Set-up:
Travelers are waiting in line to be processed by immigration officers. There are N officers, each with their own counter. No assumptions about line-switching, constant processing speed, homogeneous processing speed, or homogeneous entrance rate of travelers.
Scenario A: Travelers line up in one of 3 lines, each line leading to one of the 3 counters.
Scenario B: Travelers line up in a single line. The person at the head of the line will be directed to the next available officer.
Additional assumption: All officers will be at 100% utilization rate.
I argue that average queue time will be the same in both scenarios. Many co-workers, friends, and the following quotes disagree and believe that Scenario B (single-line) will result in a shorter average queue time.
Who's right?
"Finally, a single-line, multiple-server system has better performance in terms of waiting times than the same system with a line for each server." – Reid, Sanders; Operations Management
"Research has proven that a single line, multi-server waiting system is faster than the multiple line approach." – http://blog.lavi.com/2014/08/07/single-line-queue/
"A Long Line for a Shorter Wait at the Supermarket" – http://www.nytimes.com/2007/06/23/business/23checkout.html
Best Answer
The average waiting time (mathematical expectation) is the same but the variation of waiting time for a single line is smaller than that of multiple lines (easy to demonstrate mathematically). Thus a risk-averse person would prefer a single line. The intuition is: if you end up in a faster server your time of waiting is much shortened; if you end up in a slow server then your time of waiting is much prolonged (multiplied by the number of people ahead of you plus you). For a single line, your waiting time is simply the server's speed, it is not exacerbated.