$8.$ Grocery Store$$You are hired by Safeway to help them build a number of simulation models to better understand the customer flows and queuing processes in a grocery store setting. The pilot project at hand focuses on an off$-$peak setting where at most two checkouts are open. To better understand the necessary level of detail and the complexities involved, Safeway wants a whole battery of increasingly more realistic and complex models. For each model, Safeway wants to keep track of $($i$.$e$.,$ plot$)$ the average cycle time, queue length, and waiting time in the queue. To understand the variability, they also want to see the standard deviation of these three metrics. In addition, they would like to track the maximum waiting time and the maximum number of customers in line. Furthermore, to better understand the system dynamics, plots of the actual queue lengths over time are required features of the model. The off$-$peak setting is valid for about $4$ hours, so it is reasonable to run the simulation for $240$ minutes. Furthermore, to facilitate an easier first$-$cut comparison between the models, a fixed random seed is recommended. Because Safeway plans to use these different models later, it is important that each model sheet has a limit of one model. a$.$ In the first model, your only interest is in the queues building up at the checkout counters. Empirical investigation has indicated that it is reasonable to model the arrival process $($to the checkout counters$)$ as a Poisson process with a constant arrival intensity of three customers per minute. The service time in a checkout station is on average $30$ seconds per customer and will, in this initial model, be considered constant. Inspired by the successes of a local bank, Safeway wants to model a situation with one single line to both checkout counters. As soon as a checkout is available, the first person in the queue will go to this counter. After the customers have paid for their goods, they immediately leave the store. b$.$ On closer investigation, it is clear that the service time is not constant but rather, normally distributed with mean $=30$ seconds and standard deviation $=10$ seconds. What is the effect of the additional variability compared with the results in Part $($a$)?$ c$.$ To be able to analyze the effect of different queuing configurations, Safeway wants a model in which each checkout counter has its own queue. When a customer arrives to the checkout point, he or she will choose the shortest line. The customer is not allowed to switch queues after making the initial choice. Can you see any differences in system performance compared with the results in Part $($b$)?$ d$.$ To make the model more realistic, Safeway also wants to include the time customers spend in the store walking around and picking up their groceries. Empirical investigation has shown that there are basically two types of customers, and they need to be treated somewhat differently: Modeling and Simulating Business Processes with ExtendSim $349$ Type $1$: The light shopper who buys only a few items $($fewer than $15)$ About $60$ percent of the customers arriving to the store. The shopping time follows a triangular distribution with a most likely value of $5$ minutes, a minimum value of $2$ minutes, and a maximum value of $8$ minutes. The service times for these customers at the checkout counter are exponentially distributed with a mean of $15$ seconds. Type $2$: The heavy shopper who buys several items $($more than $15)$ Represents about $40$ percent of the arriving customers. The shopping time is triangularly distributed with a most likely value of $10$ minutes, a minimum value of $5$ minutes, and a maximum value of $15$ minutes. The service times for these customers at the checkout counter are exponentially distributed with a mean of $52$ seconds. The total arrival process to the store is still a Poisson process with a mean of three customers per minute. As for the queue configuration, Safeway feels that the setup in Part $($b$)$ with one line for each checkout is better for psychological reasons; one long line might deter customers from entering the store. Modify the simulation model to incorporate the described elements and make it more realistic. Analyze the performance of the current process using the performance measures discussed previously as well as the following additional measures: The time spent shopping $($average and standard deviation$)$ The number of customers $($average and standard deviation$)$ The separate cycle times for heavy and light shoppers $($average and standard deviation$)$ e$.$ To improve the service for the light shoppers, Safeway is thinking about dedicating one of the checkout counters to this customer group. In other words, only light shoppers are allowed to use Checkout $1.$ The other checkout $($Checkout $2)$ will handle both heavy and light shoppers. However, empirical interviews indicate that no light s