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
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Transcribed Image Text: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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