At an entrance to a toll bridge, four toll booths are open. Vehicles arrive at the bridge at an average rate of 1208 veh/h, and at the booth, drivers take an average of 10 seconds to pay their tolls. Both the arrival and departure rates can be assumed to be exponentially distributed. How would the average queue length, time in the system change if a fifth toll booth were opened? Queue Analysis - Numerical • M/M/N - Average length of queue - Average time waiting in queue - Average time spent in system A = arrival rate pm 11 P/N 10 1 NIN (1-p/NF P+0 1 2 i=P+Q = departure rate M/M/N - More Stuff - Probability of having no vehicles 1 P₁ = p p² + n! N(1-p/N) - Probability of having n vehicles p" Po P₁ =! forn SN n! P₁ = p"P NNN! - Probability of being in a queue Pop PAN= NIN(1-p/N) A = arrival rate p=²p/N<1.0 for n Σ Ν = departure rate

At an entrance to a toll bridge, four toll booths are open. Vehicles arrive at the bridge at an average rate of 1208 veh/h, and at the booth, drivers take an average of 10 seconds to pay their tolls. Both the arrival and departure rates can be assumed to be exponentially distributed. How would the average queue length, time in the system change if a fifth toll booth were opened? Queue Analysis - Numerical • M/M/N - Average length of queue - Average time waiting in queue - Average time spent in system A = arrival rate pm 11 P/N 10 1 NIN (1-p/NF P+0 1 2 i=P+Q = departure rate M/M/N - More Stuff - Probability of having no vehicles 1 P₁ = p p² + n! N(1-p/N) - Probability of having n vehicles p" Po P₁ =! forn SN n! P₁ = p"P NNN! - Probability of being in a queue Pop PAN= NIN(1-p/N) A = arrival rate p=²p/N<1.0 for n Σ Ν = departure rate

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Quantity takeoff

Quantity takeoffs (QTO) are precise estimates of the materials and labor costs required to finish a construction project. During the pre-construction phase, an estimator prepares them, breaking down the project into smaller, more manageable parts that are easier to evaluate. These measurements are used to prepare a construction scope bid. Estimators examine the drawings, specifications, and models to figure out different quantities and their costs. Since, the components, their quantity, and price are all included in the model, quantity take-off can be done automatically using Building Information Modelling (BIM). The time taken for completing a construction project can be reduced by the use of quantity takeoff. These takeoffs are also called material takeoffs and construction takeoffs.

Question

At an entrance to a toll bridge, four toll booths are open. Vehicles arrive at the bridge at an average rate of 1200
veh/h, and at the booth, drivers take an average of 10 seconds to pay their tolls. Both the arrival and departure
rates can be assumed to be exponentially distributed. How would the average queue length, time in the system
change if a fifth toll booth were opened?
Queue Analysis - Numerical
M/M/N
- Average length of queue
Ō
- Average time waiting in queue
- Average time spent in system
A = arrival rate
=
11
W=
Pop-1
1
NIN (1-p/NY
P/N<1.0
p+Ō_1
2
i=P+Q
2
μl
= departure rate
M/M/N - More Stuff
1
- Probability of having no vehicles
1
P₁
P₁ = N-10²²
pN
Σ
+
n = n! N!(1-p/N)
- Probability of having n vehicles
p"Po
for n ≤N
n!
www
P
=
P₁ =
n
- Probability of being in a queue
PAN
Pop
NIN(1-p/N)
A = arrival rate
p"Po
NT-NN!
p:
P/N<LO
for n Σ Ν
μ = departure rate

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