Question

A small factory consists of a machining center and inspection station in series. Unfinished parts arrive to the factory with exponential interarrival times having a mean of 2 minutes. Processing times at the machine are uniform on the interval [0.65, 0.70] minute, and subsequent inspection times at the inspection station are uniform on the interval [0.75, 0.80]. Ninety percent of inspected parts are "good" and are sent to shipping; 10 percent of the parts are "bad" and are sent back to the machine for rework. Both queues are assumed to have infinite capacity.

Let Yi be a random variable representing the number of parts produced during the ith hour. Generate 10000 Yis, and compute ten sample means and sample variances using 1000 Yis each, respectively. Comment on the convergence of Yito the steady-state distribution Y.

Answer 1

The convergence of sample means and variances for the generated Yi values to steady-state distribution Y can be observed as the sample size increases, demonstrating the stability and reliability of the statistical estimates in reflecting the system's behavior over time.

The process of generating 10,000 random variables Yi, each representing the number of parts produced during the ith hour in a small factory, allows for the analysis of the convergence of sample means and variances towards the steady-state distribution Y. The sample means and variances are calculated using subsets of 1,000 Yi values each. As the sample size increases, the law of large numbers suggests that the sample statistics should converge towards the true population values or, in this case, the steady-state distribution parameters.

The convergence is an indication that the statistical estimates become more reliable and representative of the underlying system behavior. In the context of this scenario, it implies that the production and inspection processes in the factory reach a stable state where the observed statistical properties stabilize over time. Monitoring the convergence of sample means and variances is crucial for assessing the accuracy and consistency of the statistical estimates and provides insights into the long-term performance and characteristics of the production system.