What Little’s Law Tells Us About Net Zero

The Little’s Law

Little’s Law is a commonly used principle in operations management that describes the relation between the flow rate (rate at which entities enter and exit from a system), the flow time (time spent in the system) and the inventory levels. Think of a pizza place. The number of orders coming in and going out for delivery per unit time would be the flow rate while the time it takes to make a pizza would be the flow time. Little’s Law gives us the following model:

Flow Rate X Flow Time = Inventory

This means the number of pizzas being made or waiting in the store at any point can be given as the product of the flow rate and flow time. Two key points one must remember when applying the Little’s Law is that the flow rate and flow time are independent of each other. The rate at which people order pizzas has nothing to do with the time it takes to make a pizza. Secondly, the Little’s Law assumes a steady state where a dynamic system converges to some fixed long run averages.

A Change in Perspective

A way of framing the net zero problem is as an inventory model of carbon emissions where the emission and absorption rates correspond to the flow rate while the atmospheric residence time of carbon emissions (~100 years) is the flow time. Multiplying these would result in the total inventory in the system as per the Little’s Law which would be the total amount of Carbon in the atmosphere.

Reverse engineering the 1.5 °C goal of the Paris Climate Agreement a range of corresponding atmospheric CO₂ concentrations and by extension corresponding Gigatonnes of Carbon (GtC) in the atmosphere can be derived. While the actual impact would depend on the effect of other gasses as well, these numbers are a good compass in modelling a range of possibilities.

While Little’s Law applies to a steady state, the assumption of Net Zero is that a steady state with respect to carbon emissions has been achieved where the emission rate equals the absorption rate leading to net zero emissions, thereby allowing us to model the situation as a long-run steady state inventory model.

What the Data Says

The literature provides a range of possible CO₂ concentrations for 1.5 °C:

• Optimistic Estimate about 425 ppm CO₂
• Moderate Estimate: around 507 ppm CO₂
• Pessimistic Estimate: up to 785 ppm CO₂

At present, the CO₂ levels are already around 430 ppm highlighting the urgency with which action must be taken.

As a standard method of calculations, one part per million (ppm) of CO₂ in the atmosphere corresponds to roughly about 2.13 (GtC). Using this conversion, the total carbon in the atmosphere under different 1.5 °C scenarios looks like this:

• 425 ppm CO₂ = ~905 GtC
• 507 ppm CO₂ = ~1,080 GtC
• 785 ppm CO₂ = ~1,672 GtC

Some policy analyses also use CO₂-equivalent (CO₂e) concentrations, which account for all greenhouse gases. For example, at ~477 ppm CO₂e, the atmosphere would contain roughly 1,016 GtC.

The residence time of Carbon is taken as 100 years for standardised models. While the short- term resident time varies between 4-10 years, accounting for longer processes like deep ocean mixing and geological sequestration we could have a long-term resident time anywhere between 100-1000 years. The standard way to model is to take an approximation of 100 years as the average resident time.

The Working of Little’s Law

In case of the Net Zero problem: we have the ultimate level of inventory commensurate with a 1.5 °C goal (~1080 GtC as a realistic scenario estimate). while also having an estimate of the flow time (~100 years). This would allow us to find the flow rate as:

Flow Rate = Inventory / Flow Time = 1080/100 = 10.8 GtC per annum

This is the maximum annual global emission to reach a steady state of net zero.

The ability to reduce the Flow time would yield great dividends given the sensitivity of the system to the flow time. We could create a range of possibilities achieved by changing this flow time in the relation mentioned above:

Flow Time (Residence Time)	Allowed Flow Rate (Annual Emissions that can be afforded)
20 Years	54 GtC per annum
50 Years	21.6 GtC per annum
75 Years	14.4 GtC per annum
100 Years	10.8 GtC per annum
150 Years	7.2 GtC per annum
200 Years	5.4 GtC per annum
500 Years	2.16 GtC per annum

A Change in Focus

The above range shows us the impact that the flow time has on our ability to make emissions globally. As of 2024 the world has already reached ~10.2 GtC emissions with net zero emissions not in sight. The ideal way to share the 10.8 GtC carbon budget is to divide it based on a score that accounts for population, income, and other HDI related growth. This would distribute the responsibility in a fair manner.

Modelling the problem with Little’s Law changes our perspective on the Net Zero issue, it also highlights a key blind spot. While most of the research debate and efforts have been around reducing emissions and increasing absorption capacities, all of these efforts attack the flow rate while the flow time remains a silent culprit.

This analysis highlights that another parallel way in which this problem can be solved is for countries to develop collaborative research on reducing the residence time of Carbon. This t would enable the world to grow without having to cap emissions for the developing world or impacting growing economies. Further, this would not only allow countries to grow and flourish but would also reduce costs spent in increasing absorption rates.

