Consider a call center where calls arrive with an exponential distribution with mean $0\mathrm{.}857$ minutes to one of our infinitely many phone trunk lines. $($We have more than enough phone lines to handle the maximum possible calls at a time.$)$

When a call arrives, the customer listens to an automated recording that takes Uniform $(0\mathrm{.}1,0\mathrm{.}6)$ minutes. This automated recording determines whether this call is a $$Tech Support$,$Sales$$ or $$Order Status$$ call. Historically $76\%$ of the calls is for $$Tech Support$,16\%$ is for $$Sales$$ and $8\%$ is for $$Order Status$.$

If the call is for $$Tech Support$$:

$$ The customer will listen to another automated recording that takes Uniform $(0\mathrm{.}1,0\mathrm{.}5)$ minutes, after which the specific product the customers calls for is determined. Historically $25\%$ of the Tech Support calls is for Product $1,34\%$ is for Product $2$ and the remaining $41\%$ is for Product $3.$

$$ Once the product type is determined, the customer will be connected to one of the Tech Support representatives specialized in that Product. There are $2,3,3$ representatives available for Product $1,$ Product $2,$ and Product $3,$ respectively. If the representatives for the product that the customer calls for are all busy, the customer will wait until one representative becomes available.

$$ The tech calls, regardless of the product types, will last for Triangular $(3,6,18)$ minutes. Upon completion, the customer will hang up$.$

If the call is for $$Sales$$:

$$ The customer will be connected to one of the $4$ sales representatives. If all are busy, the customer will wait until one becomes available.

$$ Sales calls will take Triangular $(4,15,45)$ minutes. Upon completion, the customer will hang up$.$

If the call is for $$Order Status$$:

$$ The customer will listen to another automated recording that takes Triangular $(2,3,4)$ minutes, after which the customers will have the option to speak with a sales representative. Otherwise, the customer will hang up$.$

$$ Historically, $15\%$ of the customers opt to speak with a sales representative. These customers will join a queue $($a separate queue than the sales calls$)$ of such other order status customers wishing to speak with a sales representative.

$$ This call with the sales representative takes Triangular $(2,3,4)$ minutes, after which the customer will hang up$.$

Simulate this system for $10$ hours and submit the doe file and pdf of the report containing the average total time per call type.