Given the complexity of the model, the results presented in the following sections are obtained through numerical simulations. In order to understand the results more mathematically, in this section we present a simplified baseline model which can be treated analytically. The main result, namely the existence of an equilibrium in the mixed population case, is derived here and later compared with the numerical results of the complete model.
In the baseline model, we take the simplified case of an airspace with only two airports in which the flights can only travel from one to the other. We also assume that the flights travel instantaneously between the two airports, i.e. they are loading all the sectors in between exactly at their time of departure. In this simplified setting, individual flights do not contribute to the satisfaction when they are rejected and contribute by one unit otherwise. Hence, the satisfaction is \(\mathcal {S}= n_\mathrm{a}/N\), where \(n_\mathrm{a}\) is the number of accepted flights and N is the total number of flights. This simplified setup exhibits nevertheless the main features of the full model.
Pure Populations
Consider first the case where all the companies are of type S, i.e. they shift their flight plans in time. For \(\Delta t=0\) (all waves are consecutive), shifting the flight plan does not yield any improvement, since there are flights departing in the next wave. As a result, the satisfaction is \(n^\mathrm{S}_\mathrm{a} \simeq C N_\mathrm{p} =C \lceil \frac{T_d}{\Delta t + T_0} \rceil \).
When \(\Delta t\) increases and the waves are parting from each other, S companies start to have some opportunity to shift their flight plan if they are rejected during the first waves. They allocate more and more flight plans until they hit their maximum number of flight plans \(N_\mathrm{fp}\). The number of accepted flights is thus
$$\begin{aligned} n^\mathrm{S}_\mathrm{a} \simeq \left\{ \begin{array}{ll}C\lceil \frac{T_d}{\Delta t + T_0} \rceil \left( 1 + \frac{\Delta t}{T_0}\right) &{} \quad \hbox {if} \quad \Delta t < \theta (N_\mathrm{fp}-1) \\ C\lceil \frac{T_d}{\Delta t + T_0} \rceil \left( 1 + \frac{\theta (N_\mathrm{fp}-1)}{T_0}\right) &{} \quad \hbox {if} \quad \Delta t \ge \theta (N_\mathrm{fp}-1),\end{array}\right. \end{aligned}$$
(4)
where we recall that \(\theta \) is the increment by which the flight plans can be shifted. The satisfaction of S companies is initially oscillating with \(\Delta t\), and for \(\Delta t \ge \theta (N_\mathrm{fp}-1)\) it is non increasing.
On the contrary, R companies cannot shift their flight plan and, since the flights are instantaneous, the number of accepted flights is \(n^\mathrm{R}_\mathrm{a} \simeq C N_\mathrm{p} N_\mathrm{fp}^u= C\lceil \frac{T_d}{\Delta t + T_0}\rceil N_\mathrm{fp}^u\), where \(N_\mathrm{fp}^u\) is the number of flight plans among the \(N_\mathrm{fp}\) which have no sector in common. As a consequence, the satisfaction of companies R is monotonically decreasing with \(\Delta t\).
Mixed Populations
In the mixed populations case, a further complication appears, namely that the arrival of the two types of companies is modelled as a random process and therefore different realizations of the process can lead to different values of the satisfaction. To get some intuition on the result, we consider first the case of the maximum attainable satisfaction and then consider the expected value of the satisfaction over the distribution of airlines arrival.
In the mixed population case, a fraction \(f_\mathrm{S}\) of companies are of type S and a fraction \(1-f_\mathrm{S}\) are of type R. The parameter \(f_\mathrm{S}\) will be called mixing parameter in the following. Let us consider first S companies: on the one hand, since R companies cannot shift in time, S companies are not competing with them when shifting in time and thus the periods between waves are completely available to companies of type S. On the other hand, there is a competition for the periods within waves and thus on average S companies will be able to allocate only \(N_\mathrm{p} C f_\mathrm{S}\) flights within the waves. Hence, the resulting number of flights accepted for S companies is:
$$\begin{aligned} n^\mathrm{S}_\mathrm{a}\simeq \left\{ \begin{array}{ll}C \lceil \frac{T_d}{\Delta t + T_0} \rceil \left( f_\mathrm{S} + \frac{\Delta t}{T_0}\right) &{} \quad \hbox {if} \quad \Delta t < \theta (N_\mathrm{fp}-1) \\ C \lceil \frac{T_d}{\Delta t + T_0}\rceil \left( f_\mathrm{S} + \frac{\theta (N_\mathrm{fp}-1)}{T_0}\right) &{} \quad \hbox {if} \quad \Delta t \ge \theta (N_\mathrm{fp}-1)\end{array}\right. \end{aligned}$$
(5)
Clearly this formula reduces to Eq. 4 when \(f_\mathrm{S}=1\). The behaviour of \(n^\mathrm{S}_\mathrm{a}\) is now a bit different from equation 4, since, apart from the oscillations, it is increasing with \(\Delta t\) when \(f_\mathrm{S}\simeq 0\). For intermediate values of \(f_\mathrm{S}\), the function has now a maximum in \(\Delta t\). This effect will be seen also in the full model (see the right panel of Fig. 2 in Sect. 4). Note that \(n_\mathrm{a}^\mathrm{S}\) increases with \(f_\mathrm{S}\), but slower than the total number of S companies \(N f_\mathrm{S}\), leading to a monotonic decrease in the total satisfaction with \(f_\mathrm{S}\).
