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# Carbon Emission-Based Network Capacity Planning and Pricing

Jorn Altmann, Patrick Maillé, Bruno Tuffin

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Jorn Altmann, Patrick Maillé, Bruno Tuffin. Carbon Emission-Based Network Capacity Planning and Pricing. 2025. hal-05391836

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# Carbon Emission-Based Network Capacity Planning and Pricing

Jorn Altmann

Patrick Maillé

Seoul National University

IMT Atlantique

Seoul, Korea

Rennes, France

Bruno Tuffin

Centre Inria de l'Université de Rennes

Rennes, France

## Abstract

This paper investigates how carbon emissions can be systematically incorporated into telecommunication network capacity planning and pricing decisions for network service providers to respond to their drastic CO₂ emission reduction objective. Building on queueing-based demand models inspired by Naor, we introduce a model in which user demand depends jointly on price and congestion, while operators face both monetary and environmental costs that scale with capacity and served traffic. In detail, three regulatory scenarios with their strategic interactions are examined through a game theoretic lens. The scenarios are: no intervention, a socially optimal capacity constraint, and an environmental cost cap. Numerical experiments using realistic parameters illustrate how each scenario influences optimal pricing, capacity selection, user welfare, operator revenue, and overall social welfare. Our results show that environmental constraints can substantially reduce carbon impact; They can even increase social welfare, when environmental costs are high, though often at the expense of operator's revenue. The study offers quantitative insights into how regulators and service providers can balance economic incentives with climate-driven sustainability goals for future computer networks.

## 1 Introduction

Climate change is getting awareness worldwide, leading international organizations, nations, companies, and individuals to look for ways to reduce the impact of their activities on CO₂ emissions. International initiatives are also set up by the United Nations<fnref n="1" />. Based on scientific climate models, global temperature increase needs to be limited to 1.5°C above pre-industrial levels. Due to the fact that there is already an increase of 1.2°C, the strategy is to build a society that does not produce any new emissions, resulting in net-zero emission.

With respect to policies, the European Union (EU) set rules for establishing the EU Emissions Trading System (ETS), with a focus on energy efficiency through the Energy Efficiency Directive (EED) and through the development of metrics for measuring the environmental footprint [8].

The telecommunication companies are also included in this effort, as they are believed to account for between 1.5% and 4% of total global carbon emissions [1]. With the advent of new and demanding 6G applications, the emission could even significantly rise, if no appropriate measures are taken. Although a significant portion of these emissions (over 60%) comes from the so-called Scope 3 of the value chain, which includes the manufacturing of network equipment,

<footnote n="1">¹See https://www.un.org/en/climatechange/net-zero-coalition</footnote>
hardware and smartphones, as well as the upstream and downstream energy consumed by data centers and the devices used by customers, Scope 2 emissions (about 40%) come directly from the electricity consumption to run the network.

Many mobile network operators, organized in the GSMA, the main professional association of the mobile telecommunication industry, are targeting a net zero CO2 objective by 2050 [9], some of them aiming for 2040. Improved energy efficiency through Dynamic Voltage and Frequency Scaling and network virtualization, better equipment repair and less replacement, are examples of ways to proceed [2]. But, it is also believed that such procedures will not be sufficient, as stated in the report of French environmental think tank The Shift Project [10].

One of the reasons is that over-provisioning of network capacity, which is the current net-work policy of telecommunication operators to deliver high-quality services to customer and to keep the network management simple, causes energy inefficiency. If the telecommunication operators want to achieve the net zero goals, a new paradigm for network management has to be introduced, in which over-provisioning is replaced with a capacity planning model that accepts congestion on the network. Due to this limited availability of network resources, quality may not be maximal. Of course, user satisfaction, as part of social welfare, has to be taken into account in the design of the network. While energy efficiency and conservation need to be addressed on the user side, we focus here on the actions of telecommunication providers.

Our goal in this paper is to build and analyze a model representing the interest of telecom-munication providers, users, and the corresponding CO2 emissions. The model is static, but can also be seen as a response to given costs at a given time (e.g., it could depend on the current electricity production mix).

