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