Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

Probabilistic Protocols for Node Discovery 1092 Random Protocol for Two Node Multi-channel SystemIn the random protocol RP, each node decides at random whether to talk (T )or listen (L). The two nodes thus generate an event at each time t( ) S iE =S ′ i ′such that S and S ′ are either T or L, and i and i ′ are the frequencies chosen.We use the technique described in [3,11] to analyze the RP protocol. Therandom protocol RP succeeds when, for some i, j =1, 2,...,f, either( ) ( ) ( ) ( )Ti Lj Li Tj→ or →(3)Li Tj Ti LjDefine the events A i and B i as follows:( TiA i =Li), B i =( ) LiTiA success pattern is a pair of events such that the two nodes discover eachother, e.g., A i B j , i, j ∈ 1,...,f, and thus there are 2f 2 success patterns:(4)A 1 B 1 ,...,A 1 B f ,...,A f B 1 ,...,A f B f ,B 1 A 1 ,...,B 1 A f ,...,B f A 1 ,...,B f A f .For each i, j ∈{1, 2,...,f}, the pattern A i B j (respectively B i A j ) may eitheroverlap itself, or the last event of A i B j (respectively B i A j ) may overlap with thefirst event of B j A k (respectively A j B k ) for k ∈{1, 2,...,f}. The former case1occurs with probabilityp iq ip jq jwhile the latter case occurs with probability 1p jq j.The resulting system of 2f 2 linear equations is⎡ ⎤E[N]⎢E[N]⎥[ ][ ]D U Π=U D Π.E[N]E[N]E[N]⎢ ⎥⎣ . ⎦E[N]whereΠ ′ =[π 1,1 ,...,π 1,f ,π 2,1 ,...,π f,1 ,...,π f,f ]and π i,j (respectively π i+f,j ) is the probability that A i B j (respectively B i ,A j )occurs before any other pattern. N is the run time for exactly one subrun ofRP . However, because RP always executes exactly one subrun, N is also therun time for the entire protocol. [ ] D UThe 2f 2 x2f 2 matrix is defined as follows:U D(5)