Modeling technology deployment.

The study of technological deployment or diffusion has a vast history, dating back to the early 20th century. Pioneering work by Lotka (1926) used the logistic curve to model the growth of the American railway system, laying the groundwork for subsequent research [45]. Since then, a vast body of literature has employed S-shaped curves to describe the diffusion patterns of new technologies [3,4,10,12,13,46–48].

In line with established research on technological diffusion, we will employ an S-shaped curve to model the deployment trajectory of the target technology. This approach aligns with the observation that the number of deployed technological units (or, when applicable, a directly related metric such as power capacity for energy technologies) over time (denoted by t) can be effectively captured by a logistic function:

(1)

Our choice of the logistic curve is motivated by its parsimony and recognized usefulness in modeling technological diffusion. The simple parameterization framework provided by the logistic function makes it easy to adapt the dynamic analysis to other S-curves (Gompertz function, cumulative Gaussian distribution), and the results are not qualitatively different [49]. The parameters are as follows:

• - Peak production time: this parameter corresponds to the point at which deployment speed is at its maximum, i.e. when production is at its peak. It separates the acceleration phase of the deployment process from its deceleration phase.
• K - Carrying capacity: this parameter denotes the saturation level or the target capacity of the technology. It represents the number equipment units to be renewed over the long term.
• - Characteristic deployment time: This parameter quantifies the rate or pace of the deployment process. To ensure mathematical tractability and facilitate clear calculations, we adopt the canonical form of the logistic function (as presented in Eq (1)). Consequently, is implicitly defined as half the time it takes for deployment to progress from 27% to 73% of the ultimate carrying capacity. It is noteworthy that defining based on a different growth interval would solely affect the time scale and not fundamentally alter the underlying deployment dynamics.

To track this deployment of equipment in service, the instantaneous production of equipment, i.e. the production of equipment per unit of time, should be as follows:

(2)

Technological equipment is produced to increase active capacity and follows the deployment profile given by the S-shaped curve.

Renewal constraint.

The model presented so far assumes an idealized scenario in which deployed equipment remains operational for an indefinite period. This idealized scenario, however, has only limited practical application, as technological equipment inevitably depreciates and must be replaced over time.

If this problem has little influence when we focus solely on initial purchases (as studied by [3]) or when technological substitution occurs before the equipment lifespan (as studied by [4,12,13]), it gives rise to a non-negligible production when we analyze the dynamics over a longer period of time, particularly when we reach the post-growth plateau.

While traditional technology deployment studies often neglect the critical issue of equipment renewal or industrial replacement, this topic has received attention within other fields. Early considerations of industrial replacement emerged from the domain of actuarial science, framed as an actuarial problem [50]. As Alfred Lotka (1939) succinctly described it, the problem centered on “the number of annual accessions required to maintain a body of N policyholders constant, as members drop out by death.” [23]. Building upon these actuarial foundations, Lotka himself extended the concept to industrial replacement [22,23]. This line of inquiry laid the groundwork for the development of renewal theory, a now-established branch of probability theory [51,52]. The fundamental contributions of Feller (1941, 1949) were key to the development of this field [53,54].

The field of renewal theory has evolved considerably since its first applications in industrial contexts. The emphasis has been placed on a more theoretical framework, favoring abstract mathematical questions and generalizable results. Although our current work focuses on practical results for specific industrial quantities, we will maintain links with renewal theory wherever possible and without introducing excessive complexity.

To incorporate equipment production necessitated by EoL replacements into Eq (2), we define the EoL distribution of an equipment: p_EoL. The probability that an equipment reach EoL θ years after its production is then . Furthermore, we assume that the number of equipment units produced is sufficiently large to invoke the law of large number: over equipment units produced at a fraction reaches EoL at t.

This fraction reaching EoL must then be sum for all the possible lifespan. Replacement production at years 10 is the sum of units produced at year 0 reaching EoL after 10 years, produced at year 1 reaching EoL after 9 years and so forth. This sum covers all possible lifespan values, that is to say . In practice θ never exceeds a certain maximum lifespan so the sum could run between 0 and , however setting this boundary at infinity simplifies the mathematical resolution.

Eq (2) is reformulated to account for equipment production necessitated by EoL replacements. The resulting equation is presented below:

(3)

where:

• P_tot(t): Represents the total number of equipment units produced at time t.
• P_dep(t): Represents the production of new equipment units for deployment purposes at time t.
• : Represents the EoL distribution, the probability density function of an equipment unit reaching its EoL θ years after production.
• : Represents the integral term that incorporates the production required to replace equipment produced at different points in time and reaching their EoL at t.