The paper is organized as follows. While Section 2 positions the model introduced in this paper in existing literature, Section 3 presents the model and its deviations from the existing operations research literature. Section 4 presents the analysis of different scenarios. A numerical analysis based on realistic-value parameters and a comparison of the scenarios is realized in 5. Finally, Section 6 concludes and gives insights on limitations and future work directions.

## 2 State-of-the-Art

The model introduced in this study is based on the seminal model of Naor [7], in which the network is represented by a queue. The arrival rate (that is, the demand for service) depends on the cost measured by a combination of price and congestion level, for an observable queue.

Hassin and Haviv [3] discuss several extensions, in addition to the non-observable situation, in which the arrival rate depends on mean congestion; an assumption that we will also consider. One of our extensions is to select an “optimal” capacity of the queue. Our model considers the mean service rate as the decision variable and the queue mean response time as the congestion cost. This is different to [5], in which the capacity is represented by the number of servers with fixed mean service, and the congestion is represented by the probability that all servers are busy, not taking into account the waiting time.

In [4], a model similar to ours is considered but the arrival rate depends only on the access price and not on the congestion cost (itself depending on the arrival rate, as we do), considerably simplifying the analysis since no fixed point equation has to be solved then. Furthermore, in our model, user satisfaction depends on the quality of service.

The closest model to ours is the one by Mendelson [6], which the author calls the long-run problem. It considers the impact of capacity.

Also in all the above references, the optimization does not consider any carbon-emission cost as we do in our study.
We will consider several scenarios for the parameters selection, and then will be able to make comparison in terms of what could be the “best” procedure depending on the context and quantify gains: i) the service providers selects its price and capacity to maximize its revenue, without considering carbon emissions; ii) the ISP still tries to maximize its revenue, but has a carbon-emission constraint; iii) the regulator imposes a carbon tax.

# 3 Model

## 3.1 User demand

We assume that demand depends-non-increasingly-on both the service price $p$ and to congestion (itself increasing with demand) through quality of service. Hence, a self-regulating loop, leading to demand being the (unique) solution of a fixed-point equation.

We choose, similar to [7, 6], to represent the service through a queuing system. User demand corresponds to the arrival rate $\lambda$ of the queue, and can, for example, be interpreted as the aggregate number of requests (e.g., for online content) per time unit. The dependence in price and congestion is modeled through:

* the service having a perceived cost for users, of the form $p + q\mathbb{E}[W(\lambda)]$, with $W(\lambda)$ being the service delay that depends on $\lambda$ because of congestion effects, and $q > 0$ a sensitivity parameter; and

* a distribution among all users of their valuation or willingness-to-pay for their potential requests, so that only requests valued above the perceived cost are sent.

Denoting the complementary, cumulative distribution function of valuations as $\bar{F}$, and the maximum arrival rate (i.e., the value of $\lambda$ if the perceived cost were null) as $\Lambda$, the description of the relationship between the variables leads to the mathematical writing:

$$ \lambda = \Lambda \bar{F}(p + q\mathbb{E}[W(\lambda)]). \qquad (1) $$

As the average service delay increases with congestion, the right-hand side is nonincreasing in $\lambda$. Hence, the above fixed-point equation can only have one solution. The existence of such a solution is guaranteed, if both $\bar{F}(\cdot)$ and $W(\cdot)$ are continuous, an assumption we will make in this paper<fnref n="2" />.

As in [7, 6], in this paper, we will also take $\mathbb{E}[W(\lambda)]$ as the average sojourn time in an M/M/1 queue with service rate $\mu$, i.e.,

$$ \mathbb{E}[W(\lambda)] = \frac{1}{\mu - \lambda} \quad \text{when } \lambda < \mu \text{ and } +\infty \text{ otherwise,} $$

where $\mu$ is the service capacity of the operator. That capacity will also be a decision variable in our analysis, as it implies some tradeoffs between the possibility of serving more demand, i.e., generating more revenue, and the costs in monetary and CO₂ terms associated with building and operating the infrastructure.

