Joint Routing, Scheduling And Power Control For Multihop Wireless Networks With Multiple Antennas
JJoint Routing, Scheduling And Power Control ForMultihop Wireless NetworksWith Multiple Antennas
Harish Vangala
ECE Department,Indian Institute of Science,Bangalore, India.
Rahul Meshram
ECE Department,Indian Institute of Science,Bangalore, India.
Prof. Vinod Sharma
ECE Department,Indian Institute of Science,Bangalore, India.
Abstract —We consider the problem of Joint Routing, Schedul-ing and Power-control (JRSP) problem for multihop wirelessnetworks (MHWN) with multiple antennas. We extend theproblem and a (sub-optimal) heuristic solution method for JRSPin MHWN with single antennas. We present an iterative schemeto calculate link capacities(achievable rates) in the interferenceenvironment of the network using SINR model. We then presentthe algorithm for solving the JRSP problem. This completes afeasible system model for MHWN when nodes have multipleantennas. We show that the gain we achieve by using multipleantennas in the network is linear both in optimal performanceas well as heuristic algorithmic performance.
I. I
NTRODUCTION
Multihop wireless networks are essential for ubiquitouscomputation and communication. Currently there are manyexperimental setups of multihop wireless networks aroundthe world. Ad-hoc wireless networks and sensor networksare also examples of multihop wireless networks where itis necessary to employ multiple wireless hops for even theconnectivity of the nodes deployed in a particular area. How-ever, multiple wireless hops pose many new challenges in anetwork design. But recent studies have shown that some ofthese challenges can be converted into opportunities by carefulnetwork design. Thus new communication paradigms, e.g.,opportunistic scheduling, cooperative communication, networkcoding and multiple antennas have been developed in recentyears. Exploiting these techniques together in a multihop setupto optimize the system performance is very challenging. Inthis paper we will concentrate on designing multihop networkswith multiple antennas at each node.In multihop wireless networks, unlike in the wire-linenetworks, even the concept of a link between two nodesis not properly defined. Shadowing can, irrespective of thedistance between two nodes, cause the channel to be verybad. Even when there is no shadowing, we may or maynot be able to communicate directly between two nodesdepending on the transmit power used and also if there areother users transmitting in the neighborhood. Thus, topologyof the network, transmit power, link scheduling and routing areall interrelated and for optimal performance one may need to jointly optimize power, scheduling and routing. This problemis computationally intractable and not scalable even for acentralized algorithm [1], [2], [4].Employing multiple antennas at a transmitter and/or ata receiver can provide transmit diversity, receive diversity,increase the capacity of the link and reduce BER. Thus, inwireless networks where bandwidth is scarce, it is importantto employ multiple antennas wherever feasible. This increasesthe degrees of freedom one can exploit to improve the systemperformance. Thus even for multihop network it is desirableto have multiple antennas at different nodes. However, asmentioned above, even with single antennas, jointly optimizingpower, routing and scheduling in a multihop wireless networkis very computationally intensive, with multiple antennas theproblem becomes extremely challenging. At the moment theseare very few studies available for this system. We address thisproblem in this report.II. R
ELATED W ORK
In wireless networks, it is well known that the traditionallayers of the communication network cannot be consideredin isolation. Many authors have proposed joint approaches toissues such as power control, scheduling, and routing [1], [2],[4], [5]. But there has been little effort in joint optimization ofpower, link scheduling and routing. One of the early papers inthis direction is [1]. In [2], the techniques of [1] are appliedto a different scenario where the nodes use Energy HarvestingSensors. Both [1] and [2] use
Linear Programming in theirsystem models by using the technique of Column Generation.A multihop wireless network with single antennas isstudied and solved in a similar way claiming to achievethe solutions for much larger networks than what has beenpossible thus far in [9].III. S
YSTEM M ODEL A ND T HE P ROBLEM F ORMULATION
