Dear everyone:
I have a question regarding the permutation traffic pattern.
Background
Since the seminal work of Gupta and Kumar [1], extensive efforts have been devoted to study the throughput capacity of wireless networks restricted to the so-called permutation traffic pattern. It is widely considered in the literature that under the permutation traffic pattern, the source-destination pairs are matched at random in a way that the destination sequence is a permutation of the source sequence, e.g., [1-3].
Question
My question is as follows.
A general permutation may have fixed points. An element is called a fixed point of a permutation, if it is mapped to itself under the permutation. E.g., for the permutation :(1,2,3)->(2,1,3), 3 is a fixed point of this permutation.
Since a permutation may have fixed points, the permutation traffic pattern literally allows a source to select itself as the destination of the traffic flow originated from this node. It is notable, however, it does not make sense that a node transmits to itself in a wireless network. Besides, in this case, the throughput capacity would become infinite, since the transmission to oneself does not use the wireless medium.
Therefore, the term permutation traffic pattern might be misleading in my understanding. And I am wondering whether this misleading term can be replaced by the derangement traffic pattern. Since a derangement is a permutation without any fixed point, the confusing case of talking to oneself can be avoided.
Does anyone agree with me or have a better understanding/explanation on the permutation traffic pattern?
Best,
Yin
Reference
[1] P. Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Transactions on Information Theory, 46(2):388–404, March 2000.
[2] Niesen, Urs, Piyush Gupta, and Devavrat Shah. "On capacity scaling in arbitrary wireless networks." Information Theory, IEEE Transactions on 55, no. 9 (2009): 3959-3982.
[3] Garetto, Michele, Paolo Giaccone, and Emilio Leonardi. "Capacity scaling in ad hoc networks with heterogeneous mobile nodes: The subcritical regime." IEEE/ACM Transactions on Networking (TON) 17, no. 6 (2009): 1888-1901.
I fully agree. Maybe the reason why derangement is not used is that the term is a lot less common, and people do not like 'deranged'.
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