The Mathematics of Post Office Queues

I'm sure there's probably plenty of research (anecdotal, serious, genuine and some of it even humorous) around standing and waiting in queues.  Queuing is the stereotypical British national pastime, and we seem to be quite good at it.  However, if you think that being good at it means that we enjoy it, then you're mistaken.  In my own personal opinion, the worst place to queue is at a local Post Office, and as we approach Christmas and I send out parcels and so on, I'm getting to experience more of it.  Enough of it, in fact, to have done some empirical and informal research of my own.

Firstly, let's look at some assumptions about good customer service in a queue situation:

1.  If the queue gets longer, then introduce more staff to the counters.  Or, in mathematical terms, the number of people on the counters should rise in direct relation to the number of people queuing.  Not a one-to-one relationship (one assistant per customer would be expensive to staff and lead to very wide checkouts), but that there should be a consistent rise in assistants as the queue gets longer.  This continues until, of course, the desks are fully staffed and there's no more capacity.

1b.  If the queue gets longer, then the number of staff visible doing non-customer-facing administration tasks behind the desk should decrease.

2.  Consequently, if the queue gets shorter, it's reasonable to reduce the number of assistants on the counters.

3.  If the queue gets longer, then more effort should be spent to reduce the amount of time it takes to serve each customer.  Keep things formal, keep things concise and keep the queue moving.  The amount of time spent per customer should be inversely proportional to the length of the queue.

Let's have a look at graphical representations of the ideal situations outlined above:

1.The number of assistants increases as queue length increases.  There are two lines plotted on these graphs:  one (the blue line) for a larger office where more staff are employed, and another (the pink line) where there are fewer staff employed.  The principle is the same.

2.  As queue length increases, the number of staff doing non-customer facing work decreases.  Again, there are the two lines, with the pink link showing a smaller office.

3. Amount of time spent on each customer decreases with length of customer.  For smaller branches, the amount of time spent needs to fall more quickly as there are fewer assistants to handle the queue.  One interpretation from the graph shown below (which is a purely illustrative graph, with arbitrary numbers) is that as the queue length increases from 1 to 10 people, the amount of time that should be spent on each customer should fall from around five minutes to one minute.

Now let's look at what can happen when this breaks down.

1.  Increasing the number of staff on the desks when the queue length increases
Really?  More staff?  They're on their lunch break.  They're on training.  Or...
1b.  ...worse still, they're sitting at their desk counting out money, just behind a 'position closed' sign.  They're moving around behind the other assistants, putting paper into drawers and rearranging posters; fetching and distributing their business cards.  Infuriatingly visible to the ever-lengthening queue, and incomprehensibly not serving any of the people in it.

3.  There are some questions, such as, "Do you have a landline at home?" and "Do you know when your buildings and contents insurance is due for renewal?" and "Did you know that we offer travel insurance?" that are understandable when the queue is relatively short.  After all, the desks are well-staffed, people haven't been waiting for very long and there is potential for these up-sell opportunities.  However, when the queue is long, and has been long for many minutes, and when there are only one or two assistants at the desks, these questions really aren't helpful.  

As an additional consequence:  the likelihood of a potential up-sell falls sharply as the queue length increases.  People who've been waiting in the queue for ten minutes to do a three-minute job don't have the time to book an appointment to change their broadband provider.

Here's how I picture the graph of actual time spent per customer, and how it varies with queue length.  This is for the larger office I imagined above, but again, the principles are the same for smaller and larger outlets.

What happens here is that customers actually take it upon themselves to reduce the amount of time it takes to complete their task.  Having spent so long in a queue, they now wish to move through the execution of the task more quickly.  They dispense with formalities about the weather; the time of year; their plans for the weekend, and become much more focused on their task.  They have their money ready.  They have a pen in their hand, if needed.  Out of a sense of camaraderie with their fellow queuers, they aim to get served as quickly as possible, to alleviate any further unnecessary waiting for their fellow man. Moreover, they instinctively decline any offer of discounted insurance or other special deals.

And the consequence?  Here's the related 'probability of successful up-sell' versus queue length graph.

You'll notice that the graph actually falls below zero as the queue length extends beyond around 23 people.  This seems very strange - after all, there can't be less than a 0% chance of something happening.  What happens here is that the negative effect of standing in a queue with that many people in it is that people start to tell their friends about their queuing experience, and this negatively skews any future up-sell opportunities too.  Here, then, is perhaps a more sensible approach to how to make up-sell overtures.

But seriously...

Okay, so perhaps I'm being a little obtuse.  But you get the idea.  The Royal Mail is under threat from smaller couriers and delivery providers, but it has the advantage of a large branch network and brand familiarity.  We go to the Post Office because our parents did, and we know how the delivery system works.  However, the Post Office finds itself having to diversify into various financial products to keep itself profitable.

My worry is that this diversification is detracting from their core customer experience:  I know I've potentially exaggerated how frustrating queuing can be, but it seems to me that Post Office employees are not aware of this and continue to blanket-bomb everybody with the same sales lines, which are increasingly frustrating.  As more couriers become available online, and they continue to become more efficient, building relationships with other high-street stores (Argos is an example), I strongly suspect that more people will turn to them for their delivery needs.  At our house, I would estimate that 30-40% of the parcels we receive come via Hermes, but there's also DPD, parcel2go or Interlink.

So, Royal Mail have the history and the branch network, and the face-to-face engagement with their customers.  I just wonder how long it will last.

In fact, it seems that this is an ongoing problem.  And it's been going on for years!  During my search for a cartoon to add to my article (I borrowed the one just above from the Daily Mail, I found this article which lists everything I've said:  Our too-pushy post offices: Customers forced to queue for too long before being offered services they don’t want. There are some alarming stats in there (and this is three years ago):

- One post office user in three has to queue for at least five minutes, and for some the wait is more than twice that long said Consumer Focus (then Consumer Futures, now Citizen's Advice), which tested 448 post offices in urban high streets.
- For many customers, who are used to supermarkets opening extra tills if queues develop, the small number of counters which are open is exasperating.
- For example, Consumer Futures said a typical Crown office has 7.3 counters, but only 3.8 are actually open.

(And, while I think about it:  everything I've mentioned here applies just as equally to the high street banks.  The queues; the number of assistants at desks; the number of staff walking around not-quite-so-behind-the scenes, and of course the blanket approach to upselling financial products.  Why do you think I bank online?)