Answer:
1. A two-runway (one runway dedicated to landing, one runway for dedicated to taking off) airport is being designed for propeller-driven aircraft. The average time to land an airplane is known to be 1.5 minutes with a standard deviation of 0.75 minutes. Assume that the airplanes arrivals are assumed to occur at random with exponentially distributed inter-arrival times. with average service time of 1.5 minutes. Based on the information above, answer the following questions: (a) Calculate the average waiting times and average numbers of airplanes waiting for landing for various values of arrival rates (from relatively small values to close to the service rate) and plot them as functions of the arrival rate. What arrival rate(s) would you recommend for based on plots? [Feel free to use MS Excel, MATLAB, or other computer tools to answer this part.] (b) Show that the arrival rate must be no greater than 0.5079 per minute so that the average waiting time in the sky is not to exceed 3 minutes. (c) Under the arrival rate specified in (b), show that the average number airplanes waiting in the sky for landing is 1.52 aircrafts?
2. Consider the same run-way landing problem above, assuming that the landing time is a constant 1.5 minutes. (a) Calculate the average waiting times and average numbers of airplanes waiting for landing for various values of arrival rates (from relatively small values to close to the service rate) and plot them as functions of the arrival rate. What arrival rate(s) would you recommend for based on plots? [Feel free to use MS Excel, MATLAB, or other computer tools to answer this part.] (b) Show that the arrival rate must be no greater than 0.5333 per minute so that the average waiting time in the sky is not to exceed 3 minutes. (c) Under the arrival rate specified in (b), show that the average number airplanes waiting in the sky for landing is 1.6 aircrafts? (d) Explain the difference of the results in the two problems.
3. A manufacturing cell has 4 operators (working independently in parallel) and a shared buffer space that can fit at most 6 incoming jobs. During a production period, jobs arrive to the cell with rate of 10 jobs per hour (Poisson arrival, i.e., exponential inter-arrival time). The average processing time of a job is 30 minutes (exponentially distributed service time). Answer the following questions using the analytical formulas learned in class. (a) Explain why the M/M/4/10 queueing model should be used to model this problem. (b) Show that the long-term average waiting time of a job in the buffer is 0.4554 hr or 27.3 min. (c) Show that the probability that an arriving job cannot be admitted into the cell due to full occupancy is 0.2301.
Step-by-step explanation: