Assuming the current mesh node has the knowledge on all (or some of) the neighbouring mesh nodes’ channel’s RSSI of which it can form a link with. The mesh node would only issue a trigger when its present forecasted RSSI value falls below its threshold value, and the forecasted RSSI value of a target neighbouring mesh node exceeds its threshold value. By denoting the neighbouring mesh nodes RSSI values as
\(Y_{t}^{(i)}\) where
i-1, 2,…,
M, where
M is the total number of neighbouring mesh nodes (or mesh radios in multi-radio case), the current mesh node would only issue a trigger when it current link’s RSSI
$$\hat{X}_{t + \ell \Updelta t} \le \bar{X}\,\,{\text{and}}\,\,\hat{Y}_{t + \ell \Updelta t}^{(j)} > \bar{Y}^{(j)} $$
(19)
where the index
j is defined as
$$j = \{ i:\hbox{max} \{ \hat{Y}_{t + \ell \Updelta t}^{(i)} - \bar{Y}^{(i)} ,\,0\} ,\quad i = 1,2, \ldots ,M\} $$
(20)
where
\(\bar{X}\) is the current link RSSI threshold representing the minimal QoS it must support in order to operate successfully,
\(\bar{Y}^{(i)}\) is the
ith neighbouring RSSI threshold value and
\(\hat{Y}_{t + \ell \Updelta t}^{(j)}\) is the predicted RSSI value of the
jth neighbouring mesh node if which it could form a new link with. The criteria given in (
19)–(
20) denotes that the OU-LPT method would only choose the “best” neighbouring mesh node. On the other hand if there is no better mesh node, then the scheme will not trigger a link handover event.
The Link Going Down (LGD) event is introduced to help wireless nodes to prepare for link handover or switching prior to Link Down (LD) so that switching delays and service interruptions can be minimized. Based on the forecasted RSSI values of the current link and in order to minimize the error of decision making, wireless card manufacturers like Intel [
20] would introduce a protection margin for LGD (or hysteresis factor)
\(\Updelta_{x}^{GD} \ge 0\). The purpose of having this protection margin is to augment it to the RSSI threshold value,
\(\bar{X}\) so that the current link has an enhanced threshold value,
\(\bar{X} + \Updelta_{x}^{GD}\) to ensure a better QoS. If the forecasted RSSI value is greater than the enhanced threshold value, then the system would not trigger a link handover to another mesh node. In the following Table
1 we list the trigger thresholds that are being used in this report where
\(\Updelta_{x}^{U} > \Updelta_{x}^{CU} > \Updelta_{x}^{GD} > 0.\)
Table 1
Thresholds for link handover trigger
Link-Up threshold (LU_TH) |
\(\bar{X} + \Updelta_{x}^{U}\)
|
Link-coming-up threshold (LCU_TH) |
\(\bar{X} + \Updelta_{x}^{CU}\)
|
Link-going-down threshold (LGD_TH) |
\(\bar{X} + \Updelta_{x}^{GD}\)
|
Link-down threshold (LD_TH) |
\(\bar{X}\)
|
With this protection margin
\(\Updelta_{x}^{GD}\), and for a forecasted RSSI value
\(\bar{X}_{t + \ell \Updelta t}\) the probability in making a trigger is defined as
$$P(\hat{X}_{t\,\, + \,\,\ell \Updelta t} \,\, \le \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} )\,\, = \,\,P\left( {Z\,\, \le \,\,\frac{{\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\, - \,\,E\left( {\hat{X}_{t + \ell \Updelta t} } \right)}}{{\sqrt {Var\left( {\hat{X}_{t + \ell \Updelta t} } \right)} }}} \right) $$
(21)
and if
$$E\left( {\hat{X}_{t + \ell \Updelta t} } \right)\le \bar{X} + \Updelta_{x}^{GD} \,{\text{and}}\, P\left( {\hat{X}_{t + \ell \Updelta t} \le \bar{X} + \Updelta_{x}^{GD} } \right)\ge \alpha $$
(22)
where
α ∈ (0,1) is a margin error then the current mesh node will issue a trigger at time
t to initiate a link handover to an alternative mesh node or radio.
