A simple network of nodes moving on the circle

Two simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile networks. We consider two independent walkers moving with constant speed on the discrete or continuous circle, and changing directions at indepe...

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Bibliographic Details
Published in:Random structures & algorithms 2020-09, Vol.57 (2), p.317-338
Main Authors: Cheliotis, Dimitris, Kontoyiannis, Ioannis, Loulakis, Michail, Toumpis, Stavros
Format: Article
Language:English
Subjects:
Boundary value problems
Carrier mobility
Current carriers
ergodicity
generator
Markov process
Markov processes
network routing
stochastic networks
Wireless networks
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Summary:Two simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile networks. We consider two independent walkers moving with constant speed on the discrete or continuous circle, and changing directions at independent geometric (respectively, exponential) times. One of the walkers carries a message that wishes to travel as far and as fast as possible in the clockwise direction. The message stays with its current carrier unless the two walkers meet, the carrier is moving counter‐clockwise, and the other walker is moving clockwise. Then the message jumps to the other walker. Explicit expressions are derived for the long‐term average clockwise speed and number of jumps made of the message, via the solution of associated boundary value problems. The tradeoff between speed and cost (measured as the rate of jumps) is also examined.
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20932