This question asks us to calculate the average queue length in a waiting line system. We are given the average time between customer arrivals and the average time it takes to serve a customer. This is a classic problem in queuing theory, often modeled as an M/M/1 queue assuming Poisson arrivals and exponential service times, and a single server.

Understanding Queuing System Components

In any queuing system, several key components are involved:

• Arrival Process: How customers arrive at the system. This is described by the arrival rate ($\lambda$).
• Service Process: How customers are served. This is described by the service rate ($\mu$).
• Number of Servers: How many service points are available. In this case, it's one (receptionist).
• Queue Capacity: The maximum number of customers allowed in the queue. Assuming infinite capacity unless specified.
• Queue Discipline: How customers are selected from the queue (e.g., First-Come, First-Served or FCFS). Assuming FCFS unless specified.

Given the information, we can calculate the arrival rate and service rate.

Calculating Arrival Rate ($\lambda$)

The average interval between customer arrivals is 10 minutes. This is the average inter-arrival time.

The arrival rate ($\lambda$) is the average number of customers arriving per unit of time. It is the reciprocal of the average inter-arrival time.

Average inter-arrival time $= 10$ minutes/customer

$\lambda = \frac{1}{\text{Average inter-arrival time}} = \frac{1}{10}$ customers/minute

So, the arrival rate is $1/10$ customers per minute.

Calculating Service Rate ($\mu$)

The receptionist takes an average of 6 minutes for one customer. This is the average service time.

The service rate ($\mu$) is the average number of customers served per unit of time. It is the reciprocal of the average service time.

Average service time $= 6$ minutes/customer

$\mu = \frac{1}{\text{Average service time}} = \frac{1}{6}$ customers/minute

So, the service rate is $1/6$ customers per minute.

Determining Traffic Intensity ($\rho$)

Traffic intensity, also known as utilization factor, is the ratio of the arrival rate to the service rate ($\rho = \lambda / \mu$). It represents the proportion of time the server is busy.

$\rho = \frac{\lambda}{\mu} = \frac{1/10 \text{ customers/minute}}{1/6 \text{ customers/minute}} = \frac{1}{10} \times 6 = \frac{6}{10} = \frac{3}{5}$

The traffic intensity is $3/5$. Since $\rho < 1$, the system is stable, and a steady state exists.

Calculating Average Queue Length ($L_q$)

For an M/M/1 queuing system, the formula for the average queue length ($L_q$) is:

$$L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}$$

Alternatively, it can be expressed in terms of traffic intensity ($\rho$):

$$L_q = \frac{\rho^2}{1 - \rho}$$

Let's use the formula with $\rho$ as it's often simpler after calculating $\rho$.

We have $\rho = 3/5$.

$$L_q = \frac{(3/5)^2}{1 - 3/5} = \frac{9/25}{5/5 - 3/5} = \frac{9/25}{2/5}$$

$$L_q = \frac{9}{25} \div \frac{2}{5} = \frac{9}{25} \times \frac{5}{2} = \frac{9 \times 5}{25 \times 2} = \frac{9 \times 1}{5 \times 2} = \frac{9}{10}$$

The average queue length is $9/10$ customers.

Let's verify using the formula with $\lambda$ and $\mu$ directly:

$\lambda = 1/10$, $\mu = 1/6$

$$L_q = \frac{(1/10)^2}{(1/6)(1/6 - 1/10)}$$

First, calculate the term in the parenthesis:

$$\mu - \lambda = \frac{1}{6} - \frac{1}{10} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15}$$

Now substitute back into the $L_q$ formula:

$$L_q = \frac{(1/10)^2}{(1/6)(1/15)} = \frac{1/100}{1/90}$$

$$L_q = \frac{1}{100} \div \frac{1}{90} = \frac{1}{100} \times 90 = \frac{90}{100} = \frac{9}{10}$$

Both formulas give the same result. The average queue length is $9/10$.

Summary of Calculations

Based on the calculations, the average queue length is $9/10$.

Queuing Analysis Revision

Additional Queuing Information

Queuing theory is a branch of mathematics that studies waiting lines, or queues. It helps analyze various systems where customers or jobs wait for service, such as call centers, banks, manufacturing lines, and computer networks. The goal of queuing analysis is often to balance the cost of providing service with the cost of waiting. Different queuing models (like M/M/c, M/G/1, etc.) exist to handle variations in arrival processes, service time distributions, and the number of servers.

Key performance measures in queuing systems include average queue length, average waiting time, server utilization, and the probability of waiting. These measures help in designing and managing systems efficiently to meet service level targets while controlling costs.