Location Models For Airline Hubs Behaving as M/D/C Queues By: Shuxing Cheng Yi-Chieh Han Emile White

Outline  Background  Hub network: airline, communication network  Problem statement and model building  Analytical model  Difficult point  Queuing theory background  Queuing example  Kendall notation  M/D/C queue

Airline

Hubs  A hub is a node of the network that concentrates traffic from several origins and distributes it to the final destinations.  In multi-hub networks, the traffic is concentrated at a hub and sent from there to a second hub, which distributes it to the final destinations.  The benefit of hubs is that the transportation between hubs is less expensive per unit flow than the transportation between a hub and a non-hub.  As traffic levels increase, hub airports become more congested than non-hub airports, because they receive higher traffic levels.

Airline Hub Network

Model  This paper proposes models that can be used to determine the optimal hub locations in a network system.  The optimal network is one that:  Minimizes cost of the network.  Prevents high levels of congestion at particular hubs.  However, congestion at an airport is hard to model because of certain problems.

Problem #1  The arrival rate of planes at the hub airport is highly variable throughout the day.  Airplanes follow a schedule, however:  Flights are often delayed at their origin airport or during the flight.  Weather conditions  Departing flights may be delayed, thus delaying the landing of arriving aircraft.  The actual arrival rate of airplanes at a hub airport is often non-deterministic.

Problem #2  The service rate of aircraft varies:  In the short run, it can be assumed constant.  In the long run, there will be variation in the service times due to different causes:  Type of plane that is serviced.  Weather conditions  Passengers transfer between airplanes at hubs, thus making service times dependent upon the arrival time of other flights.  The service rates of aircraft are not identical and independently distributed (i.i.d).

Problem #3  Upon arrival, airplanes must go through three processes:  Landing on a landing runway  Service at the gate  Departure through a take-off runway  The probabilistic distributions of these services are very difficult to determine.  Total time required is highly variable.

Solution  Due to these problems, approximation models are more useful.  We can use a peak hour analysis.  Traffic will be at it’s highest level.  We assume that average arrival rate and the service rate are both constant.  This allows us to model an airport hub as an M/D/C queuing system.  Poisson arrivals, deterministic service time D, and the number of servers, C.

Queuing theory  A large field within stochastic process  A mathematical tool having a direct engineering background  A lot of applications  Business analysis  Engineering system performance modeling  The next slides show several examples of different queuing theory models.

Given  Arrival rate:  Service rate:  Number of servers: N We want  Items waiting:  Waiting time:  Items queued: r  Residence time: A queuing system model

Multiserver queue

Kendall notation A queue is denoted as Q 1 /Q 2 /Q 3 /Q 4 /Q 5 Q disc  Q 1 : denotes the distribution of inter arrival times to the queue  Q 2 : denotes the distribution of service time  Q 3 : denotes the number of servers  Q 4 : denotes the maximum number of slots in the queue  Q 5 : denotes the population of the system Q1, Q2 can be assigned to M or D  M: Exponential (memoryless)  D: Constant Q4, Q5 can be omitted if they are assumed to be infinity  Q disc : the discipline that governs in which order the members of the queue are being served, it can be FCFS, LCFS, Round-Robin

M/D/C queue  M: Poisson distribution  D: The service time is constant  C: The number of servers  Queue discipline: FCFS  Relationship: Independence of arrival rate and service rate