[Tccc] Jackson Network and Queueing Theory
Flaminio Borgonovo
borgonov
Mon Nov 14 13:01:11 EST 2011
Dear Victor,
I'm afraid your argument below does not catch my point. Below You list
what is known as "Table of implications" and conclude that
<< If one lets the value of "P(B) = 1" be "true", then one must
let the value of "P(S) = 0" >>
I never constested the above statement (although I agree with Lachlan
that it is not applicable to all queues in the set \cal G)
I only contested the use of the contrapositive argument that has led
you to affirm, in theorem 1, that the following statrement is also
true
<< P(S)=1 implies P(B)=1>>. I think you cannot logically derive this
last statement from the former.
I restate below my argument, expliciting the attribute TRUE where I
have assumed it by default
Assume:
<<P(B)=1 implies P(S)=0>> is false.
Therefore we must derive:
P(B)=1 implies P(S)=1 is TRUE
Its contrapositive statement, should be
P(S)=0 implies P(B)=0, TRUE
and not, as sustained in the paper,
P(S)=1 implies P(B)=0, FALSE , or P(S)=1 implies P(B)=1 TRUE.
best regards
Flaminio
At 05.51 14/11/2011, Prof. Victor Li 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
-----Original Message----- From: Flaminio Borgonovo
Sent: Saturday, November 12, 2011 7:41 AM
To: Prof. Victor Li
Cc: tccc at lists.cs.columbia.edu ; glli at eee.hku.hk
Subject: Re: [Tccc] Jackson Network and Queueing Theory
Hi Victor and Guang-Liang,
I think the contrapositive argument in Theorem 1 of your
paper is
misused, as Lachlan suspects.
The proof of Theorem 1 shows that statement <<P(B)=1 implies
P(S)=0>>
is false. Therefore we must assume:
P(B)=1 implies P(S)=1 (bounded waiting times ----> stable
queue).
Its contrapositive statement, still true, should be
P(S)=0 implies P(B)=0, (unstable queue ----> unbounded waiting
times)
and not, as sustained in the paper,
P(S)=1 implies P(B)=1, (stable queue ----> bounded waiting
times).
Therefore P(B)=1 for stable queues is not proved. And classic
theory,
where it can be also P(S=1) and P(B)=0 (stable queue and
unbounded
waiting times), is not invalidated.
This seems to invalidate Theorem 1 and the whole paper.
Regards,
Flaminio
Prof. Flaminio Borgonovo
Dip. di Elettronica e Informazione
Politecnico di Milano
P.zza L. Da Vinci 32
20133 Milano, Italy
tel. 39-02-2399-3637
fax. 39-02-2399-3413
e-mail: borgonov at elet.polimi.it
URL [1][1] http://home.dei.polimi.it/borgonov/index.html
On 10 November 2011 18:18, Prof. Victor Li <vli at eee.hku.hk>
wrote:
> Dear colleagues,
>
> Nearly a decade ago we initiated a discussion about Jackson
networks
of queues
> on this mailing list. Since then some colleagues have enquired
about
our follow-up
> research regarding this issue. A recent paper by us is now
available
> as a technical report at the website below:
>
>
[2]http://www.eee.hku.hk/research/doc/tr/TR2011003_Queueing_Theory_
Rev
isited.pdf
>
> In this paper we consider the stability of queues. We find that
> the condition given in the literature, i.e., the traffic
intensity
is less
> than 1, is only necessary but not sufficient for a general
single-server queue to be
> stable. This shows again that product-form solutions of Jackson
networks
> are incorrect for such networks are actually unstable.
> In the paper we also give necessary and sufficient conditions
for a
G/G/1 queue to
> be stable, and discuss the implications of our results.
>
> Queueing theory has been widely used in performance analysis of
computer and
> communication systems. Colleagues who are teaching courses on
performance
> analysis or doing research in this area, and students who are
learning how to apply
> queueing theory to performance analysis, might be interested in
our
results.
> Comments on our paper are very much appreciated and can be sent
to
us by
> e-mail. Thank you very much for your attention.
>
> Best regards,
>
> Guang-Liang Li and Victor O.K. Li
> _______________________________________________
> IEEE Communications Society Tech. Committee on Computer
Communications
> (TCCC) - for discussions on computer networking and
communication.
> Tccc at lists.cs.columbia.edu
> [3]https://lists.cs.columbia.edu/cucslists/listinfo/tccc
>
References
1. [2]http://home.dei.polimi.it/borgonov/index.html
2.
[3]http://www.eee.hku.hk/research/doc/tr/TR2011003_Queueing_Theory_
Revisited.pdf
3. [4]https://lists.cs.columbia.edu/cucslists/listinfo/tccc
_______________________________________________
IEEE Communications Society Tech. Committee on Computer
Communications
(TCCC) - for discussions on computer networking and communication.
Tccc at lists.cs.columbia.edu
[5]https://lists.cs.columbia.edu/cucslists/listinfo/tccc
Prof. Flaminio Borgonovo
Dip. di Elettronica e Informazione
Politecnico di Milano
P.zza L. Da Vinci 32
20133 Milano, Italy
tel. 39-02-2399-3637
fax. 39-02-2399-3413
e-mail: borgonov at elet.polimi.it
URL [6]http://home.dei.polimi.it/borgonov/index.html
visit [7]http://www.como.polimi.it/Patria/
References
1. http://home.dei.polimi.it/borgonov/index.html
2. http://home.dei.polimi.it/borgonov/index.html
3. http://www.eee.hku.hk/research/doc/tr/TR2011003_Queueing_Theory_Revisited.pdf
4. https://lists.cs.columbia.edu/cucslists/listinfo/tccc
5. https://lists.cs.columbia.edu/cucslists/listinfo/tccc
6. http://home.dei.polimi.it/borgonov/index.html
7. http://www.como.polimi.it/Patria/
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