India too has accelerated its efforts to focus on the residence time issue. While reducing emissions and substituting non-renewable sources of energy with renewables is a priority, we are also pushing efforts on tackling the residence time. Niti Aayog’s National CCUS Mission supporting industries in capturing and storing carbon at the source along with newly developed CCUS testbeds for the cement industry, economically incentivising the private sector to invest in carbon capture technology. India is also focussing on carbon sequestration projects such as basaltic CO₂ in the Deccan traps. Innovative agricultural methods like Carbon Farming are also incentivised by the government with ISRO leading the way in funding such innovative measures.

Globally, India’s efforts could serve as an example of this bifocal strategy where the primary push is still on the emission rates, but the residence time is gradually and increasingly becoming a focus.

Attacking the Residence Time

Reducing the residence time of Carbon is a challenge that requires heavy investments, deep molecular research and global collaboration. The world may want to look at changing its focus from a unidimensional flow rate problem to a two-dimensional problem requiring the reduction of both the flow rate as well as the flow time with global responsibilities being shared accordingly. Countries should not just promise to reduce emissions but also to invest in research and development to focus on flow time reduction.

There are few key potential solutions to target the residence time of Carbon, however, each of them comes with its own costs and permanence risks (risks that the Carbon presence is not permanently addressed and that the absorbed Carbon becomes part of the atmosphere again):

Direct Air Capture: Process of removing carbon from ambient atmosphere. The absorbed carbon is stored through geological sequestration (storing carbon in rocks). If managed well, the process has little permanence risk the costs are high and the scalability is relatively lower.

Enhanced Rock Weathering: Process to make minerals naturally react with CO₂ and converting them to stable carbonates. These carbonates can be stored physically or utilised in other processes. While its more cost effective, the permanence risks are higher. Soil erosion or soil contamination can take place.

Ocean Alkalinity Enhancement: By adding alkaline chemicals to sea water, we can increase its capacity to absorb and retain CO₂. While this solution has both higher levels of permanence and relatively lower costs it may lead to unintended harm to marine biology.

Synthetic Biology: There is deep research ongoing in altering the genetic makeup of certain flora and fauna to provide them with a significantly higher capacity to absorb carbon. High costs are associated with the solution and reliable results will take time to surface.

There are many other solutions, but the right mechanisms must be decided based on the capabilities and priorities of each country while also ensuring.

Two-Tiered Equitable Goals

When speaking of creating equitable goals, we first need to divide the emissions capacity among countries equitably. This would mean Carbon allocations based on various factors primarily including, population, HDI and GDP. A weighted sum of the quantities that are added as:

x * Population + y * (1 – HDI) + z * GDP

Where x, y and z are the weights assigned to each of the quantities. This ensures that countries with higher populations are given a higher share of emission allowances but also ensures that countries contributing to global growth are not penalised unfairly and allocates emissions based on GDP as well. What this distribution would also ensure is that countries that rank poorustly on HDI would get a higher share of emission allowances so as to not restrict their potential to grow economically and improve the lives of their citizens.

Once a country level emission budget is decided. The second tier of allocation must happen between the resources being utilised in tackling the emission rates (flow rates) and the resourced focused on reducing the residence times (flow times). Given the capital intensity required to focus on R&D related to residence time reduction, poorer countries particularly in the global South must put a higher emphasis on emission reduction while the richer countries should take the bulk of the R&D responsibilities related to flow time reduction.

The governance of these norms becomes relatively easier once the goals are not decided by countries based on individual priorities but rather flow directly from the scientific data. Through this process a clear quantitative goal is created for each country that relates to the levels of acceptable carbon in the world making this metric agnostic to an individual county’s opinion.

Beyond CO₂

While Little’s Law serves as an effective model to simplify a complex reality, we must be cognisant that there are other factors preventing these numbers from being reached. These factors include other GHG’s apart from CO₂ including CH₄, N₂O, Fluorinated gases and others. To keep a common yardstick, scientists use the measure CO₂-equivalent or CO₂ethat gives the effect of all GHG’s (CO₂ and non-CO₂) in terms of an equivalent amount of CO₂ concentration. This provides for a more holistic measure of GHG emissions and would make the calculations more difficult as it would now have to model for different pollutants from varying sources with different rates of emissions and residence times.

What may also move away from the assumptions of a steady state are positive and negative feedback loops. We are assuming a model of linear improvements, however, in reality there are feedback loops which effect the stability of the system nonlinearly. For example, the melting of ice caps reduces the reflective capacity of earth (Albedo effect) which causes more heat absorption and further warming. Similarly, a higher temperature may be more conducive for growth of flora and fauna in certain regions increasing the carbon absorption capacity. Little’s Law does not model for these effects, and they may have to be added on top of the results achieved from the model.