,we prove an important property–the uniqueness of the feasiblelexicographical max-min bandwidth allocation vector.Lemma 1: Let b 1 and b 2 be two feasible bandwidth allocationvectors which are both lexicographical max-min. Thenb 1 j = b2 j for j =1, 2,...,N. In other words, the lexicographicalmax-min bandwidth allocation vector is unique.PROOF. Let S1,S 1 2,...,S 1 K 1 be a partition of the set{1, 2,...,N} such thata. For any k ∈{1,...,K} and i, j ∈ Sk 1,wehaveb1 i = b1 j .b. For any 1 ≤ k

subject to∑f vw − ∑ f uv = α k−1 + δ k ;w≠v u≠v∑LP 5(k, ¯S,β)f vw − ∑w≠v u≠v∑f vw − ∑ f uv =0;w≠v u≠v∑f xy ≤ C uv ;(x,y)∈IE uvmaximize∀v ∈ S ′ \ ⋃ k−1j=0 S′ j (13)f uv = α j ; ∀v ∈ S j ′ , 1 ≤ j 0 (or f vu > 0), then there is a back edge(v, u) (or (u, v)) inG A ;iff uv =0and f vu =0, then thereis no edge between vertex u and v in G A .Clearly, during the execution of the algorithm, for everynode v ∈ Sk ′ ,wemusthaveb v = α k . However, there maybe some node v ∉ ⋃ ki=1 S′ i such that b v = α k . Instead ofputting every node v with b v = α k into ¯S, we use the residualgraph G A to decrease the cardinality of ¯S, i.e., reduce therunning time. However, we still need to solve LP 5(k, ¯S,β)to determine the actual Sk ′ .InLP 5(k, ¯S,β), β is a small andtunable parameter. The purpose of having Constraint (17) isto force more nodes out of ¯S in each iteration. In this way,we can decrease the running time as well. Note that there isno Constraint (14) when k =1.Algorithm 1 LMMBAStep 1 k := 1; α 0 := 0; S 0 ′ := ∅;Step 2 Solve LP 4(k); α k := α k−1 + δ k ;Step 3 Construct the residual graph G A (V ′ ,E A ) accordingto the aggregated link flow allocation computed bysolving LP 4(k);¯S := ∅;forall v ∈ S ′ \ ⋃ k−1j=0 S′ jif there exists no path from v to t in G A¯S := ¯S ⋃ {v};endifendforallStep 4 Solve LP 5(k, ¯S,β); Y := ∅;forall (v ∈ ¯S such that β v > 0) doY := Y ⋃ {v};endforallStep 5 if (Y ≠ ∅)¯S := ¯S \ Y ; goto Step 4;elseSk ′ := ¯S;if ( ⋃ kj=0 S′ j = S′ )stop;elsek := k +1; goto Step 2;endifendifTheorem 1: Algorithm 1 correctly computes the feasibleunique lexicographical max-min bandwidth allocation vectorb in polynomial time.PROOF. When we solve LP 4(1), we obtain the max-minbandwidth allocation value α 1 , together with a bandwidthallocation vector b such that α 1 = min{b i |1 ≤ i ≤ N}.Note that i ∈ S 1 ′ implies b i = α 1 . However, b i = α 1does not imply i ∈ S 1. ′ So we construct the residual graphG A (V ′ ,E A ). If there is a path from v to t in G A , thenwe know that the bandwidth allocated to node v can beincreased (by a very small, but positive amount) to obtainanother feasible bandwidth allocation vector while keeping thebandwidth allocated to all other nodes unchanged. This meansthat v ∉ S 1. ′ However, when there is no path from v to t inG A , we cannot say for sure that v ∈ S 1.SowesetS ′ toinclude every such node v such that there is no path from vto t in G A . This means that S 1 ′ ⊆ S.In order to obtain S 1 ′ from S, wesolveLP 5(k, S,β), whichmaximizes the sum of the bandwidth values of the nodes inS, while keeping the bandwidths allocated to nodes not in Sat α 1 . Clearly, S is a proper superset of S 1 ′ if and only if thereexists a bandwidth allocation vector b so thatandb i ≥ α 1 , 1 ≤ i ≤ N, (19)∑b i > |S|×α 1 . (20)i∈SThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the Proceedings IEEE Infocom.

