Question: Consider a sequence of three machines each with exponentially distributed processing times. The mean processing rates for these three machines are 1, 2, and 3, jobs per hour, respectively. Consider the time between completed jobs at Machine 3 (the third machine in the series). The system operates as follows. With probability 5/8, machine three has a next job ready for processing immediately after completing a job. Sometimes there isn't a next job waiting to be processed at Machine 3 but there is a job processing on Machine 2 which when completed will be sent immediately to Machine 3.
This situation occurs 1/4 of the time. When there is not a job being processed on Machine 2 when Machine 3 needs a next job for processing, there is always a job processing on Machine 1. Thus, Machine 3 must wait for the completion of the job on Machine 1 and then its processing time on Machine 2 before it is available for processing on Machine 3. What is the mean and SCV of the time between job departures from Machine 3?