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Virginia Tech (A.A. Trani)
CEE 3604 Introduction to Transportation Engineering
Finite Source Stochastic QueuesPaving Operations
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Virginia Tech (A.A. Trani)
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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Virginia Tech (A.A. Trani)
Paving Operation Schematic
AsphaltCollection Site
Pavers
Emptytruck
Images: Caterpillar (www.cat.com)
To collectionsite
To pavingsite
Loadedtruck
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Virginia Tech (A.A. Trani)
Equations for Finite Source QueuePage 68 in Queueing Theory Handout
M = Total number of customers to be served
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Virginia Tech (A.A. Trani)
Equations for Finite Source Queue
where:
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Virginia Tech (A.A. Trani)
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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Virginia Tech (A.A. Trani)
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
Virginia Tech (A.A. Trani)
• 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
Virginia Tech (A.A. Trani) 9
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)
Virginia Tech (A.A. Trani) 10
Calculations for Paving Operation
P0 = 0.0705
Calculate probability of having zero trucks in system
Calculate other probabilities
Virginia Tech (A.A. Trani) 11
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
Virginia Tech (A.A. Trani) 12
Calculations for Paving OperationPlot of Probabilities (from 0 to 15)
Virginia Tech (A.A. Trani) 13
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
Virginia Tech (A.A. Trani)
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
Virginia Tech (A.A. Trani)
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
Virginia Tech (A.A. Trani) 16
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
Virginia Tech (A.A. Trani) 17
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)
Virginia Tech (A.A. Trani)
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