# Lead Time vs Utilization — a mathematics explanation of the chart in Phoenix Project

In the city I’m working in, Hong Kong, the average working hour is 49 hours per week.

No one would like to work overtime if they can complete their on time. Everyone would like to have a good work-life balance, to spend more time with their family and friends. Everyone would rather do more effective structured work rather than “fire fighting” with tight deadlines.

In this article, I am going to introduce a mathematical link between utilization and task wait time. You will understand that having slack time is actually an essential part of your job and watching Youtube during office hour is merely a demonstration of your time management skill.

For most office jobs, tasks arrives at random time and the size of each task varies. This variation implies that queue of tasks will occasionally build up when multiple big projects pop up. Because there is random variations in tasks, queues will build up even utilization (% time being busy) is not yet 100%. When you are close to 100% occupied, this queue would take a long time to clear. Each task would spend a substantial time waiting in the queue. The interesting piece of mathematics is the relationship between wait time and utilization:

Big surprise! When you are 100% occupied, it’s mathematically proven that any task will have infinite wait time. Size of each piece of task does not matter as the wait time would be forever when utilization is very high.

## Lesson Learned

Mathematically, it is described as the Kingman’s formula: According to Wikipedia on the formula, the mean waiting time is approximately:

In layman terms,

Waiting Time = [A formula of Utilization] ×[Variation for arrivals + variation for task length] ×[Time required for a typical task]

Therefore to reduce wait time, you can either:

2. Reduce variation. Getting tasks more regularly and tasks taking similar time.
3. Reduce the time required for a task.

[Formula of Utilization] = (Utilization %) / (100 - Utilization %)

• Utilization =90%, Output = 9
• Utilization=95%, Output = 19
• Utilization = 99%, Output = 99
• Utilization = 99.9%, Output = 999
• Utilization = 99.99%, Output = 9999

which implies that waiting time will still go through the ceiling if you do not control your utilization. Any effort in reducing variation and being more efficient in a task would not help if you do not control utilization.

## Action

Keeping 80–90% utilization as a healthy level of work, wait time would grow quickly once past the 90% utilization mark. Therefore, 10-20% of time watching Youtube is actually a good sign that everything is under control. If you keep yourself 100% busy, then you are guaranteed mathematically that you will always miss deadline. What really matter is that you can consistently deliver work in a reasonably amount of time. I used to think that slacking off is a bad thing in bosses’ eyes, but not any longer. Slacking off is an essential part of a good time management strategy.