IENG 3345 Operations Research: Stochastic and Non-Linear Models Assignment 3 Questions

IENG 3345 Question 1 (50 points): 

A maintenance person has the job of keeping two machines in working order.  The  amount  of  time  that  a  machine  works  before  breaking  down  has  an  exponential distribution  with  a mean  of  10  hours. The  time  spent  by  the  maintenance  person  to  repair  the machine has an exponential distribution with a mean of 8 hours. a) Show that this process fits the birth-and-death process by defining the states, specifying the values of the 𝜆𝑛 and 𝜇𝑛, and then constructing the rate diagram. (10 points) b) Calculate the 𝑃𝑛. (10 points) c) Calculate 𝐿, 𝐿𝑞, 𝑊, 𝑎𝑛𝑑 𝑊𝑞. (10 points) d) Determine the proportion of time that the maintenance person is busy. (10 points) e) Determine the proportion of time that any given machine is working. (10 points) 

IENG 3345 Question 2 (50 points):  

The  Centerville  International  Airport  has  two  runways,  one  used exclusively for takeoffs and the other exclusively for landings. Airplanes arrive in the Centerville airspace to request landing instructions according to a Poisson process at a mean rate of  15 per hour. The time required for an airplane to land after receiving clearance to do so has an exponential distribution with a mean of 5 minutes, and this process must be completed before giving clearance to  do  so  to  another  airplane. Airplanes  awaiting  clearance  must  circle  the airport. The  Federal Aviation Administration has a number of criteria regarding the safe level of congestion of airplanes waiting to land.

These criteria depend on a number of factors regarding the airport involved, such as the number of runways available for landing.

For Centerville, the criteria are: 

(i) the average number of airplanes waiting to receive clearance to land should not exceed 1, 
(ii) 95% of the time, the actual number of airplanes waiting to receive clearance to land should not exceed 4, and 
(iii) for 99% of the airplanes, the amount of time spent circling the airport before receiving clearance to land should not exceed 45 minutes (since exceeding this amount of time often would require rerouting the plane to another airport for an emergency landing before its fuel runs out). 

1. Identify  the  defined  system  as  a  queueing  model  (based  on  birth-and-death  process models). (5 points) 
2. Evaluate how well these criteria are currently being satisfied. (15 points) 
3. A major airline is considering adding this airport as one of its hubs. This would increase the mean arrival rate to 25 airplanes per hour. Evaluate how well the above criteria would be satisfied if this happens. (15 points)
4. To attract additional business, including the major airline mentioned, airport management is considering adding a second runway for both landings.

It is estimated that this eventually would increase the mean arrival rate to 30 airplanes per hour. Identify the changed system as  a  queueing  model.  Evaluate  how  well  the  above  criteria  would  be  satisfied  if  this happens. (15 points)