Delays

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

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Deterministic Delay
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Delay Estimation (1/2)
Greatest Delay
Total Delayed Aircraft
Greatest Queue
Runway Capacity
Delay Period
Delay Period
4
Delay Estimation (2/2)

• Area
• A/C-Hours of delay
• Demand-Capacity

D-C
Time
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Example (1/2)
End hour Operations Capacity D-C Cumul.
7 25 30 -5 0
8 30 30 0 0
9 40 30 10 10
10 50 30 20 30
11 45 30 15 45
12 15 30 -15 30
13 10 30 -20 10
14 15 30 -15 0
15 20 30 -10 0
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Example (2/2)
Area under the curve ½110½(1030)1½(304
0)1 ½(4030)1 ½(3010)1½101120 AC-hr
Avg Delay to All AC 120/250 28.8 min/AC
Avg Delay to Delayed 120/(405045151015)
41.1 min/AC
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Stochastic Delay

• Queuing theory concepts
• Probability function
• Arrival rate
• Service time
• Required data
• Arrival pattern
• Service pattern
• Service method
• Queue discipline
• Number of servers

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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 demandgtsupply)
• part c should more properly be worded increased
  to 8 minutes, not limited to 8 minutes.

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Mathematical Formulation of Delay

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M/D/1 Queuing Model

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M/D/1 Queuing Model

• q -- Arrival (or departure) rate ?
• Q -- Service rate (utilization) ??
• Average waiting time in queue

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M/D/1 Equations

• Average Time in System (hours)
• Average Queue Length (number)
• Average Time in service (hours)

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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

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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)

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Example (note different terminology)
(1/2)
End hour Operations Capacity D-C Cumul.
7 25 30 -5 0
8 30 30 0 0
9 40 30 10 10
10 50 30 20 30
11 45 30 15 45
12 15 30 -15 30
13 10 30 -20 10
14 15 30 -15 0
15 20 30 -10 0

• Arrival rate
• q 250/9 27.8 A-C/hr
• Service rate
• Q 30 A-C/hr
• Use M/M/1 model

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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)q2/Q(Q-q)27.82/30(30-27.8) 11.7 A-C
• Probability of no plane in the system
• P(0) (q/Q)01-(q/Q) 0.073
• Probability of one in the system (no line)
• P(1) (q/Q)11-(q/Q) 0.068
• Probability of two in the system (one in line)
• P(1) (q/Q)21-(q/Q) 0.063