Case Study: Optimizing Computer Terminal Rentals for Database Access in an Engineering Department $(40$ marks$)$

The director of a large engineering department is evaluating the rental of multiple computer terminals to facilitate staff access to a shared database. Following an internal survey, the director has compiled the following data:

The department processes approximately $8$ database requests each hour.

Each database search requires an average of $15$ minutes $(0\mathrm{.}25$ hours$)$ to complete.

The rental cost for a computer terminal is $$10$ per hour.

The cost incurred for keeping an engineer idle $($while waiting for a terminal$)$ is $$30$ per hour.

The director operates under the following assumptions:

The arrival of database requests adheres to a Poisson distribution.

The processing time for each request $($service time$)$ follows an exponential distribution.

a$)$ Using the provided information, determine the optimal number of computer terminals the director should rent to minimize the department's total costs. $[30$ marks$]$

b$)$ Create a flowchart illustrating your proposed queuing system. $[5$ marks$]$

c$)$ For the system proposed in part $($a$),$ calculate the expected waiting time for an engineer before they can access a terminal to initiate their database search. $[5$ marks$]$