• (40 points) A distribution center has a specific shipping operation that operates as follows:

Packages arrive to the shipping area according to a Poisson process having a rate of 60 per hour. Upon arrival, packages enter into the inspection area. The inspection area contains a single inspector who verifies the content of the package. This inspection process requires an exponentially distributed amount of time with a mean of 45 seconds. On average, one out of every six packages is in error and must be sent back into the distribution center (after inspection). The packages that pass the inspection are delivered immediately to the labeling area.

The labeling area contains a single queue for three additional label machines. The time required to place a label on the package is an exponential random variable having a mean of 90 seconds. Once the label is on the package, the package is immediately delivered to the sorting area.

Upon arrival to the sorting area, the package is immediately diverted to one of two queues (based on geographic destination). Customer demographics and sorting equipment characteristics dictate that 60% of packages are diverted to Sorting Queue 1. Both sorting queues consist of a single, automated sorting machine that processes packages individually. Both machines have a normally distributed sorting time with a mean of 30 seconds; however the standard deviation of sorting time is 3 seconds for machine 1 and 5 seconds for machine 2.

Answer the following questions about the long-run behavior of the shipping operation. Show all your work, and label each questions clearly. Write from top to down, from left to right. Work that is not label and not clear will NOT BE CONSIDERED. Attach any Excel file that you used for need to write the answers hand written or in word perfectly labeled.

• Describe the queueing (use proper notation and state the arrival and service rates) system that can be used to represent the inspection area (1.5 points)
• What is the probability that the inspection is idle (2 points)
• On average, how many packages are waiting to be inspected (2)
• Describe the stochastic process and the principle applied that governs the departure of packages from the inspection area (1)
• Describe the stochastic process and the queueing model (rates and type) that governs the arrival and service to the labeling area (1.5)
• On average, how long does a package spend in the labeling area? (10)
• What is the probability that there are two packages in the labeling area? (2.)
• What is the probability that there are four packages in the labeling area? (2.)
• Describe the stochastic process (use a small diagram) that governs the arrival and service of packages to the sorting area (1)
• On average, how many packages are in the sorting area (10)
• Suppose sorting machine 1 just became idle. On average, how long will it be until sorting machine 1 is idle again (3 points)
• Suppose the inspection area had a finite capacity of 3. How would this change your answer for part (a), (c), (f) and (k) (10 points)