We employ a parsimonious parameterization for the EoL distribution p_EoL characterized by the following parameters:

• - the average equipment lifespan before reaching EoL
• - the coefficient of variation, i.e. the variance of the equipment lifespan divided by the average lifespan.

We show that the specific choice of probability distribution function has a minimal impact on the overall system dynamics. This finding holds true as long as the two key parameters, average lifespan () and coefficient of variation () are defined (see Sect C in S1 File).

Disaggregating technological adoption through replacement waves.

Following the derivation of Eq (3), we can decompose the overall equipment production trajectory into two key constituents: the initial deployment phase P_dep, and the subsequent series of equipment replacement waves. Each technological equipment produced can be classified according to its role in this deployment and renewal dynamic:

• Initial deployment: This category includes equipment produced for the initial adoption of the technology in question. These units are often referred to as initial purchases.
• First replacement: This category represents equipment produced to replace existing units that have reached their designated EoL for the first time. These replacements are commonly known as second purchases.
• Subsequent replacements: This category encompasses equipment produced for ongoing replacements that extend beyond the initial EoL cycle. This includes units procured for third purchases, fourth purchases, and so on.

This disaggregation by replacement waves allows for a more nuanced understanding of the dynamics of technological adoption within the overall equipment production function.

The number of equipment units requiring first replacement at time t, denoted by R₁(t), can be mathematically modeled using convolution. This concept captures the cumulative effect of equipment deployed at different times reaching their EoL at t. The subsequent replacement waves, denoted by are calculated in the same manner. Eq (3) allows them to be expressed, like R₁, solely as a function of and . All computations are detailed in Sect B in S1 File and lead to the following equation:

(4)

where represents the iterated convolution of the EoL probability density function with itself.

The total equipment production for replacement at time t, P_tot(t), intuitively corresponds the sum of the initial deployment, P_dep(t), with all subsequent replacement waves, R_n(t).

Renewal process interpretation of equipment production.

Eq (3) can be reinterpreted within the framework of renewal processes. This alternative perspective offers valuable insights into the dynamics of equipment production.

By setting K = 1 in Eq (3) (without altering the underlying dynamics due to the proportionality between production and capacity), we can interpret P_dep(t) as the probability of the only equipment (K = 1) initial installation at time t. Consequently, P_tot(t) can be interpreted as the probability of either the initial installation or a replacement event occurring at time t.

This renewal process interpretation allows us to leverage established results from renewal theory regarding the existence and convergence properties of equipment production. However, a complete characterization of this production function requires a more in-depth analysis, beyond the basic framework presented here, and is carried out in the following section.

Characterizing transient dynamics.

While Renewal Steady Production (RSP) is independent of deployment speed and equipment EoL distribution (aside from its average value), these factors do influence the system transient dynamics. This section explores the transition behavior between the initial deployment regime and the eventual renewal regime.

Two types of behaviors

The deployment regime is characterized by total production, P_tot(t), being approximately equal to the deployment production, P_dep(t). This signifies a period where new equipment production dominate production needs, compared with replacements. Conversely, the renewal steady state is characterized by P_tot(t) approaching , reflecting a state where production primarily focuses on replacing equipment reaching their EoL.

Numerical simulations employing a logistic curve for the in-use capacity, Capacity(t), and a Weibull distribution for the EoL probability density function, , reveal the existence of two qualitatively distinct production behaviors. These behaviors depend on the interplay between the deployment characteristic time, , and the EoL average lifespan, .

Fig 2 illustrates the two distinct production behaviors. The upper panel (a) depicts the normalized in-use equipment (active capacity) for a fast deployment scenario (blue curve) and a slow deployment scenario (green curve). Panels (b) and (c) show the total equipment production (bold line) for each deployment scenario, which is the sum of deployment production (lighter line) and the contributions from equipment replacement waves (dashed lines).

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Fig 2. Fast deployment and Slow deployment behaviors.

Model outputs simulates a discretized version of Eq (3). The curves are computed with years and , for the fast deployment years (b) and for slow deployment years (c).

https://doi.org/10.1371/journal.pstr.0000205.g002

These behaviors can be summarized as follows:

• Fast Deployment, see Fig 2 panel (a) (blue curve) and panel (b):
  1. The system reaches its final capacity in a shorter timeframe, see Fig 2 panel (a).
  2. There is a significant overshoot in production compared to its steady renewal level, see Fig 2 panel (b).
  3. The transition from deployment to renewal steady state exhibits damped oscillations over a long period, see Fig 2 panel (b).
• Slow Deployment, see Fig 2 panel (a) (green curve) and panel (c):
  1. The system reaches its final capacity over a longer timeframe, see Fig 2 panel (a).
  2. There is no overshoot in production compared to its steady renewal level level, see Fig 2 panel (c).
  3. The transition from deployment to renewal steady state is monotonic and smooth, see Fig 2 panel (c).