This means that when S ≠ S 1, ′ we can always identify atleast one node v ∈ S (β v > 0) such that v ∉ S 1 ′ and moveit out of S. Clearly,inatmostN −|S 1| ′ iterations, we willhave S = S 1. ′ This shows that we have computed S 1 ′ and thebandwidth value for S 1 ′ (α 1 ) correctly.For general k, 1

1210MBAMMBALMMBA400350300Bandwidth864Throughput2502001501005020MBA MMBA LMMBA01 5 10 15 20 25 30Sorted Node IndexFig. 8. Network throughput (n =25,N =20,C =12,CAP =53.9)Fig. 5. Bandwidth allocation (n =35,N =30,C =3,CAP =10.9)Throughput1101009080706050Bandwidth5550454035302520MBAMMBALMMBA403020100MBA MMBA LMMBA1510501 5 10 15 20 25 30Sorted Node IndexFig. 6. Network throughput (n =35,N =30,C =3,CAP =10.9)Fig. 9. Bandwidth allocation (n =35,N =30,C =12,CAP =53.9)55504540MBAMMBALMMBA500450400350Bandwidth3530252015105Throughput300250200150100500MBA MMBA LMMBA01 5 10 15 20Sorted Node IndexFig. 10. Network throughput (n =35,N =30,C =12,CAP =53.9)Fig. 7. Bandwidth allocation (n =25,N =20,C =12,CAP =53.9)That is to say, based on the MBA scheme, some of users willnot be able to access the network at all, which is definitelynot desirable. Compared with the MBA scheme, the MMBAscheme improves the fairness somehow, without sacrificingthe network throughput too much. It guarantees every nodebe allocated some reasonable amount of bandwidth. For thecases in which the network resources are relatively sparse(C = 3,CAP = 10.9), the throughput provided by theMMBA scheme is very close to the maximum throughput,which can be seen from Fig. 4 and 6. As expected, amongthe three bandwidth allocation schemes, the LMMBA schemeachieves fairest bandwidth allocation. If the bandwidth isallocated using the LMMBA scheme, we observe from theresults that the nodes are partitioned to several groups (usuallythe number of groups is very small) and within each group,all nodes are allocate exactly the same amount of bandwidth.The throughput provided by the LMMBA scheme is alwayssmaller than that provided by the other two schemes, especiallyThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the Proceedings IEEE Infocom.

for cases in which the network resources are relatively rich(C =12,CAP =53.9).VII. CONCLUSIONSIn this paper, we have studied bandwidth allocation in multichannelmultihop wireless mesh networks with the objective ofthroughput maximization and fairness enhancement. We haveformulated three bandwidth allocation problems, namely, theMaximum throughput Bandwidth Allocation (MBA), theMaxminguaranteed Maximum throughput Bandwidth Allocation(MMBA) and the Lexicographical Max-Min Bandwidth Allocation(LMMBA) problem. We have presented LP formulationsfor the MBA and MMBA problems and an LP-based polynomialtime algorithm to optimally solve the LMMBA problem.REFERENCES[1] I. F. Akyildiz, X. Wang, W. Wang, Wireless mesh networks: a survey,Elsevier Journal of Computer Networks, Vol. 47(4), 2005, pp. 445-487.[2] P. Bahl, R. Chandra, and J. Dunagan, SSCH: Slotted seeded channelhopping for capacity improvement in ieee 802.11 ad-hoc wirelessnetworks, Proceedings of ACM MobiCom’2004, pp. 216–230.[3] M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and NetworkFlows (3rd edition), John Wiley & Sons, 2005.[4] D. P. Bertsekas, R. Gallager, Data Networks (2nd Edition), Prentice Hall,1991.[5] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge UniversityPress, 2004.[6] M. Burkhart, P. von Rickenbach, R. Wattenhofer, A. Zollinger, Doestopology control reduce interference, Proceedings of ACM Mobi-Hoc’2004, pp. 9–19.[7] R. Chandra, L. Qiu, K. Jain, M. Mahdian, Optimizing the placement ofInternet TAPs in wireless neighborhood networks, Proceedings of IEEEICNP’2004, pp. 271–282.[8] R. Draves, J. Padhye, and B. Zill, Routing in multi-radio, multi-hopwireless mesh networks, Proceedings of ACM MobiCom’2004, pp. 114–128.