Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
22.5 Routing and Resource Allocation in Collaborative Networks
When collaborative communications is used for the forwarding of messages, the routing problem even for a single message becomes much more complex. In multi-hop routing, all we needed to determine was the sequence of nodes that forward the message, and (possibly) the transmit power used by each node. The question of when, and for how long, a node should transmit was answered very simply: it should start when it had received and decoded the message from the previous node on the route, and it should stop when the subsequent node had decoded the message. When cooperative communications are used, start- and stop-time of transmission from a relay are (almost)10arbitrary parameters that need to be optimized.
In contrast to the multi-hop case an optimum solution can only be found by trying out all possible routes. For this reason, a number of heuristic algorithms have been proposed that can often come close to ideal performance. Furthermore, it must be noted that the field of “collaborative routing”
is far from mature; there is in particular a dearth of solutions for the case when multiple messages are to be transmitted through the network simultaneously.
22.5.1 Edge-Disjoint Routing and Anypath Routing
A simple form of cooperative routing is obtained if the goal of the cooperation is increased robust- ness. In that case, it is advantageous to route the message on several parallel routes through the network; these routes should have as few links and nodes in common as possible, so that the failure of a particular link cannot lead to a blockage of all routes to a destination. Edge-disjoint shortest-path routingis a way of identifying routes that do not share any links; a suitable algorithm is a minor modification of the Bellman–Ford algorithm. Note, however, that this approach does not make significant use of the broadcast effect.
Anypath routing exploits the broadcast effect to achieve diversity: each node broadcasts the data packet to a group of neighbors, called theforwarding set. As long as at least one of the nodes in the forwarding set receives the message, the next step on the route can be made; one of the successful nodes then acts as the next relay on the route. Note, however, that only a single node from a forwarding set retransmits. In other words, the broadcast effect is used to obtain selection diversity.
Thus, even if a particular link on the “nominal” shortest path goes down, the data packet can reach its destination, without the necessity of finding a new route. Anypath routing (see Figure 22.14) is thus especially useful for links that frequently change in quality.
Instead of a specific route, anypath routing results in an ensemble of possible routes; depending on the outcomes of the transmissions from the various nodes, a packet can take different routes
10Of course, a node can only start to transmit after it has decoded the message that needs to be relayed.
Source
Destination
Figure 22.14 An anypath route (gray), and one possible trajectory taken by a packet (bold).
Reproduced from Dubois-Ferriere [2006]©EPFL Switzerland.
through the network. Formally speaking, an anypath route is the union of all possible trajectories along which a packet can travel from the source to the destination. Finding the best anypath route involves a tradeoff: increasing the candidate set gives a better robustness (and might therefore help to decrease the required link margin, and thus transmit power, while retaining the same outage probability for message delivery). On the other hand, a larger forwarding set increases the danger that the packet is routed farther away from the true shortest path (which we would take if we had perfect, completely current knowledge of all the network states). Additional complications arise when the data rate is also allowed to vary – changing the data rate changes the set of nodes that are, in principle, able to correctly receive the packet from a certain preceding node. In any case, however, variations of the Bellman–Ford algorithm can find the optimum anypath route (and associated rates) in polynomial time.
22.5.2 Routing with Energy Accumulation
Another way of exploiting diversity is energy accumulation at the relay nodes. This occurs when a node stores a received signal of a packet that is too weak for decoding and combines it with another signal of the same packet that arrives later. When using energy accumulation at the nodes instead of simple multi-hopping, the optimum route changes. A simple example is given in the following.
Example 22.2 Consider a linear network with three nodes, where direct transmission from node A to node C has a path gain of 0.1, while transmission from node Ato nodeBhas a path gain of 0.19, and similarly from node 2 to node 3.
Let the threshold of the received signal energy for decodability of the signal be 1 J. Then in a classical multi-hop scenario, the optimum route is direct transmission, with transmit energy of node A equal to 10; multi-hop with route A→B→C would require 2(1/0.19)=10.53 J.
However, when the destination node can perform energy accumulation, the routeA→B →C becomes preferable: the source node 1 uses 1/0.19=5.26 J for its transmission (so that node B can decode). During that transmission, nodeC receives 0.1ã5.26=0.53 J energy from “over- hearing” the packet. Thus, it requires only 0.47 J during the second transmission, which it obtains if nodeB transmits with 0.47/0.19=2.47 J Thus, total transmission energy is 7.63 J; less than the 10 J required for direct transmission.
