1Page 1Airside Delays and CongestionAmedeo R. OdoniMassachusetts Institute of Technology1.231J/16.781J/ESD.224J Airport Systems 1.231J/16.781J/ESD.224J Airport Systems ––Fall 2007Fall 2007Page 2Airside CongestionAirside CongestionObjectives– Introduce fundamental concepts regarding airside delayTopics• The airport as a queuing system• Dynamic behavior• Long-term characteristics of airside delay• Non-linearity• Air traffic flow management• Annual capacity of an airport• Measuring delayReference: Chapters 11, 232Page 3QueuesQueuesQueuing Theory is the branch of operations research concerned with waiting lines (delays/congestion)A queuing system consists of a user source, a queue and a service facility with one or more identical parallel serversA queuing network is a set of interconnected queuing systemsFundamental parameters of a queuing system:Demand rate Capacity (service rate)Demand inter-arrival times Service timesUtilization ratio Queue discipline (FIFO, SIRO, LIFO, priorities, etc)Page 4Queuing network consisting of five queuing systemsQueuing network consisting of five queuing systemsInQueueingsystem1 Queueingsystem5Point where users make a choicePoint where users merge+Queueingsystem4 Queueingsystem2 Queueingsystem3Out3Page 5Dynamic (Dynamic (““ShortShort--RunRun””) Behavior of Queues) Behavior of QueuesDelays will occur when, over a time interval, the demand rate exceeds the service rate (“demand exceeds capacity”)Delays may also occur when the demand rate is less than the service rate -- this is due to probabilistic fluctuations in inter-arrival and/or service times (i.e., to short-term surges in demand or to slowdowns in service)These “probabilistic” (or “stochastic”) delays may be large if the demand rate is close to (although lower than) capacity over a long period of timePage 6Dynamic Behavior of Queues [2]Dynamic Behavior of Queues [2]1. The dynamic behavior of a queue can be complex and difficult to predict.2. Expected delay changes non-linearly with changes in the demand rate or the capacity.3. The closer the demand rate is to capacity, the more sensitive expected delay becomes to changes in the demand rate or the capacity.4. The time when peaks in expected delay occur may lag behind the time when demand peaks.5. The expected delay at any given time depends on the “history” of the queue prior to that time.6. The variance (variability) of delay also increases when the demand rate is close to capacity.4Page 7Example of the Dynamic Behavior of a QueueExample of the Dynamic Behavior of a Queue05101520253035401:003:005:007:009:0011:0013:0015:0017:0019:0021:0023:00Dem R1 R2 R3 R4Delays (mins)Demand (movements)301545607590105120Expected delay for four different levels of capacity (R1= capacity is 80 movements per hour; R2 = 90; R3 = 100; R4 = 110)Page 8Scheduled aircraft movements at LGA before and after slot lotterScheduled aircraft movements at LGA before and after slot lotteryy0204060801001205 7 9 11131517192123 1 3Nov, 00Aug, 0181 flights/hourScheduledmovementsper hourTime of day (e.g., 5 = 0500 – 0559)5Page 9Estimated average delay at LGA before and after slot lottery in Estimated average delay at LGA before and after slot lottery in 200120010204060801005 7 9 11 13 15 17 19 21 23 1 3Nov, 00Aug, 01Time of dayAverage delay(minsper movt)Page 10Behavior of Queuing Systems in the Behavior of Queuing Systems in the ““Long RunLong Run””The “utilization ratio”,ρ, measures the intensity of use of a queuing system:A queuing system cannot be operated in the long run with a utilization ratio which exceeds 1; the longer such a system is operated, the longer the queue length and waiting time will be.A queuing system will be able to reach a long-term equilibrium (“steady state”) in its operation, only ifρ < 1, in the long run.μλρ===capacity""demand""rateserviceratedemand6Page 11Behavior of Queuing Systems in the Behavior of Queuing Systems in the ““Long RunLong Run””[2][2]For queuing systems that reach steady state the expected queue length and expected delay are proportional to:Thus, as the demand rate approaches the service rate (or asρ →1,or as “demand approaches capacity”) the average queue length and average delay increase rapidlyThe “proportionality constant” increases with the variability of demand inter-arrival times and of service timesρ−11Page 12Delay vs. Demand and CapacityDelay vs. Demand and CapacityCapacity(ρ = 1.0)DemandExpected delay7Page 13High Sensitivity of Delay at High Levels of UtilizationHigh Sensitivity of Delay at High Levels of UtilizationCapacity(ρ = 1.0)DemandExpected delayPage 14Relationship between traffic and en route delayRelationship between traffic and en route delaySource: Eurocontrol PRC (2001)8Page 15Some statistics for the dynamic queuing exampleSome statistics for the dynamic queuing exampleCapacity(movements/hr)Maximum ofexpectedwaiting time(minutes)Expected waitingtime, all movements(minutes)Utilizationratio(24 hours)Utilizationratio(6:00–21:59)110 2 0.8 0.455 0.664100 4 1.6 0.5 0.73190 13 4.3 0.556 0.81280 39 12.8 0.625 0.913Total demand = 1200 movements per dayPage 16A More A More ““FormalFormal””SettingSettingIn a queuing system with one server, let: X = a random variable that represents the time between successive arrivals of demands at the system (“inter-arrival times” or “headways between arrivals”)λ = rate of demand arrivals per unit of time(and therefore 1/λ = E (X) = expected time between demand arrivals = “average headway between arrivals”)= variance of the time between demand arrivals T = a random variable that represents the time required to service a demand at the queuing system (“service times”)μ= service rate per unit of time (“capacity”)(and therefore 1/μ = E (T) = expected service time = “average service time”)= variance of service timesρ= λ /μ= “utilization ratio” (an indication of intensity of utilization)2Tσ2Xσ9Page 17Four Fundamental Measures of PerformanceFour Fundamental Measures of PerformanceThe quantities of interest:– L = expected number of users in queueingsystem (includes those in queue and those receiving service)– Lq= expected number of users in queue– W = expected time in queueing system per user (waiting time plus service time)– Wq= expected waiting time in queue per user4 unknowns ⇒ We need 4