Maximum Throughput and Fair Bandwidth Allocation in Multi ...

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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.
Page 4 and 5: specifies the amount of flow routed
Page 8 and 9: This means that when S ≠ S 1, ′
Page 10: for cases in which the network reso

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