Question: My-law.com is a recent start-up serving customers in search of legal services in two areas property law and family law. New customers are encouraged to send emails to the inbox my-lawyer@My-law.com and put either property law or family law in the subject line. There are always two lawyers on call, one property lawyer and the other family lawyer, available to answer emails. Whenever the property lawyer completes a conversation and becomes free, s/he picks up the oldest email with the subject property law in the inbox. The family lawyer does the same thing with the family law emails.

E-mails with property in subject line arrive at an average rate of 10 e-mails per hours. E-mails with family in the subject line also arrive at an average rate of 10 e-mails per hours. It takes one lawyer (property or family), on average, 5 minutes to write the response e-mail. Assume Ca=1 and Cs=1.

a. What is the average total time (lead time) it takes a property law customer to get a response to his/her e-mail? (Treat two flows (property and family) separately. The answer will be the same for each flow.)

b. Continuing from the previous question, suppose the firm realizes that initial inquiries for both flows (property and family) were simple enough that either lawyer could respond to any email in the same 5 minutes. To implement this, any lawyer who becomes available picks up the oldest email from the inbox irrespective of the subject line (property or family). Due to this idea, what is the average total time it takes a customer to get a response? Show all inputs.

c. Let us ignore the new idea above and go back to the original setup of property lawyer dealing only with property emails and family lawyer only with family emails. Focusing only on property law email traffic, there is an increase in demand. If the arrival rate goes up to 46 per hour and the firm hires 3 more resources (to make a total of 4 resources), what is the average wait time? Show all inputs.

d. Continuing from the previous question, further analysis shows that new traffic exhibits less variability. New measurements of the standard deviation of the property email interarrival time shows it to be 0.0155 hrs. What is the average wait time? Note that the mean of interarrival time is 1/arrival rate. Show all inputs.

e. Continuing from the previous question, the firm can increase the service rate of each of the four resources by 0.5 per hour by purchasing access to an online database at the total cost of $100 per hour. The customer waiting cost is estimated to be $25 per hour per customer and this cost is charged only on wait times. Should the firm purchase this access? Please show all work.

f. Suppose pooling is implemented at the proposal support stage and now the firm is considering if it should give priority to New Sales or Renewals. Please provide a bullet-point list of factors you will consider in making this decision and for each factor, indicate which type (New or Renewal) does it support prioritizing.