IE 306: Sample Questions for Final
All the home work questions of HW1 to HW7 are relevant for the final. As the last home works were done on the computer here are some examples how these topics could be asked in the final:
1) Consider a simple queuing network where customer enter the system with exponential interarrival times with expectation 1 minute. One server then serves the incoming people with a service time uniform between 0.3 and 0.5 minutes. After that service people leave the system with probability 80% whereas with probability 20% they have to join the queue again to wait for another service. The simulation should start with an empty system and last for 4 hours.
a) What are the entities and what are the resources and what are the events for this simple network?
b) What are two variables you can use as state variable for that system?
c) Write a C-like pseudo code of the simulation (without considering the collection of any statistics). Write the simulation main loop and one function for each eventtype. You can use the following functions:
double uniform()…. U(0,1) uniform generator
the global variable:
struct FEL fel … is the future event list
void put_on_fel(int event_type, double event_time)… put an event on the future eventlist
void get_from_fel(int *event_type, double *event_time)… the event with the smallest event_time is erased from fel. The event_type
and the event_time are stored into the variables pointed to by the pointers handed over.
initialize_system()
d) What can you do to debug the above system?
e) Try to guess the server utilization of the above system.
f) How will the server utilization change if you change the expectation of the interarrival times to 2 minutes?
2) We have coded an arena implementation of a single server queue with exponential interarrival times and exponential service times. The arrival rate is λ=10, the service rate μ=6. You want to compare the average waiting times of this system with the average waiting times of system B which has the same arrival process but uniform service times with the same rate.
a) What is the interval you should use for the uniform service times?
b) What changes are necessary in the ARENA code tocompare the two systems with common random numbers.
c) Describe the experiment you have to do for this comparison. How do you calculate a confidence interval for the difference of the mean waiting times?
3) For calculating a CI mean waiting time of customers in a steady state distribution three different methods are suggested:
Method A) Simulate 10000 customers in a single run. Calculate the average and the standard error of the 10000 waiting times to
calculate the CI.
Method B) Make 100 repetitions with 100 customers each. Calculate the average waiting times of the last 90 customers in each
repetition. Use these 100 averages to calculate a CI for the mean waiting time.
Method C) Simulate 10000 customers in a single run. Discard the first 1000 customers. Divide the other customers in 100 batches of
length 90. Calculate the 100 batch means and use them for calculating the CI of the mean waiting times.
a)Describe what are the advantages and the problems of the three methods.
b)Which method has the smallest estimation bias for the steady state mean waiting time?
c)Which method could lead to a much too short CI?
4) For a steady state simulation we have two different general approaches for calculating a CI for the mean waiting time.
a) Describe the two approaches.
b) Give the replication parameters you have to set in ARENA to realize the two approaches for a total of 10000 hours of simulation:
Number of Replications:……..Initialize between Replications
Warm-up Period: ………… statistics ….System….
Replication Length:…………(hours)
5) In an ARENA simulation you get the following output:
Output Summary for 1000 ReplicationsOUTPUTS
Identifier Average Half-width Minimum Maximum # Replications
______
patient.NumberIn 14.008 .22976 4.0000 27.000 1000
patient.NumberOut 13.722 .19993 4.0000 21.000 1000
Nurse.NumberSeized 14.008 .22976 4.0000 27.000 1000
Nurse.ScheduledUtilization .10871 .00151 .02229 .17832 1000
a) Guess the probability that the true expected number out is below 13.522. Explain why you give this result.
b) Is it possible that the utilization of the nurse is above 15 % ?
- Suppose that the number of incoming calls arriving to the call center of Meteorology Service follows Poisson distribution with a rate of 20 calls/ hour when it’s snowing and 5 calls/ hour when it’s not. There is a variable called “snow” in the simulation model, keeping track of the snow, assigned elsewhere. It takes value 1 when it’s snowing and 0 otherwise. Also note that, the call center stops answering calls after 150 calls.
We want to model the arrival process of incoming calls. Fill in the Create Module of the Arena to model this process.
- We want to record the number of entities that passed more than 10 minutes in the system until now. Write the expression for the “value” prompt of a “Count” type Record Module. (The attribute that records the creation time is called Entity.CreateTime)
- In a simulation model there are three queues called “Queue1”, “Queue2” and “Queue3” for different types of entities. We made 100 independent replications. We are interested in average and maximum value (in 100 replications) of total number of entities waiting in these queues. Without using the standard output file of Arena, define the necessary statistics using Statistics module. State the type of statistics (Time-Persistent, Tally, Counter, Output or Frequency) and write the necessary expression to obtain desired results.
Some SIMAN expressions that may be useful:
NQ(queue A): current number of entities in queue A
TAVG(Tally A): the average of the tally variable Tally A throughout the replication
TMAX(tally A): the largest observed value of the tally variable
DAVG(Statistic A): the average of the dstat (time-persistent) expression throughout the replication.
DMAX(Statistic A): the maximum value taken by the dstat (time-persistent) expression during the replication
- What is required to apply the “Common random numbers” method in Arena.
- Which Arena module (or block) is necessary to make
a)the entities wait for 10 minutes before continuing with the next step?
b)to make them wait for a resource to become available?
- Consider a portion of an Arena output given below. Using the output, find a 95% for the proportion of customer that could not enter. Also find the maximum value of average number of machines used and average value of maximum number of machines used. What is the difference between these two statistics? Explain how they are calculated.,
ARENA Simulation Results
Output Summary for 100 Replications
Project: Unnamed Project Run execution date : 12/ 7/2004
Analyst: John Smith Model revision date: 12/ 7/2004
OUTPUTS
Identifier Average Half-width Minimum Maximum # Replications
______
Average number of machines required 63.622 1.1089 54.647 80.486 100
Maximum number of machines used 98.210 1.7894 77.000 120.00 100
Proportion of customers could not enter 0.5231 0.0132 0.3852 0.7210 100