channel and to be the most forward (RS3). This last strategy attempts to exploitinstantaneouschannel state information at transmission when choos-ing candidate routes, rather than relychoos-ing on average signal-to-interference ratios. All routing strategies assume a-priori information regarding neigh-bors and destination coordinates. This assumes a form of neighneigh-bors discov-ery and route capabilities knowledge. This is clearly a form of geographic routing strategies [62, 63].
The main contributions of this chapter are:
• The study of incremental redundancy as a multiple access technique for ad hoc wireless networks
• Representation of interference and collisions statistics from the homo-geneous Poisson point process network model
• A cross-layer framework where multi-hop routing protocols are ana-lyzed and in particular the channel-driven routing strategy and tools for characterizing the spatial throughput (bit-m/dim, related to the transport capacity) from a microscopic point-of-view as a function of topological parameters (e.g node population density) and system pa-rameters (propagation, bandwith, etc.).
The outline of the chapter is as follows: In Section 4.2, we describe the sys-tem model. Section 4.3 deals with the retransmission protocols. In Section 4.4, throughput expressions are derived and we show some numerical re-sults. Finally, in Section 4.5 we draw some conclusions and point out future research directions.
4.2 System Model and Setting
4.2.1 Network and Propagation Model
We assume that nodes are distributed according to a Poisson point process on the plane with node densityσ. This topology represents an instantaneous snapshot of a mobile network of nodes. Then, for any regionSof areaA(S), the number of nodes in the region has a Poisson distribution with parameter σA(S), i.e.,
Pr[k inS] = e−σA(S)(σA(S))k
k! (4.1)
The propagation model is described by two effects: the signal attenuation due to the distancerbetween the transmitter and the receiver, proportional tor−α, whereαis the power loss exponent (positive number) ; and Rayleigh fading that causes random power variations. The envelope of the received signal is therefore, Rayleigh distributed and its power exponentially dis-tributed. The received powerPR from a mobile at distance r is expressed as:
PR=Kd0Ra2r−αP =Kd0γr−αP (4.2) where Ra is a Rayleigh distributed random variable (with unit power for simplicity), γ is an exponentially distributed random variable (with unit mean) andP is the transmit power. Kd0 represents the signal attenuation at a close-in reference distance [66]. This represents a narrowband channel with respect to the coherence bandwidth of the environment.
4.2.2 System Model and Setting
In the system we are considering, each node can transmit over a common wireless channel. Apart from the slotted transmission structure where nodes transmit packets within slots of defined duration, nodes are completely unco-ordinated (see Fig.4.3). This slotted transmission scheme requires some local frame synchronization method, for instance a form of distributed transmis-sion of pilot signals, or could be based on a common pilot from an external source (e.g. cellular infrastructure, satellite positioning systems, etc.). For the purpose of our analysis, we make the following assumptions:
• An infinite number of packets is available for each source. A packet can be seen as a separate codeword for which transmission is stopped when an acknowledgment of successful decoding is returned by the receiver. Furthermore, we assume that the ACK/NACK feedback sig-naling channel is error-free and delay-free. The sigsig-naling overhead is insignificant with respect to the data channel.
• We suppose single-user decoding where each decoder treats the signals from other users as noise. The single-user decoder for each node has perfect knowledge of the channel gain and the total interference power (i.e. noise and interfering user traffic). This can be achieved in a real system by inserting some pilot symbols.
4.2. System Model and Setting 85 time
slot
User transmission time
slot
User transmission
Figure 4.3: The slotted transmission and the channel random access. Packets are possibly coded over many slots.
• We assume a block-fading channel model. For the first two routing protocols, we assume that the fading remains constant over the whole slot and is an i.i.d process across successive slots in order to provide diversity against fading. In a real system, this can be achieved via slow frequency hopping across a large system bandwidth, where the number of frequencies is typically larger than the number of retrans-mission rounds. For the third routing strategy (RS3), we will assume a long-term static channel, in the sense that the channel remains con-stant over all the retransmission rounds of the protocol. In this case, frequency-hopping is not a possibility, since it could imply changing routes within a transmission round. Nevertheless, it is important to note that in this case, diversity against signal fading is achieved by the routing protocol, since the routes are chosen based on the instanta-neouschannel realization. This is clearly a form ofmulti-user diversity.
Diversity against interference is achieved in all cases by the retrans-mission protocol (Aloha or Incremental Redundancy) since in each slot the number of interferers is random due to the random channel-access transmission strategy.
• For each slot, each node transmits a packet with probability p and remains silent with probability 1−p such that transmit and receive nodes have spatial Poisson distributions with average node density σt=σpand σr =σ(1−p) respectively. One could incorporate in this model the possibility of having some nodes acting as pure relays. These nodes will always be in receive mode, hence increasing the receive node density. This has practical implications for wireless mesh networks, where some nodes act as gateways or repeaters between different sub-networks or different clusters. Moreover, in a slow frequency-hopping system, 1/pcould be the number of frequencies (or the number of sub-bands/carriers in an OFDMA Orthogonal Frequency Division Multiple system) when nodes transmit only on a single frequency for any time-slot coupled with a duty cycle to randomize the access to the channel.
• Each node transmits with fixed power P. Moreover, the signal model is given by:
yj,s=X
k∈Γ
q
γk,j,sP rk,j−αxk,s+nj,s (4.3) where the index s denotes the slot, yj,s the received signal at node j, xk,s the transmitted signal from node k, Γ is the set of transmitter nodes and nj,s the background noise at node j of variance σn2 = W N0 where W is the available system bandwidth (N0 is the background noise power). This represents a narrowband flat fading channel model and the generalization to frequency selective channels is straightforward but diversity features of the access protocols will be reduced. Moreover, this is a more general multiple-access scenario than the interference model considered in [9, 26], since we are at liberty to optimize the transmission probabilitypto randomize inter-ference levels, which is important in a microscopic analysis. Note that this creates a random exclusion area around each node.