Consider a manufacturing system that processes two parts (P1 and P2) on four different machines (A, B, C, and D). The processing times on each machine are random and are independent of part type. Assume that all processing times are exponentially distributed with means 1.25 minutes (Machine A), 1.60 minutes (Machine B), 1.75 minutes (Machine C), and 4.25 minutes (Machine D). P1 arrives at the rate of 16 parts per hour and P2 arrives at a rate of 14 parts per hour (arrival of both P1 and P2 are exponential). All arriving parts are processed by machine A first and then by machine B. After finishing on machine B, 85% of the parts (independent of part type) are processed by machine c and the remaining 15% are processed by machine D. After finishing on C, parts are finished and leave the system. After finishing on D, 25% of the parts (independent of part type) are finished and leave the system, while the remaining 75% return for processing on machine B, where they are treated exactly the same as the parts arriving from machine A (i. e., the processing time distribution and the probabilities of going to machines C and D after processing are the same as they are for parts arriving from A). Develop a queueing network model for this system and use your model to determine the following steady state values (please use 2 decimal places accuracy in your calculations):. a) Expected utilizations of machines A, B, C, and D
b) Expected number of parts in the entire system
c) Expected time in minutes) that parts spend in the system