In addition, for the forecasted RSSI values of neighbouring mesh nodes and for each of the
ith node we also introduce a protection margin
\(\Updelta_{y}^{(i)} \ge 0\) so as to minimize the error of false selection of a node for handover. By analogy with the probability of making a trigger for the mesh link, for each neighbouring mesh radios, we define the probability of selecting a new node:
$$P\left( {\hat{Y}_{t + \ell \Updelta t}^{(i)} \ge \bar{Y}^{(i)} + \Updelta_{{y^{(i)} }}^{CU} } \right) = P\left( {Z \ge \frac{{\bar{Y}^{(i)} + \Updelta_{{y^{(i)} }}^{CU} - E\left( {\hat{Y}_{t + \ell \Updelta t}^{(i)} } \right)}}{{\sqrt {Var\left( {\hat{Y}_{t + \ell \Updelta t}^{(i)} } \right)} }}} \right),\quad i = 1,2, \ldots ,M $$
(23)
and if
$$P\left( {\hat{Y}_{t + \ell \Updelta t}^{(i)} \ge \bar{Y}^{(i)} + \Updelta_{{y^{(i)} }}^{CU} } \right) \ge \beta $$
(24)
where
β
∈ (0,1) is a margin error, then the
ith
n can be selected to be the link handover target. By augmenting a protection margin to our OU-LPT method we can now redefine our criterion of a handover from a current mesh node to the
jth mesh node at time
t as:
$$\left\{ {\begin{array}{*{20}c} {E\left( {\hat{X}_{t + \ell \Updelta t} } \right) \le \bar{X} + \Updelta_{x}^{GD} \,and} \hfill \\ {P\left( {\hat{X}_{t + \ell \Updelta t} \le \bar{X} + \Updelta_{x}^{GD} } \right) \ge \alpha } \hfill \\ \end{array} } \right\}\,{\text{and}}\,\left\{ {\begin{array}{*{20}c} {E\left( {\hat{Y}_{t + \ell \Updelta t}^{(j)} } \right) \ge \bar{Y}^{(j)} + \Updelta_{{y^{(j)} }}^{CU} \,and} \hfill \\ {P\left( {\hat{Y}_{t + \ell \Updelta t}^{(j)} \ge \bar{Y}^{(j)} + \Updelta_{{y^{(j)} }}^{CU} } \right) \ge \beta } \hfill \\ \end{array} } \right\} $$
(25)
where the index
j is defined as
$$j = \left\{ {i:\hbox{max} \left\{ {P\left( {\hat{Y}_{t + \ell \Updelta t}^{(i)} \ge \bar{Y}^{(i)} + \Updelta_{{y^{(j)} }}^{CU} } \right),\quad i = 1,2, \ldots ,M} \right\}} \right\} $$
(26)
By analogy with statistical hypothesis testing, the procedure described above would lead us to commit a false trigger (or false positive) error. With this protection margin Δ
x
, and for a forecasted RSSI value
\(\hat{X}_{t + \ell \Updelta t}\), here we define the probability in making a false trigger (or false alarm) at time
t as
$$P\left( {\left. {\hat{X}_{t + \ell \Updelta t} \le \bar{X} + \Updelta_{x}^{GD} } \right|X_{t\ell \Updelta t} > \bar{X} + \Updelta_{x}^{GD} } \right) $$
(27)
where it is the error of committing a false trigger when the true RSSI value
\(X_{t + \ell \Updelta t}\) is greater than the enhanced threshold requirement but the forecasted RSSI value,
\(\hat{X}_{t + \ell \Updelta t}\) shows that it is lower than the threshold value plus the protection margin. From (
22) we can deduce via Kolmogorov–Smirnov goodness-of-fit test that the residuals of the smoothed and fitted RSSI values
\(\hat{\varepsilon }_{t} = X_{t} - \hat{X}_{t}\) follow
$$\hat{\varepsilon }_{t} = X_{t} - \hat{X}_{t} \sim N\left( {\mu_{{\hat{\varepsilon }}} ,\,\sigma_{{\hat{\varepsilon }}}^{2} } \right) $$
(28)
where
\(E(\hat{\varepsilon }_{t} ) = \mu_{{\hat{\varepsilon }}}\) and
\(Var(\hat{\varepsilon }_{t} ) = \sigma_{{\hat{\varepsilon }}}^{2}\). Hence we can write that
P(
false trigger) =
