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Please note that in the LoRaWAN standard case, if the base station cannot send the acknowledgement after the first receive delay, an acknowledgement can be sent after a second receive delay into another channel reserved for downlink communications.
As illustrated in Fig. 3, this acknowledgement is sent, by the base station, after a fixed receive delay denoted T d.
In other words, the receive delay between the end of the uplink packet, and the transmission of the acknowledgement in the downlink is constant.
Indeed, in this standard, an acknowledgement can be sent into the channel used for the purpose of uplink communications after a fixed receive delay.
To the best of the authors' knowledge, the collision probability in a pure ALOHA-based protocol, in which the acknowledgement is sent after a fixed receive delay in the same channel, as in the LoRaWAN standard, has never been analysed in the literature.
This occurs if another packet has been successfully sent in the interval I a =[−T d −T m −T a ;−T d −T m ] and if the channel is free at the end of the receive delay T d as illustrated in Fig. 5.
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