We have a network of half duplex nodes spread out in spacecommunicating with each other using the common wirelessmedium. Since the network uses a wireless channel of broad-casting nature, ideally every node can hear the transmissions a r X i v : . [ c s . N I] O c t raft 1 from every other node. But in practice, a node usually willhave a coverage area hence being allowed to transmit to orreceive from a limited number of other nodes. This effectivelyforms a Graph and is the starting point for the problem. Butit can be noted that interference continues to exist betweenany pair of nodes, irrespective of the graph representation.Further, the nodes may be restricted in terms of transmissionor reception towards other nodes(neighbors) in the network.This is required for a tractable model in terms of formulatingand solving the problem. We term it as transmission model ofthe system.In the given network, a subset of nodes called sources would like to transmit to their destinations , which is anothersubset of nodes. It in general needs multiple wireless hops toreach the destination. Our objective is to provide a fair (to bedefined later) data transmission rate to all those sources viamultiple intermediate nodes. A. System Model
We have a network being represented as a fully connecteddirected graph with no self-loops, denoted as G ( N , L ) . Here, N = { , , . . . , N } is the set of nodes and L = { , , . . . , L } is the set of directed links in the network. The set of sourceand destination pairs and the transmissions among them will becalled as flows . The set of flows is denoted by F = 1 , , . . . , F .Each flow f ∈ F will be associated with a source node and adestination node denoted by src ( f ) and dest ( f ) . It is allowedto have common source and/or destination nodes among flows.Each f ∈ F will also have a set of rate demands denotedby { d i , i ∈ F} , while the system may decide to allow ratesequal to { r i , i ∈ F} , depending on the availability of networkresources. The rates allowed may be lesser or greater or equalto the demanded rates. We define fairness as the least fractionin (cid:110) r i d i (cid:111) i ∈F and denote it by λ .All the nodes use multiple but equal number of antennas a for transmission or reception. The total power consumption ofa node for transmission can only be a value chosen from a setof K powerlevels (including zero), in any time slot. We restrictall the antennas to be used towards a single node at any timeinstant. That means, we use point to point transmission modelor the beam forming model of transmission in the network.Hence, a node involves in at most one active link, at any pointof time.Though this model is a simple extension of the model usedin [1] and [2] for single antennas, it is not obvious to formulateand solve the problem for multiple antennas. Further, onecannot simply state the MIMO gain in the final throughput,just by observing that each link gains in its capacity, becauseof multiple hops being involved in transmissions.Given the transmission model of the network, not everysubset of L can be activated simultaneously. We define a mode ,as a valid subset of links being activated simultaneously alongwith the amounts of powers used on the links. We represent itby an L × vector of powers used on each link. A set of modes given a schedule, achieves the link scheduling and routingsimultaneously. If such scheduling takes the power availabilityon an average at each node into account, it becomes a solutionto the JRSP problem.The fading channel gains are assumed to be known duringthe analysis being a centralized setup and are given by set of a × a matrices H ij ( ∀ i, j ∈ N ) , which represent the channelmatrices between nodes i and j . We also use the notation H (cid:48) lm ( ∀ l, m ∈ L ) to denote the channel between links l, m meaning, the channel between source node of link l anddestination node of link m . Note that H (cid:48) lm = H ij , where i = src ( l ) and j = dest ( m ) . Each scalar in the matrices istaken as a circularly Gaussian complex random variable. In[1] and [2], these were calculated by using a simple path lossmodel for a set of nodes randomly spread out over an area,based on the euclidean distances between them.Given the power spent on a link (in any mode ), we retainthe distribution of powers over the set of antennas at nodesas a degree of freedom for nodes (and hence is a variableduring link capacity calculations). Hence they use waterfilling ,whenever required. The way we compute link capacities ispresented after the mathematical statement of the problem. B. The Mathematical Formulation max λ (cid:44) (cid:18) min i (cid:26) r i d i (cid:27)(cid:19) (1a)(fairness objective)subject to : AX = r , (1b)(Flow Conservation) X · ≤ C · α , (1c)(Avg. Link Capacities) P · α ≤ P avg , (1d)(Avg. Power Control) α · , (1e)(Consistency of scheduling) X fl ≥ , ∀ f ∈ F , ∀ l ∈ L P mn ≥ , α m ≥ , ∀ n ∈ N , ∀ m ∈ M (1f) . (non-negativity of variables)The problem setup used in [1] and [2] is quite generaland the problem formulation under multiple antennas can bemade similar. Except that the terms used in it come from adifferent network scenario in multiple antennas. Its a conven-tional Multicommodity Flow problem from network theory andoperations research literature, which is modified and exploitedto suit the JRSP problem. The problem’s objective is to findthe solution to the JRSP problem, that gives maximum fairness among the users. The mathematical statement of JRSP problemcontains the well known flow conservation constraints, link
