Thursday, 26 June 2014

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.

Wednesday, 25 June 2014

The game of buying a house and stamp duty

My title is an awkward one, I know, and I tried to think of a more punchy headline.

Earlier this year, the average price of a house in the UK reached £250,000.  Now averages in this case do not mean a great deal, as house prices do not follow a Normal (Gaussian) distribution.  But that figure is important, symbolically.  You see, house sales in the UK attract a tax, Stamp Duty Land Tax (SDLT), paid by the buyer as a percentage of the total value paid for the property.  And  SDLT is a cliff-edge tax.  If the transaction is less than £125K, SDLT is 0%.  The next band is up to £250K, and is 1%.  The next band goes to £500K and is 3%, and then is 4% up to £1M, 5% up to £2M and 7% over £2M.  The cliff-edge means that the SDLT on a property costing £250,000 is £2500.  If the property costs one pound more, £250,001, SDLT is £7500.03p.  Hence the importance of reaching that average price.

There are similar cliff-edges at £125K (£0 to £1250), at £500K (£15000 to £20000) and £1M and £2M, but it is the size of the edge at £250K which causes most angst.  Needless to say, there are many ploys of avoiding the jump in SDLT.  It is illegal, for instance to offer £250K for a house and then offer £10K for the bedroom curtains.  Because solicitors are involved with the purchase, they are aware of the need to keep within the law, and ask purchasers to sign documents that confirm that there are no inappropriate side deals involved.  [It is not clear whether one purchaser got away with his offer to pay for the £30,000 wedding of the vendor's daughter to reduce the house price from £520K to £490K and the SDLT from £20800 to £14700.]

All these cliff-edges lead to some strategies for vendors setting the asking price for a property, and for purchasers making offers.  There are very few properties priced close to a cliff-edge threshold.  You won't find many houses with an asking price of £255K.  If you do, then it is probably a sign that the vendor will accept £250K but not a penny less.  In fact, the gap above the cliff-edge is quite large.  Our local paper lists very few between £250K and £280K.  Because of its good economy, Exeter has a vibrant housing market, and our average house prices are very close to £250K (according to one website, over the last year, the mean was £234K, and the rate of increase has pushed it higher).  Over the past year, over fifty houses with an Exeter address sold for exactly £250K, and just 8 sold for between £250,001 and £260,000 ... including one at £253,000.  (The skewness of the house price distribution in Exeter is shown by the observation that of 2367 property sales in the last year, 1714 were under £250K, 2296 under £500K.)

So, the estate agents in Exeter, faced with house prices hovering around the cliff-edge, have to give careful advice about the strategy for pricing. 

But, why did anyone pay £253,000 for a property?  Here is the other side to the situation.  The purchaser also has to devise a strategy.  Many houses are sold by formal or informal tender, and an attractive property may be sold for more than the asking price.  The vendor and agent know this - and it is in their interests to encourage it.  So, what offer does a buyer make for a house listed at £240K, knowing the cliff-edge?  I suspect that the property at £253K was the result of a determined buyer, deciding that he/she really wanted the property, and being prepared to pay the extra £5K tax as part of the deal. The prospective buyer doesn't know how many other bids there will be (unless the agent tells them, nor what bids have been made.)  And the agent and vendor must agree a time to set a deadline for all offers.

I keep wondering how OR could actually help buyers and sellers with property close to one of the cliff-edges.  But I know that OR, through game theory and bidding strategy theory, can offer ways of thinking about the matter.