Abstract
In this paper we propose a generic queueing model that can be used to evaluate the performance of a wireless sensor node that uses energy harvesting. The alteration of such a device between the transmit and sleep mode (or between consuming energy and harvesting energy), is modeled by means of a finite capacity queueing system with repeated server vacations. The duration of a service, resp. vacation, is determined by the available energy at the start of the service, resp. vacation. Therefor we introduce in the model a variable that keeps track of the available energy. The system occupancy and the available energy are observed at inspection instants (i.e., the end of a service or of a vacation), resulting in a discrete-time Markov Chain. We derive closed form formulas for the system occupancy distribution at inspection instants and at arbitrary time instants together with the Laplace transform of the waiting time distribution. The possible use of the model to evaluate the system’s performance for various parameter values is illustrated by means of a number of examples.
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Flemish FWO project G0B7915N, “Modelling and Control of Energy Harvesting Wireless Sensor Networks”.
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Appendices
Appendix 1
We derive explicit formulas for \(\left( {\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}}\) and \(\left( {\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}}\).
In a similar way as in [18], it is possible to show that for \(n = 0, \ldots ,N - 1\)
and hence
and the probability of having a system occupancy equal to n at an arbitrary instant of a vacation interval is given by
For n = N, we obtain
which leads to
In the same way, for n = 1, …, N − 1
and hence
The probability of having a system occupancy equal to n at an arbitrary instant of a service interval is given by
For n = N, we obtain
Appendix 2
Since
by taking the derivative
we obtain
Using the results for \(\left( {\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( \theta \right) \cdot \overline{\varvec{e}}\) and \(\left( {\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( \theta \right) \cdot \overline{\varvec{e}}\) in (41) and (44) and the fact that
and.
\(\mathop \sum \limits_{n = 0}^{N - 1} \overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} \left( 0 \right) \cdot \left( {\mathop \sum \limits_{l = 1}^{n + 1} E\left[ {S_{{F^{l} \left( {G\left( 1 \right)} \right)}} } \right], \ldots ,\mathop \sum \limits_{l = 1}^{n + 1} E\left[ {S_{{F^{l} \left( {G\left( {i_{max} } \right)} \right)}} } \right]} \right)^{^{\prime}} = \mathop \sum \limits_{n = 0}^{N - 1} \overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} \left( 0 \right) \cdot \left( {\left( {n + 1} \right) \cdot E\left[ {S_{1} } \right], \ldots ,\left( {n + 1} \right) \cdot E\left[ {S_{{i_{max} }} } \right]} \right) ^{^{\prime}}\) we obtain Little’s well known result
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Blondia, C. A queueing model for a wireless sensor node using energy harvesting. Telecommun Syst 77, 335–349 (2021). https://doi.org/10.1007/s11235-021-00758-1
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DOI: https://doi.org/10.1007/s11235-021-00758-1