Adjusting for differences in wait time target and actual waited time (Scaling or Indexing)

Our job is relatively easy if we have only one queue with the same target wait time for all . We always serve the person with the longest wait. However, in order to uphold the equity principle, we need to have different wait time targets based on the need. If we have 30 individuals waiting for service and if they are distributed across 5 different target time frames, it would be difficult for a human mind to identify who needs to go first while considering equity, equality and fairness. In this case, we need to interpret (adjust) wait times in light of the patient's priority determined by the wait time target. This is called scaling or indexing.

In the example below, there are 12 people waiting for service. Their current wait times and their target wait times are presented. The question is who should go next. The longest waiting person have been there for 56 minutes (case numbers provided as identifiers not for ranking). Even with 12 people, we need to find a method to identify who should go first. Who do you think should go first?

Case Number	Waited Time (min)	Wait Time Target (min)
1	16	5
2	26	10
3	15	20
4	20	30
5	53	45
6	11	10
7	8	30
8	56	20
9	43	45
10	18	5
11	45	20
12	33	20

If we purely use First In First Out principle, we can reorder based on their wait time and case #8 has the longest wait time (first in) should go first. But the need or severity of case #8 is different than the need of case #10. It is identified in the target wait time that the case #10 should get service within 5 minutes while the case #8 should get service within 20 minutes. Both cases are above their wait time target, case #8 by 36 minutes and case #10 by 13 minutes. It looks like it is justified that case #8 should receive the service first because s/he waited the longest and also it has the most minutes above its target. However, we can only be sure of our selection between the patients where the wait time targets is same. Certainly between the cases #8, 11, 12, and 3, the case #8 should receive the service first because it waited the longest. But how can we compare the rest to #8?

Case Number	Waited Time (min)	Wait Time Target (min)
8	56	20
5	53	45
11	45	20
9	43	45
12	33	20
2	26	10
4	20	30
10	18	5
1	16	5
3	15	20
6	11	10
7	8	30

In order to ensure equity and fairness, we need to use a system to select the person based on the considerations of need and wait time, where scaling or indexing is useful.

Scaling is adjusting the wait time of an individual based on their need. Indexing is the same concept but interprets wait time based on the wait time target. The theory behind both concepts is the same.

What is the value of each time unit for the individual in comparison to others'. If the maximum wait time target is 45 minutes, each minute is 9 times more valuable (or important) for a patient with 5 minute wait time target. In scaling, we multiply each waited minute with 9 for the people with 5 minute wait time target and 3 for the people with 15 minute wait time target. In this case, case #10 would have a scaled (or adjusted) wait time of 162 minutes (18 x 9).

Similarly in indexing we divide the wait time by the wait time target. This will give us the length of wait as a ratio of the wait time target. This is probably easier to understand and, more importantly, easier to explain. Table below show the order of recommended service order for people on the waitlist based on the indexing methodology. Case #10 and #1 climbed to the top of the list.

Case Number	Waited Time (min)	Wait Time Target (min)	Waited Time Indexed to Target
10	18	5	3.6
1	16	5	3.2
8	56	20	2.8
2	26	10	2.6
11	45	20	2.3
12	33	20	1.7
5	53	45	1.2
6	11	10	1.1
9	43	45	1.0
3	15	20	0.8
4	20	30	0.7
7	8	30	0.3

With scaling or indexing, we eliminated differences between the wait time targets and waited time, and brought all cases to the same level. We can now use the Queue Discipline Ratio (explained in the previous blog entry) effectively for all cases. We don't have several distributions with different wait times. In the indexing model, 1 will always represents the target. We can divide the average indexed value for people who received their service with the average indexed value for people who are currently waiting. Our objective is to achieve the third graph for equitable and fair delivery of service (when there is a waitlist).