1. Location models for airline hubs behaving as M/D/c queues
Shuxing Cheng
Emile White
Yi-Chieh Han

2. Outline A hub location model Solve the model
Assumptions
Equations
Solve the model
Steps must be taken in synthesis

3. M/D/c model

4. Model assumptions Use a peak-hour analysis (Worst-case analysis )
Due to improve the performance at peak hours
Reduce queuing effects during off-peak time
Concentrate on the landing runways
Different runways for landing and take-off
Take-off runways are assumed to behave in the same way as landing runways
Only delayed by the service time at the gate
Follow a Poisson distribution
Arrivals of flights to a hub located at node k follow a Poisson distribution during peak hours
Peterson et al. (1995) confirms this conclusion using real data
A deterministic service time at the same node
Use M/D/c model

5. First hub location model Campbell’s [1994] formulationk i j m

6. P[queue length at node k > b] < θq , for all k
Constraints
The new probabilistic constraint forces the probability of more than b airplanes waiting on queue to be less than of equal to θq (Eq. 5).
P[queue length at node k > b] < θq , for all k
This equation must be rewritten in an analytic form for the model to be possible to solve.
Let ps be the steady state probability of s customers being in the system with c servers. Our new equation for this probability is (Eq. 7): or

7. The expression for the probabilities of ps are derived from the arrival rate of the flights, λ, the number of runways at the same hub, c, and a service time of T = 1/μ, where μ is the service rate.
To compute our individual probability estimates (Eq. 8), we define a parameter vi, which is denoted as (Eq. 9):
This can then be used to find the limiting distribution for ps.
Note, the limiting distribution for ps exists only when c > λT. Otherwise, the queue length tends to infinity.
The procedure to find the limiting distribution for ps is very long and difficult. The procedure defines several new variables, involves the calculation of various roots of different variables and equations. As a result, we only discussed a few key ideas.
Upon completion of these mathematical derivations, we can conclude that the constraint outlined in Eq. 7 would be defined as Eq. 20 (refer to the paper).

8. We solve this by intuition!
aij : known average rate of airplanes traveling from i to j through hub k;
The first item: airplanes arriving directly from the origin node i;
The second item: airplanes from another hub m;
Both and vs are not known, the Eq(20)(refer to the paper) is unsolvable!
We solve this by intuition!
When the increases, the queue length increases, the increases for large s.
The decreases for smaller s.
The sum of decreases.
cannot be satisfied!

9. The problem is: How to get this ?
There must exist a continuous range for :
That satisfy the equation:
So for each node k, we have
The problem is: How to get this ?
It can only solved by numerical method.

10. We can formalize our model further:
Find the candidate node, estimate the service time T; the number of runways C.
For each candidate node, with given T and C, find
Initializing the arrival rate
Solving Eq.(11) for roots
Solving Eq.(9) and (10) to get
Checking if Eq.(20)(which is equivalent to Eq.(19)) holds.
If the equation holds as equality,then stop; otherwise, change the value.
Once the values of are given, we then solve the following model

11. Min
s.t