Criterion for the existence of a production Overshoot

The phenomenon of production overshoot is a hallmark of fast deployment scenarios. In simpler terms, when the deployment process is rapid, the production must surpass its steady renewal level to meet the swiftly growing active equipment capacity.

Formally, the fast deployment regime is characterized by:

(6)

A detailed analysis presented in Sect C in S1 File leads to two main conclusions. The first is that the occurrence of an overshoot—and, more generally of the fast/slow deployment behavior—depends solely on the ratio between and . If this ratio, exceeds a certain critical threshold , the deployment time is sufficiently long for overproduction to be unnecessary. Conversely, overproduction is required to reach the final capacity rapidly enough.

Secondly, this critical ratio depends only weakly on the variance, as captured by , and even on the type of distribution used for p_EoL. Across a wide range of values and any p_EoL function, the critical ratio, denoted by , that separates fast behavior from slow behavior lies within the range [0.27, 0.34]. As is implicitly defined through the canonical form of the logistic function, these r_c values can be hard to read. In a more practical form, this corresponds to a maximum active equipment capacity growth over one average equipment lifespan of around 70% of the final capacity.

Furthermore, Eq (S6) suggests that the intensity of the production overshoot is directly proportional to the ratio with additional, gradually diminishing corrective terms accounting for contributions from subsequent replacement waves.

Characterization of the transient dynamics towards Renewal Steady State.

The second critical industrial concern is the duration of the transient phase before reaching the RSS. In essence, the question is: For how long will production oscillate after peak deployment is achieved?

As observed in Fig 2, the production oscillations exhibit a damped behavior, with annual production converging exponentially towards the renewal steady production, RSP. This convergence phenomenon can be rigorously proven by applying specific techniques from renewal theory, as outlined in classical works by Leadbetter (1964) [55] and Cox (1970) [56]. In our specific case, these techniques will be employed to construct an explicit solution to Eq (3), thereby formally demonstrating the exponential convergence. This somewhat technical derivation is presented in Sect D in S1 File. The full solution proves to be relatively complex and opaque, but an analytical approximation—developed in Sect D in S1 File (see Eq S14) and shown in Fig 3a—provides a high-quality surrogate. This first residue approximation enables a quantitative analysis of the effects of deployment on the transition duration.

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Fig 3. Effectiveness of the analytical approximation (Eq. S14) and corresponding transition time.

(a) Comparison of the analytical first residue approximation given by (Eq S14) with the simulated output for the fixed values = 5 y, = 30 y, and different values of . (b) Transition time to reach RSS (Renewal Steady State) as a function of and . There is a misleading scientific inaccuracy, the inequalities should be normalized : “|P_tot(t)/RSP−1|<0.05” and “”.

https://doi.org/10.1371/journal.pstr.0000205.g003

Our analysis reveals two key effects influencing the convergence towards the RSS:

• Impact of coefficient of variation of EoL (CV_EoL): A lower CV_EoL implies a narrower distribution of equipment EoL lifespan. In practical terms, this signifies that most equipment units are likely to reach EoL close to the average lifespan, . Consequently, the production peak will be replicated, albeit with damped intensity, at intervals of approximately years due to concentrated replacement waves. Conversely, a higher CV_EoL broadens the EoL distribution, leading to more dispersed replacement peaks and a faster convergence towards the steady-state production level (higher damping rate).
• Approximation of damping rate using Normal distribution: The real parts of the roots of Eq (S9) for various EoL probability density functions (pdfs) appear to be very similar. This implies that the damping rate calculated analytically for a normal distribution serves as a good approximation regardless of the actual EoL pdf. The accuracy of this approximation is particularly pronounced for low CV_EoL values. However, for higher CV_EoL values, the Gamma distribution might exhibit a slightly higher damping rate. While the precise reason for this discrepancy is not entirely clear, it could potentially be explained by a more nuanced analysis considering the skewness of the distribution. Nonetheless, the objective here is to explain the dynamics using the simplest and most efficient parameterization.

Fig 3a demonstrates that the first residue approximation (Eq S14) rapidly gains accuracy. This occurs because subsequent terms in the solution exhibit faster damping due to their roots having lower real parts. This observation allows us to develop an approximate estimate for the transition duration.