[9] L.R. Ford and D.R. Fulkerson, Flows in Networks, Princeton UniversityPress, 1962.[10] V. Gambiroza, B. Sadeghi, E. W. Knightly, End-to-end performance andfairness in multihop wireless backhaul networks, Proceedings of ACMMobiCom’2004, pp. 289–301.[11] P. Gupta, P. R. Kumar, The capacity of wireless networks, IEEETransactions on Information Theory, Vol. 46(2), 2000, pp. 388–404.[12] Y. T. Hou, Y. Shi, H. D. Sherali, Rate allocation in wireless sensornetworks with network lifetime requirement, Proceedings of ACM MobiHoc’2004,pp. 67–77.[13] X. L. Huang, B. Bensaou, On max-min fairness and scheduling inwireless ad hoc networks: Analytical framework and implementation,Proceedings of MobiHoc’2001, pp. 221–231.[14] IEEE 802.11 Working Group, Wireless LAN Medium Access Control(MAC) and Physical Layer (PHY) Specifications, 1997.[15] IEEE 802.11a Working Group, Wireless LAN Medium Access Control(MAC) and Physical Layer (PHY) Specifications - Amendment 1: HighspeedPhysical Layer in the 5 GHz band, 1999.[16] K. Jain, J. Padhye, V. Padmanabhan and L. Qiu, Impact on interferenceon multihop wireless network performance, Proceedings of ACMMobiCom’2003, pp. 66–80.[17] J. M. Kleinberg, Y. Rabani, and E. Tardos, Fairness in routing and loadbalancing, Proceedings of IEEE FOCS’1999, pp. 568–578.[18] P. Kyasanur, N. H. Vaidya, Routing and interface assignment in multichannelmulti-interface wireless networks, Proceedings of WCNC’2005,pp. 2051–2056.[19] H. Luo, S. Lu, A topology-independent fair queueing model in ad hocwireless networks, Proceedings of IEEE ICNP’2000, pp. 325–335.[20] H. Luo, P. Medvedev, J. Cheng, S. Lu, A self-coordinating approachto distributed fair queueing in ad hoc wireless networks,Proceedings ofIEEE INFOCOM’2001, pp. 1370–1379.[21] N. Megiddo, Optimal flows in networks with multiple sources and sinks.Mathematical Programming, Vol. 7(3), 1974, pp. 97–107.[22] K. Moaveninejad, X. Y. Li, Low-interference topology control forwireless ad hoc networks, Journal of Ad Hoc and Sensor WirelessNetworks, Accepted for publication.[23] C. E. Perkins, E. M. Royer, S. R. Das, Quality of service for ad hocon-demand distance vector routing (work in progress), IETF InternetDraft, 2000.[24] A. Raniwala, T. Chiueh, Architecture and algorithms for an IEEE802.11-based multi-channel wireless mesh network, Proceedings ofIEEE INFOCOM’2005, pp. 2223–2234.[25] A. Raniwala, K. Gopalan, T. Chiueh, Centralized channel assignmentand routing algorithms for multi-channel wireless mesh networks, ACMMobile Computing and Communications Review (MC2R), Vol. 8(2),2004, pp. 50–65.[26] J. So, N. H. Vaidya, Multi-channel MAC for ad hoc networks: Handlingmulti-channel hidden terminals using a single transceiver, Proceedingsof ACM MobiHoc’2004, pp. 222–233.[27] J. So, N. H. Vaidya, Routing and channel assignment in multichannelmulti-hop wireless networks with single network interface,UIUC Technical Report, 2005. Available at: http://www.crhc.uiuc.edu/wireless/groupPubs.html[28] J. Tang, G. Xue, C. Chandler, W. Zhang, Interference-aware routing inmulti-hop wireless networks using directional antennas, Proceedings ofIEEE INFOCOM’2005, pp. 751–760.[29] J. Tang, G. Xue, W. Zhang, Interference-aware topology control andQoS routing in multi-channel wireless mesh networks, Proceedings ofACM MobiHoc’2005, pp. 68–77.[30] L. Tassiulas, S. Sarkar, Maxmin fair scheduling in wireless networks,Proceedings of IEEE INFOCOM’2002, pp. 763–772.[31] Q. Xue, A. Ganz, Ad hoc QoS on-demand routing (AQOR) in mobilead hoc networks, Journal of Parallel Distributed Computing, Vol. 63(2),2003, pp. 154–165.[32] H. Zhai, J. Wang, Y. Fang, Distributed packet scheduling for multihopflows in ad hoc networks, Proceedings of IEEE WCNC’2004, pp. 1081–1086.This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the Proceedings IEEE Infocom.