Math

My-law is a recent start-up trying to cater to customers in search of legal services who are intimidated by the idea of talking to a lawyer or simply too lazy to enter a law office. Unlike traditional law firms, My-law allows for extensive interaction between lawyers and their customers via telephone and the Internet. This process is used in the upfront part of the customer interaction, largely consisting of answering some basic customer questions prior to entering a formal relationship.

In order to allow customers to interact with the firm's lawyers, customers are encouraged to send email to my-law. From there, the incoming emails are distributed to the lawyer who is currently "on call." Given the broad skills of the lawyers, each lawyer can respond to each incoming request.

E-mail arrive from 8am to 6pm at a rate of 10 email per hour (coefficient of variation for the arrivals is 1). At each moment in time, there is exactly one lawyer "on call," that is, sitting at his or her desk waiting for incoming emails. It takes the lawyer, on average, 5 minutes to write a response email.The standard deviation of this is 4 minutes.

1. What is the average time a customer has to wait before his/her call is answered by the on-call lawyer?

Would this be 25.5 minutes

2. When not responding to emails, the lawyer on call is encouraged to actively pursue cases that potentially could lead to large settlements. How much time can a My-law lawyer dedicate to this activity during an 10-hour shift (assume the lawyer can instantly switch between emails and work on a settlement)?

Would this be 100 minutes to pursue high value cases

To increase the responsiveness of the firm, the board of My-law proposes a new operating policy. Under the new policy, the standard deviation for responding would be 0.5 minutes. The average writing time would remain unchanged.

3. How would the amount of time a lawyer can dedicate to the search for large settlement cases change with this new operating policy?

I don't know where to start.

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