Q$5$: Waiting Lines$$Model for a Single$-$Server System

Background:

Assume you are managing a service system where a single server currently handles an arrival rate of $8$ customers per hour $($ $=8)$ and services $10$ customers per hour $($ $=10).$ You are considering doubling the service rate $($to $20$ customers per hour$)$ while maintaining a single$-$server system. Your task is to analyze the impact on waiting times and customer queue lengths.

Tasks:

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A$.$ Do you intuitively expect that doubling the service rate in a single$-$server system will cut the average waiting time in line by half? Why or why not?

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Provide your reasoning based on queuing theory principles, particularly in terms of how increasing the service rate affects the average waiting time $($Wq$)$ and system time $($Ws$).$

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B$.$ If the service rate is doubled, what impact will it have on the average number of customers in the queue $($Lq$)?$ Calculate the average number of consumers waiting in line before and after the service rate change.

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Use relevant queuing formulas $($e$.$g$.,$ Lq and Wq$)$ to calculate the average number of customers in the queue before and after doubling the service rate.

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C$.$ Reflect on the results from part B$.$ What are the implications for the manager?

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Discuss the practical implications of the changes to the system, specifically how changes in service rate $($$)$ affect the number of customers in the queue and waiting times.

From both a theoretical and practical perspective, explain how small increases in service rate could have a significant impact on operational efficiency and customer satisfaction.