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**Cross-Layer Performance Analysis for Decentralized Multihop Wireless Networks**

Tarik Tabet Mobile Communications Laboratory Ecole Polytechnique Fédérale de Lausanne Lausanne, Switzerland Raymond Knopp Mobile Communications Department Eurecom Institute Sophia-Antipolis, France

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Motivation/Goal Rapidly-deployable small-scale multihop wireless networks (e.g. broadband hotspots for emergency/disaster relief) Motivated by Cross-Layer mechanisms (PHY/MAC/Routing) aiming at maximizing the spectral efficiency of the network Tool for characterizing the transport capacity of such networks (bit m/s) from a “microscopic” point-of-view as a function of topological parameters (e.g. node population density) and system parameters (propagation, bandwidth,

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**Interplay between Tools**

Interference statistics Multiple-Access/Coding for Decentralized Channels (Collision Channels) Massey+Mathys (IT-1985) Caire + Tuninetti (IT-2000) Random Network Topologies (Spatial Processes) Kleinrock+Nelson (COM-84) Sousa (IT 1992) Baccelli (2003) Achievable link throughput/delay Geographic Routing Strategies and performance metrics Baccelli et al (2003)

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**Network and propagation model**

Nodes are spatially distributed in the plane according to a homogeneous spatial Poisson process[NelsonK84]. The number of nodes in a region S of area A(S) is: The propagation model is described by the signal attenuation due to the distance r between the transmitter and the receiver and the Rayleigh fading (narrowband/flat):

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**System model and setting**

Slotted transmission structure. Messages are potentially coded across many slots (interference diversity). Slotted Gaussian Collision channel with fading. An infinite number of packets available for each source (stability of protocols is ignored in this study, no queuing). ACK/NACK feedback signaling channel is error-free and delay-free. Single-user decoding.

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**System model and setting…**

Block-fading channel model. Signal strength Slot duration t

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**System Model and Setting**

time slot User transmission

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**System model and setting…**

For routing based on average SINR (RS1,RS2), the fading is an i.i.d process across successive slots -> diversity against fading with coding. In a real system, this can be achieved via slow frequency hopping across a large system bandwidth. For (RS3), we assume a long-term static channel (i.e. constant over slots) -> diversity against fading is realized by a form of multiuser diversity in the routing scheme. 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 p and 1-p) respectively. Each node transmits with fixed power P.

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MAC protocols Nodes access the channel at random and employ simple protocols to retransmit the erroneously received packets. Two retransmission protocols: Slotted Aloha and Incremental Redundancy. Analytical techniques are based on [CaireT01] adapted to this channel interference scenario. Spatial Poisson process characterization leads to a new representation of interference and collisions between concurrent transmissions for the Gaussian collision channel with fading.

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MAC protocols… We compare these strategies to the generalization of the collision channel without feedback or delay constraints [MasseyM85]. As it will be seen later, the measure of success of a transmission will be an achievable ergodic throughput of this channel. 1 2 3 4 5 collision Decoding interval

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MAC protocols… In the case of coded slotted aloha or incremental redundancy, the decoding stops after the reception of an ACK on the feedback channel leading to a finite average delay. When packets from different nodes collide, it may still be possible to successfully decode the packet with the strongest received signal power, which is known as the "capture effect”. The coded slotted aloha could be generalized to an M -slotted aloha where each codeword is split in M parts and transmitted in M slots (in order to benefit from some diversity). We consider that the M blocks are independent and M is fixed. M controls the average delay.

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**Signal model The signal model is given by:**

where the index s denotes the slot, yj,s the received signal at node j, xk,s the transmitted signal from node k, nj,s the background noise at node j. Link performance of different schemes is characterized by the instantaneous average mutual information (information outage probability)

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**Information outage probability**

The instantaneous average mutual information for a (i, j) pair of nodes: where

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**Information outage probability…**

We need to compute the MGF of V [Sousa92] the p.d.f of the distance between a transmitter and a receiver is given by:

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**Information outage probability…**

We finally obtain : and is the Gamma function.

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**Incremental Redundancy**

It adjusts the code rate by incrementally transmitting redundancy information until decoding is successful. Node k encodes its message information of b bits each independently of other nodes by using a channel code with code book where N is the slot length and L is the accumulated number of slots. Codewords are divided into L sub-blocks of length N, and we let for denote the punctured code of length obtained from by deleting the last sub-blocks.

