ANALYSING AND MODELING PERFORMANCES OF A LONG-HAUL AIR ROUTE NETWORK

This paper deals with analyzing and modelling performances of a long-haul air route network operating as the queuing network. The network consists of the routes/tracks with flight levels serving aircraft/flights as the “service channels”. The main network performances are the “ultimate” and “practical” capacity of “service channels”, the aircraft/flight demand, delays before entering and total time of aircraft/flights spending in the network, and the related generalized costs including those of airlines, air passengers, policy makers and society. The analytical models of the particular network performances and three routing or assignment models/procedures for matching the aircraft/flight demand to capacity are developed and applied to the long-haul air route network in the North Atlantic airspace between Europe and North America. The results have indicated that. the network capacity has been strongly dependent on the number of routes/tracks and flight levels, i.e., “service channels” and their “ultimate” and/or “practical” capacity. The “ultimate” capacity has been mainly influenced by the ATC (Air Traffic Control) separation rules applied between aircraft/flights operating in the same directions. The “practical” capacity has been strongly influenced by the “ultimate” capacity and the average delays imposed on aircraft/flights before entering the network. The rather superior and close to optimal model/procedure for matching demand to capacity has been routing or assignment of the aircraft/flights demand in proportion to the “ultimate” or “practical” capacity of particular “service channels” minimizing the total generalized costs of the actors/stakeholders involved.


INTRODUCTION
The ATC (Air Traffic Control) is considered as one of the main components of the air transport system together with airports and airlines. This includes: i) the controlled airspace established over the particular countries, continents and oceans; ii) technical/technological components such as radio-emissions and related costs/externalities. The examples of spatial configuration of the above-mentioned long-haul air route network are shown on Fig. 1 (a, b). a) Airline-based network -Transpacific airlines [10] b) ATC-based network -Routes/tracks in the North Atlantic airspace [11,12] Fig. 1. Examples of the spatial schemes of long-haul air route networks Fig. 1a shows the scheme of the long-haul network over Pacific Ocean consisting of the routes of airline flights between the origin and destination during a given period of time in terms of intensity, structure, time and space distribution in the network; and iii) the models/procedures for matching demand to capacity aiming at minimizing the generalized costs of the actors/stakeholders involved.

MODELLING PERFORMANCES OF A LONG-HAUL AIR ROUTE NETWORK
3.1 Literature review. The research on analysing and modelling performances of the air route networks similar to that presented in the given context has been scarce and actually not existing in an explicit form. Therefore, this rather short literature review presents the research related to analysing performances of air route networks similar to that presented in this paper and some studies by the air transport industry. For example, this has been the longstanding, exhaustive, and rather matured academic research on the analytical and simulation modelling of the airport and airspace "ultimate" and "practical" capacity based on the stochastic and deterministic queuing theory [2,17,19,20]. The queuing networks have been also the subject of intensive academic research. The analytical models for estimating performances of these networks such as the demand, capacity, average customers delay and the total customers time in the network and related costs, different routing procedures enabling minimization of both previous individually (per customer) and the system (all customers), and prioritizing of particular categories of customers using different criteria have been under focus. The applications of these models have primarily been considered for the computer networks [9,18,21,22].
The research closely related to operations of the long-haul air route networks has mainly dealt with optimization of the aircraft/flight trajectories subject to different criteria. In particular, the effects of new technologies contributing to the aircraft/flight precise guidance, reducing the ATC separation rules, and the impacts of weather intended to more efficient and safer (conflictfree) operations have been under focus in the transatlantic airspace as the most congested overwater airspace in the world [23][24][25][26][27]. Some research has also dealt with the analysis of fuel efficiency of airline flights in the transatlantic airspace [28].
The relevant studies carried out by the aviation industry stakeholders have mostly included the long-term forecasting of the aircraft/flight demand and potential effects of the ATC innovative technologies on the capacity and efficiency of operations of the given (Transatlantic) air route network [29,30].

Configuration of the network.
The simplified spatial configuration of the long-haul air route network synthesized from that on Fig. 1b and shown on Fig. 2 (a, b) is considered for analysis and modelling performances.  As can be seen, the network consists of (N) routes/tracks of the approximately same average length (d i ) each with (M i ) available FLs (Flight Levels). The aircraft/flights between (K) origin and (L) destination airports are handled in the network. They access the network along (K) routes connecting their origin airports with the entry points of the network defined as the geographical WPs (Way Points). After passing through the network, these aircraft/flights leave the network along (L) routes connecting the corresponding network exit WPs and their destination airports.

