Fundamentals of Transportation/Queueing/Solution3

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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:

Part A

Probability of 5 cars,

Part B

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

Part C

Expected number of vehicles on ramp in peak hour.

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.