Dear Flaminio,
An implication is equivalent to its contrapositive. If two implications are not equivalent, then their contrapositives are not necessarily equivalent. Please allow us to continue using our notations in the following clarification. We use X and Y to represent P(B) = 1 and P(S) = 0, respectively. From V(T, F) = F we have V'(T, F) = F where V' is the contrapositive of V, and V'(T, F) = F means "the value of V'(X' Y') is F when (X', Y') = (T, F) with X' and Y' representing P(S) = 1 and P(B) = 0, respectively. Your argument concerns the contrapositive of V(X, X'). Since V(X, X') and V(X, Y) are not equivalent, the contrapositive of V(X, Y) and the contrapositive of V'(X, X') are not equivalent. You also said that you agreed with Lachlan that V(X,Y) = F is not applicable to all queues in the set \cal G. This is incorrect. Please see the reply to Lachlan in the previous email. Best regards, Guang-Liang and Victor From: Flaminio Borgonovo Sent: Tuesday, November 15, 2011 2:01 AM To: Prof. Victor Li ; Flaminio Borgonovo ; [email protected] Cc: [email protected] ; [email protected] Subject: Re: [Tccc] Jackson Network and Queueing Theory 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: [email protected] ; [email protected] 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: [email protected] URL [1] http://home.dei.polimi.it/borgonov/index.html On 10 November 2011 18:18, Prof. Victor Li <[email protected]> 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. > [email protected] > [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. [email protected] 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: [email protected] URL http://home.dei.polimi.it/borgonov/index.html visit http://www.como.polimi.it/Patria/ _______________________________________________ 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