Assumptions.
In addition to the above-mentioned configuration, modelling of the network performances similarly as that of the queuing network is based on the following assumptions [2,18]:  The network consists of the fixed set of routes/tracks of approximately equal length each with several FLs (Flight Levels); they represent the longest (second in order) segments of threesegment long-haul routes spreading between the origin and destination airports of aircraft/flights where cruising phase of flights is performed. The first in order segments enable access of the aircraft/flights from the origin airports to the entries of particular network "service channels". The last third in order segments enable the aircraft/flights reaching the destination airports after leaving the network "service channels";  The routes/tracks of the network approximately parallel to each other are separated by the ATC specified minimum lateral distance(s);  The routes/tracks and their FLs operate as the "service channels" independently of each other;  The aircraft/flights routed or assigned to the particular "service channels" stay there all the time; they are handled by the constant capacity of "service channels" based on the ATC time-based horizontal and vertical minimum separation rules; those on the same FLs maintain approximately the same speed thus eliminating the potential overtaking conflicts and needs for their resolving by changing FLs and/or route/track;  The intensity of aircraft/flight demand is usually lower than the capacity of particular "service channel(s)" and that of the corresponding ATC controllers; however some delays can be imposed on particular aircraft/flights before entering the network due to the inherent randomness of their arrivals at the entry WPs of particular "service channels'"; depending on their length these delays can be realized at the origin airports just before the aircraft/flight departures in the scope of the "ground holding" procedures and/or along the access routes to the entry WPs"; and  Different models/procedures for routing or assigning the aircraft/flights to the particular "service channels" depending on their expected performances can be applied under given conditions.

3.4
The models of performances. The models of performances of a given air route network are developed based on the above-mentioned assumptions.

"Ultimate" capacity.
The "ultimate" capacity of a given air route network is defined as the maximum number of aircraft/flights, which can be served during the specified period of time under conditions of the constant demand for service [2,20,31]. The "ultimate" capacity of the route/track, i.e., the "service channel" (i), can be estimated as follows: ( ) ∑ ⁄ ( ) ⁄ (1a) where τ ji/min (Δt) is the ATC minimum time-based separation rules between the aircraft/flights on the FL (j) of the "service channel" (i) during time (Δt) (min). Under conditions of the constant demand of "perfectly packed" aircraft/flights separated by the ATC minimum separation rules requesting service in the particular "service channels", the "ultimate" capacity of the network from Eq. 1a is estimated as follows: where all symbols are analogous to those in the previous Eqs.

"
Practical" capacity. The "practical" capacity of a given air route network can be expressed by the maximum number of aircraft/flights handled during a given period of time under conditions of imposing an average delay on each of them before entering the network. If the aircraft/flights arrive at the "service channel" (i) according to the Poisson processes and are served by the constant "ultimate" capacity (1a), each "service channel" can be considered to operate as M/G/1 queuing system. Under the steady-state conditions when the intensity of aircraft/flight demand λ i (Δt) remains always lower than the "service channel's" "ultimate" capacity μ i (Δt), i.e., ρ i (Δt) = λ i (Δt)/μ i (Δt) < 1, the average delay imposed on of aircraft/flights before entering it is estimated as follows [2,18,32,33]: where σ i/S is the standard deviation of the service time of aircraft/flights on route/track (i) independent of time (Δt) (min/ac).
After specifying the maximum average delay in (2a) as ̅ ( ) ( ), the intensity of aircraft/flights representing the "practical" capacity of the "service channel" (i) operating under conditions ρ i (Δt) < 1 can be estimated as follows: where all symbols are analogous to those in the previous Eqs.

