Location models for airline hubs
behaving as M/D/c queues
Shuxing Cheng
Emile White
Yi-Chieh Han
Outline
A hub location model
• Assumptions
• Equations
Solve the model
Steps must be taken in synthesis
M/D/c model
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
First hub location model
Campbell’s [1994] formulation
k
i
m
j
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):

p
s  b  c 1
s
 q
bc
or
1   ps   q
s 0
 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):
c 1
v
i 0
i
 c  T
 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).
k   aij xijkm   aij xijmk
i
j
m
i
j
m
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 k 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 Ps increases for large s.
The Ps decreases for smaller s.
bc
The sum of
bc
P
s 0
s
 P decreases.
s 0
s
 1   q cannot be satisfied!
There must exist a continuous range for  :   max
bc
That satisfy the equation:
P
s 0
So for each node k, we have
s
 1  q
 a
i
j
m
ij
xijkm   aij xijmk  max,k
i
j
m
The problem is: How to get this  max,k ?
It can only solved by numerical method.
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  max,k
• Initializing the arrival rate 
• Solving Eq.(11) for roots Z j
• Solving Eq.(9) and (10) to get v j
• 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  max,k are given, we then solve the following model
Min
 c
i
s.t
j
k
 x
k
ijkm
m
ijkm
1
xijkm   f k yk
k
i , j
m
xijkm  yk
i , j ,k ,m ,
xijkm  ym
i , j ,k ,m ,
 a
i
j
ij
m
yk  0 ,1 ,
xijkm   aij xijmk  max,k
i
j
m
0  xijkm  1
i , j ,k ,m.