CS485 Sylabus

### Performance Analysis of the slotted Aloha protocol

• Reminder: Slotted Aloha

• Recall: Slotted Aloha:

• Time is divided into slots

• A transmission can only begin at the start of a slot:

• When a message msg arrives:

the message msg must wait until the start of a slot to be transmitted

• Choice of time unit

• Fact:

• Normally, we use:

 time unit = second

• However:

 Expressions can often be simplified using a different time unit

• Time unit used in Aloha analysis:

 1 time unit = length of 1 message                     = length of 1 slot

• 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 ```

• Converting S and G for changing time unit:

• G = # messages that arrives per time unit

 We may need to convert the value when the time unit changes

• Example:

• Suppose G = 5 arrivals per sec
• Length of message = 2 sec

Then:

 ``` # arrivals = 5 in 1 sec (5/sec) = 10 in 2 sec = 10 per "length of message" G = 10 / time unit (with message length as time unit) ```

• Performance analysis of the slotted Aloha protocol

• When is a transmission successful:

• Consider a transmission x that arrives prior to a slot s:

• A tranmission y that arrive during the slot "s−1" will collide with the transmission x:

• Transmissions that do not arrive during the slot "s−1" will not collide with the transmission x:

Or:

• Conclussion:

• A transmission x in slot s is successful if and only if:

 There are no (zero) message arrivals during the slot s − 1

• Throughput of the slotted 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 1 time unit (t.u.) ] Recall Poisson arrival process: e-Gτ (Gτ)k Prob[ k arrivals in τ t.u. ] = ------------- (G = arrival rate) k! (G × 1)0 e(−G × 1) Prob[ 0 arrivals in 1 t.u. ] = ---------------- 0! = e−G Hence: Throughput S = p × G = e−G × G = G e−G ```

• Summary: throughput of the slotted Aloha protocol

 ``` S = G × e−G (Slotted Aloha) ```

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

• Maximum throughput of the Slotted Aloha protocol

• Previously:

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

• The maximum can be found through differentiation:

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

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

 ``` S(1.0) = 1.0 × e-1.0 = 1.0 × 0.368 = 0.37 ```

Maximum channel utilization with slotted Alpoha = 37% (not very good)