[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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