1 CEE 8207 Summer 2013 L#6 Queue. 2 Queueing System Provide a mean to estimate important measures of...
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Transcript of 1 CEE 8207 Summer 2013 L#6 Queue. 2 Queueing System Provide a mean to estimate important measures of...
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1CEE 8207 Summer 2013 L#6 Queue
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2CEE 8207 Summer 2013 L#6 Queue
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Queueing System Provide a mean to estimate important measures
of Highway Performance Travel time Speed
Affects Roadway Design Required left-turn bay length
Queueing System Components
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Arrival Process/Pattern
Average Arrival Rate Statistical distribution of interarrival times or
headways Traffic
usually Random arrival Poisson Distribution
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Service Process/Pattern &# of Servers Service Process/Pattern
Average Service Rate vs . Service rate is .
Distribution of Service Time
# of Servers # of channels to perform service i.e – Grocery Cashier
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Server Capacity & Queue Discipline Server Capacity
How much can . i.e.
Queue discipline Means by which next customer/entity is selected
FIFO – First In, First Out i.e:
LIFO – Last In, First Out i.e:
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Queue Process (cont’) Kendal’s Notation
To describe Queue Process in easy way i / j / n
i: arrival process j: service process n: # of servers
For arrival process and service process If Random → M If Deterministic → D
In Traffic Arrival Rate = . Service Rate = .
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8CEE 8207 Summer 2013 L#6 Queue
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Example of Queue Process Signalized Intersection
Arrival Process/Pattern Vehicle, Random Arrival
Service Process/Pattern Green Signal Timing (i.e; 1 veh/2sec when green)
Service Channel, # of Servers # of lanes for corresponding movement
System Capacity .
Queue Discipline .
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10CEE 8207 Summer 2013 L#6 Queue
Queue Dia-gram
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• An incident occurs on a freeway• Freeway Capacity before the incident : 4,000 veh/hour• Constant Flow : 2,900 veh/hour of morning commute• Due to the incident,
• 8:00 – Freeway closed• 8:12 – Partial open 2,000 veh/hour• 8:31 – Freeway fully open
• Assume D/D/1 Queue, FIFO
• STEP I – Determine Arrival Pattern
Queue Example
• STEP II – Determine Service Pattern
• STEP III – Cumulative Concept
Find• Time of Queue Dissipation• Longest Queue Length• Total Delay• Average Delay per Vehicle• Longest Wait of any vehicle
• STEP IV – Queue Diagram Can find all the answers
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Step I – Arrival Pattern
We need to determine which one is the arrival process in this problem set
Constant Morning Commute Flow 2,900 veh/hour = 48.33 veh/min
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Step II – Service Pattern Freeway capacity is Service Rate
4,000 veh/hour Due to the incident
8:00 – 8:12 Freeway Closed → 0 veh/hour 8:13 – 8:30 Partial Open with 2,000 veh/hour =
33.33 veh/min 8:31 – Normal Operation with 4,000 veh/hour =
66.67 veh/min
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Step III – Cumulative Concept Need to express Step I and II in Equation Arrival Pattern
No change 48.33t t : time in min
Service Pattern Some changes due to the incident 0 t ≤ 12 33.33(t- 12) 12 < t ≤ 31 633.33 + 66.67(t-31) t > 31
Remember that there is only ONE server – free-way lane
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Step IV – Queue Diagram
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Step IV (2)
Queue Dissipation Point
Arrival # = Service #
633.33+66.67(t-31) = 48.33t
t = 78.16 min 78.16
3777.5
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Step IV (3)Max Delay and Queue LengthMax diff btw arrival and service lines
Data on Points (t, # of vehi-cles)Point A = (31, 633.33)Point B = (31, 1498.33)Point C = (13.1, 633.33)
t = 31 min, the 633.33rd ve-hicle
Longest Queue
Longest Delay
AC
B
Longest Queue = 1498.33 – 633.33 = 865 (vehicles)
Longest Delay = 31 – 13.1 = 17.9 (minutes)
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Step IV (4)
Total Delay, Average Delay
Area btw arrival and service lines
A & C : TriangleB: (Trapezoid) – (Triangle)
Triangle Area = 0.5×(length)×(height)Trapezoid Area = 0.5×(height) ×(Length1 + Length2)
B
Total Delay = 37,604.2 veh-min
C
A
Average Delay = Total Delay / total # of Vehicles= 37,604.2 / 3777.5 = 9.95 min/veh
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