The problem of finding the optimum route involves the issues of finding (i) what nodes should participate, and (ii) in what sequence, and with what power. Unfortunately, the former problem is NP-hard, i.e., it can be solved exactly only by trying out all possible node combinations, and
Relaying, Multi-Hop, and Cooperative Communications 553
picking the best one.11 A number of heuristic algorithms exist for finding the best route. Some of those algorithms start out with the optimum multi-hop route (which is determined by the Dijkstra or Bellman–Ford algorithm), and then add on nodes that reduce the overall energy consumption.
Another type of algorithm builds up the route from scratch, starting from the source node. When it adds the next relay on the path, it reduces the signal energy still required at all the other nodes; this energy reduction depends on how much energy those other nodes can “overhear” when the new node transmits. In either case, the energy savings from the energy accumulation (and associated routing) increase as the density of nodes increases: the closer the nodes, the more energy a node can “overhear,” (see Figure 22.15).
0.78 0.76 0.74 0.72 0.70 0.68 0.66 0.64
node number
50 100 150 200 250 300 350 400
RPAR Energy/εP Energy
Energy consumption ratio of RPAR over SP under different nodes density RPAR/SP
Figure 22.15 Energy savings of routing taking into account energy accumulating compared to shortest-path algorithm when network density increases.In this figure:RPAR, Relay PAth Routing; SP, Shortest Path.
Reproduced from Chen et al. [2005]©IEEE.
A somewhat different case of energy accumulation occurs when multiple nodes transmit, syn- chronously, in parallel, to effect higher receive power:
1. Parallel nodes use orthogonal transmission (Section 22.3.3) or distributed space–time coding (Section 22.3.4) for transmission.
2. Parallel nodes use distributed beamforming (Section 22.3).
Also in this case, finding the optimum route is NP-hard. A heuristic method for finding a route is to subsume the nodes acting in parallel into a “super-node,” and then try to find the best route of supernodes.
11For the case of cooperative broadcasting, the first problem is easy (since all nodes participate in the transmission), but the determination of the correct order of node participation is NP-hard.
22.5.3 Fountain Codes
When Fountain codes are used, relay nodes can exploit “overhearing” the signals intended for other relay nodes in an even more efficient manner: they accumulate mutual information, instead of energy, as discussed in Section 22.3.6. Still, routing with mutual-information accumulation shares two important properties with energy accumulation: (i) finding an optimum route is NP-hard, and (ii) for heuristic algorithms, it is useful to break down the problem into two subproblems:
determination of the physical route or order of nodes through which packets propagate, and the allocation of resources (time, power) among the nodes. Under the assumption that each node has a fixed transmission power, the determination of the optimum resource allocation (time) can be done by a Linear Program (LP) for a specific routing order. A simple algorithm then revises the routing order based on the results of the LP. Iterating between the two subproblems (resource allocation and routing order) yields a very efficient approach to good route finding even in very large networks.
The LP can be set up the following way: by the end of thek time interval, defined as the time at which thekth node decodes the transmitted packet, the total information flow to thek-th node from thek−1 nodes ahead of it in the route must exceed the packet payload ofB bits. Formally,
k−1
i=0
k n=0
Ai,nCi,k≥B (22.43)
whereAi,nis the resource (time or bandwidth) allocated to TXiin thenth time interval,12andCi,k is the data rate (a function of channel quality) from nodei to node k. These constraints, together with the goal “minimization of total energy” (or other goals) constitute an LP that can be solved by standard software packages. The solution of the LP is subsequently used to update the route:
if the start time of thek+1 time interval becomes identical to that of thek-th time interval, the sequence of thek-th andk+1-th node on the route are swapped; if a relay node swaps its place in the decoding sequence with the destination, it is not used at all (it would only become active after the destination has already decoded the message).
22.5.4 Other Collaborative Routing Problems
Different Optimization Criteria
In the previous parts of this section, we always used “overall energy consumption” as a criterion for optimization of a route. However, other criteria can be used in practice. To give but a few examples:
• Network lifetime maximization: network lifetime is usually defined as the time during which all nodes have sufficient energy to operate properly. Nodes in the center of a network are in particular danger of running out of energy, since they have the highest likelihood of acting as relays.
• Message delay: while the use of many hops through the network can decrease the energy con- sumption, it also increases the latency; in particular when the communication rate in the network is fixed13.
• Network throughput: when multiple messages are being transmitted, then the resulting interfer- ence decreases the overall throughput of the network. The amount of interference depends on the route as well as on the particular collaboration scheme.
12Note that the duration of a time interval isnot fixed, but variable, and actually an output of the LP. It just denotes a time during which the transmission parameters are constant.