Fig 3b depicts the duration of the transient phase between the peak deployment level and the RSS, characterized by a constant capacity of K and annual production of RSP. In Fig 3b, the transition duration is precisely defined as the time needed for both and to fall below 5% of their respective steady-state values (RSP and K). It is important to note that this 5% threshold is an adjustable parameter that influences the measured duration; a higher threshold will yield a shorter apparent transition time.

Fig 3b underscores the existence of a trade-off for a given EoL distribution (characterized by the pair ) between the time required to reach the target capacity (dotted line) and the overall transition duration. Achieving the target capacity swiftly necessitates fast deployment, which leads to a significant deployment peak, inducing production oscillations and consequently, a longer transition period.

The fast deployment dilemma.

The previous model, despite its simple parameterization, captures an essential issue of a deployment to renewal dynamic regarding the annual equipment unit production: the existence of a trade-off between the time to reach the target capacity and the time to reach a steady renewal production.

On the one hand:

• The time to reach target capacity is proportional to the characteristic time of deployment time (as this is what measures this parameter).
• The deployment production peak intensity is inversely proportional to .

On the other hand:

• The annual production in a renewal steady state (RSP) is inversely proportional to the average lifespan of an equipment .

So there is a point at which reaching target capacity more quickly means that deployment peak production exceeds renewal steady production and thus creates oscillation in the production. This is the fast deployment dilemma, after this critical point (), a gain in time on reaching the target capacity costs a longer transition and more intense oscillations, and vice versa.

The benefits of reaching target capacity quickly are often obvious as the technology answer a particular need or demand. In the other hand the cost of this fast deployment, understood as a annual production undergoing overshoot and oscillations, can be underestimated. Indeed as it will be shown in the following sections the overshoot and the oscillations can be of huge magnitude, and such an equipment production dynamic can lead to different risks.

The role of lifespan variance.

Previous findings must be nuanced by the significant role of equipment lifespan variance, captured in the model by the parameter . Indeed Fig 3a shows that if the deployment peak remains unchanged, oscillations can be significantly damped for some values.

The dynamic is the following: an equipment lifespan precisely determined will result in concentrated replacement waves, echoes of the production peak, and thus a long and fluctuating transition. On the contrary, a large will spread out the replacement waves and the transition will be short and smooth.

Increasing thus seems a way out the fast deployment dilemma, reducing the cost to only an important overshoot. However tuning would not be that easy in practice as it must be done at a constant : for every equipment decommissioned in advance, an other equipment decommission must be delayed. We can easily imagine the problems and difficulties of implementing such a strategy.

Key modeling features.

The modeling assumptions and parameters are summed up in Table 1. Two key assumptions underpin the model:

1. Sustained technological capacity: Following deployment, a certain level of technological capacity is maintained. This implies a period of incremental evolution with no disruptive technological breakthrough rendering the deployed technology obsolete. Mathematically, the evolution of active capacity is represented by an S-shaped curve.
2. Equipment lifespan distribution: The active lifespan of individual equipment before reaching EoL follows a bell-shaped probability distribution centered around an average value. This distribution reflects inherent variations in equipment lifespans.

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Table 1. Modeling assumptions and parameters.

As shown in Sects C and D in S1 File the results are robust to different choices of S-shaped and Bell-shaped curves, other than logistic and Weibull.

https://doi.org/10.1371/journal.pstr.0000205.t001

The model uses a logistic curve to capture the S-shaped trajectory of active capacity over time. This curve is parameterized by two key elements:

• Final active capacity K: This parameter represents the long-term, steady-state level of maintained technological capacity. It represents the ultimate level of equipment deployed and maintained for this particular technology.
• Characteristic deployment time : this parameter reflects the timescale associated with achieving the final level of active capacity, K. It essentially captures the speed of deployment for the technology.

The model uses a weibull distribution to represent the variation in equipment lifespans before reaching their EoL. This distribution is characterized by two parameters:

• Average equipment lifespan : this parameter represents the average time an equipment unit remains active before requiring replacement. It reflects the typical duration of service for individual equipment items.
• Coefficient of variation : This parameter quantifies the degree of spread around the average lifespan (). A higher indicates greater variability in equipment lifespans. In simpler terms, it reflects the level of dispersion in how long individual equipment units last before needing replacement.

Results highlighted: Industrial deployment dynamics.

Within the framework outlined and based on the adopted assumptions, the model predicts two qualitatively distinct behaviors for equipment production arising from the interplay between deployment speed and average equipment lifespan. These behaviors are summarized in Table 2 for clarity.

