[Tccc] Jackson Network and Queueing Theory

Prof. Victor Li vli
Sun Nov 13 23:51:16 EST 2011


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]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. http://home.dei.polimi.it/borgonov/index.html
   2. 
http://www.eee.hku.hk/research/doc/tr/TR2011003_Queueing_Theory_Revisited.pdf
   3. 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
https://lists.cs.columbia.edu/cucslists/listinfo/tccc 





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