Advanced Mathematics and Numerical Modeling of IoT

Table 2: Notations.

Notation Description Value

푇PS Preamble sensing time 15.6 milliseconds

푇WS Wake-up sensing time 183 milliseconds

푇FS Frame sensing time 183 milliseconds

푇BS Beacon sending time 90 milliseconds

푇푆 Sleep duration Variable milliseconds

푇푃 Preamble transmission time for device trigger

(i)Long preamble (B-MAC)

(ii)Short preamble (X-MAC)

(iii)Variable (Wise-MAC)

푇WP Wake-up packet time for trigger 90 milliseconds

푇DP Data packet time for trigger 150 milliseconds

푇BL Beacon listen time Random

푇퐵 Back-off period 1 milliseconds

푇퐿 Listen period Variable

푇DT Data transmission time 200 milliseconds

푇ACK ACK transmission time 90 milliseconds

푇DPT Data pending time Random

푇DC Duty-cycle time

=푇PS+푇푆 (preamble-based)

=푇WS+푇푆 (wake-up packet-based)

=푇FS+푇푆 (packet-based)

= beacon interval (receiver-initiated)

푇tot-active Total active duration 푇퐴-PPS+푇퐴-aggre (i)

푇tot-sleep Total sleep duration 푇interval−푇tot-active (ii)

푇interval Request interval 1 hour (=3600 seconds = 3600000 milliseconds)

퐼퐴 Current consumption in active state 0.0061944 mA

퐼푆 Current consumption in active state 0.0000083 mA

푇퐴-aggre Active duration during data aggregation time

푇퐴-PPS Active duration in a duty cycle

푇AW Acknowledgement waiting period 100 milliseconds

푉 Supply voltage 3.5 V

devices are aggregated in the concentrator and transmitted
to the server.

3.3. Energy Model.For M2 M networks, energy conservation
is one of the most critical challenges, as it is in sensor
networks. It is important to note that in order to save energy,
each device should remain active only for required duration,
and the rest of the time should go to sleep. Therefore, when
calculating the energy consumption of each device, we need
to know the total active duration,푇tot-active(i) and the total
sleep duration,푇tot-sleep(ii) in a request interval,푇interval.By
using (i) and (ii), the energy consumption for each device can
be expressed as follows:

4. Numerical Analysis

Now,wenumericallyanalyzeeachLPLprotocolinterms
of M2M communication models. In particular, we focus on
data aggregation time, which is the total time required to
aggregate data from all devices with respect to a request.
Table 2presents notations used for our numerical analysis.

4.1. B-MAC.B-MAC is a representative LPL protocol utiliz-

ing a preamble for the receiver trigger. As shown inFigure 1,

each device repeats a short time wake-up for푇PSto detect

the preamble transmission and then sleeps for푇푠,per푇DC.

A sender that wants to send data first transmits a long

preamble for푇푃to trigger the receiver that is performing

periodic preamble sensing (PPS) before data transmission.

Each preamble transmission can be detected by all devices

within communication range of a sender, and all nodes that

detect the preamble transmission, as well as the intended

receiver,havetoremainactivefor푇퐿,untilthepreamble

transmission ends.

4.1.1. Periodic Report.Since each device should send its

data to the concentrator on the predetermined schedule, the

report time of each device is as follows:

푇resp=푇퐵+푇푃+푇DT+푇ACK. (1)

Therefore, the total report time of푛nodes is

푇B-MAC(푃)=푛∗푇resp=푛∗(푇퐵+푇푃+푇DT+푇ACK). (2)