This paper is submitted to NCC-2012 conference proceedings.. raft 1 capacity constraints and average power constraints. This formsa simple linear programming problem (LPP) as given in (1).In the problem statement (1), ’ A ’ refers to the node versuslink representation of the graph which takes the form of an N × L matrix with elements a ij taking signed binary values , ± , as below: a ij = , if node i is not a part of link j +1 , if node i is the source to link j, − , if node i is the dest to link j.X refers to an L × F matrix with values X fl representingthe average effective rate of transmission chosen on link l corresponding to flow f . r refers to an N × F matrix, which contains the netoutwards data rate at each node corresponding to each flow.i.e. r ij = , if node i is not a source or dest. of flow j + r j , if node i is the source to flow j, − r j , if node i is the destination to flow j. M = { , , ..., M } refers to the set of all possible modes in the network. Each mode m ∈ M will take a representationof an L dimensional column vector. The modes set includesan idle state (i.e. a ) as well. α represents the vector of time fractions given to each ofthe modes. C is an M × L vector, where an element C ml represents thecapacity of a link l ∈ L in mode m ∈ M . i.e., each column of C is a vector of same dimension as a mode, but link powersare replaced with link capacities. P refers to an N × M matrix called power profile matrix with values P mn representing the amount of power spent bynode n in mode m ∈ M . It corresponds to the power spenton the single active link in the neighborhood of a node asa transmitter, at any time slot. P avg is an N × vector,representing the power availability at each node, on an average.Note that the variables in the problem are r i ( ∀ i ∈ F ) , X and α while rest all( A, P, C and P avg ) are constants once thenetwork and set of all modes M are fixed. Also note that,its a pure linear program since the objective can alternatelybe written as a set of linear inequalities introducing one newvariable.With the set of assumptions we made in the system model,given a network( A ) with average power values( P avg ), thisproblem statement becomes universally applicable for anynumber of antennas( a ) on nodes, given the set of modes M and their capacities C . But with multiple antennas, we seefollowing issues.1) The link capacity calculations are interdependent (a jointoptimization problem) due to interference. We would havea capacity region of points being vectors of link capacityelements.2) The capacity region of such a vector interference channel is still an open problem.3) We need a computationally feasible method to decidelink capacities in a mode (columns of C ), atleast as a goodachievable rate point. Because, we have to calculate the samefor a large number of modes. C. Link Capacity computation
Let m = ( m , m , . . . , m L ) T be the given vector of L powers spent on links. To compute the link capacities, oneshould solve the problem of waterfilling on each of the links.But, the effective noise that each user sees is a sum of AWGnoise and the interference from other users. Assuming eachsource emits independent Gaussian symbols, we need thecovariance matrix of the effective noise, for which we needcovariance matrices of each transmitter in the network. This isclearly not known at the time of computation of link capacities.Because, link capacity computation itself is water flling i.e.an optimization decision over all transmit covariance matricespossible at all the links.We propose the following iterative scheme to greedilycalculate the capacities of links in the network which areactive simultaneously. We continue the iterations till we seethe convergence in sum rate of all links in the network. Thesum rate increases monotonically after each iteration.We use the following notation. C i is the i th link capacity. K i is the transmit covariance matrix used on link i while m i is the power spent on link i . sumrate corresponds tothe sum of all link capacities active in the network. Thefunction “waterfill(.)” performs waterfilling on any specifiedlink and outputs two values namely link capacity and optimumtransmit covariance matrix to achieve that capacity under theinterference value at that instant. It takes three argumentsnamely link index at which waterfilling is to be performed, setof all transmit covariance matrices and channels( H (cid:48) ij ) betweenlink i and every other link j in the network. (i.e., source ofa link to the destination of other link — H (cid:48) ij = H pq where p = src ( i ) and q = dest ( j ) ). Algorithm 1
Iterative Waterfilling algorithm for finding link-capacities in a point-to-point modeinitialize sumrate = −∞ , C i = 0 , K i = ( m i /a ) ∗ I ∀ i = { , , . . . , L } while | (cid:80) i C i − sumrate | ≥ (cid:15) do sumrate = (cid:80) i C i for i = 1 to L doif link i is active then [ K i , C i ] = waterfill( i ; σ ; { K j , H (cid:48) ji ∀ j ∈ L} ) end ifend forend while D. Complexity Issues
