[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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