In the diagram below we show 3 nodes transmitting packets using slotted Aloha. A

Question

In the diagram below we show 3 nodes transmitting packets using slotted Aloha.

A. What is the measured utilization of the channel over 10 slots (from slot 0 to slot 9)?

B. What is the optimal probability of transmission if there are 3 nodes (N=3) having data to transmit all the time?

C. What is the long-term average normalized throughput S (N, p*) of the slotted Aloha system with N=3 nodes and the optimal transmission probability as discussed in B.? How much does the long term average throughput differ from the measured throughput in the figure below?

Explanation / Answer

What is the measured utilization of the channel over 10 slots (from slot 0 to slot 9)?
Pr[0] is the probability of a slot that does not contain frame, i.e., idle frame slot. Pr[0]=0.1. Pr[0]=G0e-G/0!=1*e-G=0.1. ln(e-G)=ln(0.1). -G=-2.30259. G=2.30259.
Therefore the measured utilization of channel over 10 slots (from slot 0 to slot 9) load is 2.30259.

The throughput is S=Ge-G=2.30259*e-2.30259=0.230258. The throughput is 23.0258%, below the optimal 36.8%.
The optimal operating condition for slotted ALOHA is G=05. Here G=2.30259 > 0.5. Therefore it is an overloaded channel.

What is the optimal probability of transmission if there are 3 nodes (N=3) having data to transmit all the time?
If a node starts transmission at the beginning of a mini-slot, by the end of the mini- slot either
– No collision occurred and the rest of the transmission will be uninterrupted
– A collision occurred, but by the end of the mini-slot the channel would be idle again
Hence a collision at most affects one mini-slot

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