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A service station consists of two back-to-back stations S1 and S2. Customers follow this workflow: Customers first arrive at S1​, are served, and then proceed to S2​. After service at S2​, with probability 0.1, a customer is sent back to S1​ for rework (reentrant flow). Otherwise, with probability 0.9, the customer leaves the system. Assume the following conditions: The arrival process follows a Poisson distribution with rate 10 per hour. The service times at S1 and S2​ follow exponential distributions with means 5 minutes and 6 minutes​, respectively. S1 has a single server and S2 has two servers. Both have infinite queue capacity.

What is the distribution of the interarrival time of Type A customers? Specify the distribution and its mean in minutes. Distribution: [dist] Mean: [mean]

Find the marginal distribution of X: P(X=0) = [p0] P(X=1) = [p1] P(X=2) = [p2]  

Complete the following roadmap matrix R of X(t): R =  [r00] [r01] [r02] [r10] [r11] [r12] [r20] [r21] [r22]

Are X and Y independent? Choose your answer below and state the reason on the scratch paper.

Customers arrive at a service desk according to a Poisson process with rate

A more detailed data study reveals that the arrival rate of all types depends on the elapsed time in hours since the service desk opens taking the following form: 

What law was specifically challenged by the Freedom Riders? 

According to our textbook author, explain two contributing factors to U.S. prosperity during the 1950’s. 

Given that there are total 12 customer who arrived in the first hour, what is the probability that 4 of them arrived in the first 15 minutes? Compute four digits after the decimal point. Show your work on the scratch paper.