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**Incremental Redundancy…**

If successful decoding occurs at the l-th transmission, the effective coding rate for the current codeword is R/l bit/dim where R=b/N. For the sake of computing information theoretic quantities, we let Following the analysis in [CaireT01], we define the throughput as: bit/dim where is the mean delay measured in slots for the transmission of an information message.

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**Incremental Redundancy…**

In incremental redundancy, the receiver has memory of the past signals since it accumulates mutual information. The probability of successful decoding after l transmitted slots is given by: where and we used the fact that for The throughput is:

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**Slotted Aloha Again the throughput is defined as: bit/dim.**

In Aloha, the receiver has no memory of the past signals, and the probability of successful decoding after l transmitted slots is given by:

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Slotted Aloha… The mean delay is then given by: And the throughput:

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**Spatial throughput expressions**

The spatial throughput is expressed as a function of the product of the number of the simultaneously successful transmissions per unit space by the average jump made by each transmission [NelsonK84]. We carry out its optimization with respect to the channel access probability p (MAC) and target Spectral Efficiency R (PHY) The relationship between the spatial throughput and the Gupta-Kumar transport capacity is described in [Baccelli03].

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**Expected forward progress**

The expected forward progress of a packet in the direction of its final destination F, is the distance Z between the transmitter and the receiver (an intermediate node) projected onto a line towards the final destination and the transmission to that receiver is successful.

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Routing strategies In the following we consider three routing strategies: one that maximizes the expected forward progress by moving the packet to the node most forward towards the final destination (RS1). the second moves the packet to the closest node in range. Similar strategy is considered in [GrossT02] where the transmit range is on the order of , n being the number of nodes in an unit area, and in [YuenY03] in the context of mobile infostations networks (RS2). The third where the next hop is selected to exploit the best channel and to be the most forward -> attempt to exploit instantaneous channel state information at transmission when choosing candidate routes (RS3).

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**Spatial throughput The spatial throughput is defined as**

where is the expected forward progress for strategy , and (for u=RS1, RS2), for slotted Aloha and for incremental redundancy.

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**Maximal expected forward progress (RS1)**

We are looking for a receiver that maximizes the forward progress, and we consider a sender centric transmission model We obtain finally

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**Closest node in range (RS2)**

The p.d.f of the minimum distance between the transmitter and the receiver among all the receive node distances is : The expected forward progress for this strategy is:

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**Channel driven maximal forward progress (RS3)**

Exploit multi-user diversity for slowly varying channels in dense networks. Instantaneous CSI is available at the transmitter in order to select the next hop which is the most forward and with the best channel. The forward progress is: The outage probability is conditioned on the knowledge of the channel at the transmitter. We make use of a result on stable distributions with exponent 1/2.

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**Channel driven maximal forward progress RS3…**

The MGF of V is given by:

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**Channel driven maximal forward progress RS3…**

The outage probability is then: Which leads to an expected forward progress for RS3 where :

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Numerical results The spatial throughput is expressed as a function of different system parameters: the transmit SNR P/N0the target information rate R, the transmit probability p, and the node density . The optimal throughput vs. the target information rate is derived by maximizing over the channel access probability p.

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**Numerical Example: SNR = 5dB, s=1 node/m2**

Increasing average delay Spatial Throughput (bit – m /dim/m2) Rate R

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Numerical results… The channel driven strategy performs substantially better than the other strategies by exploiting transmissions only to nodes with instantaneously good channels. This suggests that routing should be based on the instantaneous channel strength of the link, which could require fast route updates (in comparison to existing routing protocols for ad hoc networks) if the channel changes rapidly. For the maximal expected forward progress, routing is based on a spatial empirical average of the SINR's at the transmitter among the nodes in its proximity. This is reasonably simple for slowly varying channels and could be included in existing routing protocols.

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Future directions Receiver-oriented MAC where scheduling (SINR-based) of transmitters at the receiver. MIMO and directional antennas: interference mitigation + spectral efficiency increase. Analysis of multi-user detection techniques. Non-uniform or event-driven traffic

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Conclusions We derived formulas for the spatial throughput for simple MAC protocols and transmission strategies for random networks described by a spatial Poisson point process. ->Allows for an operational assessment of an ad hoc deployment. It is shown that coding and retransmission protocols are a viable and simple solution for providing fully decentralized multiple-access communications in ad hoc wireless networks despite harsh propagation characteristics (interference from nearby competing nodes). Random exclusion and a decentralized protocol allow for the mitigation of the interference coming from other nodes. A routing protocol aiming to maximize the expected forward progress and exploiting multi-user diversity is shown to significantly outperform other routing schemes.

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