1. Delays  Deterministic Assumes “error free” type case Delay only when demand (known) exceeds capacity (known)  Stochastic Delay may occur any time Random arrivals and departures per stat function

2. Deterministic Delay

3. Delay Estimation (1/2) Greatest Delay Greatest Queue Total Delayed Aircraft Delay Period Runway Capacity Delay Period

4. Delay Estimation (2/2)  Area A/C-Hours of delay  Demand-Capacity D-C Time

5. Example (1/2) End hour OperationsCapacity D-CCumul. 72530 -50 830 00 94030 10 5030 2030 114530 1545 121530 -1530 131030 -2010 141530 -150 152030 -100

6. Example (2/2) Area under the curve = ½*1*10+½*(10+30)*1+½*(30+40)*1+ +½*(40+30)*1 +½*(30+10)*1+½*10*1=120 AC-hr Avg Delay to All AC = 120/250 = 28.8 min/AC Avg Delay to Delayed = 120/(40+50+45+15+10+15) = 41.1 min/AC

7. Stochastic Delay  Queuing theory concepts Probability function Arrival rate Service time  Required data Arrival pattern Service pattern Service method Queue discipline Number of servers

8. Delay Equations (random/poisson arrivals, uniform service dist means variance = 0)  See p. 304  To use for HW prob 16, must compute average hourly demand (blows up if demand>supply)  part c should more properly be worded “increased” to 8 minutes, not “limited” to 8 minutes.

9. Mathematical Formulation of Delay

10. M/D/1 Queuing Model

11.  q -- Arrival (or departure) rate = λ Q -- Service rate (utilization) =  Average waiting time in queue M/D/1 Queuing Model

12. M/D/1 Equations  Average Time in System (hours)  Average Queue Length (number)  Average Time in service (hours)

13. M/M/1 Queuing Models  M -- Exponentially distributed arrival and departure times and one departure channel (server, e.g., runway)  1 – One runway  q – Arrival (or departure) rate Q -- Service rate  From statistics recall: Exponential distribution:

14. M/M/1 Queuing Models  Average waiting time in queue  Average time in system  Average queue length  Probability of k units in system: P(k)= (q/Q) k [1-(q/Q)]

15. Example (note different terminology) (1/2)  Arrival rate q = 250/9 = 27.8 A-C/hr  Service rate Q = 30 A-C/hr  Use M/M/1 model End hour OperationsCapacity D-CCumul. 72530 -50 830 00 94030 10 5030 2030 114530 1545 121530 -1530 131030 -2010 141530 -150 152030 -100

16. Example (2/2)  Average wait time E(w)=q/[Q(Q-q)]=27.8/[30(30-27.8)] = 0.42 hr/A-C  Average queue length E(m)=q 2 /[Q(Q-q)]=27.8 2 /[30(30-27.8)] = 11.7 A-C  Probability of no plane in the system P(0) = (q/Q) 0 [1-(q/Q)] = 0.073  Probability of one in the system (no line) P(1) = (q/Q) 1 [1-(q/Q)] = 0.068  Probability of two in the system (one in line) P(1) = (q/Q) 2 [1-(q/Q)] = 0.063