Summarizing, for a given service capacity $\mu$ and price $p$, the demand level $\lambda$ will satisfy (as the unique solution of a fixed-point equation)

$$ \lambda = \Lambda \bar{F} \left( p + \frac{q}{\mu - \lambda} \right). \qquad (2) $$

Note that the solution of the above equation depends both on $p$ and $\mu$. Hence, we will sometimes write $\lambda(p, \mu)$ to make that dependence explicit.

<footnote n="2">²In case of non-continuous $\bar{F}$ and/or $W$, one can consider the value $\lambda = \max\{x \ge 0 : x \le \Lambda \bar{F}(p + q\mathbb{E}[W(x)])\}$, which is in $[0, \Lambda]$.</footnote>
## 3.2 Service provider

The service provider (or operator) is assumed to try to maximize its overall monetary revenue, that is

$$R(p, \mu) = p\lambda(p, \mu) - C(\mu, \lambda), \quad (3)$$

with $C(\mu, \lambda)$ being the monetary cost per time unit associated with offering a capacity $\mu$ (cost of building and maintaining the infrastructure, amortized over time) and serving demand $\lambda$ (mostly the energy cost).

For each of those components, we consider that there is a fixed cost $c_0$, just to have the system available, and a constant marginal cost afterwards. Hence, the affine functions prepresent

* building and maintenance costs of the form that is affine in $\mu$ and independent of $\lambda$,

* operating (energy) costs that is affine in $\lambda$, with a fixed value itself that is affine in $\mu$, as keeping a larger system running consumes more energy.

Overall, this leads to the following overall cost for the operator:

$$C(\mu, \lambda) = c_0 + c_{\mu}\mu + c_{\lambda}\lambda \quad (4)$$

with $c_0, c_{\mu}, c_{\lambda}$ three non-negative parameters.

## 3.3 Environmental costs

As providing the network service also comes with environmental costs (in terms of CO₂ emission, other pollution effects and use of natural resources), we consider the environmental cost $E(\mu, \lambda)$ in the model. Like for monetary costs incurred by the operator, we can decompose those environmental costs into those corresponding to the infrastructure (building and maintenance) and to the operation (mostly energy costs), and we expect those environmental costs to be affine, leading to a form similar to (4) but with different parameter values:

$$E(\mu, \lambda) = e_0 + e_{\mu}\mu + e_{\lambda}\lambda \quad (5)$$

with $e_0, e_{\mu}, e_{\lambda}$ three non-negative parameters.

## 3.4 User and social welfare

To define user welfare, as in [5] we can consider the total value generated by the service for all users selecting to subscribe:

$$V(p, \mu) := \Lambda \int_{p+\frac{q}{\mu-\lambda}}^{\infty} v dF(v) = \Lambda \left( \left( p+\frac{q}{\mu-\lambda} \right) \bar{F} \left( p+\frac{q}{\mu-\lambda} \right) + \int_{p+\frac{q}{\mu-\lambda}}^{\infty} \bar{F}(v) dv \right).$$

User welfare (UW) is the cumulated benefit for all users, i.e., the difference between the value of the service and its perceived cost. Aggregating over all users leads to

$$UW = V(p, \mu) - \lambda \left( p + \frac{q}{\mu - \lambda} \right). \quad (6)$$

We finally define social welfare (SW) as the overall value of the service to society, that is the sum of the utilities of all actors minus the environmental costs, i.e.,

$$SW = UW + R(p, \mu) - E(\mu, \lambda) \quad (7)$$

$$= V(p, \mu) - \lambda \frac{q}{\mu - \lambda} - C(p, \mu) - E(\mu, \lambda). \quad (8)$$
# 4 Game-theoretical analysis

## 4.1 Scenarios to compare

Our model focuses on two decision variables, namely the service price $p$ and the service capacity $\mu$, and is in that sense very close to [5]. However the queuing model is slightly different. In particular, the authors of [5] consider an M/M/∞ queue with constrained capacity and focus on asymptotic situations, in which capacity and potential demand proportionally increase. Additionally, we consider objectives that include the environmental impacts of decisions.