The number of constraints in the problem is ( N F + L + F + 1) while number of variables is ( F + LF + M ) . If N is This paper is submitted to NCC-2012 conference proceedings.. raft 1 taken as the size of the network, N ≤ L ≤ N ( N − sincegraph is assumed to be directed and fully connected, havingno self loops. Further F ≤ N ( N − .Though one can derive an upper bound on M as K L (asuper exponential number) by arguing that M is less thanall possible choices of powerlevels among L links, it can beseen via bruteforce enumerating simulations that the numberof possible modes in a network of N nodes is exponential in N (see fig.1). In anycase, the number of variables in the problemgrows enormously with N and is much larger compared tonumber of constraints in the network. Typically, this numberturns out to be more than a million for a network as small as11 nodes and 4 powerlevels. Number of Nodes (N)
Average Number of Modes (M)
P2P
K=2K=4K=6 K=8K=10
Fig. 1. A typical number of modes(M) as a function of number of nodes(N)
Hence, its a prohibitively large problem to be solveddirectly for networks more than roughly 10 nodes. Notethat, we have ignored the calculations required to get linkcapacities. IV. T HE H EURISTIC S OLUTION
One definitely needs to have a solution procedure, that isfeasible for solving the network, upto atleast reasonable sizeof the network. [1] proposes a heuristic column generationmethod to solve this problem and is claimed to solve networksupto approximately 30 nodes. [2] applies the similar thing withslight variation, on the networks of energy harvesting sensornodes. We see that we can still solve the problem upto almostthe same level, even when the network has multiple antennasand hence the complexity is much larger.The solution method proceeds as follows. First it can beseen by using the analysis of number of basic variables that,no more than N + L + 1 number of modes are necessaryto obtain the optimal basic feasible solution. Then as per the column generation procedure, solution starts by considering asmaller problem called Master Problem with just N + L + 1 randomly chosen mode variables( α (cid:48) ) from α , rest all being thesame to the original optimization problem in (1).It also considers a Sub Problem that chooses a new modecalled a good mode from the remaining modes, considering which we would improve the performance of the masterproblem.
Sub Problem: max m ∈M\M (cid:48) θ ( m ) (cid:44) u T C m − v T P m − βsubject to : θ ( m ) ≥ . Such a mode will replace an already chosen mode variablewhich is given zero scheduling time in the optimal solutionof
Master problem. This starts next iteration, in which masterproblem attains an improved solution.The same after iterating for sufficient number of iterations,converges to a (sub)optimal solution and an (sub)optimalchoice of set of N + L + 1 modes from M .In this procedure, sub-problem is the stage, which requiresan optimum choice of variable by searching exhaustivelyamong auxiliary function values evaluated at M − ( N + L +1) modes, which is again computationally infeasible. This can beavoided by using a heuristic algorithm for sub-problem, from[1] and [2]. This comes at the cost of sub-optimality to thesolution. But it was shown via simulation examples that, thesolution that we see via this heuristic, is close to the optimalsolution.The heuristic algorithm for MHWN with MIMO extendedfrom its single antenna version is presented below as Algo-rithm 2, in a ready to implement algorithmic form. In this, nextlevel ( x ) is the notation used for denoting the power valuewhich is least among the power values that are greater than x from power level set. And interf erers ( l ) is the notationused to denote the set of all the neighbouring links to link l that share a node with link l . Algorithm 2
The Heuristic Algorithm to obtain an efficient sub-optimal solution to the sub-problem
Initialize mode vector goodmode = 0 , lasttheta = θ ( goodmode ) , AllowedSet = L while AllowedSet (cid:54) = φ dofor i = 1 to L doif i ∈ AllowedSet then m = goodmode m i = nextlevel ( m i ) θ i = θ ( m ) end ifend for Let besttheta (cid:44) max i θ i &Let the bestlink l (cid:44) arg max i θ i if besttheta ≤ lasttheta then break the loop. else lasttheta = besttheta m = goodmode if m l = 0 then AllowedSet = AllowedSet \ interferers ( l ) end if m l = nextlevel ( m l ) goodmode = m end ifend while Declare goodmode as the solution to sub problemThis paper is submitted to NCC-2012 conference proceedings.. raft 1 Since we solve the problem (1) with a heuristic solutionat the sub problem level of column generation, we call thismethod as
