Operations Management Average Customer Wait

Hi there: Please ONLY DO THIS IF YOU ARE POSITIVE of the outcome. Thank you!!! (Please show work and show spreadsheets so I can see how you came to these conclusions)

-------My-law.com 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.com 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 call 1-800-my-lawyer. From there, the incoming calls 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.

Call arrivals follow a Poisson process with a rate of 10 calls per hour. At each moment in time, there is exactly one lawyer "on call," that is, sitting at his or her desk waiting for incoming calls. It takes the lawyer, on average, 5 minutes to answer the caller's questions, and this time follows the exponential distribution.

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

2. When not responding to phone calls, the lawyer on call is encouraged to actively pursue cases that potentially could lead to large settlements. How much time can a My-law.com lawyer dedicate to this activity during an 8-hour shift (assume the lawyer can instantly switch between calls and work on a settlement)?

To increase the responsiveness of the firm, the board of My-law.com proposes a new operating policy. Under the new policy, the average response time would be reduced to 4 minutes and it still follows the exponential distribution.

3. How would the average time a customer has to wait before his/her call is answered by the on-call lawyer change?

4. How would the amount of time a lawyer can dedicate to the search for large settlement cases during an 8-hour shift change with this new operating policy?

Switch data suggest that callers are very patient and rarely hang up due to waiting. However, the firm is worried that long waits may reduce the likelihood of future relationship. It is estimated that customers don't mind being on hold for 10 minutes or less. In such cases they have a 30%
probability of following up the initial phone call with additional engagement with the firm. On the other hand, if they are put on hold for more than 10 minutes, their likelihood of future engagement is cut in half.

5. How would the new operating policy impact the overall follow-up probability?