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.