The three situations we aim at comparing, in order to get some insights about possible regulatory interventions, are summarized below:

* **Scenario 1 – No intervention (NI):** In this scenario, the operator selects $p$ and $\mu$ to maximize its revenue $R(p, \mu)$, as expressed in (3). Define $p_{NI}$ and $\mu_{NI}$ as the revenue-maximizing price and capacity, leading to an arrival rate $\lambda_{NI}$.

* **Scenario 2 – Social Welfare Constrained capacity (SWC):** In the SWC scenario, to limit the environmental cost of the service, the regulator imposes the capacity to maximize SWC but lets the provider decide the price for revenue maximization.

* **Scenario 3 – Constrained environmental cost (CEC):** The regulator imposes a constraint $K$ on the environmental cost of the service, of the form $E(\mu, \lambda) \le K$. The operator can still select both $p$ and $\mu$ to maximize its revenue, as long as the constraint is satisfied. A problem for the regulator is then to choose $K$, anticipating the operator's reaction. To fit the goal of reaching an environmental cost reduction, the constraint $K$ can be set as a percentage $\alpha$ of the maximum revenue, $K$ can be set as:

$$ K = \alpha E(\mu_{NI}, \lambda_{NI}) \text{ with } 0 < \alpha < 1. \qquad (9) $$

## 4.2 Steps of the game-theoretical analysis

The decisions are taken in different order depending on the considered scenario among the ones described in Section 4.1. As in [6], we can decompose the maximization into the short-run problem, for which the processing capacity $\mu$ is fixed and the service provider selects $p$, and the long-run problem, in which $\mu$ is selected.

But given that the regulator and the service providers are strategic, we consider that they make their decisions by anticipating the future actions of the others. Hence, we analyze the game using backward induction:

1. Given $p$ and $\mu$, the user demand $\lambda(p, \mu)$ has been shown to exist (Section 3.1) and to be the solution of the fixed-point equation (2). It is unique, if $F$ is an absolutely continuous distribution. Recall also Footnote 2 that in the case of a non-unique solution of the fixed-point equation, we consider $\lambda(p, \mu)$ has the max solution of the equation.

2. For any $\mu$, the service price $p$ is chosen to maximize the provider revenue for the NI and SWC scenarios, using (i.e., anticipating) the resulting user demand rate $\lambda$. In the CEC scenario, $p$ is still chosen to maximize the provider revenue, but under the constraint that the environmental cost is at most $K$.

3. At the highest level, $\mu$ is determined, but again, using (i.e., anticipating) the resulting $p$ and $\lambda$:

* In the NI scenario, $\mu$ is $\mu_{NI}$ maximizing the provider revenue;
* In the SWC scenario, the regulator imposes $\mu$, to anticipatively (as described below) maximize social welfare $SW$;

* In the CEC scenario, $\mu$ is selected to maximize revenue, provided the environmental cost remain below $K$.

Reference [6] offers a formulation and an interpretation of the revenue-maximizing price $p^*(\mu)$ by expressing the revenue in terms of $\lambda$, but in a simplified case without costs. The long-term problem is then the maximization of $R(p^*(\mu), \mu)$, possibly under constraint according to the scenario.

In the CEC scenario, limiting the environmental cost described in (5) means, for the short-term problem with fixed $\mu$, potentially requiring the prices to be sufficiently large, since from (1) $\lambda$ is a decreasing function of $p$. Indeed and in more detail, reducing the environmental may mean reducing $\lambda$ due to its contribution $e_{\lambda}\lambda$ to the cost; which may require a sufficiently high price. This limits the range of feasible price values. For the long-term problem, the constraint (5) leads to an upper bound on the range of capacities. We will observe those restrictions in the numerical experiments in the next section.

