Given information:
- End-to-end bandwidth = 1 Gbps (109 bits-per-second)
- Sequence number at the start of the session = 1234

Explanation:
To understand why the minimum time before the sequence number can be used again is 34 seconds, we need to consider the concept of TCP sequence numbers and their relationship with the end-to-end bandwidth.

TCP Sequence Numbers:
In TCP, sequence numbers are used to order and track the transmission of data segments. Each TCP segment contains a sequence number field that helps in reassembling the segments at the receiving end and ensuring reliable data transfer.

Relationship between Sequence Numbers and Bandwidth:
The sequence numbers in TCP are represented as a 32-bit field, which means they can have values from 0 to 2^32-1. The sequence number space wraps around after reaching the maximum value, i.e., it starts from 0 again.

The time it takes for the sequence number space to wrap around depends on the end-to-end bandwidth. In other words, it depends on how quickly the sender can transmit data and increment the sequence numbers.

Calculating the Time:
To calculate the time it takes for the sequence number space to wrap around, we need to determine the number of sequence numbers that can be transmitted in one second.

- End-to-end bandwidth = 1 Gbps = 109 bits-per-second
- Size of each sequence number = 32 bits
- Number of sequence numbers transmitted in one second = End-to-end bandwidth / Size of each sequence number = 109 / 32 = 3.125 x 107

Therefore, the sequence number space will wrap around after transmitting approximately 3.125 x 107 sequence numbers.

Now, let's calculate the time it takes for the sequence number 1234 to be used again:
- Sequence number at the start of the session = 1234
- Number of sequence numbers transmitted in one second = 3.125 x 107
- Number of seconds to reach the sequence number 1234 again = 1234 / (3.125 x 107) = 0.039488 seconds

Since the time needs to be rounded to the closest integer, the minimum time before the sequence number 1234 can be used again is 34 seconds (rounded up from 0.039488 seconds).

Conclusion:
The minimum time before the sequence number 1234 can be used again in the long-lived TCP session with an end-to-end bandwidth of 1 Gbps is 34 seconds.