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Table 2. Fast or slow characterization of deployment dynamics: Influence of parameters on production behavior.

https://doi.org/10.1371/journal.pstr.0000205.t002

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Table 3. This table summarizes the parameters used in the nuclear deployment model.

The parameters regarding EoL are estimated from available data. The range of parameters between brackets is used to measure model sensitivity.

https://doi.org/10.1371/journal.pstr.0000205.t003

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Table 4. This table summarizes the parameters used in the smartphone deployment model.

The parameters regarding EoL are estimated from available data. The range of parameters between brackets is used to measure model sensitivity.

https://doi.org/10.1371/journal.pstr.0000205.t004

1. Fast deployment scenario:
   This scenario occurs when deployment is swift relative to the average equipment lifespan. Specifically, more than 60% to 70% of the final active capacity becomes operational within a single average equipment lifespan (). During this rapid deployment phase, equipment production surpasses its renewal steady level, resulting in an initial overshoot. This overshoot is subsequently followed by damped oscillations around the steady-state production value.
2. Slow deployment scenario:
   This scenario arises when deployment unfolds at a slower pace. In this case, less than 60% to 70% of the final capacity is deployed within a single average equipment lifespan. Consequently, equipment production exhibits a monotonic rise until it reaches its steady-state value associated with long-term renewal.

Nuclear power plants - A fast deployment example.

The global deployment of nuclear power plants exemplifies the deployment to renewal dynamics explored in this analysis. Since the 1990s, global nuclear capacity, measured in Gigawatt (GW) to reflect total power generation capability, has exhibited minimal growth [27]. This indicator corresponds directly to the technological service, energy production, which is not the case for the number of active power plants. Indeed, Nuclear power plants are still undergoing a phase of continuous technological improvements and the nominal power output of a single reactor has more than double between the 1970s and the 2010s.

For this case study, production data will represent the annual capacity newly connected to the electric grid to be coherent with the capacity measurement. Determining a definitive average lifespan for nuclear facilities is challenging, and the literature lacks a universally accepted estimate. Based on available data, we adopt a conservative value of 50 years for the equipment average lifespan (). It is important to emphasize that this value corresponds to the average lifespan; however, the model is based on a Weibull distribution that accounts for heterogeneity across plants, regions, and other sources of variability. Furthermore, the sensitivity analysis enables the assessment of how variations in both the average lifetime and its variance affect the model outcomes.

The annual production curve generated by the model (light blue line, lower panel of Fig 4) exhibits overshoot and subsequent damped oscillations. This behavior is indicative of a fast deployment scenario, as the characteristic deployment time () of 4.35 years is significantly lower than the critical value of (approximately 13.5 years, with r_c = 0.27 and years).

The historical production data (blue dots, lower panel of Fig 4) demonstrates good agreement with the model predicted production trajectory based on the historical capacity data. A significant overshoot is observed, with the deployment peak exceeding steady renewal production by nearly a factor of three. Additionally, production exhibits a decline of almost 90% between 1982 and the 2000s. These findings indicate that this relatively simple model effectively captures some key drivers of the deployment to renewal dynamics in nuclear power plant deployment.

The case of nuclear power plant: Specific insights and limitations.

The application of the model to nuclear power plant deployment presents three key considerations compared to a generic technological deployment scenario.

Limited population and averaging: due to the relatively small number of active nuclear power plants (fewer than 500 globally), the modeled production curve reflects the expected value of added capacity rather than the exact annual GW increase. Consequently, the model predictions are compared to 10-year averaged historical data (blue dots) instead of raw annual data (red dots in Fig 4). This is further necessitated by the variability in power output capacity between individual plants, which can vary in the ratio of one to two.

Plant refurbishment and lifespan extension: The model incorporates the concept of an EoL distribution but does not explicitly account for the possibility of extending operational lifespan through plant refurbishment. While not explicitly modeled, the chosen average lifespan of 50 years appears to yield reasonable results when compared to historical data.

Nuclear skills gap and fast deployment risks: An additional point of interest is the current state of the nuclear power industry, which is experiencing its first period of declining activity. Concerns regarding a potential nuclear skills gap have been raised in the literature [58,59]. This situation highlights the potential risks associated with fast deployment strategies in the nuclear sector, as a skilled workforce is crucial for plant operation and maintenance.

Smartphones - A slow deployment example.