Likewise, R companies face competition within each wave and only \(N_\mathrm{p} C (1-f_\mathrm{S})\) of the best flight plans are allocated by them in average. On the contrary, their suboptimal rerouted flight plans are specific to them and thus face no competition from S companies. This is true under the condition that the rerouted flight plans do not cross the best flight plan at any point—because it is also used by S companies. Hence, the number of accepted flights for R companies is:
$$\begin{aligned} n_\mathrm{a}^\mathrm{R} \simeq N_\mathrm{p} C (1-f_\mathrm{S}) + N_\mathrm{p} C (N_\mathrm{fp}^u - 1) = C \lceil \frac{T_d}{\Delta t + T_0} \rceil (N_\mathrm{fp}^u - f_\mathrm{S}). \end{aligned}$$
(6)
On the contrary of \(n_\mathrm{a}^\mathrm{S}\), this quantity decreases monotonously with \(\Delta t\). It also decreases with \(f_\mathrm{S}\), but slower than the total number \(N (1-f_\mathrm{S})\) of R companies. As a consequence, the satisfaction of R companies increases with \(f_\mathrm{S}\), i.e. decreases with its own fraction \(1-f_\mathrm{S}\).
We are now interested in seeing how the relative satisfaction of S and R companies depend on the wave structure. It is clear that there is always a root for \(\Delta \mathcal {S}\) as a function of \(f_\mathrm{S}\). In fact,
$$\begin{aligned} \Delta \mathcal {S}\propto \left\{ \begin{array}{ll} C \lceil \frac{T_d}{\Delta t + T_0} \rceil \left( 1 + \frac{1}{f_\mathrm{S}}\frac{\Delta t}{T_0} - \frac{N_\mathrm{fp}^u - f_\mathrm{S}}{1-f_\mathrm{S}} \right) &{} \quad \hbox {if} \quad \Delta t< \theta (N_\mathrm{fp}-1) \\ C \lceil \frac{T_d}{\Delta t + T_0} \rceil \left( 1 + \frac{1}{f_\mathrm{S}} (N_\mathrm{fp}-1)\frac{\theta }{T_0} - \frac{N_\mathrm{fp}^u - f_\mathrm{S}}{1-f_\mathrm{S}} \right) &{} \quad \hbox {if} \quad \Delta t\ge \theta (N_\mathrm{fp}-1)\end{array}\right. \end{aligned}$$
(7)
For \(\Delta t< \theta (N_\mathrm{fp}-1)\), there is a single value of \(f_\mathrm{S}\) for which \(\Delta \mathcal {S}= 0\), which is given by:
$$\begin{aligned} f_\mathrm{S}^* = \frac{\Delta t/T_0}{\Delta t/T_0 + N_\mathrm{fp}^u -1}. \end{aligned}$$
Since \(N_\mathrm{fp}^u\) is at least equal to 1, and usually larger, the value of \(f_\mathrm{S}^*\) is somewhere between 0 and 1, which means that there is always an equilibrium. Note, however, that for \(\Delta t = 0\) the equilibrium is in fact \(f_\mathrm{S} = 0\). For \(\Delta t \ge \theta (N_\mathrm{fp}-1)\), the value of the root is given by the same expression, replacing \(\Delta t/T_0\) by \((N_\mathrm{fp}-1)/(\theta /T_0)\). In this case, there is always a root in (0, 1) when \(N_\mathrm{fp}^u>1\).
Interestingly, the equilibrium value depends on \(\Delta t\), which means that different wave structures lead to different relative advantages of companies of type S with respect to those of type R. More specifically, the \(f_\mathrm{S}^*\) increases monotonically with \(\Delta t\), until it reaches a plateau for \(\Delta t/T_0 \ge \theta (N_\mathrm{fp}-1)\) where it does not depend on \(\Delta t\) anymore.
Effect of Randomness on the Arrival of Companies
The above equations give a first good approximation of the satisfaction for each type of company, but it is easy to see that they are incorrect in some cases, in particular for small values of \(\Delta t\). Indeed, the arrival of companies is random, forcing us to reevaluate the simple average behaviour described above. For instance, when \(\Delta t=0\), allocating the first C companies in a wave to S or to R companies is very different. In both cases, C best flight plans are allocated, but the subsequent S companies are blocked because they cannot shift their flight plan, whereas the subsequent R companies can reroute. In other words, when allocating C flights of type S and then C flights of type R, 2C flights are accepted. If C flights of type R are allocated first and then C flights of type S, only C flights are accepted in total. To capture this effect, it is necessary to go into the details of the allocation by computing expected values using probability distributions.
