A very public set of queues in series

When our class was introduced to the art and craft of simulation model building, the problem we were given was to model a laundrette.  The reason for this exercise was to study a familiar situation (at least for most students), where there were queues in sequence and non-standard queues as well.  Some customers would use more than one washing machine (one type of server), some would use more than one dryer, some would want both washer and dryer, while others would only use a washer.  The service times on the machines were to all intents and purposes constant, in contrast to the systems we had met in queue theory courses, where an exponential service time was assumed (or in special cases, Gamma or Erlang).  And, if the instructor was especially devious, we would be reminded that this laundrette would not reach a steady state, because the arrival rate of customers was time-dependent.  Our aim in the simulation course was to produce a model and explore the consequences of varying the numbers of washing machines and tumble-dryers.  As I recall, there were various assumptions made about the behaviour of customers who had to queue; generally, it was assumed that every customer would be served.  The doors of the launderette would be closed to customers at a particular moment, but those “inside” would complete their service.  Given the arrival pattern of customers, costs of washes, drying, running the machines and the shop, what would be the best number of each machine to have?

Memories of this exercise came to me when I encountered another everyday situation with queues in sequence and many of the above “features” as well.  I have written earlier about queues in public toilets, and my encounter was in two of these.  Sometime a student project could address the provision of equipment (WCs, urinals, basins, soap dispensers and hand dryers – all “servers” of the sequence of queues) in such facilities, perhaps restricted to those in supermarkets or motorway service stations.  (Just give me the credit for the idea!) 

Many places use a “hole in the wall” which provides soap, water and drying air, either at the press of a button, or automatically in sequence.  One gentlemen’s toilet in Exeter has four urinals, two WCs and two holes in the wall.  Unfortunately, the holes in the wall are programmed to give soap at the press of a button, then 10 seconds of water, followed by 30 seconds of drying air.  Even if a user removes his hands from the orifice, the hot air keeps on coming.  So, the service time at each opening is well over 40 seconds, probably nearer a minute.  It is no wonder that many gentlemen do not bother to use the hand-cleaning machinery.  They would have to queue in the small space, or stand helplessly waiting for the hot air to come to an end so they could start their service.  (It is left as an exercise in human observation to determine how long a male needs to spend at a urinal, and compare that with the time at the hole in the wall.) 

On holiday, I met the opposite extreme; the blast of hot air was a mere four seconds.  Again the user was frustrated, this time at having to repeatedly press a button to get more drying time.  By comparison, one manufacturer claims that their very fast hand dryers take “Only ten seconds”.

So, I suspect that in neither case has anyone considered the implications of the queues and the congestion and frustration caused at each one.  

In contrast, office buildings have to provide facilities according to formulae based on the number of workers in the building – and those formulae have been derived from queue models and (in many cases) simulation.