This subsection presents a second case study exhibiting a qualitatively different production behavior: global smartphone deployment. Here, annual production data is estimated based on the number of smartphones sold worldwide each year, acknowledging potential limitations in data source credibility (see Data section). Determining the average lifespan of a smartphone is also challenging. However, based on available production data, a value of 3.7 years appears to yield the most consistent results. It is important to recognize that the “true” average lifespan may reside within a range around 3 years. This characteristic positions smartphone deployment as a slow deployment scenario within the framework of the model.

The analysis reveals a slow deployment behavior for smartphones despite a potentially shorter deployment characteristic time () compared to the nuclear power plant case. This seemingly counterintuitive observation can be attributed to the critical ratio . In the context of smartphones, the average equipment lifespan () is significantly shorter than that of nuclear power plants. Consequently, even with a potentially faster deployment process (lower ), the critical ratio remains high, leading to slow deployment dynamics according to the model predictions.

The model agreement with historical data should not be seen as an uncompromising validation of its predictive capabilities. Unreliable access to production data and the arbitrary choice of an average lifespan make the quantitative aspect of this comparison more fragile. The strength of the model lies in its ability to explain the qualitative difference between the two production profiles with coherent parameters, in particular the lifetime of around 3 years. In the case of smartphones, the average lifespan is short enough for production not to fluctuate.

The case of smartphones: Specific insights and limitations.

Unlike nuclear power plants, the significantly larger population of smartphones allows for the confident application of the law of large numbers. However, a discrepancy is observed between the historical data on item production (number of new smartphones sold annually, represented by gray dots in Fig 5) and the model output. While a production plateau observed in the 2020s suggests an average equipment lifespan () of 4.5 years, this value results in a poor fit for the historical data in the 2010s. Conversely, a of slightly less than 4 years yields a good fit for the 2010s data, but the model overestimates the steady-state production level (RSP).

This discrepancy can be addressed by incorporating the effect of second-hand sales (represented by green dots in Fig 5), which became increasingly significant in the late 2010s. Two potential modelizations of this effect are presented:

Static model: We assume a constant equipment lifespan (). When a smartphone is sold second-hand, it is treated as new production within the model. This approach results in the green curve, which exhibits a good fit with the data.

Dynamic model: We consider the rise of second-hand sales to effectively extend the average equipment lifespan () over time. This dynamic effect would necessitate a shift from the green curve to the gray curve during the 2010s to reflect the increasing contribution of second-hand devices.

The impact of second-hand markets highlights a potential future extension of the model. Incorporating a time-dependent EoL distribution, where the average can increase over time, could offer a more nuanced representation of deployment dynamics in markets with significant second-hand activity.

Other case studies.

In this section, we present several additional case studies to enrich the insights drawn from the analysis and to demonstrate the applicability and generality of the model.

The case of iPhones: A subcase of smartphones.

iPhones, a subcategory of smartphones, also appear to follow a slow deployment profile. Interestingly, the deployment trajectory of iPhones closely mirrors that of smartphones in general, with no significant differences in the quantitative deployment parameters ( years, ). However, the average service lifetime that best fits the data is approximately six years—twice that of the one found for smartphones. This may be explained by the existence of a more developed second-hand market. For iPhones, we only have access to new device sales, meaning that the modeled lifetime corresponds to the end of service rather than resale. This estimated six-year lifetime aligns well with the end-of-support dates for iOS updates (https://endoflife.date/iphone), beyond which devices become effectively obsolete due to incompatibility with newer applications.

This example is valuable in two respects. First, it demonstrates the model’s robustness to changes in scale—from an entire technology to products of a specific brand within that technology. Second, it echoes the point raised in the introduction: although the iPhone has undergone substantial technological evolution from its first release in 2007 to the iPhone 16 in 2025, its overall production dynamics remain consistent with the model. This behavior is crucial for understanding industrial, economic, and material consumption implications.

The case of passenger cars: France & China.

The active passenger car fleet is well suited to our modeling framework. The number of newly registered vehicles each year serves as a clear proxy for Production, while the size of the active fleet—though somewhat more challenging to quantify—can be approximated using metrics such as the number of active automobile insurance policies, and represents the technology capacity. A notable advantage of this technology is that such statistics are typically made available at the national level by public authorities. We were able to obtain data on the active passenger car fleet and annual vehicle registrations for both China and France.

Historical capacity data closely follow an S-shaped curve for both China (Fig 7a) and France (Fig 7b). In the French case, the war period (1939–1945) and several data discontinuities (1940, 1948, and 2010) introduce some irregularities, yet a clear inflection point can be observed around the 1980s, and the size of the active fleet stabilizes at around 35 million vehicles. In contrast, the deployment in China appears to occur over a much shorter time frame, with a inflection around 2017. While estimating the precise saturation level is inherently uncertain, the steepness of the inflection suggests that saturation is likely to occur below 400 million vehicles.