13With adaptive modulation and coding, a short link allows the use of a higher communication rate, so that the overall time for a message to reach the destination might actually decrease when many short (instead of one long) link is used.
Relaying, Multi-Hop, and Cooperative Communications 555
Routing with Selfish Nodes
Up to now, we have considered routing with the goal of decreasing theoverall energy consumption of the network. For either centralized or distributed routing algorithms, it is assumed that the nodes will obey an algorithm that maximizes the social benefits, not the individual benefits of the nodes.
This assumption works well in ad hoc networks in industrial or military/security settings, where all nodes are under the control of a single operator. However, in ad hoc networks made up of, e.g., laptops of individual users, the situation is different: every user asks “what is the benefit for me,”
or, in other words “why should I exhaust my battery in order to forward messages for somebody else?” There have to be proper incentives for users (typically, that their messages will also be forwarded by somebody else). Designing proper rules that maximize benefits for each user, while at the same time discouraging “rule breaking,” is thus an interesting problem of networking.
It is easy to see from the above description thatgame theory can be applied to this problem.
One class of game-theoretic approaches uses “virtual payments and credits”: whenever a node acts as relay for somebody else’s message, it receives a “virtual credit”; it can then spend it as payment to other nodes for forwarding its own message when the need arises. A second class of game- theoretic algorithms uses “enforcement” of good behavior either by a watchdog (centralized), or by “reputation-based” algorithms: a node that does not forward messages gets a bad reputation, which negatively affects its own ability to ask other nodes to forward messages for it. In either case, antisocial behavior is discouraged by appropriate punishment through the other nodes.
Clustering and Partitioning
A cluster of nodes can use cooperative communications to bridge larger distances (via collaborative beamforming) than a single node can achieve. Thus, there is a better connectivity of a network, i.e., the probability that a node is isolated (cannot be reached by any route) is smaller for a network allowing collaborative beamforming than for a noncollaborative (multi-hop) network. An example for this is shown in Figure 22.16.
Link range A
Multi-hop Cooperative cluster
A ≥ N–1log(N) + c(N) A ≥ N–14π(4log N) (log log N + log2)
α + 2α 2
α + 2
(a) (b)
Figure 22.16 Improved connectivity through use of cooperative clusters: collaborative beamforming of the cluster in the center increases the possible range.Adenotes the required range of radio coverage to achieve connectivity with high probability;αis the path loss exponent.
Reproduced from Scaglione et al. [2006]©IEEE.
Another case where clustering of nodes comes in handy is in networks where the number of hops is restricted (e.g., to two hops), but there are still a large number of nodes. In that case, it
is required to find suitable cooperation nodes; in other words, which nodes should be “paired up”
for the forwarding of a message. Choosing such node pairs can be considered a special case of so-called “matching problems on graphs,” for which there is a rich literature in computer science and operations research. Particular examples include (i) minimal weighted matching, (ii) greedy matching, and (iii) random matching [Scaglione et al. 2006].
22.5.5 Scaling Laws
Cooperation between nodes leads to a change in the scaling of the network throughput as the node density increases. In Section 22.4.11, we saw that for multi-hop transmission, the throughput per node tends to zero as the node density increases. For cooperative communications, the feasible throughput per node is at least constant; in other words, the aggregate network throughput increases linearly. More precisely, it has been shown that the network throughput cannot increase faster than Nlog(N ); and furthermore there is a known, constructive scheme that achieves a network throughput that scales asN.
This well-scaling scheme is hierarchical cooperation, which as its basic building block contains a three-phase cooperation scheme based on clustering:
1. In the first phase, the source node transmits the information to surrounding nodes. To be more precise, we divide the area containing nodes into cells, and the source node sends the information to the nodes located in the cell. Note that by application of the cellular principle, and an appropriate reuse distance, transmission from source node to surrounding nodes can happen in many cells in parallel. The source then divides the information intoM blocks, and sends one such block to a particular node in the cluster (this does not exploit the broadcast effect).14 2. In the second phase, the cluster of nodes performs MIMO transmission to the cell (cluster) in
which the destination node is located. Each node independently encodes the information block it received in the first phase, thus providing (distributed) spatial multiplexing. The nodes in the receiving cluster quantize the received signal.
3. In the third phase, the nodes in the receiving cluster send this quantized information within the cluster; by appropriate decoding of the spatial-multiplexing signal, a node can recover the original information.
This three-phase cooperation scheme can now be applied in a recursive manner, if distances need to be covered that are larger than what can be achieved (under given power constraints) with a single application of the three-phase scheme. The recursion starts with a small area, and is applied over consecutively larger areas until it can encompass the whole network area (see Figure 22.17).