Moreover, another effect plays a role for high values of \(\Delta t\). In this case, only the first time periods after the wave can be reached by all the flights in the wave, but the last time period can only be reached by those which are departing late in the wave. However, many of these flights have already been allocated during the first time periods. The number of time periods which can be reached by all the flights is \(m = \lceil (\theta (N_\mathrm{fp}-1)/T_0)\rceil - 1\). For \(\Delta t/T_0 < m\), all the time periods are fully allocated, but after that the last one is only filled with the remaining flights S which are late enough, which correspond to a fraction \((1-m+\theta (N_\mathrm{fp}-1)/T_0)\) of flights.
Let us denote \(n = N/N_\mathrm{p}\) the deterministic number of flights in each wave. Then, given that \(n_\mathrm{S}\) flights of type S are to be allocated in a wave, and that \(n_\mathrm{S}'\) of them are to be allocated in the first C flights, for a single wave:
$$\begin{aligned} n_\mathrm{a}^\mathrm{S} \simeq \left\{ \begin{array}{ll} n_\mathrm{S}' + C \Delta t/T_0 &{} \quad \hbox {if} \quad \Delta t/T_0 < m \\ n_\mathrm{S}' + C (m-1) + \Gamma &{} \quad \hbox {if} \quad \Delta t/T_0 \ge m\end{array}\right. \end{aligned}$$
(8)
where
$$\begin{aligned} \Gamma =\min (C, (n_\mathrm{S}-(n_\mathrm{S}'+C(m-1))) (1-m+\theta (N_\mathrm{fp}-1)/T_0)) \end{aligned}$$
(9)
and with the extra condition that \(n_\mathrm{a}^\mathrm{S}<n_\mathrm{S}\).
The probability to have \(n_\mathrm{S}\) flights among the n in the wave is described by the binomial distribution with parameter \(f_\mathrm{S}\), and the probability that \(n_\mathrm{S}'\) of the first C flights is of type S is described by the hypergeometric distribution. Therefore, the expected value of the number of accepted flights is
$$\begin{aligned} \mathbb {E}[n_\mathrm{a}^\mathrm{S}] = \sum _{n_\mathrm{S}=0}^n \sum _{n_\mathrm{S}'=0}^C n_\mathrm{a}^\mathrm{S}(n_\mathrm{S}, n_\mathrm{S}') B(n_\mathrm{S}; n, f_\mathrm{S}) H(n_\mathrm{S}'; n, n_\mathrm{S}, C) \end{aligned}$$
and the expected satisfaction of company S is:
$$\begin{aligned} \mathcal {S}^\mathrm{S} = N_\mathrm{p} \times \mathbb {E}[n_\mathrm{a}^\mathrm{S}]/(N f_\mathrm{S}). \end{aligned}$$
Following the same reasoning, the number of accepted R companies given the number \(n_\mathrm{R}\) R companies in a wave and the number \(n_\mathrm{R}'\) of them in the first C allocated ones is given by \(n_\mathrm{a}^\mathrm{R} = n_\mathrm{R}' + (N_\mathrm{fp}^u-1)C\), and its expected value is
$$\begin{aligned} \mathbb {E}[n_\mathrm{a}^\mathrm{R}] = \sum _{n_\mathrm{R}=0}^n \sum _{n_\mathrm{R}'=0}^C n_\mathrm{a}^\mathrm{R}(n_\mathrm{R}, n_\mathrm{R}') B(n_\mathrm{R}; n, 1-f_\mathrm{S}) H(n_\mathrm{R}'; n, n_\mathrm{R}, C), \end{aligned}$$
with the corresponding expected satisfaction:
$$\begin{aligned} \mathcal {S}^\mathrm{R} = N_\mathrm{p} \times \mathbb {E}[n_\mathrm{a}^\mathrm{R}]/(N (1-f_\mathrm{S})). \end{aligned}$$
Figure 1 illustrates the analytical results of the baseline model with their numerical simulations. The left panel shows the difference of satisfaction between the two types of companies as a function of \(f_\mathrm{S}\) for different values of \(\Delta t\), whereas the right panel shows the corresponding total satisfaction. As explained more in detail in the following sections, the roots of the curves in the left panel represent stable equilibrium points for the system, which are distinct from the optimal points of the system—the maxima of the curves in the right panel.
The agreement between the analytical computations and the simulations is very good, showing that the approximations made have a negligible effect. The larger discrepancies are close to \(\Delta t = 0\), where the analytical model predicts a constant difference in satisfaction, whereas the simulations produce a slightly decreasing function. However, both are well below the zero line and as a consequence the equilibrium is located at \(f_\mathrm{S}=0\).