Regarding vehicle lifespans, the parameter that yields the best agreement between model output and registration data corresponds to an average lifetime of approximately 17 years for both France and China. This estimate aligns well with estimated median fleet age and vehicle survival rates [60–62], and notably provides consistent results across both countries.

For France, additional data from INSEE (french national institute for statistics and economic studies) on used car registrations enable further analysis of the second-hand market, akin to the smartphone case study. When all registrations—new and used—are considered, the apparent production volume is nearly four times higher, corresponding to an average duration of ownership (rather than technical lifespan) of just under 5 years.

A notable discrepancy arises between the model output and historical data for both new and used vehicle registrations (see Fig 7b, red curve), particularly between 1960 and 1980, during which observed production growth significantly exceeds model predictions. A similar, though less pronounced, disparity is also observed for new vehicle registrations. One key contributing factor is the assumption of a constant average service lifetime. Over a long time series and for a technology subject to significant evolution, this assumption is likely too restrictive. This observation suggests a potential refinement of the model, in which technological progress is incorporated through a gradual increase in the average product lifetime over time. Historical data appear to support this interpretation, indicating a substantial extension of ownership duration—from approximately 3 years between 1960 and 1980 (upper bound of the trajectory envelope) to values approaching 5 years in more recent decades.

Finally, the main contribution of this case study is to support the existence of two qualitatively distinct deployment regimes. The French example aligns well with a slow deployment profile—smooth, monotonic growth that stabilizes at a renewal-level production rate. In contrast, the Chinese case exhibits a peak in the late 2010s, coinciding precisely with the capacity inflection point. This temporal alignment strongly suggests that the observed decline in production is indeed constrained by the saturation of system capacity. The deployment duration in China is significantly shorter than in France (approximately 4 years vs. 14 years), and short enough to exhibit characteristics of rapid deployment. The ratio of deployment time to average lifespan () falls within the critical range distinguishing rapid from slow deployment regimes.

The Chinese passenger car case study also provides a valuable foundation for the subsequent industrial analysis. Indeed, the same manufacturers and production facilities are typically responsible for producing and assembling the components required for vehicle construction. Even as technological progress drives the transition toward electric vehicles, such a shift does not eliminate the reliance on key materials such as steel, glass, rubber, plastic, and textiles.

Durable consumer goods and energy technologies.

Fig 8 consolidates the previously discussed examples along with additional cases involving household appliances and energy transition technologies. For the newly included technologies, historical production data were not available. Therefore, their positions in the (, ) plane were determined by estimating through logistic curve fitting on historical capacity data, and was heuristically approximated. The corresponding historical capacity data are compiled in section Materials and methods. For photovoltaic and wind technologies, projected capacity values for 2030, 2040, and 2050 as provided by the International Energy Agency (IEA) are used; these points are therefore partially based on future scenarios rather than solely on historical data fitting (see Sect E in S1 file).

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Fig 8. This figure displays various technologies positioned according to their deployment parameter () and average service lifespan (), overlaid on the background of Fig 3b, which illustrates a measure of transition time.

Since Fig 3b is computed for —a value not shared by all technologies but of a relatively minor importance (see Sect C in S1 File)—the displayed times should primarily be interpreted as an indication of how fast or slow deployment occurs relative to the lifespan of the technology, rather than as a direct measure of transition time.

https://doi.org/10.1371/journal.pstr.0000205.g008

As discussed previously, the boundary between (, ) pairs associated with slow and fast deployment regimes is approximately defined by a line with a slope of 0.3—essentially the diagonal extending from the lower left to the upper right of the diagram. Above or to the left of this boundary lies the domain of slow deployments, where the average lifespan is sufficiently short relative to the deployment time, preventing the emergence of a deployment peak. Conversely, below or to the right of this line lies the domain of rapid deployments, where the deployment time is sufficiently short relative to the average lifespan to generate a production peak and potentially oscillatory behavior.

Most technologies fall within a characteristic deployment time between 3 and 8 years. Given that the majority also exhibit average lifetimes under 20 years, this generally results in slow deployment dynamics. This trend may be attributed to the nature of these technologies as consumer goods, which are primarily purchased by individuals. In such cases, the relationship between production and capacity may be more nuanced than captured by the current model, particularly because producers can influence product longevity through design and marketing strategies. Nuclear power stands out as a clear example of fast deployment—both globally and in France—driven by long average lifespans and strong political commitment during its deployment. The case of passenger vehicles in China lies near the boundary between the two regimes, positioned close to the critical diagonal; as shown in Fig 7a, it exhibits a deployment peak, albeit far less pronounced than that observed for nuclear energy in Fig 4.

