Need to develop a formula for index/indicator measuring wait time for tellers

Question

I have data that looks like this:

        Tell_1  Tell_2  Tell_3
        0       0      -8
       -3       0       0
        0       0       0
       -4      -2       0
        0       0      -2
      -14      -4      -1
        0       0       0
       -1       0      -1

Index:  I1      I2      I3

'----------------------------------------------

This data represents the wait time of each customer at a teller. The value 0 means the customer did not wait at all. The value -x means the customer had to wait x minutes before being served.

How can I develop an teller performance index (a mathematical function that takes the wait times an produce 1 value) that shows the teller performance ($I1$, $I2$, $I3$,...)?

One way is to just sum each column to obtain -22,-6, and -12. This is only good at showing wait times. However, this does not show how many customers were served immediately (which is a good thing). As a result this index is no good.

Another approach that I considered was to assume a max wait time value, say 100, then calculate the index per teller as $\sum(x_i+100)$ to get: 778, 794, and 788. But how good is this one?

I need the index to reflect both the wait times and the number of customers that were served immediately.

Note: This is not a homework, also, it is not a real situation.

Thanks for your help.

Answer 1

I would recommend looking at the average wait time per customer. While this does not directly count the number of customers who did not have to wait at all, that number definitely affects the measure.

Teller #1 has an average wait time per customer of 2.75 minutes.

Teller #2 has an average wait time per customer of 0.75 minutes.

Teller #3 has an average wait time per customer of 1.50 minutes.

This would suggest that perhaps Teller #2 is more efficient.

Answer 2

I think you'll have a hard time finding a single number that will do what you want. Any reason why you can have a vector (more than one index)?

But this may do some of what you want.

Each customer a teller sees is worth, say, $20$ points. The wait time subtracts from this $1$ point per minute, down to zero. If the customer is seen without any wait, then there's a bonus of $10$ points.

By this formula, the three tellers in order would have a score of $178,214,188$.

So there's a premium on no wait time, and the score increases with the number of customers seen.

Answer 3

You could use the weighted average of the average wait time of those who had to wait and the proportion of customers served immediately. This would not have as nice as interpretation as others but it would be very flexible and allow you to decide the weight between the time of those who wait and number of customers served immediately.

You would need to pick $\alpha$ very carefully based on what you want.

$Index = \alpha*AVG + (1-\alpha)*p$

For example if $\alpha = .5$ then

Tel_1: $.5*22/4 + .5*4/8 = 3$

Tel_2: $.5*3/2 + .5*2/8 = .875$

Tel_3: $.5*12/4 + .5*4/8 = 1.75$

The lower the index the better where $0$ means no one had to wait