$($

My$-$law.com$)$

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 send e$-$mails to my$-$lawyer@My$-$la

w$.$com. From there, the incoming e$-$mails 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$-$mails arrive from $8$ a$.$m$.$ to $6$ p$.$m$.$ at a rate of $10$ e$-$mails 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 e$-$mails. It takes the lawyer, on average,

$5$ minutes to write the response e$-$mail. The standard deviation of this is $4$ minutes.

a$.$ What is the average time a customer has to wait for the response to his$/$her e$-$mail, ignoring any transmission times? Note: This

includes the time it takes the lawyer to start writing the e$-$mail and the actual writing time. $[[9\mathrm{.}6]$

b$.$ How many e$-$mails will a lawyer have received at the end of a $10-$hour day? $[9\mathrm{.}6]$

c$.$ When not responding to e$-$mails, the lawyer on call is encouraged to actively pursue cases that potentially could lead to large

settlements. How much time in a $10-$hour day can a

My$-$law.com lawyer dedicate to this activity $($assume the lawyer can instantly

switch between e$-$mails and work on a settlement$)?[9\mathrm{.}6]$