Matching demand to capacity.
Matching demand to capacity in the given context can generally be carried out by three models/procedures for routing or assigning the particular aircraft/flights to the network air routes/tracks and their FLs (i.e., "service channels"): i) user-optimizing deterministic; ii) useroptimizing stochastic; and iii) the system optimizing [34]. For such a purpose the utility of each aircraft/flight to be maximized under given conditions needs to be specified. This utility is usually expressed by the generalized aircraft/flight costs and can be maximized by minimizing them. These generalized costs can include the airline operating and air passenger time costs while onboard, and the internalized costs of impacts of GHG (Green House Gases) emissions on the environment. They all directly depend on the total aircraft flying time, i.e., the average delay, through the network. Thus, minimizing this time minimizes the generalized aircraft/flight costs and maximizes their corresponding utilities given the other factors constant.

Model I: User-optimizing deterministic assignment procedure.
The user-optimizing deterministic assignment procedure actually starts by submitting the flight plans to the ATC service provider(s) some time in advance. They usually aim at optimizing own (individual) above-mentioned utilities under perfectly expected conditions in the network. This will make the utilities of all aircraft flights approximately equal if assigned to the available routes/tracks and their FLs (i.e., "service channels"). In other words, the utilities of all aircraft/flights are expected to be equal independently on the assigned route/track, i.e., "service channel". This also implies that none aircraft/flight can increase its utility if changes the assigned route/track, i.e., "service channel. If the capacities and total travel times through the network are assumed to be equal for all available routes/tracks, i.e., "service channels", the user-optimizing deterministic procedure implies the uniform routing or assigning aircraft/flights to each of them. For the "service channel" (i), the assigned demand is equal to: is the intensity of aircraft/flights demand requesting service, i.e., passing through the network during time (Δt); K(Δt), L(Δt) is the number of origin and destination airports of the aircraft/flights demand during time (Δt); and kl (Δt) is the intensity of aircraft/flights demand between the origin airport (k) and destination airport (l) during time (Δt).
The other symbols are analogous to those in the previous Eqs.
From (2a) and (3), the average aircraft/flight delays before entering the network can be estimated. Since both demand and capacity at each route/track, i.e., "service channel" are equal, the corresponding average delays will also be equal implying that eventual shifting the "channel" would not increase the utility of corresponding aircraft/flight(s). Otherwise, the utilities of aircraft/flights will be exclusively influenced by different capacities of routes//tracks, i.e., service channels".

Model II: User-optimizing stochastic assignment procedure.
The user-optimizing stochastic assignment model/procedure implies the probabilistic choice of route/tracks, i.e., "service channel" by aircraft/flights at the time close to their departure times. The choice is influenced by the inherent randomness of factors influencing the expected utility of aircraft/flights. One of the main causes of such randomness can be uncertainty in predicting weather in the network. Again, each aircraft/flight aims at optimizing its own (individual) utility if the ATC accepts and enables chosen route/track under given conditions.
The user-optimizing stochastic assignment model/procedure routes or assigns the aircraft/flights to the "service channel" (i) according to the MNL (Multinomial Logit) model as follows: is the delay-free average time of aircraft/flight on the route/track, i.e., in the "service channel" (i), during time (Δt) (h); and d i is the length of route/track, i.e., "service channel"(i) (nm; km). The other symbols are analogous to those in the previous Eqs.

Model III:
System-optimizing assignment procedure. The systemoptimizing assignment model/procedure is applied when the ATC acts as the single decision-making entity. In approving the submitted flight plans and realizing the corresponding flights, the ATC aims on the one hand at optimizing their total above-mentioned utilities and on the other its own utility, the latter in terms of maximizing utilization of the available network capacity. The procedure implies an intuitively reasonable assignment of the expected aircraft/flight demand in direct proportion to the "ultimate" or "practical" capacity of particular (available) routes/tracks, i.e., "service channels". The experience so far with optimization of the communication networks operating as the queuing networks has indicated that this is not optimal but close to the optimal assignment enabling minimization of the total aircraft/flights service time, i.e., the total average delay, in the network and consequent abovementioned utilities as the total generalized costs of the actors/stakeholders involved [18]. This model/procedure is as follows: where the other symbols area analogous to those in the previous Eqs. This model/procedure can also be useful for the practical purposes when the capacities of particular routes/tracks, i.e., "service channels", are not as expected. In most cases they can be compromised in terms of availability of routes/tracks due the fast and intensive-changing weather (head/tail wind, storms, volcanic eruptions). In (5), the "practical" capacities of tracks/routes, i.e., "service channels" instead of their "ultimate" counterparts estimated by (2b) can also be used.
3.6 Evaluation of the models/procedures for matching demand to Capacity. The above-mentioned models/procedures for matching demand to capacity are evaluated based on the above-mentioned generalized costs of aircraft/flights including those of airlines, air passengers onboard, and impacts of GHG emissions on the environment (externalities). These costs directly or indirectly mainly depend on the aircraft/flights total time spending in the network, i.e., the total average delay. In this case, the extra aircraft/flight generalized costs compared to their delay-free time-based counterparts are used for evaluation.

