MIDTERM EE/GG 550

                                                      Fall 2003

• Develop a STELLA model for the two problem sets below, and hand in your results BEFORE class on Tuesday October 21.  That is the model is due at/before 9:30 AM on Tuesday October 21.  Name your model yournameEQ1.stm and yournameEQ2.stm.

• Each school day your exam is late will result in the loss of 10 points of a total of 100 points.

• Do not discuss the take-home exam or your model with anyone except your instructor. Failure to comply with this rule does violate the Code of Academic Conduct and will be reported. However, please do not hesitate to ask your instructor questions if you have any.  Also, if you feel completely stuck, take a break, go for a walk, get ice cream and then come back to the exam.

• Remember before doing anything on the computer map the model on paper.

• Document your model within the STELLA program as you do in lab.  As always you do not need to list the assumptions that are given to you but if you feel you need to make additional assumptions to solve the problem set, list and justify your assumptions.

• Be careful to note the units of each variable.

Question 1.

You have decided to learn how to brew beer.  You go online to find what seems to be the easiest recipe and learn the following facts. After you add most ingredients to boiling water you have what is called a wort which has a boiling temperature of 80 degrees Celsius. You boil the wort mixture for 1 hour, and then you let the temperature of the wort slowly decline to below 35 degrees Celsius. Only when the temperature has declined below 35 degrees can you add yeast to the mixture. The yeast will allow fermentation to take place. Since you will be brewing after work at night - you are interested in learning at what time you can add the yeast to the mixture – that is if you are able to do it before going to sleep. Thus what you want to figure out is how long it will take the wort to decline below 35 degrees if you keep it in your pantry.   You decide to create a STELLA model to figure this out. 

Assume the following:

• Initial temperature of the wort is 80 degrees (boiling point) - and assume that you will boil the mixture for 1 hour.
• Cooling occurs as a function of the difference between indoor temperature in your pantry and the temp of the wort as well as a factor of proportionality.
• The factor of proportionality equals 0.15.
• The temperature of your pantry is 20 degrees and stays constant for the entire simulation.
• Begin your simulation at 6PM in the afternoon – since that is the usual time you get home from work and run your model with the unit of time equal to hours but with a DT = 0.125.
• Since there are lots of things you need to do at home you cannot be checking on your wort every second, but you decide that a reasonable interval between checking the temperature of the wort is every 30 minutes.

Task

1. Create a simulation model of the decline in temperature over the course of 24 hours.  Create a signal that will let you know when it is time to check the wort (every 30 minutes) - and when it is time to add yeast to the wort (after temp has declined below 35 degrees). Those two conditions should be tied into one signal. (Hint: initially you may want to do this separately as two signals). Remember that you only will be checking the wort every half hour (e.g. at 1 and then again at 1:30 etc.) and thus you only can add the yeast at those same intervals – e.g. at 1 and then the next chance is at 1:30.  (20 points)

2. Assume that you decide to increase the speed of the cooling process by leaving the wort to cool down outside on your porch.  However, this means that the temperature outside the wort does not stay constant as it did in your pantry. Assume that you will begin the boiling process at 6PM (initial temperature of the wort is still 80 degrees), when the outside temperature is 18 degrees Celsius. The outside temperature drops linearly over time, half a degree (1/2 degree Celsius) every half hour until sunrise at 6AM. At 6AM the outside temperature at the porch will begin increasing again but now at a rate of 1 degree every half hour, until it hits a maximum of 20 degrees and stays there until 6PM the next evening. (20 points)

                                                     i.     Create a simulation model of the temperature of the wort during the course of the next 24 hours. Assume that you put the wort outside immediately after the 1 hour of boiling and that you begin your simulation at 6PM in the afternoon.

                                                      ii.     As before create a signal that will let you know when to add yeast to the wort and when to check up on the wort. Make sure that you are only checking the temperature of the wort every 30 minutes and thus you can only add the yeast at the whole or half hour.  At what time would you now add yeast to the mixture?

Total 40 points

Question 2.

Description.

You have been hired by the San Diego zoo to figure out the best schedule for their safari buses given zoo attendance. 

Assume that there are four bus stations in total on the bus route (station 1, station 2, station 3 and station 4).  Buses run from station 1 to station 2 to station 3 and end at station 4. Then they go directly to station 1 again.

