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A datagram network allows routers to drop packets whenever they need to. The probability of a router discarding a packet is p. Consider the case of a source host connected to the source router, which is connected to the destination router, and then to the destination host. If either of the routers discards a packet, the source host eventually times out and tries again. If both host-router and router-router lines are counted as hops, what is the mean number of hops a packet makes per transmission?

C. 3p+3
D. p+3

in Computer Networks by Junior (579 points)
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Each packet may make 1, 2 or 3 hops. For 1 hop, the first router drops it and the probability is $p$. For 2 hops, it goes through first router but not the second and the probability is $(1-p)p$. For 3 hops, it goes through both routers and the probability is $(1-p)(1-p)$. Mean hops per transmission is given by $1 \times p + 2 \times (1-p)p + 3 \times (1-p)(1-p)$ which simplifies to $p^2 - 3p +3$

can you explain me little bit more ?? I'm very weak in that subject that'Y

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