# American Institute of Mathematical Sciences

2011, 1(4): 763-780. doi: 10.3934/naco.2011.1.763

## Markovian characterization of node lifetime in a time-driven wireless sensor network

 1 Department of Mathematics and Computer Science, University of Balearic Islands, 07122, Palma, Spain

Received  June 2011 Revised  August 2011 Published  November 2011

While feeling honoured for being invited to write a paper dedicated to Prof. Yutaka Takahashi, I was enthusiastically wondering how to connect my current research on sensor networks to his excellent professional profile. The question or, better, the answer, was not simple. Considering, for instance, the field of Markov chains, as far as I know there are hardly works in literature that use this well-known modelling paradigm to represent the operational states of a sensor network. However, in a very recent work on time-driven sensor networks, I proposed the exponential randomization of the sense-and-transmit process, in order to avoid tight synchronization requirements while preserving good expectations in terms of lifetime and reconstruction quality. But$\ldots{}$oh, I said exponential, that's the connection! $\ldots{}$ So, specifically, in this paper a Markov chain is constructed to characterize the activity of a node in a time-driven sensor network based on stochastic (exponential) sampling. Since this activity can be translated to energy consumption, the exact solution to the Markov chain yields the complete statistical distribution of node lifetime. The effects of several parameters on the average and variance of this lifetime are also analyzed in detail.
Citation: Sebastià Galmés. Markovian characterization of node lifetime in a time-driven wireless sensor network. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 763-780. doi: 10.3934/naco.2011.1.763
##### References:
 [1] K. Akkaya and M. Younis, A survey on routing protocols for wireless sensor networks, Ad Hoc Networks, 3 (2005), 325-349. [2] A. V. Balakrishnan, On the problem of time jitter in sampling, IRE Trans. Inf. Theory, 8 (1962), 226-236. [3] G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, "Queueing Networks and Markov Chains," 2nd edition, Wiley, New York, 1998. doi: 10.1002/0471200581. [4] R. L. Cook, Stochastic sampling in computer graphics, ACM Trans. on Graphics, 5 (1986), 51-72. doi: 10.1145/7529.8927. [5] S. Galmés, System design issues in time-driven sensor networks based on stochastic sampling, Simulation Modelling, Practice and Theory, 19 (2011), 1530-1543. doi: 10.1016/j.simpat.2011.04.006. [6] S. Galmés and R. Puigjaner, Randomized data-gathering protocol for time-driven sensor networks, Computer Networks Journal, 55 (2011), 3863-3885. doi: 10.1016/j.comnet.2011.08.002. [7] J. R. Higgins, "Sampling Theory in Fourier and Signal Analysis: Foundations," Oxford University Press, Oxford, 1996. [8] H. Karl and A. Willig, "Protocols and Architectures for Wireless Sensor Networks," Wiley, New York, 2005. [9] B. Krishnamachari, "Networking Wireless Sensors," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511541025. [10] A. Kumar, P. Ishwar and K. Ramchandran, On distributed sampling of smooth non-bandlimited fields, in ''Proc. of the Third International Symposium on Information Processing in Sensor Networks," Berkeley, CA, USA, (2004), 89-98. [11] A. Kumar, P. Ishwar and K. Ramchandran, On distributed sampling of bandlimited and non-bandlimited sensor fields, in ''Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing," Montreal, Canada, (2004), III-925-III-928. [12] , "OMNeT++ Community Site," OMNeT++ 3.x documentation and tutorials,, 2005. Available from: , (). [13] A. V. Oppenheim, A. S. Willsky and I. T. Young, "Signals and Systems," Prentice-Hall, New Jersey, 1983. [14] G. Reise and G. Matz, Distributed sampling and reconstruction of non-bandlimited fields in sensor networks based on shift-invariant spaces, in ''Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing," Taipei, Taiwan, (2009), 2061-2064. [15] M. L. Santamaría, S. Galmés and R. Puigjaner, Simulated annealing approach to optimizing the lifetime of sparse time-driven sensor networks, in ''Proc. of 2009 IEEE International Symposium on Modeling, Analysis & Simulation of Computer and Telecommunication Systems," London, UK, (2009), 193-202. [16] I. Stojmenovic, "Handbook of Sensor Networks: Algorithms and Architectures," Wiley, New York, 2005. [17] S. Tilak, N. Abu-Ghazaleh and W. R. Heinzelman, A taxonomy of wireless micro-sensor network models, ACM Mobile Computing and Communications Review (MC2R), 6 (2002), 28-36. [18] Y. Yu, V. K. Prasanna and B. Krishnamachari, Energy minimization for real-time data gathering in wireless sensor networks, IEEE Trans. on Wireless Communications, 5 (2006), 3087-3096. doi: 10.1109/TWC.2006.04709.

