A queueing model for a wireless sensor node using energy harvesting

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.

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 $n = 0, \ldots ,N - 1$

$$ \begin{aligned} (\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} )_{i} \left( \theta \right) = & \sigma_{i} \cdot \left\{ {\frac{1}{{\left( {\lambda - \theta } \right)V_{i} }}[ V_{i}^{*} \left( \vartheta \right)\left( {\frac{\lambda }{\lambda - \theta }} \right)^{n} - \mathop \sum \limits_{k = 0}^{n} h_{k,i} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} } \right\} \\ = & \frac{1}{{\left( {\lambda - \theta } \right)D}} \cdot (\overline{p}_{0} )_{i} \cdot \left\{ { V_{i}^{*} \left( \vartheta \right) \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n} - \mathop \sum \limits_{k = 0}^{n} h_{k,i} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} } \right\}, \\ \end{aligned} $$

and hence

$$ \left( {\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( \theta \right) \cdot \overline{\varvec{e}} = \frac{1}{{\left( {\lambda - \theta } \right)D}} \cdot \{ \overline{\varvec{p}}_{0} \cdot \overline{\varvec{V}}^{\varvec{*}} \left( \vartheta \right) \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n} - \mathop \sum \limits_{k = 0}^{n} \overline{\varvec{p}}_{0} \cdot {\varvec{B}}_{{\varvec{k}}} \cdot \overline{\varvec{e}} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} $$

(40)

and the probability of having a system occupancy equal to n at an arbitrary instant of a vacation interval is given by

$$ \left( {\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}} = \frac{1}{\lambda \cdot D} \cdot \overline{\varvec{p}}_{0} \cdot \left[ {\overline{\varvec{e}} - \mathop \sum \limits_{k = 0}^{n} {\varvec{B}}_{{\varvec{k}}} \cdot \overline{\varvec{e}}} \right] $$

(41)

For n = N, we obtain

$$ (\overline{\varvec{\omega }}_{{\varvec{N}}}^{\varvec{*}} )_{i} \left( \theta \right) = \sigma_{i} \cdot \left\{ {\frac{1}{{\left( {\lambda - \theta } \right)V_{i} }}\mathop \sum \limits_{n = N}^{\infty } \left[ {V_{i}^{*} \left( \vartheta \right)\left( {\frac{\lambda }{\lambda - \theta }} \right)^{n} - \mathop \sum \limits_{k = 0}^{n} h_{k,i} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} } \right]} \right\} $$

which leads to

$$ \left( {\overline{\varvec{\omega }}_{{\varvec{N}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}} = \frac{1}{D} \cdot \overline{\varvec{p}}_{0} \cdot \overline{{{\varvec{E}}\left[ {\varvec{V}} \right]}} - \mathop \sum \limits_{n = 0}^{N - 1} \left( {\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}} $$

(42)

In the same way, for n = 1, …, N − 1

$$ \begin{aligned} (\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} )_{i} \left( \theta \right) = & \rho_{i} \cdot \left\{ {\mathop \sum \limits_{k = 1}^{n} \frac{{(\overline{\varvec{p}}_{{\varvec{k}}} )_{i} }}{{\mathop \sum \nolimits_{l = 1}^{N} (\overline{\varvec{p}}_{{\varvec{l}}} )_{i} }} \cdot \frac{1}{{\left( {\lambda - \theta } \right)S_{i} }} \cdot \left[ {S_{i}^{*} \left( \vartheta \right) \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} - \mathop \sum \limits_{l = 0}^{n - k} g_{l,i} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k - l} } \right]} \right\}, \\ = & \frac{1}{{\left( {\lambda - \theta } \right)D}} \cdot \left\{ {\mathop \sum \limits_{k = 1}^{n} (\overline{\varvec{p}}_{{\varvec{k}}} )_{j} \cdot [ S_{i}^{*} \left( \vartheta \right) \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} - \mathop \sum \limits_{l = 0}^{n - k} g_{l,i} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k - l} } \right\} \\ \end{aligned} $$

and hence

$$ \begin{aligned} (\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} )(\theta ) \cdot \overline{e} = & \frac{1}{{\left( {\lambda - \theta } \right)D}} \cdot \left\{ {\sum\limits_{k = 1}^{n} {\overline{\varvec{p}}_{{\varvec{k}}} \cdot \overline{\varvec{S }}^{*} (\vartheta ) \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} - \sum\limits_{k = 1}^{n} {\sum\limits_{l = 1}^{k} {\overline{\varvec{p}}_{{\varvec{l}}} \cdot {\varvec{A}_{\varvec{k - l}}} \cdot \overline{e} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} } } } } \right\} \\ = & \frac{1}{(\lambda - \theta )D} \cdot \left\{ {\sum\limits_{k = 1}^{n} {\overline{p}_{k} \cdot \overline{S}^{*} (\vartheta ) \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} - \sum\limits_{k = 1}^{n} {\left( {\overline{p}_{k - 1} - \overline{p}_{0} \cdot {\varvec{B}}_{\varvec{k - 1}} } \right) \cdot \overline{e} \cdot \left( {\frac{\lambda }{\lambda - \theta }} \right)^{n - k} } } } \right\} \\ \end{aligned} $$