Heuristic Column Generation (HCG). It should benoted that, this solution is sub-optimal for the sub-problemand hence the final solution that we obtain by using this issub-optimal.Further, it can be predicted and also observed in sim-ulations presented in next section that, the final solution(maximum fairness) varies over a range of values, dependingupon the initially selected random set of modes. This issuewas not discussed much in the earlier papers using this HCG.Hence we in this paper, propose to choose the best valueamong solutions achieved by solving the problem for multiplenumber of initial points, analogous to the traditional tabusearch methods used for non-convex optimization.In next chapter, we solve the problem for several networkswith multiple antennas and get the final solutions. We firstshow for appropriate networks that the HCG gives a sub-optimal solution but is very close to the optimal solution.Then we see that we can solve the problem upto 30 nodes,almost similar to the single antennas case, as claimed in earlierpapers. Further, we show that the MIMO does improve thefinal performance of the network and further that the gain islinear with the number of antennas( a ) on nodes.V. SIMULATION RESULTS
Consider a network of 15 nodes as shown in fig-2. Consider
Fig. 2. A network of 15 nodes the following network parameters:Number of Nodes N = 15 Number of Directed Links L = 60 Number of antennas on each node a = 5 Power levels used by each node = [0 , ;Channels are generated as L number of × matrices of unitvariance, circularly Gaussian complex random numbers.source,destination pairs(flows) are = (7,13), (10,5), (11,8)The rate demands from the flows = [10 , , unitsNodes Avg power availability = 3 units, uniform to all.(Note: In this scenario, we have approximately 0.5 millionmodes allowed in the network)By formulating the problem (1) and solving it directly (op-timally), we get the following solutions. Allowed user rates after solving the problem are: r = 5 . units, r = 8 . units r = 11 . unitsHence the optimum fairness that can be achieved is: λ = 0 . A dominant mode which is given fairly good amount oftime fraction is shown in fig-3 with numbered bold arrows.
Fig. 3. A dominant mode in the network to achieve optimal throughput
Now we use Heuristic algorithm to solve the problem andsee the relative performance. This is shown in fig-4. Note thatwe use multiple initial points. we get the following solutions.Allowed user rates after solving the problem are: r = 5 . units, r = 8 . units r = 11 . unitsHence the optimum fairness that can be achieved is: λ = 0 . F a i r n ess P e r f o r m a n ce Heuristic SolutionsOptimal Solution
Fig. 4. Fairness Values achieved using Heuristic Algm, Compared withOptimal for the network of 15 nodes
Next, we consider a network of 30 nodes and the followingparameters.Number of Nodes N = 30 Number of Directed Links L = 110 Number of antennas on each node a = 4 Source,destination pairs(flows) are:= (2,30), (5,10), (1,30), (4,6)The rate demands from the flows = [10 , , , unitsNodes Avg power availability = 3 units, uniform to all. This paper is submitted to NCC-2012 conference proceedings.. raft 1 Fig. 5. A network of 30 nodes Formulating the problem (1) and solving it using Heuristicfor multiple initial points, we see the following results fig-6.Note that, this problem cannot be solved directly and hencethe optimal performance achievable is not known. F a i r n ess P e r f o r m a n ce Fig. 6. Heuristic algorithmic performance for network of N=30
Allowed user rates after solving the problem are: r = 2 . units, r = 13 . units r = 5 . units, r = 16 . unitsHence the optimum fairness that can be achieved is: λ = 0 . So we demonstrated that proposed system model with JRSPproblem is solvable upto reasonable size of the networkand that the extended heuristic solution gives close enoughsolutions to the optimal values. We now show the performancegain achieved using multiple antennas in the system.We consider the same 15 node network and the problemshown in fig-2. We vary the number of antennas on eachnode( a ) from one to ten and see how the optimal performance(fairness) varies. Note that the channel matrices observe achange in dimensions as well as values, for each case. Wethen consider the same network problem but solve it usingthe Heuristic column generation. We vary the number ofantennas ( a ) from one to ten and see the performance (fairness)variation. Both these are shown in fig-7 We see that bothoptimal as well as Heuristic algorithmic performances improvelinearly as we increase the number of antennas on nodes. F a i r ne ss P e r f o r m an c e Optimal Fairness PerformanceHeuristic Fairness Performance
Fig. 7. MIMO gain in Optimal and Heuristic algorithmic performance of anetwork with 15 nodes
VI. C
ONCLUSION
We have proposed a feasible system model for multihopwireless networks with multiple antennas extended from theMHWN with single antennas. We have shown that the problemis solvable upto reasonale size of the network almost similarto the network with single antennas. We then have shown thatthe performance gain that we achieve using multiple numberof antennas on nodes is linear.R
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