A datagram network allows routers to drop packets occasionally when their buffer
ID: 3646018 • Letter: A
Question
A datagram network allows routers to drop packets occasionally when their buffers become overloaded. The probability that a router will drop a packet is p. Consider a network topology in which the source host is connected to the source router, which is connected to an intermediate router, which is connected to the destination router, and then to the destination host. If any of the three routers discard a packet, the source host will eventually time-out and retransmit the packet again. Consider both host-router and router-router connections as hops. All lines are noise-less (error-free), and hosts cannot drop packets.(a) What is the mean number of hops that a packet makes for each transmission?
(b) What is mean number of transmissions required for successful delivery of a packet to the destination host?
Explanation / Answer
Answer (A) 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 1 x p + 2 x (1-p)p + 3 x (1-p)(1-p) which simplifies to p**2 - 3p +3 Answer (B) The probability of successful transmission all the way is (1-p)**2 Let us denote it by w. The average number of transmissions per packet is given by w + 2w(1-w) + 3w(1-w)**2 + + nw(1-w)**(n-1) + .... which reduces to 1/w, that is, 1/(1-p)**2 mean hops per packet = mean hops per transmission x mean number of transmissions which is ( pxp - 3p + 3 ) / ((1-p) x (1-p))