(43)

The probability of having a system occupancy equal to n at an arbitrary instant of a service interval is given by

$$ \left( {\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}} = \frac{1}{\lambda \cdot D} \cdot \left[ {\overline{\varvec{p}}_{{\varvec{n}}} \cdot \overline{\varvec{e}} - \overline{\varvec{p}}_{0} \cdot \overline{\varvec{e}} + \mathop \sum \limits_{k = 0}^{n - 1} \overline{\varvec{p}}_{0} \cdot {\varvec{B}}_{{\varvec{k}}} \cdot \overline{\varvec{e}}} \right] $$

(44)

For n = N, we obtain

$$ \left( {\overline{\varvec{\pi }}_{{\varvec{N}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}} = \frac{1}{D} \cdot \mathop \sum \limits_{n = 1}^{N - 1} \overline{\varvec{p}}_{{\varvec{k}}} \cdot \overline{{{\varvec{E}}\left[ {\varvec{S}} \right]}} - \mathop \sum \limits_{n = 0}^{N - 1} \left( {\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} } \right)\left( 0 \right) \cdot \overline{\varvec{e}} $$

(45)

Appendix 2

Since

$$ \overline{\varvec{W}}^{*} (\theta ) = \frac{1}{{1 - P_{b} }}\left[ {\sum\limits_{n = 1}^{N - 1} {\overline{\varvec{\pi }}_{{\varvec{n}}}^{*} (\theta ) \cdot Diag\left( {\prod\nolimits_{k = 1}^{n} {S_{{F^{k} (1)}}^{*} (\theta ), \ldots ,\prod\nolimits_{k = 1}^{n} {S_{{F^{k} (i_{\max } )}}^{*} (\theta )} } } \right) + \sum\limits_{n = 0}^{N - 1} {\overline{\varvec{\omega }}_{{\varvec{n}}}^{*} (\theta ) \cdot Diag\left( {\prod\nolimits_{k = 1}^{n + 1} {S_{{F^{k} (G(1))}}^{*} (\theta ), \ldots ,\prod\nolimits_{k = 1}^{n + 1} {S_{{F^{k} (G(i_{\max } ))}}^{*} (\theta )} } } \right)} } } \right] $$

by taking the derivative

$$ E\left[ W \right] = - \frac{{d\overline{\varvec{W}}^{*} \left( \theta \right)}}{d\theta }|_{\theta = 0} \cdot \overline{\varvec{e}} $$

we obtain

$$ \begin{aligned} E[W] = & \frac{1}{{1 - P_{b} }}\left\{ { - \sum\limits_{{n = 1}}^{{N - 1}} {\frac{{d{\overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} } (\theta )}}{{d\theta }}|_{{\theta = 0}} \cdot \overline{\varvec{e}} - \sum\limits_{{n = 0}}^{{N - 1}} {\frac{{d{\overline{\varvec{\omega }}_{{\varvec{n}}}^{\varvec{*}} } (\theta )}}{{d\theta }}|_{{\theta = 0}} \cdot \bar{e}} } } \right. \\ & \; + \sum\limits_{{n = 1}}^{{N - 1}} {\bar{\pi }_{n}^{*} (0) \cdot \left( {\sum\nolimits_{{l = 1}}^{n} {E\left[ {S_{{F^{l} (1)}} } \right]} , \ldots ,\sum\nolimits_{{l = 1}}^{n} {E\left[ {S_{{F^{l} (i_{{\max }} )}} } \right]} } \right)} ^{'} \\ & \left. { \; + \sum\limits_{{n = 0}}^{{N - 1}} {\bar{\omega }_{n}^{*} (0) \cdot \left( {\sum\nolimits_{{l = 1}}^{{n + 1}} {E\left[ {S_{{F^{l} \left( {G(1)} \right)}} } \right], \ldots ,\sum\nolimits_{{l = 1}}^{{n + 1}} {E\left[ {S_{{F^{l} \left( {G\left( {i_{{\max }} } \right)} \right)}} } \right]} } } \right)^{\prime} ]} } \right\} \\ \end{aligned} $$

(46)

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

$$ \mathop \sum \limits_{n = 1}^{N - 1} \overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} \left( 0 \right) \cdot \left( {\mathop \sum \limits_{l = 1}^{n} E\left[ {S_{{F^{l} \left( 1 \right)}} } \right], \ldots ,\mathop \sum \limits_{l = 1}^{n} E\left[ {S_{{F^{l} \left( {i_{{{\text{max}}}} } \right)}} } \right]} \right)^{^{\prime}} = \mathop \sum \limits_{n = 1}^{N - 1} \overline{\varvec{\pi }}_{{\varvec{n}}}^{\varvec{*}} \left( 0 \right) \cdot \left( {n \cdot E\left[ {S_{1} } \right], \ldots ,n \cdot E\left[ {S_{{i_{{{\text{max}}}} }} } \right]} \right) ^{^{\prime}} $$

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

$$ E\left[ W \right] = \frac{1}{{1 - P_{b} }} \cdot \frac{1}{\lambda } \cdot E\left[ \eta \right] $$

(47)