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##### References:
 [1] K. Akkaya and M. Younis, A survey on routing protocols for wireless sensor networks, Ad Hoc Networks, 3 (2005), 325-349. [2] A. V. Balakrishnan, On the problem of time jitter in sampling, IRE Trans. Inf. Theory, 8 (1962), 226-236. [3] G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, "Queueing Networks and Markov Chains," 2nd edition, Wiley, New York, 1998. doi: 10.1002/0471200581. [4] R. L. Cook, Stochastic sampling in computer graphics, ACM Trans. on Graphics, 5 (1986), 51-72. doi: 10.1145/7529.8927. [5] S. Galmés, System design issues in time-driven sensor networks based on stochastic sampling, Simulation Modelling, Practice and Theory, 19 (2011), 1530-1543. doi: 10.1016/j.simpat.2011.04.006. [6] S. Galmés and R. Puigjaner, Randomized data-gathering protocol for time-driven sensor networks, Computer Networks Journal, 55 (2011), 3863-3885. doi: 10.1016/j.comnet.2011.08.002. [7] J. R. Higgins, "Sampling Theory in Fourier and Signal Analysis: Foundations," Oxford University Press, Oxford, 1996. [8] H. Karl and A. Willig, "Protocols and Architectures for Wireless Sensor Networks," Wiley, New York, 2005. [9] B. Krishnamachari, "Networking Wireless Sensors," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511541025. [10] A. Kumar, P. Ishwar and K. Ramchandran, On distributed sampling of smooth non-bandlimited fields, in ''Proc. of the Third International Symposium on Information Processing in Sensor Networks," Berkeley, CA, USA, (2004), 89-98. [11] A. Kumar, P. Ishwar and K. Ramchandran, On distributed sampling of bandlimited and non-bandlimited sensor fields, in ''Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing," Montreal, Canada, (2004), III-925-III-928. [12] , "OMNeT++ Community Site," OMNeT++ 3.x documentation and tutorials,, 2005. Available from: , (). [13] A. V. Oppenheim, A. S. Willsky and I. T. Young, "Signals and Systems," Prentice-Hall, New Jersey, 1983. [14] G. Reise and G. Matz, Distributed sampling and reconstruction of non-bandlimited fields in sensor networks based on shift-invariant spaces, in ''Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing," Taipei, Taiwan, (2009), 2061-2064. [15] M. L. Santamaría, S. Galmés and R. Puigjaner, Simulated annealing approach to optimizing the lifetime of sparse time-driven sensor networks, in ''Proc. of 2009 IEEE International Symposium on Modeling, Analysis & Simulation of Computer and Telecommunication Systems," London, UK, (2009), 193-202. [16] I. Stojmenovic, "Handbook of Sensor Networks: Algorithms and Architectures," Wiley, New York, 2005. [17] S. Tilak, N. Abu-Ghazaleh and W. R. Heinzelman, A taxonomy of wireless micro-sensor network models, ACM Mobile Computing and Communications Review (MC2R), 6 (2002), 28-36. [18] Y. Yu, V. K. Prasanna and B. Krishnamachari, Energy minimization for real-time data gathering in wireless sensor networks, IEEE Trans. on Wireless Communications, 5 (2006), 3087-3096. doi: 10.1109/TWC.2006.04709.
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