Airline operating costs.
When served on the route/track, i.e., "service channel" (i), the average airline operating costs can be estimated by the regression equation using the empirical data as follows [2,35,36]: is the total average aircraft/flight time, i.e., the total average delay of spending on the route/track, i.e., "service channel" (i), during time (Δt).
The other symbols are analogous to those in (6a). The time (t i (Δt) in (6a) is the sum of average delay before entering and the delay-free time of staying of the aircraft/flights on the route track, i.e., "service channel" (i), as follows: where all symbols are analogous to those in the previous Eqs. The total average aircraft/flight time, i.e., the total average delay of spending in the network from Eq. 6a is equal as follows: where all symbols are analogous to those in the previous Eqs.

Cost of passenger time.
The average costs of time of air passengers onboard an aircraft/flight served on the route/track, i.e., in the "service channel" (i) of the network is estimated as follows [2], [37], [38]: where α i is the average value of passenger time while onboard of an aircraft/flight in the "service channel", i.e., route/track (i) ($US/h). The other symbols are analogous to those in Eq. 6a.

Environmental costs / externalities.
The environmental costs/externalities relate to the impacts of GHG emissions from the consumed fuel of aircraft/flights served in the network.
 Fuel consumption The average fuel consumed by an aircraft/flight served on the route/track, i.e., in the "service channel" (i) is estimated as follows [28], [39]: is the average fuel consumed by an aircraft/flight of the seating capacity (S i ) served on the route/track, i.e., in the "service channel" of length (d i ) during time (Δt) (ton/flight); SFC i is the average specific fuel consumption of an aircraft/flight served on the route/track, i.e., in "service channel" i.e., (i) (kg/pkm); and LF i is the average load factor of an aircraft/flight served on the route/track, i.e., in the "service channel"(i).
The other symbols are analogous to those in the previous Eqs.  GHG emissions The average quantity of GHG emissions of an aircraft/flight served on the route/track, i.e., in the "service channel" (i), based on Eq. 8a is estimated as follows [2,40]: is Carbon Dioxide equivalent (tonCO 2e /ton of fuel).
 Cost/externalities of GHG emissions The average costs/externalities of GHG emissions of an aircraft/flight served on the route/track, i.e., in the "service channel" (i) i.e., based on (8b) are estimated as follows [2,41]: is the avearge cost of GHG emissions based on the GWP (Global Warming Potential) ($US/tonCO 2e ). The other symbols are analogous to those in the previous Eqs. (6), (7), (8), the total average costs of an aircraft/flight served on the route/track, i.e., in the "service channel" (i), is equal as follows:

Total generalized costs.From
Under the assumption that these average costs are approximately equal for all aircraft/flights served in the network during the specified period of time, the total extra generalized costs of realized flights based on the difference between their total and delay-free time counterparts are estimated as follows: The other symbols are analogous to those in the previous Eqs.

