A new city is to be built with many skyscrapers of height $H$. For this project, an infinite amount of material is available, but only a fixed number of workers. At each moment, the workers are distributed equally between all construction sites. If all workers were focused on a single construction site, the building would grow with velocity $v_{max}$ and thus be completed after time $T=H/v_{max}$. However, new construction sites are opened after each time intervall $\Delta t$. When there are $n_{act}$ active (unfinished) sites, the growth velocity, for each of them, drops to $v=v_{max}/n_{act}$. On the other hand, each time a building is completed, all of its workers are set free and immediately distributed among the remaining sites. $a^2$

(1)## Problems:

- How will $n_{act}(t)$ change with time ?
- What will be its asymptotic (long time) behaviour ?
- How is the height $h_k(t)$ of the $k$-th building growing ?
- How long is the lifetime $\tau_k$ of the $k$-th construction site ?

## Comments and Solutions:

- The $k$-th construction site is started at
**opening time**$t^{(0)}_k=(k-1)\Delta t$ and finished at the**closing time**$t^{(1)}_k$, so its**lifetime**is $\tau_k=t^{(1)}_k-t^{(0)}_k$.

- If $T<\Delta t$, each building is finished before the next construction site is opened. In this trivial case, $n_{act}=1$ and $\tau_k=T$.

- If $T>\Delta t$, we expect that the overall progress becomes slower and slower because the
**limited resource**of the workers is devided into more and more sites.

- To handle the problem more easily, we adopt a
**continuum approximation**, i.e. we ignore the step-like changes of $n_{act}(t)$ and treat is as a gradually evolving variable.

- Then $n_{act}(t)$ increases with the site
**opening rate**$R_{open}=1/\Delta t$ and decreases with the site**closing rate**$R_{close}(t)$, so that $\frac{d}{dt}n_{act}(t)=R_{open}-R_{close}(t)$.

- Since we don't know $R_{close}(t)$ at the moment, let us first ignore it (
**infinite lifetime limit**). Note that this situation prevails in the initial phase, before the first building is finished. It could also be realized by increasing the goal height $H$ to infinity.

- Then the number of sites increases linearly with time, $n_{act}(t)=1+R_{open}t=1+(t/\Delta t)$.

- The
**growth velocity**at time $t$ is therefore $v(t)=\frac{v_{max}}{1+(t/\Delta t)}$.

- The height of building number $k$ as a function of time is $h_k(t)=\Theta(t-t^{(0)}_k) \int_{t^{(0)}_k}^t \frac{v_{max}}{1+(t^{\prime}/\Delta t)} dt^{\prime}$.

- Evaluating the integral yields $h_k(t)=\Theta(t-t^{(0)}_k) (v_{max}\Delta t) \log{\left[ \frac{1+(t/\Delta t)}{1+(t^{(0)}_k/\Delta t)}} \right]$. This shows that each building grows logarithmically.

- Now let us assume that buildings stop growing after reaching their goal height $H$, but that the workers of the finished building are not available anymore for other construction sites (
**no-return approximation**).

- In this case, we can compute the closing time $t^{(1)}_k$ of each site $k$ by noting that $h_k(t^{(1)}_k)=H$. We thus obtain $t^{(1)}_k = ?$*
- Using Poisson-distributed opening times $t^{(0)}_k$ does
**not**change the behaviour - Also the implementation of the returning workers seems not to alter the behaviour of the model