Rate-Distortion Bounds on Location-Based Routing Protocol Overheads in Mobile Ad Hoc Networks Nabhendra Bisnik and Alhussein Abouzeid Department of Electrical Computers and System Engineering Rensselaer Polytechnic Institute Troy, NY 12180 bisnin@rpi.edu, abouzeid@ecse.rpi.edu Abstract—We present an information theoretic analysis of the minimum routing overhead incurred for reliable routing of packets using location-based routing. We formulate the mini- mum routing overhead problem as a rate-distortion problem and derive a lower bound on the minimum routing overhead incurred. We also characterize the deficit in transport capacity caused by the routing overheads. It is observed that for high mobility and packet arrival rates, the routing overheads may consume the entire capacity of a network. I. I NTRODUCTION The time varying topology of mobile networks makes efficient routing of packets a challenging problem. Under high node mobility, existing routing protocols for either incur a large overhead in order to cope with the mobility or yield low delivery ratio. Understanding the lower bounds on the amount of protocol overhead incurred for reliable routing of packets is important for development of efficient routing pro- tocols, and for understanding the effective network capacity available to the network users. In this paper we present an analytical information-theoretic framework for characterizing the lower limit on over- head incurred by geographical routing protocols in a one- dimensional mobile wireless. In geographic routing each node maintains its location information at one or more location servers (e.g. see [3]). When a source wants to forward a packet to a destination, it queries an appropriate location server for location of the destination. The location server replies to the source node with the available location information. Thereafter, the source and intermediate nodes forward the packet according to the location of the des- tination. The mobility model considered in this paper is Brownian motion with variance σ 2 . We divide the geographic routing overheads into two cate- gories: (i) Location update overhead: The overhead incurred in updating the location servers such that the location errors in the reply to location queries is less than ǫ, and (ii) Beacon overhead: The overhead incurred in beacon transmission such that the probability that a node has consistent neigh- borhood information when it needs to forward a packet is greater than 1 - δ. We formulate the problems of finding the minimum values of the above-mentioned overheads as rate distortion problems [2]. For location update overheads, the distortion measure used is squared error in the location stored at the location servers (squared error distortion error). For beacon overheads, the distortion measure is the probability that a perceived neighbor is not a actual neighbor ( Hamming distortion measure). Using a rate-distortion formulation, we present lower bounds on the minimum geographic routing overhead incurred in terms of node mobility, packet arrival process, and reliability criteria ǫ and δ. We also discuss the residual capacity of mobile ad hoc networks after taking into account the routing overheads. II. LOCATION UPDATE OVERHEAD Let X i (t) and ˆ X i (t) denote the actual location of node i and the location of the node available at the location server at time t. Let T i (j ) denote the time at which j th packet destined to node i arrives in the network and let f S (t) denote the pdf of the packet inter-arrival time (T i (j ) - T i (j - 1)). It is required that the expected deviation of ˆ X i (t) from X i (t) should be less than ǫ 2 . Let P N denote the family of probability density functions for which D iN = 1 N N  j=1 E[D i (T j ] ≤ ǫ (1) where D i (t)= |X i (t) - ˆ X i (t)|. Definition 1: R N (ǫ 2 ) is defined as the minimum rate (in terms of bits per packet) at which a destination must transmit its location information to its location server such that D iN ≤ ǫ 2 . According to [2], R N (ǫ 2 ) is given by R N (ǫ 2 )= min PN∈PN(ǫ 2 ) 1 N I PN (X N i ; ˆ X N i ) (2) where I PN (X N i ; ˆ X N i ) is the mutual information between X N i and ˆ X N i , X N i = {X i (T 1 ),X i (T 1 ),...,X i (T N )}, and ˆ X N i = { ˆ X i (T 1 ), ˆ X i (T 1 ),..., ˆ X i (T N )}. The minimum rate at which a destination must update its location information such that a large fraction of packets are delivered, represented by R(ǫ 2 ), is given by R(ǫ 2 )= lim N→∞ min R N (ǫ 2 ) (3) Theorem 1: In order to ensure high delivery ratio, the lower bound on the location update rate (in bits per packet)