MAC Protocol – Slotted ALOHA Probability and Efficiency

mac

Given:

There are two active nodes, A and B, in a slotted ALOHA network.

Node A transmit in each slot with probability a while Node B transmit
with probability b.

The first slot is numbered Slot 1, the second slot is numbered Slot 2, etc.

What is the probability that Node A succeeds for the first time in Slot 6? What is the efficiency of the system?

My reasoning/solution (that is incorrect):

  1. For node A to succeed for the first time in slot 6, this means I need to calculate the probability (Node A succeed for the first time in slot 6) = Pr(Node A fail in slot 1,2,3,4,5) * Pr(Node A succeed in slot 6) = (1-a)(1-a)(1-a)(1-a)(1-a)a
    However this seems to be wrong, do I need to factor in node B in this case?
    Should if I need to account for node B, IP will need to calculate:
    Pr(Node A fail in slots 1-5, Node B can either succeed or fail in slots 1-5) * Pr(node A succeeds in slot 6 but node B fails)?
    If so, how do I calculate it? A bit stumped on this question.

  2. Efficiency of the system can be calculated by the probability that some node succeeds in a slot? How do I calculate this when both nodes have different probabilities? I am only taught how to calculate if they have the same probability.

Best Answer

Given the correct answer and what is written in answer of @Zac67 I interpret it as follows.

Probability of X := first packet of A being transmitted successfully only in slot 6

P (X) = P ( [X1 := no packet of A is transmitted in slots 1-5] and [X2 := packet of A is transmitted in slot 6]) = /both events are independent/ P (X1) * P(X2) = /all 5 slots are independent/ P (X3 := no packet of A is transmitted in a slot)^5 * P(X2)

P (X3) = P ([A does not transmit] or [A transmits, but there is a collision]) = P ([A does not transmit]) + P([A transmits, but there is a collision]) = (1-a) + (ab)

P (X2) = P ([A transmits but B not]) = P (A transmits) * P (B does not transmit) = a * (1-b).

Thus P(X) = (1 - a + ab)^5 * a (1 - b)

Efficiency

If we define efficiency a the probability that there is a successful transmission in a slot, then

P ([successful transmission]) = P ([only one of A or B transmits]) = P ([A transmits but B not] or [B transmits but A not]) = /* events are mutually exclusive */ P ([A transmits but B not]) + P([B transmits but A not]) = a ( 1 - b) + b (1 - a) = a + b - 2ab.

I am not sure how to generify (word) this formula for more than 2 systems.

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