4
$\begingroup$

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

  • 0
    McDonald's had tested this and their data says that multiple lines results in shorter average wait times. However, some line waiters find multiple lines more stressful as they start to worry about being in the slower line. – 2017-01-27
  • 1
    I think your additional assumption weighs the scales somewhat — one likely advantage of Scenario B is that it does a better job of keeping the utilization rate up. – 2017-01-27
  • 0
    They are certainly not always the same, as you can show directly from a simple M/M/1 type example. In general, you can upper-bound the total "unfinished work" $U_{single}(t)$ at any time $t$ in the single line system by $U_{single}(t) \leq U_{other}(t) + (n-1)L_{max}$, where $U_{other}(t)$ is the total unfinished work using any other (possibly multi-line) approach, $n$ is the number of (possibly time-varying) servers, and $L_{max}$ is the size of the largest job. This is equation (3) here: http://www-bcf.usc.edu/~mjneely/pdf_papers/routing_to_parallel_queues_ciss2002.pdf – 2017-01-27
  • 1
    How can you assume $100$ percent utilization rate if there might be a point at which there are no more travelers? Note that in most such systems, it is never the case in Scenario B that a customer is waiting *and* a server is idle, but it is possible in Scenario A. – 2017-01-27
  • 0
    In light of that, I wonder, @DougM, how that result obtains. Maybe the servers work faster because they're competing (in a friendly way, one hopes) to clear their lines faster than their colleagues. – 2017-01-27
  • 0
    @BrianTung : If the McDonald's data holds, it must be due to non-modeled factors like what you mention, or: (i) customers don't bump into each other as much walking to/from the different servers if they are in lines close to each server, (ii) customers in multiple lines can see the menu options better while in line, and hence order more quickly. – 2017-01-27
  • 0
    McD experiment - I don't propose that multi-line will be *faster* in my scenario - though it certainly could be if we factor in behavioral factors as Brian/Michael mentioned. Michael - The paper will take me a while to digest. But I wonder if my "additional assumption" will allow us to refine that inequality to Usingle(t)=Uother(t) My logic: In both scenario, the input rate (rate at which people enter the system of queue(s)) and the output rate (aggregate rate at which 3 counters process customers) are the same. Avg wait time can be determined by those rates. Thus avg wait time is the same. – 2017-01-28

1 Answers 1

0

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.