$$P(\hat{X}_{t\,\, + \,\,\ell \Updelta t} \,\, \le \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\,|\,\,X_{t\,\, + \,\,\ell \Updelta t} \,\, > \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} ) = \,\frac{{\int_{\,\, - \infty }^{{\,\,x\,\, = \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} }} {\left[ {\,1\,\, - \,\,\Upphi \left( {\frac{{\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\, - \,\,x\,\, - \,\,\mu_{{\hat{\varepsilon }}} }}{{\sigma_{{\hat{\varepsilon }}} }}} \right)} \right]} \,\,f_{{\hat{X}}} \left( x \right)\,\,dx\,}}{{1\,\, - \,\,\int_{\,\, - \infty }^{\,\,\,\infty } {\Upphi \left( {\frac{{\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\, - \,\,x\,\, - \,\,\mu_{{\hat{\varepsilon }}} }}{{\sigma_{{\hat{\varepsilon }}} }}} \right)\,f_{{\hat{X}}} \left( x \right)\,\,dx} }}\, $$
(29)
where
Z ~
N(0,1),
\(\Upphi ( \cdot )\) denotes the cumulative standard normal distribution function and
\(f_{{\hat{X}}} (x) = \frac{1}{{\sigma_{{\hat{X}_{t + \ell \Updelta t} }} \sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{x - \mu_{{\hat{X}_{t + \ell \Updelta t} }} }}{{\sigma_{{\hat{X}_{t + \ell \Updelta t} }} }}} \right)^{2} }}\) is the probability density function (pdf) of the forecasted RSSI values. On the other hand we can also define the probability of making a false non-trigger (or missed trigger) as
$$P(\hat{X}_{t\,\, + \,\,\ell \Updelta t} \,\, > \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\,|\,\,X_{t\,\, + \,\,\ell \Updelta t} \,\, \le \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} ) = \,\frac{{\int_{{x\, = \,\,\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\,}}^{\infty } {\Upphi \left( {\frac{{\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\, - \,\,x\,\, - \,\,\mu_{{\hat{\varepsilon }}} }}{{\sigma_{{\hat{\varepsilon }}} }}} \right)} \,f_{{\hat{X}}} \left( x \right)\,\,dx\,}}{{\int_{\,\, - \infty }^{\,\,\,\infty } {\Upphi \left( {\frac{{\bar{X}\,\, + \,\,\Updelta_{x}^{GD} \,\, - \,\,x\,\, - \,\,\mu_{{\hat{\varepsilon }}} }}{{\sigma_{{\hat{\varepsilon }}} }}} \right)\,f_{{\hat{X}}} \left( x \right)\,\,dx} }}\, $$
(30)
which is the error when the true RSSI value
\(X_{t + \ell \Updelta t}\) is less than the enhanced threshold requirement but the forecasted RSSI value,
\(\hat{X}_{t + \ell \Updelta t}\) shows that it is greater than the threshold value plus the protection margin. For a complete derivation of these two results, please refer to the “
Appendix”.
In addition, for the forecasted RSSI values of neighbouring mesh nodes, by analogy with the probabilities of making a false trigger of the current mesh node, for each neighbouring mesh node, we define the probability of making false selection at time
t of an
ith node as
$$P\left( {\left. {\hat{Y}_{t + \ell \Updelta t}^{(i)} > \bar{Y}^{(i)} + \Updelta_{{y^{(i)} }}^{CU} } \right|Y_{t + \ell \Updelta t}^{(i)} \le \bar{Y}^{(i)} + \Updelta_{{y^{(i)} }}^{CU} } \right),\quad i = 1,2, \ldots ,M $$
(31)
where
\(\hat{Y}_{t + \ell \Updelta t}^{(i)}\) and
\(\bar{Y}^{(i)}\) are the
ith mesh node’s forecasted RSSI value for leads
\(\ell \ge 1\) and its RSSI threshold value respectively. Furthermore by deducing the residuals
\(Y_{t}^{(i)} - \hat{Y}_{t}^{(i)}\) as
$$\hat{\varepsilon }_{t}^{(i)} = Y_{t}^{(i)} - \hat{Y}_{t}^{(i)} \sim N\left( {\mu_{\varepsilon }^{(i)} ,\,\left( {\sigma_{\varepsilon }^{(i)} } \right)^{2} } \right) $$
(32)
where
\(E\left( {\hat{\varepsilon }_{t}^{(i)} } \right) = \mu_{{\hat{\varepsilon }}}^{t}\),
