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CEE 3604 Introduction to Transportation Engineering

Finite Source Stochastic QueuesPaving Operations

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Example # 6 Paving Operations

• A paving operation can be represented using a finite source queue

• In the paving operation, pavers are the servers and the trucks carrying the paving material are the customers (i.e., sometimes called calling units)

• Pavers move slowly and for this problem can be represented as static

• Trucks load and unload material at two locations: a) paver site and b) at the material collection site

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Paving Operation Schematic

AsphaltCollection Site

Pavers

Emptytruck

Images: Caterpillar (www.cat.com)

To collectionsite

To pavingsite

Loadedtruck

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Equations for Finite Source QueuePage 68 in Queueing Theory Handout

M = Total number of customers to be served

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Equations for Finite Source Queue

where:

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Numerical Example (Paving Operation)

AsphaltCollection Site

Two Pavers

Empty truckspeed = 75 km/hr

Loaded truckspeed = 60 km/hr

15 trucks used to haul asphalt

15 km 15 km

Service time4 minutes

Queueing System

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Paving Operation Example

• Definition of terms:

• T1 = total transit or cycle time (min)

• T2 = service time (min)

• = 1/ T1 (inverse of cycle time)

• = 1/T2 (inverse of service time)

• M = total number of entities to be served (trucks)

• S = no. of servers (i.e., pavers)

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λµ

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• represents the arrival rate per customer to the queueing system

• The arrival rate considers the cycle time it takes for the customer to return to the system

• represents the service rate per server (like in the infinite source queueing system)

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Observations

λ

µ

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Calculations for Paving Operation

T1 =15km

75 km/hr+

15km60 km/hr

⎡⎣⎢

⎤⎦⎥= 0.45 hours

T2 =4

60= 0.667 hours

Travel times to and from paving site

λ =1

0.45= 2.22 per hour

µ =1

0.0667= 15 per hour

Calculate arrival and service rates

The same customer arrival rate to paving site

Trucks per hour perpaver (per server)

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Calculations for Paving Operation

P0 = 0.0705

Calculate probability of having zero trucks in system

Calculate other probabilities

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Calculations for Paving OperationFor P1 (one truck in queueing system)

P1 =M !

(M − n)!n!(λ − µ)n P0

P1 =15!

(15 −1)!1!(2.2 −15)1(0.0705) = 0.15663

For P3 (three trucks in queueing system)

P3 =M !

(M − n)!s!sn− s(λ − µ)n P0

P3 =15!

(15 − 3)!2!23−2(2.2 −15)3(0.0705) = 0.15641

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Calculations for Paving OperationPlot of Probabilities (from 0 to 15)

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Calculations for Paving OperationCalculation of L and Lq

Note: Is best to make a table with values of Pn and n* Pn to estimate L

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Trucks (n) Pn n*Pn0 0.0705 0.0000

1 0.1566 0.1566

2 0.1624 0.3249

3 0.1564 0.4692

4 0.1390 0.5561

5 0.1133 0.5664

6 0.0839 0.5035

7 0.0559 0.3916

8 0.0332 0.2652

9 0.0172 0.1547

10 0.0076 0.0764

11 0.0028 0.0311

12 0.0008 0.0101

13 0.0002 0.0024

14 0.0000 0.0004

15 0.0000 0.0000

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Calculation of L (expected no. of trucks in queueing system)

L =n=0

M

∑ nPn L = 3.51

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Trucks (n) Pn (n-s) (n-s)Pn

2 0.1624 0.0000 0.0000

3 0.1564 1.0000 0.1564

4 0.1390 2.0000 0.2781

5 0.1133 3.0000 0.3399

6 0.0839 4.0000 0.3357

7 0.0559 5.0000 0.2797

8 0.0332 6.0000 0.1989

9 0.0172 7.0000 0.1203

10 0.0076 8.0000 0.0611

11 0.0028 9.0000 0.0255

12 0.0008 10.0000 0.0084

13 0.0002 11.0000 0.0020

14 0.0000 12.0000 0.0003

15 0.0000 13.0000 0.0000

1.806315

Calculation of Lq (expected no. of trucks in queue)

Lq =n=2

M

∑ (n − s)Pn

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Calculations for Paving OperationCalculation of W and Wq (waiting times)

λ = λ(M − L)

λ = 2.22(15 − 3.51) = 25.54 trucks /hr

where:

Implies 25.54 trucks arrive to the queueing system per hour on average

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Calculations for Paving OperationCalculation of W and Wq (waiting times)

W =Lλ=

3.51 trucks25.54 trucks/hr

= 0.1375hours

Wq =Lqλ

=1.81 trucks

25.54 trucks/hr= 0.071 hours

On average a truck waits 4.25 minutes in the queue waiting for an empty paver

The time a truck spends in the queueing system is 8.25 minutes (including service time of 4 minutes)

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Other Metrics to Measure the Performance of the System

• Expected number of trucks working (K)

• Operating efficiency of a truck (K/M)

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K = M − L = 15 − 3.51 = 11.49 trucks

K / M = 11.49 /15 = 0.767 (dim)

• Expected number of pavers working (L-Lq)

• Operating efficiency of the paver (R/S)

R = L − Lq = 3.51−1.81 = 1.70 pavers

R / S = 1.7 pavers2 pavers

= 0.85pavers