[Tccc] Jackson Network and Queueing Theory

Flaminio Borgonovo borgonov
Wed Nov 23 06:31:21 EST 2011


   Dear Victor and Guang-Liang,
   following our messages on your paper, I, and the collegue that signs
   below,  have discussed it thouroughly, and we have agreed that,
   besides logical issues in Theorem 1, there are other primary
   objections to your proof, originated by (5).
   a)  Theorem 1 derives from relation (5), that we have interpreted as
   (roughly speaking):
   - if waiting times are unbounded the queue is unstable
   Since a variable can be unbounded even if finite, as it is for the
   negative exponential, (5) implies that queues with waiting times with
   such distributions cannot be stable. In our view, the thesis of
   Theorem 1 is already here: unbounded W_n implies unstable queue.
   Therefore (contrapositive) stable queue implies bounded W_n. Therefore
   we must  clarify the origin of (5).
   b) We have looked at Loynes'article, that you cite as [3], and we have
   not found  assertion (5), that you refer to [3]. The only result that
   resembles (5) is, in Loynes' article, Theorem 1, which asserts that "a
   queue is stable iff
   lim_n sup sum U_k < infty",
   which is completely different from your (5). The condition above, in
   fact, is not contradicted by negative exponential service and
   interarrival times with E[U_k]<0. In this case, in fact, E[sum
   U_k]=-infty.
   Can you explain these objections?
   Regards
   Flaminio and Luca Barletta

   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
   visit  [2]http://www.como.polimi.it/Patria/
   

References

   1. http://home.dei.polimi.it/borgonov/index.html
   2. http://www.como.polimi.it/Patria/



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