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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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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Queue Dia-gram

• 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 . . . in this problem set

Constant . 2,900 veh/hour = .

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Step II – Service Pattern

. . : Service Rate .

Due to the incident 8:00 – 8:12 8:13 – 8:30 8:31 after

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Step III – Cumulative Concept Need to express Step I and II in Equation Arrival Pattern

Service Pattern

Remember that there is –

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Step IV – Queue Diagram

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Step IV (2)

Queue Dissipation Point

78.16

3777.5

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Step IV (3)Max Delay and Queue Length

Max diff btw arrival and service lines

Longest Queue

Longest Delay

AC

B

Longest Queue =

Longest Delay =

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Step IV (4)

Total Delay, Average Delay

Area btw arrival and service lines

A & C : Triangle

B: (Trapezoid) – (Triangle)

Triangle Area = 0.5×(length)×(height)

Trapezoid Area = 0.5×(height) ×(Length1 + Length2)

B

Total Delay =

C

A

Average Delay

= Total Delay / total # of Vehicles

=

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