Distributed Fair Transmit Power Adjustment for Vehicular Ad Hoc ...

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The second area of research related to our work is theone addressing fairness to share the wireless medium. In thiscategory we can find strategies that consider only unicastcommunications and i) assign a portion of the estimated bandwidthto each flow, such as [17], or ii) provide a schedulingmechanism to achieve its fairness criteria, e.g., [18].Recently, due to the increasing attention of researchers onVANETs, some studies have tried to apply these methodologiesto vehicular environments. For example, [19] addressespower control in VANETs with the goal of producing a connectednetwork topology; or [20], which describes a schemebased on a utility fair function to share the broadcast medium.In the latter, a scheduling approach is proposed that can beperfectly valid for non-safety VANETs’ applications, however,it does not satisfy all the safety constraints presented inSec. II.IV. THE DISTRIBUTED FPAV ALGORITHMBased on the rationale outlined in Section II and on theBMMTxP (Beaconing Max-Min Tx Power) problem as givenin [12] we define the D-FPAV algorithm and formally provethat it achieves fairness.A. The BMMTxP problem and D-FPAVHere, a short version of the problem statement and definitionspresented in [12] are reproduced, which are required toprove Theorem 1. We refer the reader to Section 3.2 of [12] fora complete explanation with the correspondent justifications.Assume a set of nodes N = {u 1 ,...,u n } is moving alonga road modeled as a line 1 of unit length, i.e. R = [0,1], andthat nodes can be modeled as points x(i) ∈ [0,1].Each of the network nodes sends a beacon with a predefinedbeaconing frequency F , using a certain transmit powerp ∈ [0,P max ].Definition 1 (Power assignment): Given a set of nodesN = {u 1 ,...,u n }, a power assignment PA is a functionthat assigns to every network node u i , with i = 1,...,n, aratio PA(i) ∈ [0,1]. The power used by node u i to send thebeacon is PA(i) · P max .Definition 2 (Carrier Sensing Range): Given a power assignmentPA and any node u i ∈ N, the carrier sensing rangeof u i under PA, denoted CS(PA,i), is defined as intersectionbetween the commonly known CS range 2 of node u i at powerPA(i) ·P max and the deployment region R. The CS range ofnode u i at maximum power is denoted CS MAX (i).Given a power assignment PA, the network load generatedby the beaconing activity under PA is defined as follows.Definition 3 (Beaconing load under PA): Given a set ofnodes N and a power assignment PA for the nodes in N,1 Modeling the road as a line is a reasonable simplification in our case sincewe assume the communication ranges of the nodes to be much larger thanthe width of the road.2 The CS (Carrier Sense) range, in ideal conditions, is the distance to whicha node’s transmissions can be sensed, or also considered as the distance towhich a node can interfere with other transmissions.u 1u 2u 3u 4u 5u 6u 7u 850mFig. 2. Node deployment used in the example of D-FPAV execution.the beaconing network load at node u i under PA is definedas:BL(PA,i) = |{u j ∈ N,j ≠ i : u i ∈ CS(PA,j)}| ,where CS(PA,j) is the carrier sensing range of node u j underpower assignment PA.We remark that the above definition of beaconing load can beeasily extended to account for beacon messages of differentsize, and for different beaconing frequencies in the network.The framework for distributed power control discussed belowcan be applied also with a more general definition of beaconingload.Definition 4: (Beaconing Max-Min Tx power Problem(BMMTxP)): Given a set of nodes N = {u 1 ,...,u n } inR = [0,1], determine a power assignment PA such that theminimum of the transmit powers used by nodes for beaconingis maximized, and the network load experienced at the nodesremains below the beaconing threshold MBL. Formally,⎧⎨⎩max PA∈PA (min ui∈N PA(i))subject toBL(PA,i) ≤ MBL ∀i ∈ {1,...,n}where PA is the set of all possible power assignments.In [12], we presented FPAV, an algorithm based on ‘waterfilling’ able to compute PA when global knowledge wasassumed. In order to facilitate reading and due to completenesswe describe the essentials of the FPAV algorithm. Fig. 3presents FPAV which works as follows: every node startswith the minimum transmit power assigned; then, all thenodes increase their transmit power simultaneously of the sameamount ǫ · P max as long as the condition on the beaconingnetwork load (MBL) is satisfied. At the end of FPAV all nodeshave increased their power the same number of steps k and endup with a power of p = (kǫ) ·P max . Note that in [12] we alsoproposed a ‘second stage’ of the FPAV algorithm in order toachieve per-node maximality. However, it is not included in thepresent study because further simulations showed a marginalgain in many cases due to the high network dynamics. In thefollowing we present a fully distributed, asynchronous, andlocalized algorithm called D-FPAV for solving BMMTxP.D-FPAV is summarized in Figure 4. A node u i is continuouslycollecting the information about the status (currentposition, velocity, direction, and so on) of all the nodes withinits carrier sensing range at maximum power CS MAX (i).These are the only nodes that node u i can affect when sendingits beacon. The actual mechanism to acquire the required stateinformation is discussed in the following subsection. Basedon this information, node u i makes use of FPAV to computethe maximum common value P i of the transmit power for all,
