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

110 G. Alonso et al.– D is a diagonal f 2 x f 2 1matrix with the ((i, j), (i, j))-th entry equal to .p 2 i q2 i– U is an f 2 x f 2 matrix formed by a column of f matrices, i.e., U =[V,V,...,V] T where V is defined as⎡V =⎢⎣With the condition1p 1q 11p 2q 2···10 0 ··· 01p f q f0 0 ··· 0 ··· 0 0 ··· 01 1p 1q 1 p 2q 2···p f q f··· 0 0 ··· 0.............1 1 10 0 ··· 0 0 0 ··· 0 ···p 1q 1 p 2q 2···⎤⎥⎦p f q f∑i=fj=f∑π ij = 1 (6)i=1 j=1the resulting system of linear equations has 2f 2 + 1 unknowns and 2f 2 +1equations. Solving this system of equations gives us E[N], the expected runtimeof RP.Theorem 1 (RP). The expected run time for the RP protocol is:1+ ∑ fj=1 p jq j( ∑f) 2. (7)2j=1 p jq j3 Conditional Protocolfor Two Node Multi-channel SystemsThe conditional protocol CP is implemented as a series of two-phase subrunswhere phase 1 consists of a finite number of random steps and phase 2 consistsof a single step.In phase 1 of a subrun of the CP protocol, a node follows the random protocolRP until 1) the node talks (T ) or 2) the node listens (L) and hears the othernode’s broadcast.Phase 2 of a subrun of the CP protocol consists of a single step. The behaviourof a node in phase 2 is conditional on the way in which phase 1 ended.If a node talked (T ) at the end of phase 1, then it will listen (L) in the phase2 in an attempt to determine if the other node heard its broadcast. If a nodelistened (L) and heard the other node’s broadcast at the end of phase 1, then itwill talk (T ) in phase 2 in an attempt to answer the other node’s broadcast.If a subrun is successful, then node discovery occurs in phase 2 and the CPprotocol terminates. If a subrun is unsuccessful, however, then another subrunis executed, i.e., phase 1 and phase 2 are repeated, until node discovery occurs.Let the probability of success in a subrun of the CP protocol be denotedby Pr[success in subrun]. Since the subruns of the CP protocol are independenttrials, the number of subruns of the CP protocol is a geometric random variable