Understanding Barber Shop Queueing Dynamics

This question asks us to calculate the expected waiting time for a customer at a barber shop. The key information provided is that customers arrive following a Poisson process, and we know the average time between customer arrivals and the average time a customer spends being served (in the barber's chair).

Identifying the Queueing Model

This scenario is a classic example that can be analyzed using queueing theory. Specifically, it fits the M/M/1 queueing model:

• The first 'M' represents the Poisson arrival process.
• The second 'M' represents the assumption of exponential service times (this is a standard assumption in basic M/M/1 models, inferred from the average service time).
• The '1' signifies that there is a single server, which is the barber.

Key Parameters from the Question

Let's define the parameters based on the information given:

• Mean Inter-arrival Time: 20 minutes. This is the average time between consecutive customer arrivals.
• Average Service Time: 15 minutes. This is the average time the barber spends with one customer.

Calculating Arrival Rate ($\lambda$) and Service Rate ($\mu$)

To use the queueing formulas, we need the arrival rate ($\lambda$) and service rate ($\mu$) in terms of customers per unit of time (we'll use minutes).

• Arrival Rate ($\lambda$): If the average time between arrivals is 20 minutes, then the rate of arrivals is the reciprocal:

  $$ \lambda = \frac{1}{\text{Mean Inter-arrival Time}} = \frac{1}{20} \text{ customers per minute} $$

• Service Rate ($\mu$): If the average time to serve a customer is 15 minutes, then the rate at which the barber can serve customers is:

  $$ \mu = \frac{1}{\text{Average Service Time}} = \frac{1}{15} \text{ customers per minute} $$

Calculating Server Utilization ($\rho$)

Server utilization, denoted by $\rho$, represents the fraction of time the server (the barber) is busy serving customers. It is calculated as the ratio of the arrival rate to the service rate:

$$ \rho = \frac{\lambda}{\mu} $$

Substituting the values:

$$ \rho = \frac{1/20}{1/15} = \frac{15}{20} = \frac{3}{4} = 0.75 $$

This result indicates that the barber is occupied 75% of the time.

Determining Expected Wait Time ($W_q$)

The question asks for the expected time a customer waits *for their turn*, which means the time spent waiting in the queue before service begins. For an M/M/1 queue, this is denoted as $W_q$. The formula is:

$$ W_q = \frac{\lambda}{\mu(\mu - \lambda)} $$

Let's calculate the components needed for the formula:

1. Calculate $(\mu - \lambda)$: This represents the net service capacity.

   $$ \mu - \lambda = \frac{1}{15} - \frac{1}{20} $$

   To subtract these fractions, we find a common denominator, which is 60:

   $$ \mu - \lambda = \frac{4}{60} - \frac{3}{60} = \frac{1}{60} \text{ customers per minute} $$

2. Calculate the denominator $\mu(\mu - \lambda)$:

   $$ \mu(\mu - \lambda) = \left(\frac{1}{15}\right) \times \left(\frac{1}{60}\right) = \frac{1}{900} $$

   The units here are effectively (customers/min) * (customers/min) = customers²/min², but in the context of the formula, it relates to the rate at which queue length decreases.

3. Calculate $W_q$: Now, substitute the values into the formula:

   $$ W_q = \frac{\lambda}{\mu(\mu - \lambda)} = \frac{1/20}{1/900} $$

   Dividing by a fraction is the same as multiplying by its reciprocal:

   $$ W_q = \frac{1}{20} \times 900 = \frac{900}{20} = 45 \text{ minutes} $$

Alternatively, we can use the formula involving utilization ($\rho$):

$$ W_q = \frac{\rho}{\mu(1 - \rho)} $$

1. Calculate $(1 - \rho)$:

   $$ 1 - \rho = 1 - 0.75 = 0.25 $$

2. Calculate the denominator $\mu(1 - \rho)$:

   $$ \mu(1 - \rho) = \left(\frac{1}{15}\right) \times (0.25) = \frac{1}{15} \times \frac{1}{4} = \frac{1}{60} $$

3. Calculate $W_q$:

   $$ W_q = \frac{\rho}{\mu(1 - \rho)} = \frac{0.75}{1/60} = 0.75 \times 60 = \frac{3}{4} \times 60 = 45 \text{ minutes} $$

Final Answer Explanation

Both methods confirm that the expected waiting time for a customer before they start receiving service at the barber shop is 45 minutes.