$5.$ The Holix Corporation has two separate processing centers for credit applications. Each credit application passes through three steps: step $1-$background credit check, step $2-$credit risk profile analysis, and step $3-$credit level approval and communication to client.The processing time for each of the stages is $20$ minutes for credit check, $50$ minutes for credit risk profile analysis, and $14$ minutes for credit approval and communication. The set$-$up time between successive applications for each step is $10$ minutes. While both processing center take similar steps, they each deal with different types of applications: processing center $1$ evaluates residential customers and processing center $2$ evaluates commercial customers.

$5$a$.$ Each of the two processing centers is organized in exactly the same manner and receives the same average number of applications per day. For example, processing center $1$ gets about $30$ applications per $8-$hour day. The first step in center $1$ has $2$ employees working in parallel. The second step has $4$ employees working in parallel, and the third step has $2$ employees working in parallel. What is the average lead time to process an application through all three steps in each processing center?

$5$b$.$ Holix is considering merging these two operations into one system. The new system faces a total arrival rate of $60$ applications per $8-$hour day. The first step would now have $4$ employees operating in parallel, the second would have $8,$ and the third would have $4.$ Processing time per step per employee would remain the same. However, the set$-$up tine is anticipated to increase to $11\mathrm{.}5$ minutes. What is the average lead time to process an application through all three steps in this merged processing center? Why does this merger of processing centers generate the lead time observed?

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