Configuration and operation of the network.
The ATC-based longhaul air route network considered in this paper is established in the NAT HLA (North Atlantic High-Level Airspace) divided into 6 ACCs (Area Control Centres): Bodo, Reykjavik, Gander, Shanwick, New York East, and Santa Maria Oceanic [42]. The airspace is completely overwater and consequently without the ground-based navigational facilities and radar coverage preventing the ATC radar-monitoring and controlling of aircraft/flights. While in this airspace, As shown on Fig. 1b the network consists of the set of almost parallel air routes/tracks with the specified number of FLs (Flight Levels) called OTS (Organized Track System) spreading between two continents. These routes/tracks generally coincide with the great-circles, i.e., the shortest distances between any two points on the globe implying performing the orthodrome-based air navigation. Starting from the year 2015, these routes/tracks have been laterally separated by the standard distance of: g = 30 nm called RLSM (Reduced Lateral Separation Minima) instead of the previously: g = 60 nm (i.e., from 1 to ½ degree of latitude).
Supported by SLOP (Strategic Lateral Offset Procedure), this separation still guarantees the safe aircraft deviating around the route/track centerlines of about or one or two nm (nmnautical mile). The network, i.e., OTS set up 24h in advance and based on the prevailing weather (primary wind) conditions aims at reducing the impacts of headwinds and increasing benefits from tailwinds as much as possible including the airline preferences submitted in advance. In general, using the OTS is not mandatory but highly recommended [23,42,43].
The sets of WPs along each route/track enable checking the aircraft/flights position where the course, speed, and/or altitude can change. Under such conditions, the aircraft/flights have to perform RNAV (Area Navigation) by using the traditional compass and/or the satellite navigation systems such as GPS (Global Position System) [44].
For reporting their positions, the aircraft/flights use the satellite communication CPDLC (Controller-Pilot Data Link Communications), HF (High Frequency) link and/or alternatively ADS-C & ADS-B (Automatic Dependent Surveillance) system. In the latest case, the controller-pilot-controller voice communication is replaced by the automatic downlink transfer of the position reports and the other flight information if necessary [23,43,[45][46][47].
Before entering the network, the aircraft/flights contact the ATC Oceanic Center requesting the already assigned routes/tracks including the estimated time of arrival at their entry gates (WPs). This enables the ATC controllers to estimate and establish the required separation between the aircraft/flights and issue the corresponding clearances to pilots. The assigned routes/tracks can coincide or be different from the initial ones, but the aircraft/flights have to follow these assigned the latest.
After entering the network, the aircraft have to report their position when crossing the WPs along routes/tracks including predicting the time of crossing the next and the successive WPs ahead as shown on Fig. 1b. In this way, the ATC controllers can "monitor" the safe separation between aircraft/flights while in the network [1,43,48].

Developments of air traffic.
The above-mentioned network serves the air transport market between Europe, North, and South America as one of the busiest in the world. Fig. 3 (a, b) shows some relevant development of the air traffic in this market over time.  [53][54][55]. Fig. 3b shows the daily number of flights impacted by COVID-9 pandemic disease during the nine months of the year 2020 (March-October). As can be seen, just after the closure of the airspace and network (March 2020), the average number of daily flights decreased for about 85 % compared to its counterpart in the year 2019. Later it has been gradually recovering but not more than up to about 30 % of its counterpart in October 2019 [51].
The developments before the impact of COVID-19 pandemic disease questioned the sufficiency of capacity of this network to handle generally expecting growing aircraft/flight demand safely, effectively, and efficiently. While the COVID-19 crisis has impacted the air passenger demand and corresponding airline capacity hardly, they are expected to return to 2019 level by 2024 and then continue to grow at the rate similar to that before the pandemic. This expectation is based on the similarity with the recovery patterns from the previous crisis [16,56]. Under such conditions, it is reasonable to expect that the performances of this network will again come to the research and practice agenda.

The air route network and traffic pattern.
The application of the above-mentioned models to the long-haul air transport network on Fig. 1b  (Westbound tracks and traffic). The considered network, i.e., the OTS to handle the westbound traffic between Europe and North America is assumed to consist of: N(Δt) = 6 routes/tracks (A, B, C, D, E, F), each with M i (Δt) = 11 most preferred FSs (Flight level(s)) (FL 310 -FL 410). These aircraft/flights typically depart from Europe during the daylight between early morning and late afternoon (11:30h-19:00h UTC (Coordinated Universal Time) at 30 0 W) in order to arrive at North America between early afternoon and late evening, i.e., during daylight. The opposite eastbound flights are scheduled to depart from North America to Europe in the evenings (01:00h UTC to 08:00h UTC on the North American side at 30 0 W), thus enabling passengers to arrive at their European destinations in the morning.

