$($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.

$9\mathrm{.}13$

Practice

Problems

Variability and Its Impact on Process Performance: Waiting Time Problems $201$

In order to allow customers to interact with the firm$$s lawyers, customers are encouraged

to send e$-$mails to my$-$lawyer@My$-$law.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$)?[9\mathrm{.}6]\backslash$