Skyscraper Model

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)
\begin{equation} a^2 \end{equation}

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
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