Question

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-
[email protected] 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.
b. How many e-mails will a lawyer have received at the end of a 10-hour day?
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 on 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)? To increase the responsiveness of the firm, the board of My-law.com proposes a new operating policy. Under the new policy, the response would be highly standardized, reducing the standard deviation for writing the response e-mail to 0.5 minute. The average writing time would remain unchanged.
d. How would the amount of time a lawyer can dedicate to the search for large settlement cases change with this new operating policy?
e. How would the average time a customer has to wait for the response to his/her e-mail change? Note: This includes the time until the lawyer starts writing the e-mail and the actual writing time.

Answer 1

Answer / Explanation:

Before we start solving this question, we should take note of the parameters below:

λ = 10 emails per hour → inter arrival time (a) = 10 emails / 60 minutes → a = 6 minute,

Therefore, CVₐ = 1

Where processing time (p) = 5 minute,

Therefore, CVp = 4 minute / 5 minute = 0.8 minute

Hence,

(a) To calculate the average processing time a customer has to wait (Ts), we have:

Waiting Time = Processing X [ (5/6) / ( 1 - 5/6)] X [ 1² + 0.8²) /2 ]

Solving the above equation further, we arrive at 20.5 Minutes.

Therefore,

Average processing time = Waiting time + processing time

                                            = 5 minutes  + 20.5 minutes

                                             = 25.5 minutes.

(b) To calculate how many emails a lawyer will receive at the end of 10 hours day, we have:

Emails per day = 10 emails per hour X 10 hours

                           = 100 Emails per day.

(c) Idle Time = Probability that there are no customer in the system = unknown.

Therefore, the Idle time = ( 1 - amount of utilization) x 10 hours

Calculating further, we have 1.66 hours.

To increase the responsiveness of the firm, the board of My-law.com proposes a new operating policy. Under the new policy, the response would be highly standardized, reducing the standard deviation for writing the response e-mail to 0.5 minute. The average writing time would remain unchanged.

(d) Hence, to know the amount of time a  lawyer can dedicate to the search for large settlement cases change with this new operating policy,

The average amount of time would not change since the utilization is not dependent on the variance or standard deviation but depends on the average processing and inter arrival time.

(e) How would the average time a customer has to wait for the response to his/her e-mail change? Note: This includes the time until the lawyer starts writing the e-mail and the actual writing time.

To do this, we recall out precious gotten parameter:

Hence, Processing Time (p) = 5 minutes

CVp = 0.5 minutes / 5 minute = 0.1

Waiting time = 5 minute X [ 5/6 (1 - 5/6)] X [ (1² + 0.1² ) / 2 ] = 12.63 minute

Total Response time = waiting time + processing time = 17.63 minutes.