IAdvanced Manufacturing Systems Engineering Assignment 2 Queueing Network Models Names:

1. Consider a manufacturing system with a single station with three servers. Jobs arrive following an exponential distribution with a rate of 7 jobs per hour, while the service rate of each of the three servers follows an exponential distribution with a rate of 10 jobs per hour. If the total number of jobs that can be in the system at any time is 5, use this information to answer the following questions.
   1. Draw and label a state space diagram, showing all states and the rates of transition between states.
   2. Write a complete set of balance equations
   3. Solve your system of equations to determine the long run probability of the system being in each possible state
   4. What is the average number ofjobs in the system at any point in time?
   5. What would be the effect on the state probabilities if the arrival rate reduces to 4 jobs per hour?
2. Consider a simple barber shop with one (1) "service chair" and two (2) "waiting chairs." Customers to this barber shop arrive with exponential arrival events depending on the state of the system. If the system is empty, customers arrive at rate m. If there is a customer in the chair, but no one in the waiting chairs (a total of one customer in the system), the arrival rate slows to 0.8K. If there is one customer in service and one waiting (a total of two customers in the system), the arrival rate slows further to 0.6m. Once there are three customers in the system, an incoming customer decides to turn away.

The barber also changes his work rate according to the system state. Service times are also exponential. If there is one customer in the chair and no customer waiting, the barber works at rate p. If there is one customer waiting (a total of two customers in the system), the barber speeds up to rate 1.1 g; if there are two waiting (a total of three customers in the system), he works at rate 1.2p. If is estimated to be 3 customers per hour, and is estimated at 4 customers per hour;

1. Draw and label a state space diagram, showing all states and the rates of transition between states.
2. Write a complete set of balance equations
3. Solve your system of equations to determine the long run probability of the system being in each possible state
4. What is the average number ofjobs in the system at any point in time?

3. Consider a manufacturing system represented by the diagram shown below. There are two stations each with a single machine. The arrival rate to the manufacturing system is m. When jobs arrive, they are processed at station l, with a service rate of Pl. Once jobs are processed at station l, they proceed to station 2, to be processed at a service rate of 1.12. However, there is a single waiting spot at each station for a single job to wait if the server at the station is occupied. Once, a job is already waiting for an occupied server, an additional job cannot be allowed to join the queue. Draw and label a state space diagram, showing all states and the rates of transition between states. Note: Here, I just want to see the state diagram only. You are not required to solve it.

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assignment-2-....docx