Station 1 is the station where the buses are cleaned and then they leave to pick up visitors. Visitors arrive and are able to catch the bus at Stations 2 and 3 (both close to separate entrances) where each bus stops first at station 2 and then at station 3 to pick up passengers. Station 4 (the souvenir shop) is the final stop on the bus route. There all passengers must leave the bus and then buses directly travel back to Station 1.  Visitors arrive and wait for the safari buses only at Station 2 and Station 3, there they board and then everyone leaves each bus at Station 4.  Thus no one gets off the bus at Stations 2 and 3.

The buses pull out from Station 1 into Station 2 at specific intervals - that is the frequency of departure is an exogenous parameter, controlled by you – and your initial hunch is that the best departure schedule is one bus every 5 minutes.

There is no limit to the maximum number of buses that can be at Station 1.  However, no bus can pull out from Station 1 to Station 2 if Station 1 is empty. 

No more than 1 bus can enter Stations 2, 3 and 4 at a time. Thus e.g. if a bus is present in Station 2, buses that already have departed from Station 1, and have reached Station 2, must wait until Station 2 is empty. The same rule applies to Stations 3 and 4.

The travel time of each bus between stations depends on its speed from station to station, (which is assumed to be a uniform 30 miles per hour) and the distance between stations.  The distance between station 1 and 2 is 2 miles, the distance between stations 2 and 3 is 2.5 miles, the distance between 3 and 4 is 4 miles and the distance between stations 4 and 1 is 5.5 miles.

In addition, the total travel time of each bus also depends upon the time it takes for the passengers to board the bus at each station – or DWELL time, and the exit time at station 4.  Thus each DWELL time at each station is a function of the time it takes for each passenger to board the bus (call it e.g. BOARD TIME), which is 6 sec per passenger and the total number of passengers that has accumulated at each station between bus arrivals. When a bus pulls into a station all passengers waiting at that time - and only those - will board the bus.  Assume that there is no capacity limit to how many passengers each bus can accept.

Assume that a constant number of visitors arrive at stations 2 and 3. The rate of passenger arrival in station 2 is 2 passengers per minute and for station 3 it is 3 passengers per minute.

At Station 4 all passengers on each bus will exit. It takes 6 seconds per passenger to exit just as it takes 6 seconds per passenger to enter the buses at Stations 2 and 3.

In the beginning of the simulation (@9AM), Station 1 has 9 buses, all other Stations are empty but one bus left Station 1, 2 minutes ago - thus there are 10 buses total in the system.  In the beginning no visitors are waiting in the system as the zoo just opened. Thus initially there are no passengers waiting in the system, but begin accumulating immediately at 9AM.

TASK:

1. Build a STELLA model of this bus system assuming that buses leave station 1 at a frequency of 1 bus every 5 minutes.  Run the model for 5 hours (beginning at 9AM) with a DT of 1 at the timescale equal to minutes. (40 points)

2. Graph the flux of buses in Station 1 and 2 vs. time and Stations 3 and 4 vs. time.  Graph the DWELL time at Stations 2 and 3 vs. time and interpret what you see. Estimate the total number of visitors that are able to complete their bus trip over the course of those 5 hours and the cumulative average travel time per bus during the course of the simulation, which is the amount of time that lapses between the initial departure from station 1 until it enters station 1 again. (10 points)

Be very careful to define every single unit in your model - and run your model at the timescale of minutes.  Make sure you do not deal with fractions of buses nor people.

3. What is the optimal frequency at which the buses should pull from station 1, where optimal is the frequency that will minimize cumulative average travel time per bus. What is the cumulative average travel time per bus at this optimal frequency? Estimate the total number of visitors that are able to complete their bus trip over the course of those 5 hours. (5 points)

4. Incorporate a rush hour into this model. Assume that between 11 and 12 noon 4 visitors will arrive every minute at station 2 and 6 passengers at station 3. Will the optimal frequency of bus departure change? (5 points)

Useful hints:  In essence you need to model simultaneously but separately the flow of buses and flow of passengers, where one flow influences the other.  E.g. to know the time buses stay at station 4, you must know the total amount of passengers on each bus entering station 4 – and of course you need to know when each bus enters station 4.  Similarly to know how long each bus stays at stations 2 and 3 you need to know how many people have accumulated at each station.  Keep in mind that the state of an oven can be measured by OSTATE() and a specific space on a conveyor or on a queue is measured by QELEM().

Total 60 points