[Tccc] Jackson Network and Queueing Theory

Lachlan Andrew lachlan.andrew
Tue Nov 15 18:42:39 EST 2011


Greetings Victor,

On 16 November 2011 01:10, Prof. Victor Li <vli at eee.hku.hk> wrote:
>
> Let X represent P(B) = 1. Let Y represent P(S) = 0.
> 1) Let the value of X be T (denote this by X = T).

You are giving two definitions of  X.  You can't do that.  Otherwise
you could prove that black = white by saying "Let X = black.  Let X =
white.  Then black = X = white".

Please choose your definition of X, and then we can continue the discussion.

> In your comments based on an M/M/1 queue,
> there seems to be a misunderstanding about V(T, F) = F. To see V(T, F) = F,
> one assumes X = T. Even if X = T does not hold, V(T, F) = F
> is still correct. Moreover,
> your comment concerns only values (X, Y) = (u, z) where u and z are in {T,
> F} such that V(u, z) = T. This makes the comment irrelevant to V(T, F) = F.

Of course V(T,F)=F.  I don't think I've ever disputed that (even
though I've said many wrong things in this thread ):

My argument again is:
   a) For some queues in  G,   (P(B)=1)=F.
   b) if  X=F  then V(X,F)=V(X,T)=T.
   c) Hence, there exists a queue in G, such that  V(P(B)=1, P(S)=0) = T.
   d) Hence, it is not true that for all queues in G, V(P(B)=1, P(S)=0) = F.

Please tell me which step you disagree with.

You are right that my comments are about cases where  V(u,z)=T.  I
agree this is irrelevant to V(T,F)=F, but it is central to the flaw in
the first sentence of your proof: for particular statements u,z, you
claim that    V(u,z)=F for all queues, and I have argued that V(u,z)=T
for some queues.

> As to the second point raised in your e-mail, we would like to address the
> difference between
> the contexts of 'If one lets the value of ? "P(B) = 1" be "true" ' in our
> previous e-mail
> and the context of "Thus P(B) = 1 is true" in line 8 of the proof.
> In the former we discuss V(T, F) = F. In the latter we prove P(B) = 1 for a
> stable queue
> as shown by the first 8 lines of the proof.

Fair enough.  For clarity, I propose we reserve the word "let" for
definitions, rather to mean "if this happens to be true".

Cheers,
Lachlan


-- 
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




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