Webflux is an Internet$-$based DVD rental business specializing in hard$-$ to$-$find, obscure films. Its operating model is as follows. When a customer finds a film on the Webflux Web site and decides to watch it$,$ she puts it in the virtual shopping cart. If a DVD is available, it is shipped immediately $($assume it can be shipped during weekends and holidays, too$).$ If not available, the film remains in the customer's shopping cart until a rented DVD is returned to Webflux, at which point it is shipped to the customer if she is next in line to receive it$.$ Webflux maintains an internal queue for each film and a returned DVD is shipped to the first customer in the queue $($first$-$in$,$ first$-$out$).$

Webflux has one copy of the $1990$ film Sundown, the Vampire in Retreat, starring David Carradine and Bruce Campbell. The average time between requests for the DVD is $10$ days, with a coefficient of variation of $1.$ On average, a customer keeps the DVD for $5$ days before returning it$.$ It also takes $1$ day to ship the DVD to the customer and $1$ day to ship it from the customer back to Webflux. The standard deviation of the time between shipping the DVD out from Webflux and receiving it back is $7$ days $($i$.$e$.,$ it takes on aver$-$ age $7$ days to $($a$)$ ship it$,($b$)$ have it with the customer, and $($c$)$ ship it back$)$; hence, the coefficient of variation of this time is $1.$

a$.$ What is the average time that a customer has to wait to receive Sundown, the Vampire in Retreat DVD after the request? Recall it takes $1$ day for a shipped DVD to arrive at a customer address $($i$.$e$.,$ in your answer, you have to include the $1-$ day shipping time$).$

b$.$ On average, how many customers are in Webflux's internal queue for Sundown? Assume customers do not cancel their items in their shopping carts.

Thanks to David Carradine's renewed fame after the recent success of Kill Bill Vol. I and II which he starred in$,$ the demand for Sundown has spiked. Now the average interarrival time for the DVD requests at Webflux is $3$ days. Other numbers $($coefficient of variation, time in a customer's possession, shipping time$)$ remain unchanged. For the following question only, assume sales are lost for customers who encounter stockouts; that is those who cannot find a DVD on the Webflux Web site simply navigate away without putting it in the shopping cart.

c$.$ To satisfy the increased demand, Webflux is considering acquiring a second copy of the Sundown DVD$.$ If Webflux owns a total of two copies of Sundown DVDs $($whether in Webflux's internal stock, in customer's possession, or in transit$),$ what percentage of the customers are turned away because of a stockout?