Question

Automatic repeat request (ARQ) is an error-control method for data transmission that uses
acknowledgements to achieve reliable data transmission over an unreliable link. If the sender
does not receive an acknowledgment before a timeout, it re-transmits the same packet to the

destination. This process repeats itself until the sender obtains confirmation that the infor-
mation was successfully delivered. Suppose that the probability of successfully transmitting

a packet is 0.25, independent from one attempt to another.
(a) 1 pt – What is the probability of a transmission failure on the first attempt?

(b) 1 pt – What is the probability of being successful within the first 3 attempts?
(c) 1 pt – What is the probability that a packet requires exactly k attempts, where k is an
integer, for successful delivery of the packet to the destination?

(d) 1 pt – Compute the probability that the packet is successfully delivered to the destination on an even attempt (i.e., the attempt 2, or 4, or 6, or 8, or ...).

(e) 1 pt – Are the events in parts (d) and (e) independent? Support your claim through a
mathematical argument.

Answer 1

(a) Probability of transmission failure on the first attempt = 0.75.
(b) Probability of being successful within the first 3 attempts = 0.75*0.75*0.25 = 0.1406.
(c) Probability that a packet requires exactly k attempts = (0.75)^k-1 * 0.25, where k = 1, 2, 3, .....
(d) Probability that the packet is successfully delivered to the destination on an even attempt = (0.75*0.25) + (0.75³ * 0.25) + (0.75⁵ * 0.25) + (0.75⁷ * 0.25) .......... = = 0.4286.

(e) (There is no event specified in part (e)).