(d) Mobile devices transmit onto the uplink of a shared radio channel whenever they like (i.e. pure Aloha). . The transmission rate is 2Mbps and each
station transmits packets of exactly 1000 bits. In 1sec, an average of 40 devices attempt to transmit onto the channel with one packet each, and their
transmission attempts follow a Poisson distribution. (i) Determine the maximum time period when packets are vulnerable to collisions on the channel.
(ii) Calculate the normalised offered load to the network (G), the probability (\gamma) of no collisions during the vulnerability period, and from this
the probability of a collision on the radio channel.

Question

(d) Mobile devices transmit onto the uplink of a shared radio channel whenever they like (i.e. pure Aloha). . The transmission rate is 2Mbps and each
station transmits packets of exactly 1000 bits. In 1sec, an average of 40 devices attempt to transmit onto the channel with one packet each, and their
transmission attempts follow a Poisson distribution. (i) Determine the maximum time period when packets are vulnerable to collisions on the channel.
(ii) Calculate the normalised offered load to the network (G), the probability (\gamma) of no collisions during the vulnerability period, and from this
the probability of a collision on the radio channel.

Accepted Answer

⚹ In the pure Aloha protocol, the maximum time period when packets are vulnerable to collisions is twice the packet transmission time ⚹ ⚹ To calculate this, we first determine the packet transmission time ⚹ ⚹ Given that the transmission rate is 2 Mbps (megabits per second) and the packet size is 1000 bits, the packet transmission time ($T_p$) is calculated as follows: $T_p = \frac{ ext{packet size}}{ ext{transmission rate}} = \frac{1000 ext{ bits}}{2 imes 10^6 ext{ bits/sec}} = 0.0005 ext{ sec}$ ⚹ ⚹ Therefore, the maximum time period when packets are vulnerable to collisions ($T_v$) is $T_v = 2 imes T_p = 2 imes 0.0005 ext{ sec} = 0.001 ext{ sec}$ ⚹ ⚹ Now, let's calculate the normalized offered load to the network (G) ⚹ ⚹ The normalized offered load (G) is the average number of packets generated by the system during one packet time ⚹ ⚹ Given that an average of 40 devices attempt to transmit one packet each in 1 second, we can calculate G as follows: $G = ext{average number of packets per second} imes T_p = 40 imes 0.0005 = 0.02$ ⚹ ⚹ Next, we calculate the probability ($\gamma$) of no collisions during the vulnerability period ⚹ ⚹ In pure Aloha, the probability of no collisions for a given packet during the vulnerability period is given by $\gamma = e^{-2G}$ ⚹ ⚹ Substituting the value of G, we get $\gamma = e^{-2 imes 0.02} = e^{-0.04} \approx 0.9608$ ⚹ ⚹ Finally, the probability of a collision on the radio channel is simply 1 minus the probability of no collision ⚹ ⚹ Therefore, the probability of a collision is $1 - \gamma = 1 - 0.9608 = 0.0392$ ⚹ ⚹ To summarize, the maximum time period when packets are vulnerable to collisions on the channel is 0.001 sec, the normalized offered load to the network (G) is 0.02, the probability of no collisions during the vulnerability period ($\gamma$) is approximately 0.9608, and the probability of a collision on the radio channel is approximately 0.0392 ⚹

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