Greetings Victor, What you say is correct, but does not address the argument that I sent you off-list. For the benefit of the list, I'll repeat the argument here.
If "P(B)=1" is false (as it is for many interesting queues, and in particular for M/M/1 queues), then, as you correctly say, V( [P(B)=1], [P(S)=0] ) = V(F, [P(S)=0]) = T This contradicts your claim in the first sentence of the proof of Theorem 1 that V( [P(B)=1], [P(S)=0] ) = F for all queues in G. As a second point, you say 'If one lets the value of "P(B) = 1" be "true" ', but line 8 of the proof says that P(B)=1 is a conclusion, not a hypothesis. Cheers, Lachlan On 14 November 2011 15:51, Prof. Victor Li <[email protected]> wrote: > Dear Lachlan and Flaminio, > > Thanks for your comments. Let V(X, Y) stand for > "X implies Y", a logical implication statement in general. > Write (X, Y) = (T, F) to mean "the value of X is T (true) and > the value of Y is F (false)". Let V(T, F) = F represent > "the value of V(X, Y) is F when (X, Y) = (T, F)". > Smilarly, V(F, F) = V(F, T) = V(T, T) = T. > In fact, V(T, F) = F and V(F, F) = V(F, T) = V(T, T) = T > correspond to the truth table values of the implication, and > V(T, F) = F is completely determined by (X, Y) = (T, F) > regardless of what X or Y means. Any other assumption > is unnecessary for V(T, F) = F. > Unless one is reasoning with a different logic, > V(T, F) = F is just fine both in general and in particular > when X and Y represent P(B) = 1 and P(S) = 0, respectively. > If one lets the value of "P(B) = 1" be "true", then one must > let the value of "P(S) = 0" be "false" because a queue with a.s. > bounded waiting time is not unstable. > > > Best regards, > > Guang-Liang and Victor -- Lachlan Andrew Centre for Advanced Internet Architectures (CAIA) Swinburne University of Technology, Melbourne, Australia <http://caia.swin.edu.au/cv/landrew> Ph +61 3 9214 4837 _______________________________________________ IEEE Communications Society Tech. Committee on Computer Communications (TCCC) - for discussions on computer networking and communication. [email protected] https://lists.cs.columbia.edu/cucslists/listinfo/tccc
