In order to allow customers to interact with the firm’s lawyers, customers are encouraged to send e-mails to m..r@My-law.com. From there, the incoming e-mails are distributed to one of the three lawyers who are currently “on call”. Given the broad skills of the lawyers, each lawyer can respond to each incoming request.
E-mails arrive from 8am to 6pm at a rate of 10 e-mails per hour. Interarrival times are exponentially distributed. It takes the lawyer, on average, 15 minutes to write the response e-mail. The standard deviation of this is 20 minutes.
a) (6 points) What is the average time a customer has to wait for the response to his/her e-mail?
| a(avg. inter-arrival time) | p (avg. service time) | m (number of servers) | u (avg. utilization) | CVa(coef. of var.- arrival) | CVp(coef. of var.- service) |
To calculate the average time a customer has to wait for a response to their email in the context of My-law.com, we need to apply principles from queuing theory, specifically the M/M/1 queuing model. In this model:
λ (lambda) represents the arrival rate of customers.
μ (mu) represents the service rate of the system (in this case, the time it takes for a lawyer to respond to an email).
ρ (rho) represents the utilization of the system.
W represents the average time a customer has to wait.
L represents the average number of customers in the system.
Let’s break down the given information and calculate these parameters:
λ (Arrival Rate): E-mails arrive at a rate of 10 e-mails per hour. Since the service time is given in minutes, we need to convert this to emails per minute. Thus, λ = 10 e-mails per hour / 60 minutes per hour = 1/6 emails per minute.
μ (Service Rate): The average service time to write a response email is 15 minutes. We’ll convert this to emails per minute as well: μ = 1 email / 15 minutes = 1/15 emails per minute.
ρ (Utilization): Utilization represents the fraction of time the server is busy. It is calculated as ρ = λ / μ.ρ = (1/6) / (1/15) = 15/6 = 2.5.
W (Average Waiting Time): We can use Little’s Law to calculate the average waiting time:W = L / λ
Here, L represents the average number of customers in the system, which can be calculated as:
L = ρ / (1 – ρ)
L = (2.5) / (1 – 2.5) = -2.5 / -1.5 = 5/3 customers
Now, plug L and λ into the waiting time formula:
W = (5/3) / (1/6) = (5/3) * 6 = 10 minutes per email.
So, the average time a customer has to wait for a response to their email is 10 minutes.
In summary, My-law.com experiences an average response time of 10 minutes per email, assuming an arrival rate of 10 emails per hour and a service rate of 15 minutes per email. This information is crucial for managing customer expectations and optimizing the efficiency of the legal service startup’s operations.
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