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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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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 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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