# 5 Numerical analysis and comparison of scenarios

We consider an exponential distribution of valuations with parameter $\tau$, so that $\bar{F}(x) = e^{-\tau x}$ for $x \ge 0$. The default parameter values we use in what follows are given in Table 1.

<table><caption>Table 1: Default parameter values for numerical applications</caption><thead><tr><th>τ</th><th>q</th><th>Λ</th><th>c<sub>0</sub></th><th>c<sub>μ</sub></th><th>c<sub>λ</sub></th><th>e<sub>0</sub></th><th>e<sub>μ</sub></th><th>e<sub>λ</sub></th><th>α</th></tr></thead><tbody><tr><td>0.1</td><td>1</td><td>1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.01</td><td>0.3</td><td>0.1</td><td>0.2</td></tr></tbody></table>

## 5.1 Determination of objective-maximizing parameters

<figure><img src="image_3.png" /><figcaption>Figure 1: Short-term problem: impact of price $p$ for $\mu = 1$</figcaption></figure>
Figure 1 displays the provider revenue in terms of its service price $p$, using the corresponding demand rate $\lambda$, and the environmental cost. We can note that there is a unique revenue-maximizing price, while for our parameters the environmental cost varies only slightly with the price.

<figure><img src="image_4.png" /><figcaption>Figure 2: Long-term problem: impact of capacity $\mu$ on price $p$</figcaption></figure>

Figure 2 displays that optimal price in terms of $\mu$ when maximizing the provider revenue (3). We show the NI (no intervention) situation when there is no constraint imposed to the provider, and the CEC (constrained environmental cost) one when the environmental cost is restricted (here to a proportion 0.2 of what would occur without intervention). We can see that the revenue-maximizing price without constraint decreases with $\mu$: the larger $\mu$, the more demand the provider has an interest to serve. On the other hand, as soon as $\mu$ reaches 0.38, the price in the constrained case increases: this is required to limit the demand and satisfy the environmental cost constraint. When $\mu$ is above 0.43, no price is feasible in the sense that $\mu$ alone is then too large in (5) with respect to the contraint 9.

Figure 3 displays the provider revenue in terms of $\mu$ for the corresponding revenue-maximizing prices. For small values of $\mu$, the revenues are the same under both scenarios: with the revenue-maximizing price the environmental constraint is satisfied, hence that constraint has no impact, and the chosen price is the same. But when $\mu$ reaches around 0.38, the moment when the revenue-maximizing price is below the required minimum price to satisfy the environmental constraint, then the revenue decreases in the CEC scenario (up to the moment when $mu$ is too large for the constraint to be feasible). Figure 4 displays the corresponding SW. We can also see that there is a $\mu$ maximizing social welfare even when the provider is maximizing its price.

## 5.2 Comparison of outputs between the three considered regulation scenarios

We now compare the outputs for the three scenarios, NI, CEC and SWC, in terms of some parameters.

Figure 5 displays the provider revenue, demand $\lambda$, service price $p$, capacity $\mu$, UW, SW and environment cost in terms of the user sensitivity $q$ to congestion: the larger $q$, the more negatively congestion is perceived (in particular with respect to the service price $p$). As $q$ increases, in the fully revenue-oriented scenario NI and as well as SWC, the provider revenue
<figure><img src="image_5.png" /><figcaption>Figure 3: Long-term problem: impact of capacity $\mu$ on the provider revenue</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Long-term problem: impact of capacity $\mu$ on SW</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5: Long-term problem: impact of parameter *q*</figcaption></figure>
and the demand decreases with $q$. The capacity $\mu$ increases, illustrating the need to increase capacity when congestion repels users. It is also interesting to note that, at least with our default parameters, the differences in values between NI and SWC scenarios are limited in terms of SW, provider revenue and demand, but the gain in environmental cost with SWC is significative. The CEC scenario of course, by assumption, significantly reduces the environmental cost, but at the expense of UW and provider revenue. But UW and demand seem here to increase with $q$ for that scenario, contrary to the other scenarios. That counterintuitive observation may be interpreted as follows: since the NI capacity $\mu$, and with it the environmental cost, increase with $q$, the CEC environmental cost constraint becomes slacker (as it is a proportion of the NI environmental cost) when $q$ is larger. This allows the operator to lower its prices (which were lower-bounded to satisfy the constraint) and serve more users.