The ATC separation rules.
The aircraft flying along the given route/tracks on the same FLs are longitudinally separated by the ATC minimum time-based separation rules of: τ ji/min (Δt) = 10 min. The ATC minimum vertical separation rules between the closest FLs are: h = 1000 ft thanks to RVSM (Reduced Vertical Separation Minima) program implemented in the year 2004. Fig. 4 (a, b) shows the simplified scheme of application of these ATC separation rules (nm -nautical mile; ft -feet) [1,42]

Capacity 4.4.1.1 "Ultimate" capacity.
Based on the ATC minimum longitudinal separation time of: τ ji/min (Δt) = 10 min, the "ultimate" capacity of the route/track (i) with M i = 11 FLs during the period of (Δt) = 1 h is equal to: If this route/track "ultimate" capacity is equal for all (N(Δt)=7) routes/tracks and their equal number of available FLs, (M i ( ) = 11; i = 1, 2, …, 6), the total network "ultimate" capacity will be: If the constant intensity of aircraft/flight demand takes place in the westbound direction during the period: Δτ = 8 h (for example between 11:30 -19:30 UTC or GMT), the total "ultimate" capacity of the network will be: Similarly, the "ultimate" capacity of the network handling the eastbound traffic under given conditions represented by the given/constant configuration of the air route network and constant demand for service can be estimated.
The average length of 20 busiest routes between Europe and North America has been D(Δt) = 3620 nm, of which the oceanic segment (the second in order as mentioned above) has been d i = 1325 nm (i.e., about 37 % of the to‫ؘ‬ tal length) (i = 1, 2, …, 7; M i = 11) [58].
Then, under conditions of the constant demand for service, the total maximum number of aircraft/flights, which can simultaneously be handled in the network under given conditions can be:

"Practical" capacity
Based on the "ultimate" capacity, the "practical" capacity of the given air route network is estimated by specifying the maximum average delay imposed on each aircraft/flight before entering the network (2b). Fig. 5 shows dependence of the network's "practical" capacity on the maximum average delays imposed on an aircraft/flight before entering the network, the standard deviation of average service time, i.e., the ATC minimum time-based separation rules, and the "ultimate" capacity. Fig. 5. Relationship between the "practical" capacity of the given air route network, the average delay imposed on an aircraft/flight and the standard deviation of its service time As can be seen, if the average aircraft service time, i.e., the ATC minimum time-based separation rules are almost perfectly adjusted (i.e., without any deviations), then independently on the imposed delays on the aircraft/flights before entering the network, the corresponding "practical" capacity will remain very close to its "ultimate" counterpart.
If the deviations from the aircraft/flight service time increase even for a couple of minutes, the "practical" capacity will substantively decrease. At the same time, it will increase at decreasing rate with increasing of the average imposed delays and consequently come closer to its "ultimate" counterpart.

Matching demand to capacity 4.4.2.1 Scenario of the network available "ultimate" capacity.
The above-mentioned three models of matching demand to capacity are applied by using a part of data related to the North Atlantic air route network shown on Fig.  1b. The scenario of available "ultimate" capacity of the air route network is given in Table 1. On average, during that period, γ(Δt) = 500 ac/8h have requested service in the network (Based on Fig. 3b, Month 1-March 2020). As mentioned above, the total potential "ultimate" capacity of the given network is:μ a (Δt) = 3168 ac/8h. This implies that the share of available "ultimate" capacity to which the aircraft/flight demand of γ(Δt) = 500 ac/8h can be assigned is: u(Δt) = μ a (Δt)/μ(Δt) = 1040/3168 = 0.328 or ≈ 32.8 %.

Assignment of demand to capacity.
By taking into account the available "ultimate" capacity and the total delay-free average flying time along the particular routes/tracks, i.e., "service channels", of the given network, the Fig. 7. Relationship between the average delay of an aircraft/flight, demand/capacity ratio of particular routes/tracks, and routing or assignment models/procedures in the given example When the assignment Model I and Model II are applied, the corresponding average delays will increase more than proportionally with increasing of the demand/capacity ratio of the route/tracks due to the almost uniform assigned demand on the one hand and their lower "ultimate" capacity on the other. Model III will produce generally lower and more uniform distribution of the average delays of aircraft/flights among particular routes/tracks than its two counterparts. The average aircraft/flight delay is generally the highest on the route/track with the lowest "ultimate" capacity independently on the applied assignment model/procedure.