The case of PV technology appears particularly noteworthy. Similar to nuclear energy, when incorporating the IEA’s projected milestones, PV falls within the fast deployment regime. This is primarily due to its relatively long average lifespan combined with strong drivers supporting a swift energy transition. The potential emergence of a production peak and subsequent oscillations raises important questions regarding the long-term sustainability of this technology. These issues will be explored in future work, which will refine the present model specifically for application to photovoltaic systems.

Endogeneous production oscillations.

Our study demonstrates that oscillations in economic activity, such as business cycles, can arise endogenously from the dynamics of capital installation and renewal within specific industries. This finding contrasts with traditional explanations of cyclical industries, which often attribute fluctuations to exogenous factors in the broader macroeconomic environment.

Specifically, we have identified that the timing of major capital investments and subsequent renewal or replacement cycles can create periodic fluctuations in industry-specific economic activity. These endogenous cycles are driven by the internal dynamics of the industry itself, rather than being primarily responsive to external economic forces.

It is important to note, however, that the existence of these industry-specific, endogenous cycles does not preclude the influence of macroeconomic factors. In reality, both endogenous and exogenous cycles likely coexist and interact. The endogenous cycles we have identified may be modulated, amplified, or attenuated by broader economic trends, resulting in a complex interplay between industry-specific dynamics and macroeconomic conditions.

On wether and how to mitigate oscillations.

Oscillation dynamics raise questions of production capacity sizing, which have repercussions on all sub-systems, whether material (number of plants, raw material requirements, energy supply system) or non-material (workforce, training path, regulation). Being able to cope with production peaks means operating below capacity during troughs. This low capacity utilization periods question the viability of the production actor, who risks to become stranded assets, and could therefore slow down investment. Employments specific to this technology will also fluctuate generating unemployment period, which can lead to skill maintaining issues. Consequently, it is crucial to consider the possibility of such a dynamic and adapt industrial deployment strategies accordingly.

At first glance, the potential actions of the industry seem to be of different kinds: to mitigate, bypass or adapt to this oscillating production dynamic. Reducing equipment service lifespan, lowering deployment speed or broadening EoL distribution can all help mitigate oscillations. However, these solutions raise questions of consumption and waste, in the case of the former, and of feasibility, in the case of the latter two. In a regionalized approach, an import/export policy can help bypass these oscillations. For example, size capacity to the production peak and then export, or size capacity to the renewal steady production and import to manage the peak. Finally, the industry can adapt to such oscillations through structured mechanisms of knowledge transfer and human capital management, as illustrated in recommendations from the nuclear sector [58,63]. In practice, this entails strategies aimed at retaining experienced workers in order to preserve tacit knowledge, strengthening educational and training programs to build a sustainable workforce pipeline, and implementing targeted initiatives to attract new entrants into careers in this industrial sector [58]. Adaptation can also occur through closer cooperation with other stakeholders, particularly the public sector, to stabilize investment flows, subsidies, and regulatory frameworks. Notably with aim of adapting dynamically production with demand and avoiding situations of overcapacity, as occurred in the French nuclear sector during the 1990s [64], and as may also be the case for photovoltaics [65], where substantial investments and subsidies have driven rapid growth in production that is not always aligned with contracting demand or with more moderate demand growth.

The fast deployment profile has been primarily discussed in terms of the induced production oscillations and the associated risks. From this perspective, it may seem natural to conclude that a slow deployment regime is preferable, and that one could ‘return’ to such a regime either by slowing deployment or by reducing the effective lifespan of equipment. In practice, however, this depends entirely on the nature of the technology and on what is considered desirable—particularly from the standpoint of producers, governments, or consumers. Both slowing deployment and reducing lifespan can entail significant costs and risks. For instance, in the case of low-carbon energy production technologies (such as PV or wind power), slowing deployment would increase cumulative greenhouse gas emissions, while reducing equipment lifetime would significantly raise cumulative mineral extraction. In such cases, the “fast deployment dilemma” becomes a multidimensional trade-off, in which different stakeholders may pursue conflicting objectives. While this discussion extends beyond the scope of the model presented here, the model’s results can nevertheless inform the evaluation of such trade-offs and contribute to the design of mitigation and adaptation strategies that address oscillations without compromising deployment speed or material consumption.