CPU-on-Demand (CPUD) offers real-time high-performance computing services. CPUD owns 1 supercomputer that can be accessed through the Internet. Their customers send jobs that arrive on average every 7 hours. The standard deviation of the interarrival times is 4 hours. Executing each job takes on average 3 hours on the supercomputer and the standard deviation of the processing time is 3.6 hours. (Round your answer to 1 decimal place.) How long does a customer have to wait to have a job completed?

This is a classic queueing theory problem. To solve this, we'll use the **M/G/1 queueing model**, where:

- M = Markovian arrival process (Poisson process with exponentially distributed interarrival times)
- G = General service time distribution (with arbitrary distribution for service time)
- 1 = Single server (supercomputer)

Given:
- Mean interarrival time (λ) = 7 hours
- Standard deviation of interarrival time = 4 hours
- Mean service time (τ) = 3 hours
- Standard deviation of service time = 3.6 hours

The formula to calculate the **average waiting time** in a queue (Wq) for an M/G/1 queueing model is given by:

\[
W_q = \frac{\lambda^2 \cdot \sigma_s^2 + \tau^2 \cdot (1 - \rho)}{2(1 - \rho)}
\]
where:
- \(W_q\) = average time waiting in queue
- \(\rho = \lambda/\mu\)

Learn how to calculate the average waiting time for job completion in CPUD's high-performance computing service using the M/G/1 queueing model. This problem involves queueing theory, probability distributions, and statistics, including interarrival and service time analysis. Suitable for advanced college-level students and professionals.

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