Figure 6 shows the same outputs in terms of the user sensitivity $\tau$ to total perceived cost (recall that $\bar{F}(x) = e^{-\tau x}$ is the proportion of users willing to accept a total cost $x$). The larger the sensitivity parameter $\tau$, the worse all the outputs. There is notably a value of $\tau$ above which the regulator is imposing $\mu = 0$ (hence no service) to ensure that SW is not becoming negative due to environmental costs exceeding the gain for users and provider. The same type of thing happens in the CEC scenario from a slightly larger value of $\tau$: then due to the constraints on $p$ and $\mu$, the provider revenue would become negative, hence the preference for not providing service at all.

Figures 7 and 8 display the outputs in terms of the environmental cost parameters $e_\mu$ and $e_\lambda$, respectively. The NI scenario is not sensitive to those parameters, as the objective there is provider revenue, regardless of environmental cost. In Figure 8, $e_\lambda$ has no significative impact either in the SWC scenario, since $\lambda$ is not changing with $e_\lambda$. More impact, but still a minor one, is experienced in the CEC scenario. Interestingly, Figure 7 shows a threshold on $e_\mu$ above which SW is larger for the CEC scenario than for the NI scenario: imposing a stringent constraint on the environmental cost may be beneficial to society as a whole if the environmental cost of capacity is large. In such cases, while a constrained scenario seems more desirable than the laisser-faire situation, having it accepted by operators and users might be difficult as both operator revenue and user welfare are higher in the NI case.

# 6 Conclusions

This paper develops a carbon-aware model for network capacity planning and pricing, bringing together user behavior, operator decisions, and environmental policies within a single framework. It is motivated by the requirement that the telecommunication providers significantly reduce their carbon emissions. By representing service quality through an M/M/1 queue and embedding both monetary and CO₂-related costs in the social welfare optimization problem, this study evaluates how price and capacity decisions evolve under three scenarios: no regulation, a capacity constraint, and a cap on environmental cost. Using backward induction and numerical analysis, we examine how parameters, such as user sensitivity and environmental cost, shape system outcomes. The comparison highlights the interplay between prices and capacity across three scenarios.

The work contributes a novel integration of environmental cost modeling into classical queue-based pricing and capacity frameworks. Unlike prior studies, the model explicitly incorporates affine CO₂-related costs for both infrastructure and operation, enabling quantitative assessment of carbon-aware regulation. The paper also expands the policy perspective by analyzing how operators and regulators set their strategies. The numerical experiments provide quantitative insights into how environmental constraints affect operator revenue, user welfare, and social welfare. Notably, the results reveal conditions under which environmental caps not only reduce
<figure><img src="image_8.png" /><figcaption>Figure 6: Long-term problem: impact of parameter τ</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 7: Long-term problem: impact of parameter $e_\mu$</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 8: Long-term problem: impact of parameter $e_λ$</figcaption></figure>
emissions but can increase social welfare when capacity-related environmental costs are high, an important finding for policymaking in sustainable telecommunications.

Several simplifying assumptions limit the scope of the present model. The queueing system is static and assumes exponential service times and valuations, whereas real-world traffic patterns and user heterogeneity are more complex. Environmental costs are modeled as affine in capacity and demand, though in practice they may exhibit nonlinearities or depend on energy-mix dynamics. The study also focuses on a single provider without competition, and does not model time-varying electricity carbon intensity. Consequently, future work may extend the framework to multi-operator markets, incorporate dynamic decision-making, explore data-driven environmental models, and investigate user-side behavioral responses to environmentally motivated congestion.

## Acknowledgement

This work has been supported by the Inria Associate Teams programme through the CONNECT project.

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