CS485 Sylabus

### Performance Analysis of the unslotted Aloha protocol

• Reminder: unslotted Aloha

• Recall: unslotted Aloha:

• When a message msg arrives:

it will be transmitted at the same time as its arrival moment:

• Reminder: Throughput

• Throughput S:

 ``` S = p × G Where: S = throughput G = offered load (arrival rate) p = fraction of transmissions that are successful = probability that a transmission is successful ```

• Performance analysis of the unslotted Aloha protocol

• When is a transmission successful:

• Consider a transmission x that arrives at time t:

• A tranmission y that arrive during the transmission of x will collide with the transmission x:

• A tranmission y that arrive 1 time unit before t will also collide with the transmission x:

• Transmissions that arrive outside the critical interval (see figure below) will not collide with the transmission x.

Before:

Or after:

• Conclussion:

• A transmission x is successful if and only if:

 There are no (zero) message arrivals during 2 time units

• Throughput of the unslotted Aloha:

 ``` Throughput S = #successful transmissions per time unit (= slot) = Total # transmissions × fraction successful = G × fraction successful = p × G p = fraction of successful transmission = probabability that a transmission in slot is successful = probability that no (0) message arrives in previous slot = Prob[ 0 arrivals in 2 time unit (t.u.) ] Recall Poisson arrival process: e-Gτ (Gτ)k Prob[ k arrivals in τ t.u. ] = ------------- (G = arrival rate) k! (G × 2)0 e(−G × 2) Prob[ 0 arrivals in 2 t.u. ] = ---------------- 0! = e−2G Hence: Throughput S = p × G = e−2G × G = G e−2G ```

• Summary: throughput of the unslotted Aloha protocol

 ``` S = G × e−2G (Unslotted Aloha) ```

• Graph of Slotted Aloha's throughput (S) vs. offered load (G):

• Maximum throughput of the Slotted Aloha protocol

• Previously:

 ``` Throughput S = G × e−2G (Slotted Aloha) ```

• The maximum can be found through differentiation:

 ``` S = G × e−2G S' = e−2G - 2G × e−2G Solve: S' = 0 ==> e−2G - 2G × e−2G = 0 <==> 1 - 2G = 0 <==> 2G = 1 <==> G = 1/2 = 0.5 ```

• The maximum achievable throughput in the unslotted Aloha network is:

 ``` S(G) = G × e−2G ; max for G = 0.5 S(0.5) = 0.5 × e-1.0 = 0.5 × 0.368 = 0.18 ```

Maximum channel utilization with unslotted Alpoha = 18% (not very good)