Simulation I
Queuing Theory
Sample Calculations
1
Farm
In a dairy barn there are two water troughs (i.e. drinking places). From each
trough only one cow can drink at the same time. When both troughs are
occupied new arriving cows wait patiently for their turn. It takes an exponential
time to drink with mean 3 minutes. Cows arrive at the water troughs according
to a Poisson process with rate 20 cows per hour.
Use P0 = 1/3 to perform the following calculations:
a) Compute the average number of troughs that are occupied (i.e.,“busy”)
b) Compute the fraction of time the “service facility” is busy.
c) What is the fraction of cows finding both troughs occupied on arrival?
2
Student Dorm
In the ViennaLive student dorm in Vienna, there are 3 gyms. Because of the
COVID-19 restrictions, only 1 student can train at the same time. If all gyms
are busy, the residents put their name in the waiting list and wait for residents
to finish their workout. It takes an exponential time to train with a mean of
45 minutes. Residents arrive at the gyms according to a Poisson process with a
rate of 3 per hour. Use P0 = 0.07476636 for the following calculations:
a) Calculate the average number of residents in the system.
b) Calculate the average waiting time of the residents.
c) How would these calculations change if the dorm would allow 2 people in
each gym? In this case, use P0 = 0.105081469 for your calculations.
3
Burger Burger
In the Burger Burger restaurant, there are 2 servers. The restaurant does only
pickup orders now. Each server can work with only one customer. In case they
are busy, the customers stand in one common line for the 2 servers. The time
for each customer speaking to the server is exponentially distributed with an
average of 7.5 minutes. The customers arrive at Burger Burger according to a
Poisson process with a rate of 12 customers per hour. Use P0 = 0.142857143 to
calculate the following:
1
a) Determine the mean number of customers waiting in line and the mean
waiting time.
b) Determine the expected amount of time customers spend in the restaurant
in total.
c) What is the number of servers busy taking an order from customers on
average?
4
Blue Chip Life Insurance
In the Blue Chip Life Insurance Company, the deposit and withdrawal functions
associated with a certain investment product are separated between two clerks,
Clara and Clarence. Deposit slips arrive randomly (a Poisson process) at Clara’s
desk at a mean rate of 16 per hour. Withdrawal slips arrive randomly (a Poisson
process) at Clarence’s desk at a mean rate of 14 per hour. The time required
to process either transaction has an exponential distribution with a mean of 3
minutes. To reduce the expected waiting time in the system for both deposit
slips and withdrawal slips, the actuarial department has made the following
recommendations: (1) Train each clerk to handle both deposits and withdrawals,
and (2) put both deposit and withdrawal slips into a single queue that is accessed
by both clerks.
a) Determine the expected waiting time in the system under current procedures for each type of slip. Then combine these results to calculate the
expected waiting time in the system for a random arrival of either type of
slip.
b) If the recommendations are adopted, determine the expected waiting time
in the system for arriving slips. Hint: Use P0 = 0.142857143 for your
calculations.
5
Car Wash
Janet is planning to open a small car-wash operation, and she must decide how
much space to provide for waiting cars. Janet estimates that customers would
arrive randomly (i.e., a Poisson input process) with a mean rate of 1 every 4
minutes, unless the waiting area is full, in which case the arriving customers
would take their cars elsewhere. The time that can be attributed to washing
one car has an exponential distribution with a mean of 3 minutes.
Compare the expected fraction of potential customers that will be lost because of inadequate waiting space if
a) 0 spaces (not including the car being washed),
b) 2 spaces, and
c) 4 spaces were
provided.
2
6
Miller Manufacturing
Miller Manufacturing owns 10 identical machines used for the production of
colored nylon. Machine breakdowns occur following a Poisson distribution with
an average of 0.01 breakdowns occurring per operating hour per machine. The
company looses $100 each hour a machine is not working.
The company employs one technician to fix these machines whenever they
break down. Service times to repair the machines are exponentially distributed
with an average of 8 hours in order to repair one machine.
Management wants to analyze the impact of adding another service technician (service technicians are paid $20 per hour).
a) Compute the expected total costs that incur per hour if one technician is
employed.
Hint: Use P0 = 0.322 as a basis for your calculations.
b) Compute the expected total costs that incur per hour if two technicians
are employed.
Hint: Use L = 0.811 as a basis for your calculations.
3