Analyzing small business inventory and operations

See the attachment.
Process Analysis Queuing
You have taken a job as manager of Cupcake Heaven in Old Town Alexandria, Virginia. It is located on a busy street with lots of walk-in traffic. The owner, Sue, is thrilled at the prospect of you helping her analyze her operations. Cupcake Heaven currently operates in the following manner.

The baker arrives each morning at 5 am. It takes her 30 min to mix and bake a batch of cupcakes. A batch is 50 cupcakes. While a batch bakes, she is cleaning up from the first batch and therefore cannot start a second batch until the first has finished baking. The decorator, who frosts and decorates the cupcakes, arrives at 5:30 am. It takes him 40 minutes to decorate a batch. The baker finishes baking at 10 am. She then finishes cleaning up, makes coffee for customers, takes inventory, and places a replenishment order to be delivered later that day for the next day's baking. The store opens at 10 am and remains open until 8 pm. Some days the cupcakes sell out by 2 pm, other days they have leftovers. Averaging the demand for an entire day, they have found demand to be about 40 cupcakes per hour with a standard deviation of 5.

Sue wants you to answer the following questions for her.

1. What is the current daily capacity of the baking/decorating process? Is it sufficient to meet the daily average demand?
2. What is the maximum inventory level of cupcakes (in batches) waiting to be decorated?
3. What is the maximum length of time for which a batch of cupcakes will wait (in inventory) to be decorated? What time is the decorator finished decorating?
4. Now consider the customer's service experience. Customers start arriving at 10 am. However, the weekday is really only busy at lunch time 11:30-1:30 and after school 2:30-4:30. Weekends are different, with pretty steady customer arrivals all day. Customers during the weekday busy periods tend to buy just one cupcake, while on weekends the average order is for 3 cupcakes. When you ask, Sue is not sure what the arrival process is during the weekday busy periods but knows on average she sells 150 cupcakes during each two hour busy period. It takes the cashier about 30 seconds to write up and ring up an order and accept payment, but sometimes customers don't know what they want and other times, they pay with cash which takes longer. On average, Sue estimates a cashier takes about 45 seconds to ring someone up, but it is fairly random. Sue has space inside for 5 people in line and 5 people waiting for their cupcakes. In addition, there is a line demarcated by ropes outside that can hold 5 more people.

Fill out the following table based on weekday lunch time two-hour rush. Show your work and and include appropriate units.

λ
Average interarrival time
μ
Average service time
Utilization
S
Ca
Cs
Lq
Ls
Wq
Ws
How much free time do(es) the server(s) have for clean up, refilling napkin holders etc. during the two hour period?

Now turn to the ordering/packaging process. The packaging process though can be lengthy. A customer tells the cashier what he wants, the cashier marks it on a slip of paper, and puts it on the packaging table behind the counter. This part on average takes 30 seconds. The packaging person gets the cupcake (or cupcakes) from the correct tray (15 seconds), folds the package container (10 seconds), places the cupcake in the package and closes the package (15 seconds), then calls the customer's name and waits for the customer to get the package (20 seconds). Ignore variability in the process and assume that there is one cashier and one packaging person.

5. Draw a flow chart, label each step and determine the capacity of each step.
6. Where is the system capacity? What is the cycle time of the process?
7. If the packaging person goes on break and the cashier has to do all the steps, what is the cycle time of the process?