Evaluation of the assignment models/procedures.
The abovementioned routing or assignment procedures/models are evaluated by the average extra generalized costs of airlines, air passengers, and environmental externalities, all depending on the total average aircraft/flight delays, i.e., total time spending in the network. The reference case has been the costs based on the average delay-free aircraft/flight time spending in the network. These average extra costs are estimated for the single aircraft/flight and 500 flights/8h (i.e., during the above-mentioned east-west daily shift) served in the network independently on the assigned route/track.
 Airline operating costs For particular routing or assignment procedures/models, the average delays on Fig. 8 and the aircraft/flight delay-free time spending in the network, the total corresponding times independently on the route/track are estimated by (6b), (6c), respectively. Then, the airline operational costs per an aircraft/flight are estimated by the regression equation as follows [35,36]: , where all symbols are analogous to those in (6a). These and the corresponding airline extra costs compared to the reference delay-free time costs (Model/procedure 0) are estimated and given in Table 2.  Air passenger time costs The air passenger time costs are estimated by (7) and given in Table 3. Extra costs ($US) -̅ (500 flights) 0 + 363000 + 308500 + 254000 1) Based on 50 % medium-and 50 % high-income passngers on board and their 50 % business and 50 % leisure trips [37], [38]; 2) ̅ = 303 seats/flight; ̅̅̅ = 0.81 per flight [28] As can be seen, these extra costs per an aircraft/flight and for 500 flights/8h are higher than that of the airlines. Again, these costs at Model III are lower than that at the Model I and Model II for about 30 % and 18 %, respectively.

 Total generalized costs
The total generalized extra costs are estimated from the corresponding components in Tables 2, 3, 4, and given in Table 5.  Table 5 indicates that despite the total extra costs per single aircraft/flight are relatively low, those per 500 aircraft/flights handled during the period of 8h, i.e., during the east-west daily shift, can be rather substantive. As expected, Model III (System-optimizing procedure) is shown superior with about 30 % and 18 % lower generalized costs than that of Model I (User-optimizing deterministic procedure) and Model II (User-optimizing stochastic procedure), respectively. This indicates the crucial role of the ATC as the central air traffic control/management entity enabling optimizing the system generalized costs of all aircraft/flights served in the network under given conditions.

CONCLUSIONS
This paper has presented analyzing and modelling performances of the long-haul air route network operating as the queuing network in the large airspace according to "what-if" scenario(s). These performances have been the network capacity consisting of the capacities of particular routes/tracks as the "service channels", the aircraft/flight demand, and their relationships influencing the aircraft/flight total average delays and related generalized costs of airlines, air passengers, and impacts on the environment/externalities. The analytical models for estimating the "ultimate" and "practical" capacity of the particular routes/tracks as the "service channels", three models/procedures for matching the aircraft/flight demand to the "channels" capacity -user-optimizing deterministic, user-optimizing stochastic, and the system optimizing, and the models for estimating particular generalized costs have been developed. They have been applied to the air route network established in the North Atlantic airspace between Europe and North America.
As expected, the "ultimate" capacity of the given network has been mainly influenced and increasing with increasing of the number of available routes/tracks and their flight levels as the "service channels" given the ATC minimum longitudinal time-based separation rules between the aircraft/flights operating on the same flight level(s). The "practical" capacity has been lower than its "ultimate" counterparts, but with decreasing gap with increasing of the average delays imposed on the aircraft/flights before entering the network. Among three routing or assignment models/procedures for matching demand to capacity, the system-optimizing model assigning the aircraft/flights in proportion to the "ultimate" capacity of particular routes/tracks, i.e., "service channels" has appeared superior. It has produced the lowest total average delays and related extra generalized costs compared to its other two counterparts. This has confirmed that the system-optimizing assignment model/procedure elaborated for the communication networks could also be useful for achieving close to optimal matching demand to capacity in the given context. Despite the total extra costs of an aircraft/flight assigned by all three models/procedures have been relatively low, they have shown to be substantive for the number of aircraft/flights handled in the network during the daily-shift period.
Further research could relate to more detailed analysis of operations of the given network by increasing the number of different operating scenarios, approaches to estimating the network's "ultimate" and "practical" capacity (for example applying simulation vs analytical approach), models/procedures for matching demand to capacity, and methods for estimating the generalized costs of the main actors/stakeholders involved. The influence of weather could be of particular interest for more realistic estimation of the above-mentioned performances under different "what-if" scenarios.