\(Var(\hat{\varepsilon }_{t}^{(i)} ) = \left( {\sigma_{{\hat{\varepsilon }}}^{(i)} } \right)^{2}\) and
\(Y_{t}^{(i)}\) is the smoothed RSSI value at time
t for
ith neighbouring mesh node. Hence in analogy with (
31) we can write that
P(
false node selection) =
$$P(\hat{Y}_{t\,\, + \,\,\ell \Updelta t}^{(i)} \,\, > \,\,\bar{Y}_{{}}^{(i)} \,\, + \,\,\Updelta_{{y^{\left( i \right)} }}^{CU} \,\,|\,\,Y_{t\,\, + \,\,\ell \Updelta t}^{(i)} \,\, \le \,\,\bar{Y}_{{}}^{(i)} \,\, + \,\,\Updelta_{{y^{\left( i \right)} }}^{CU} \,)\, = \,\frac{{\int_{{\,\,y\,\, = \,\,\bar{Y}_{{}}^{(i)} \,\, + \,\,\Updelta_{{y^{\left( i \right)} }}^{CU} }}^{\,\infty \,} {\Upphi \left( {\frac{{\bar{Y}^{(i)} \,\, + \,\,\Updelta_{{y^{\left( i \right)} }}^{CU} \,\, - \,\,y\,\, - \,\,\mu_{{\hat{\varepsilon }}}^{(i)} }}{{\sigma_{{\hat{\varepsilon }}}^{(i)} }}} \right)} \,\,f_{{\hat{Y}^{(i)} }} \left( y \right)\,\,dy\,}}{{\int_{\,\, - \infty }^{\,\,\,\infty } {\Upphi \left( {\frac{{\bar{Y}^{(i)} \,\, + \,\,\Updelta_{{y^{\left( i \right)} }}^{CU} \,\, - \,\,y\,\, - \,\,\mu_{{\hat{\varepsilon }}}^{(i)} }}{{\sigma_{{\hat{\varepsilon }}}^{(i)} }}} \right)\,\,f_{{\hat{Y}^{(i)} }} \left( y \right)\,\,dy\,} }} $$
(33)
where
Z ~
N(0,1),
\(\Upphi ( \cdot )\) denotes the cumulative standard normal distribution function and
\(f_{{\hat{Y}^{(i)} }} (y) = \frac{1}{{\sigma_{{\hat{Y}_{t + \ell \Updelta t}^{(i)} }} \sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{y - \mu_{{\hat{Y}_{t + \ell \Updelta t}^{(i)} }} }}{{\sigma_{{\hat{Y}_{t + \ell \Updelta t}^{(i)} }} }}} \right)^{2} }}\) is the probability density function (pdf) of the forecasted RSSI values of the neighbouring
ith mesh node with mean
\(\mu_{{\hat{Y}_{t + \ell \Updelta t}^{(i)} }}\) and variance
\(\sigma_{{\hat{Y}_{t + \ell \Updelta t}^{(i)} }}^{2}\). Hence we can now redefine our criterion of a handover at time
t from a current mesh node to the
jth mesh node as:
$$\left\{ {\begin{array}{*{20}c} {E\left( {\hat{X}_{t + \ell \Updelta t} } \right) \le \bar{X} + \Updelta_{x}^{GD} \,and} \\ {P\left( {\hat{X}_{t + \ell \Updelta t} \le \bar{X} + \Updelta_{x}^{GD} } \right) \ge \alpha \,\,and} \\ {P\left( {\left. {\hat{X}_{t + \ell \Updelta t} \le \bar{X} + \Updelta_{x}^{GD} } \right|X_{t + \ell \Updelta t} > \bar{X} + \Updelta_{x}^{GD} } \right) \le \bar{\alpha }} \\ \end{array} } \right\} $$
(34)
and
$$\left\{ {\begin{array}{*{20}c} {E\left( {\hat{Y}_{t + \ell \Updelta t}^{(j)} } \right) \ge \bar{Y}^{(j)} + \Updelta_{{y^{(j)} }}^{CU} \,and} \\ {P\left( {\hat{Y}_{t + \ell \Updelta t}^{(j)} \ge \bar{Y}^{(j)} + \Updelta_{{y^{(j)} }}^{CU} } \right) \ge \beta \,\,and} \\ {P\left( {\left. {\hat{Y}_{t + \ell \Updelta t}^{(j)} \ge \bar{Y}^{(j)} + \Updelta_{{y^{(j)} }}^{CU} } \right|\hat{Y}_{t + \ell \Updelta t}^{(j)} < \bar{Y}^{(j)} + \Updelta_{{y^{(j)} }}^{CU} } \right) \le \bar{\beta }} \\ \end{array} } \right\} $$
(35)
where
\(\bar{\alpha },\,\bar{\beta } \in \,(0,1)\) and the index
j is defined as
$$j = \left\{ {i:\hbox{max} \left\{ {P\left( {\hat{Y}_{t + \ell \Updelta t}^{(i)} \ge \bar{Y}^{(i)} + \Updelta_{{y^{(i)} }}^{CU} } \right),\quad i = 1,2, \ldots ,M} \right\}} \right\} $$
(36)
In order to reduce the probability of making false trigger and the probability of selecting the wrong AP to a wider margin, we can modify the above decision criteria to the following scheme: