More than 100 flights cancelled and delays of sometimes over 20 hours with consequences for over 76,000 travelers. That was the sad result of Vueling’s planning policies according to an article in last week’s Dutch Financial Times and (in English) in El Pais. Experts stated that Vueling scheduled more flights than they even had airplanes and staff. In between flights they apparently scheduled no safety time meaning that even the slightest hiccup leads to a snowball effect all subsequent flights. In the end, Vueling’s personnel therefore cannot take the required breaks resulting in even further delays and cancellations when the plane needs to be grounded and the personnel needs to rest. This case represents a good example of Kingman’s formula, part of queuing theory, and of how not using these insights results in poor planning, overburdening (Muri) and unevenness (Mura) and in the end disastrous results for lead times and service.

Kingman’s Formula

John Kingman (1939) is a British professor, known, among other things, for his studies into queuing theory. 1961, when pursuing his Ph. D., he published a paper about queue behavior in heavy traffic. This paper presented a formula now known as the Kingman formula. The formula tries to explain waiting time and, without going into too many details of Kingman’s formula, in essence it states that waiting time is a function of:

average processing time;

variation in cycle time (i.e., departures from the queue);

variation in the takt time or demand rate (i.e., arrivals in the queue); and

the utilization (i.e., load/capacity or arrival rate/departure rate).

Let’s explore this formula, how it relates to Muri (overburdening) and Mura (unevenness), and how Vueling (and others) can benefit from Kingman’s insights.

Utilization and the Muri Effect

Kingman’s formula states that the higher the utilization, the longer the lead time. This is an exponential relationship. The closer you get to 100% utilization, the faster your lead time will increase and ultimately will rocket straight through the roof. It’s a clear example of how overburdening resources (Muri in Lean lingo) leads to serious problems, not only for the resources involved, but also for your customers.

So the first lesson for Vueling and –to be honest– quite some others, is to not plan at 100% utilization. In fact you plan for poor service when you do so.

When we drill down somewhat further, we see that utilization is defined as load/capacity. Load is defined as the demand rate, and capacity as the departure rate. Now when we state that we should not plan at 100% utilization, we in fact say that the demand rate should not exceed the departure rate. In Lean lingo: takt time should not be smaller than cycle time. A lesson for sales and commercial functions therefore is to never sell more than what can be produced. Now this seems obvious, but in practice this rule is hardly ever respected. The focus typically is on top line and top line only. So, our second lesson is to never sell more than what can be sold. Did you read that, Vueling?

Furthermore, to make sure we don’t go beyond a certain maximum utilization level, we better also ensure we are planning based upon realistic cycle times, and not some theoretical values. This is our third lesson: don’t plan using theoretical cycle times.

And knowing that our actual capacity to produce is unfortunately restricted by a lot of waste (remember Ohno: “capacity = work + waste”), it is clear that we can reduce our utilization, shorten our lead times and increase our service by reducing the waste that obstructs us to achieve our true potential. The fourth lesson therefore is to reduce utilization by continuously reducing waste.

Variation and the Mura Effect

The second important element in Kingman’s formula is variation; variation in arrival rate (or takt time) and variation in departure rate (or cycle time). Kingman’s formula postulates that the higher the variation in these parameters, the earlier the lead time lengthening effect of utilization will be present. In other words, at a given level of utilization, your lead time and inventories will be higher when you experience more variation. A theoretical foundation for the negative effects of uneven workloads referred to as Mura in Lean.

So let’s drill down somewhat more here as well. First about the variation in demand rate or takt time. It is key to create level demand for the resources and to do so we have to start with level selling, something I discussed in one of my earlier blog posts. And when we are not able to perfectly level our sales, we need to translate the remaining, uneven demand into a level demand for production. This is the territory of mixed-level loading or Heijunka in Lean. I also wrote about this before in a blog post about EPEI. Level selling and Heijunka reduce variation on the demand side. Also, work should be balanced across the various processing steps to reduce Mura. This is known as line balancing or Yamazumi. Finally, work should be released in a fashion that we do not overburden resources. Pull flow principles materialized via Kanban help leveling the load in real time, on which I wrote a several posts as well. The less we are able to do this, the more risk we take for long and variable lead times and inventories. So lesson five is to level sell and to level load (through Heijunka, Yamazumi and Kanban).

Now let’s turn to the other side. To reduce the lead time lengthening effect of cycle time variation, we need to standardize our work methods and continuously detect and eradicate all causes of variation from the standard. This implies disciplined and rigorous standardization, problem detection and eradication of root causes. This summarizes lesson six: standardize, detect and eradicate problems.

Now of course in reality it will difficult if not impossible to perfectly level demand. And at the same time I haven’t yet seen the perfect operation. So you will experience variation. You will need to face this variation, and buffer for it while still necessary. And the buffer should be adapted to the level of problems you experience. And subsequently you should never stop eradicating problems, day-after-day-after-day…

When you have a well-running, well-industrialized, high-volume, low mix process you might plan at 95% utilization. But when your process requires continuous debugging, is often confronted with unscheduled down time and needs to produce a large portfolio of different products you would be a fool to plan that one at the same level of utilization. So ensure sufficient “breathing space” for your process to deal with its problems, eradicate your problems and subsequently reduce your buffers.

In Kingman’s words, adapt your utilization to the level of variation you experience. This is lesson seven for Vueling. When you plan with zero buffers, you are living in a dream. Wake up.

Experiencing Kingman

To illustrate the effects represented by Kingman’s formula, there are some nice simulators around on the web. You can find a nice example of such a simulator at http://www.traffic-simulation.de/.

In this simulation, you can play with utilization (particularly the “density” parameter in the model) and variation (using the mix between passenger cars and trucks using “truck fraction”) and evaluate the effects on road congestion.

Kingman’s Seven Lessons

To avoid long and variable lead times and the poor and unreliable service that goes with it, we should eliminate Muri (overburdening) and Mura (unevenness). And Kingman’s formula from queuing theory helps us identifying the levers that we can use to improve. They can be summarized in the following seven lessons:

Plan at less than 100% utilization Sell only what can be sold Plan using realistic cycle times Reduce utilization by continuously reducing waste Level sell and level load (through Heijunka, Yamazumi and Kanban) Standardize, and continuously detect and eradicate problems to reduce variation Adapt your utilization to the level of variation you experience

So next time you are planning, plan to succeed. Don’t plan to fail.