How long do I need to wait to see a GP? \[M]
Waiting to see the doctor can be an extremely time consuming process - it's a real pet hate. One thing I hate even more though is politicians meddling - setting arbitrary (moving-)targets (but investing no money) to improve service. This causes a great deal of bureaucracy and often-times makes things even worse... Furthermore, it allows me to blog-post on why constructing arbitrary goals and metrics to measure quality without thinking about how they effect the thing you're trying to measure (i.e. almost all the time) is futile at best and moronic at worst.
To highlight what happens when politicians meddle, I've written an extended simulation experiment... Suppose we are interested in how long patients have to wait over the course of one day (24 hours) to see the doctor in a hypothetical surgery, in which we have one doctor who takes 15 minutes per patient. Patients arrive at random, but on average at 20 minute intervals (for those who like statistics - the inter-arrival times are exponentially distributed), and the doctor is incredibly industrious and sees patients immediately if possible and on a first-come-first-serve basis otherwise.
You may naturally think that this doctor has a good deal and waiting in his surgery will be a relatively short and painless experience as patient arrivals are less frequent than the time he has to see them - however, the random nature of arrivals under even this simple model mean that there are periods where the surgery is very busy and you can wait long periods of time. In the graph on the left hand side of the following figure I have plotted the time at which patients have arrived (x-axis) against how long they have waited (y-axis). In total there are 16 patients who have not had to wait at all, but one patient who had to wait 85 minutes. The average wait is 22 minutes.
Now politicians love waiting time targets, and so to curry favour with the public may decide to make policy that doctors should be financial penalised per patient (say £10) if the patient has been waiting in excess of some arbitrary target (say 45 minutes for arguments sake - the red horizontal line above). In this particular doctors surgery on this particular week there have been 15 patients in total who have had to wait longer than 45 minutes (and so there would be a £150 financial penalty). Unfortunately without further investment (and recall this is a particularly industrious doctor who is working at maximum capacity), due to the random arrival times of patients the only way to avoid excessive penalties is to modify (detrimentally) behaviour...
Now, the optimal strategy for this particular doctor to minimise possible financial penalties is to continue to operate a first-come-first-serve practice, but if the patient has waited longer than 45 minutes then they should defer seeing that patient until the surgery is empty of patients who have waited under 45 minutes (they have already been penalised for this patient, and having them wait longer has no further penalty). In the right hand graph in the figure above I have considered the same patients with the same arrival times but served according to this modified strategy. Indeed, it has been effective in that now only 6 patients (versus 15) have exceeded the 45 minute threshold (saving £90 in financial penalties!). However, note that the patients that do wait over 45 minutes now wait considerable periods of time (with one patient waiting 2 1/2 hours!). Indeed, the average wait time for patients is exactly the same, the only difference is the amount of time patients need to wait is more variable (which is surely a bad thing if you are planning to go to the doctors).
Comically, if the metric chosen to measure performance was to instead promote having the lowest possible variation in waiting times (which to me seems more sensible), then to achieve this outcome (and assuming the doctor deploys the optimal strategy) then there should be no waiting time target. Indeed, lowering the waiting time target (as politicians love to do to show increasing performance), makes things ever more variable / worse. In the following figure I have considered the same surgery over a one month period with no target and a target of 20 minutes waiting time (with the same financial penalties). Note that patients now have to wait up to 10 hours...!
Now, I have considered distribution of patient waiting times for one realisation of patient arrivals over the course of a year at the surgery under the optimal no-target (blue) and 20 minute target (red) strategies. The crosses mark the longest patient wait under either strategy.
Note that the effect of the target is to in effect fatten the tail of the distribution. Both strategies have the same average wait time, but the target strategy has a wait time which is far more variable.
In summary: If you are a politician (or HR, or other goal-setter), please think very carefully at what you are intending to measure and what you are trying to achieve before sticking your oar in...