Fundamentals of Transportation/Queueing/Solution3

TProblem

Problem:

In this problem we apply the properties of capacitated queues to an expressway ramp. Note the following:

• Ramp will hold 15 vehicles
• Vehicles can enter expressway at 1 vehicle every 6 seconds
• Vehicles arrive at ramp at 1 vehicle every 8 seconds

Determine:

(A) Probability of 5 cars,

(B) Percent of Time Ramp is Full,

(C) Expected number of vehicles on ramp in peak hour.

Example

Solution:

$\lambda ={\frac {3600}{8}}=450\,\!$

$\mu ={\frac {3600}{6}}=600\,\!$

$\rho ={\frac {\lambda }{\mu }}=0.75\,\!$

Part A

Probability of 5 cars,

$P(n)={\frac {\left({1-\rho }\right)}{1-\rho ^{N+1}}}\left(\rho \right)^{n}={\frac {\left({1-0.75}\right)}{1-0.75^{16}}}\left({0.75}\right)^{5}=0.06=6\%\,\!$

Part B

Percent of time ramp is full (i.e. 15 cars),

$P(n)={\frac {\left({1-\rho }\right)}{1-\rho ^{N+1}}}\left(\rho \right)^{n}={\frac {\left({1-0.75}\right)}{1-0.75^{16}}}\left({0.75}\right)^{15}=0.0033=0.33\%\,\!$

Part C

Expected number of vehicles on ramp in peak hour.

$E(n)={\frac {\left(\rho \right)}{\left({1-\rho }\right)^{}}}{\frac {1-\left({N+1}\right)\left(\rho \right)^{N}+N\rho ^{N+1}}{1-\rho ^{N+1}}}={\frac {\left({0.75}\right)}{\left({1-0.75}\right)^{}}}{\frac {1-\left({15+1}\right)\left(\rho \right)^{15}+15\rho ^{16}}{1-\rho ^{16}}}=2.81\approx 3\,\!$

Conclusion, ramp is large enough to hold most queues, though 12 seconds an hour, there will be some ramp spillover. It is a policy question as to whether that is acceptable.