Algorithm FPAV:INPUT: a set of nodes N = {u 1 ,...,u n } in [0,1]OUTPUT: a power assignment PA which is an(ǫ · P max -approximation of an) optimal solutionto BMMTxP∀u i ∈ N, set PA(i) = 0while (BL(PA) ≤ MBL) do∀u i ∈ N, PA(i) = PA(i) + ǫend while∀u i ∈ N, PA(i) = PA(i) − ǫFig. 3. The FPAV algorithm.Algorithm D-FPAV: (algorithm for node u i )INPUT: status of all the nodes in CS MAX (i)OUTPUT: a power setting PA(i) for node u i , such thatthe resulting power assignment is an optimalsolution to BMMTxP1. Based on the status of the nodes in CS MAX (i),compute the maximum common tx power levelP i s.t. the MBL threshold is not violated at anynode in CS MAX (i)2a.Broadcast P i to all nodes in CS MAX (i)2b. Receive the messages with the power level fromnodes u j such that u i ∈ CS MAX (j); store thereceived values in P j3. Compute the final power level:PA(i) = min { P i ,min j:ui∈CS MAX(j){P j } }Fig. 4. The D-FPAV algorithm.nodes in CS MAX (i) such that the condition on the MBL is notviolated (step 1). Note that this computation is based on localinformation only (the status of all the nodes in CS MAX (i)),and it might be infeasible (i.e., it might violate the conditionon MBL at some node) globally. To account for this, nodeu i broadcasts the computed common power level P i to allnodes in CS MAX (i) (step 2a). In the meanwhile, node u ireceives the same information from the nodes u j such thatu i ∈ CS MAX (j) (step 2b). After having received the powerlevels computed by the nodes in its vicinity, node u i cancompute the final transmit power level, which is set to theminimum among the value P i computed by the node itselfand the values received from nodes in the vicinity (step 4).Setting the final power level to the minimum possible level isnecessary in order to guarantee the feasibility of the computedpower assignment (see theorem below).Theorem 1: Assume the CS ranges of the nodes are symmetric,i.e. u i ∈ CS MAX (j) ⇔ u j ∈ CS MAX (i). Then,algorithm D-FPAV computes an optimal solution to BMMTxP.Proof: First, we have to show that the power assignmentcomputed by D-FPAV is a feasible solution to BMMTxP.Assume the contrary, i.e. assume there exists node u i suchthat BL(PA,i) > MBL, where PA is the power assignmentcomputed by D-FPAV. This means that node u i has too manyinterferers, all of which are located in CS MAX (i) (assumingsymmetric CS ranges). Let u j ,...,u j+h , for some h > 0,be these interferers, and let PA i be the power assignmentcomputed by node u i for all the nodes in CS MAX (i). Instep 1 of D-FPAV, u i computes an optimal solution PA ito BMMTxP restricted to CS MAX (i). Assuming symmetricCS ranges, this solution includes a power setting for theinterferers u j ,...,u j+h , and this power setting is such thatBL(PA i ,i) ≤ MBL. At step 2 of D-FPAV, the power settingPA i is broadcasted to all the nodes in CS MAX (i), whichincludes all the interferers u j ,...,u j+h . Hence, each of theinterferers receives from node u i a power setting PA i suchthat the condition on the beaconing load is not violated atnode u i . Since the final power setting of the interferers is atmost PA i (this follows from the minimum operation executedat step 3 of D-FPAV), and assuming a monotonic CS range,we have that the beaconing load at node u i cannot exceed theMBL threshold – contradiction. This proves that the powerassignment computed by D-FPAV is a feasible solution toBMMTxP.Let us now prove that the computed power assignment isoptimal. Let PA be the power assignment computed byD-FPAV, and let p min be the minimum of the node powerlevels in PA. Assume PA is not optimal, i.e. that thereexists another feasible solution PA ′ to BMMTxP such thatthe minimum of the node power levels in PA ′ is p ′ > p min .Without loss of generality, assume that PA ′ sets the powerlevel of all the nodes to p ′ . Since PA ′ is feasible, we havethat BL(PA ′ ,i) ≤ MBL ∀i ∈ 1,...,n. Hence, every nodeu i in the network computes a power setting P i ≥ p ′ at step 1 ofD-FPAV. Consequently, the final power setting of every nodein the network as computed by D-FPAV is at least p ′ > p min ,which contradicts our initial assumption. It follows that thesolution computed by D-FPAV is optimal.Theorem 2: Algorithm D-FPAV has O(n) message complexity.The straightforward proof of the theorem is omitted.Let us use the scenario of Figure 2 to present D-FPAVexecution with a toy example. We have eight cars, denotedu 1 ,...,u 8 , which are placed on a 1km long road, with relativedistance varying from 50m to 200m. For the sake of clarity,we assume that the carrier sensing range can be representedas a segment centered at the transmitting node, and that themaximum CS range is initially 400m. We also assume thatall nodes send beacons of the same size with equal frequency,being the maximum beaconing load MBL such that any nodecan be in the CS range of at most two other nodes.We summarize D-FPAV execution with the matrix reportedin Table I, where the row corresponding to node u i reportsthe values of the transmit power (actually, what is reported in
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