Math Problem Statement

Customers send emails to a help desk of an online retailer ever two minutes (on average), the standard deviation of the inter arrival time is also 2 min. There are three employees answering emails and it takes an average of 4 min to write a response. The standard deviation of processing time is 2 min. How long does each customer wait to be served

Solution

To calculate the average wait time for each customer at the help desk, we can model this situation using queueing theory. Specifically, we'll apply the M/M/c queue model where:

  • M indicates that arrivals follow a Poisson process (which is valid given the exponential inter-arrival times).
  • M indicates that service times are exponentially distributed.
  • c indicates the number of servers (employees), which is 3 in this case.

Parameters:

  1. Arrival Rate (λ\lambda):

    • Average inter-arrival time = 2 minutes, so: λ=1Average Inter-arrival Time=12 customers/minute=0.5 customers/minute\lambda = \frac{1}{\text{Average Inter-arrival Time}} = \frac{1}{2} \text{ customers/minute} = 0.5 \text{ customers/minute}
  2. Service Rate (μ\mu):

    • Average service time = 4 minutes, so: μ=1Average Service Time=14 customers/minute=0.25 customers/minute\mu = \frac{1}{\text{Average Service Time}} = \frac{1}{4} \text{ customers/minute} = 0.25 \text{ customers/minute}
  3. Number of Servers (cc):

    • c=3c = 3

Traffic Intensity (ρ\rho):

ρ=λcμ=0.530.25=0.50.75=230.67\rho = \frac{\lambda}{c \cdot \mu} = \frac{0.5}{3 \cdot 0.25} = \frac{0.5}{0.75} = \frac{2}{3} \approx 0.67

Average Number of Customers in the System (LL):

For an M/M/c queue, the average number of customers in the system can be calculated using: L=λμ11ρL = \frac{\lambda}{\mu} \cdot \frac{1}{1 - \rho} However, this is the formula for c=1c=1. For c>1c > 1, we use a different method involving Erlang's B formula and some approximations.

Average Waiting Time in the System (WW):

The average time a customer spends in the system (waiting + service) can be approximated by: W=LλW = \frac{L}{\lambda}

Given the complexity of calculating LL for multiple servers, a practical approach is to use numerical or simulation methods for precise values. However, we can estimate WW using the following approximation:

Average Wait Time in Queue (WqW_q):

The average time a customer waits in the queue can be approximated by: Wq=ρ2μ(1ρ)W_q = \frac{\rho^2}{\mu (1 - \rho)}

Calculating WqW_q:

  1. ρ2=(23)2=49\rho^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}
  2. 1ρ=123=131 - \rho = 1 - \frac{2}{3} = \frac{1}{3}
  3. μ=0.25\mu = 0.25

Thus: Wq=490.2513=49112=4912=4895.33 minutesW_q = \frac{\frac{4}{9}}{0.25 \cdot \frac{1}{3}} = \frac{\frac{4}{9}}{\frac{1}{12}} = \frac{4}{9} \cdot 12 = \frac{48}{9} \approx 5.33 \text{ minutes}

Conclusion:

Each customer waits, on average, approximately 5.33 minutes to be served.

Questions for Further Exploration:

  1. How would the average wait time change if the number of employees increased?
  2. What happens to the average wait time if the arrival rate of emails increases?
  3. How can we calculate the probability that all employees are busy at the same time?
  4. What impact does a decrease in the average service time have on customer satisfaction?
  5. Can we derive a formula for the average wait time with different arrival and service rates?

Tip:

To optimize customer wait times, consider implementing a ticketing system or increasing the number of employees during peak hours.

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Math Problem Analysis

Mathematical Concepts

Queueing Theory
Poisson Process
Exponential Distribution

Formulas

\( \lambda = \frac{1}{\text{Average Inter-arrival Time}} \)
\( \mu = \frac{1}{\text{Average Service Time}} \)
\( \rho = \frac{\lambda}{c \cdot \mu} \)
\( W_q = \frac{\rho^2}{\mu (1 - \rho)} \)

Theorems

Erlang's B Formula
Little's Law

Suitable Grade Level

Grades 11-12