Description: As we discussed in class, sometimes you wait in line because the system simply doesn't have enough capacity. Other times you wait in line because of randomness in the system. In this lab you will consider how these two effects manifest themselves on toll roads.

Instructions: Answer all of the questions below. You may use any books or notes at your disposal but you should should work entirely on your own.

1. At one extreme, you can imagine that people would wait in line at toll plazas as illustrated below:
   

   That is, people are uniformly distributed across lanes and wait in the lane they were driving in (i.e., they don't change lanes). If this were the case, and the mean arrival rate of vehicles in each lane were less than the mean service rate in each lane (which is generally true during off-peak hours), then we could determine the steady state queue length as follows:

   

   where denotes the mean arrival rate over all lanes, denotes the mean service rate in each lane, and denotes the number of lanes.

   Suppose that the mean arrival rate is 700 vehicles per hour, the mean service rate is 250 vehicles per hour, and that there are 5 lanes.

   1. What is the steady state queue length?

   2. What is the expected waiting time (per vehicle) in the steady state?

   3. How would this number compare to the average waiting time at an actual toll plaza during off-peak hours (with the same arrival rate, service rate, and number of lanes)?

   
   

   
   

2. At the other extreme,you can imagine that people would wait in line at toll plazas as illustrated below:
   

   That is, people wait in a single lane, and proceed to the first available toll booth. If this were the case, and the mean arrival rate of vehicles were less than the mean service rate across all lanes (which is generally true during off-peak hours), then we could determine the steady state queue length as follows:

   

   where:

   

   Suppose that the mean arrival rate is 700 vehicles per hour, the mean service rate is 250 vehicles per hour, and that there are 5 lanes.

   1. What is the steady state queue length?

   2. What is the expected waiting time (per vehicle) in the steady state?

   3. How would this number compare to the average waiting time at an actual toll plaza during off-peak hours (with the same arrival rate, service rate, and number of lanes)?

   
   

   
   

3. During peak periods, the arrival rate of vehicles at a toll plaza generally increases for a period of time, peaks at a value that is larger than the service rate at the plaza, and then decreases. In addition, there are generally several different payment options and many different types of drivers. Systems of this kind are fairly difficult to model analytically and, as a result, are often modeled using simulation. The on-line Toll Plaza Simulator is one example of such a model.
   

   Use the Toll Plaza Simulator to design the best toll plaza that you can. (Note: Your grade on this question will be based on how well you do relative to the other members of the class.) You should first use the Toll Plaza Simulator in its animated mode to try and learn something about the way the system behaves. Then, you should run a `repeated simulation' to evaluate your final design.

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