This problem involves queueing theory, specifically the M/M/1 queue model. In this model:

• The arrival rate follows a Poisson distribution.
• The service time follows an exponential distribution.
• There is a single server (the telephone booth in this case).

Given Data:

• Arrival rate $\lambda = 15$ callers per hour.
• Service time is 15 minutes per call, which is equivalent to 0.25 hours.
  • Therefore, the service rate $\mu = \frac{1}{\text{service time}} = \frac{1}{0.25} = 4$ calls per hour.

Step 1: Expected number of callers in the booth

The expected number of callers (average number in the system) can be calculated as: $L = \frac{\lambda}{\mu - \lambda}$ Where:

• $L$ is the expected number of callers in the system.
• $\lambda = 15$ callers/hour.
• $\mu = 4$ calls/hour.

However, in this case, $\lambda = 15$ is larger than $\mu = 4$, which means the system is unstable. This implies the number of callers in the system would grow without bound, as more callers arrive than can be served. Therefore, under the current setup, the booth will always be congested and an equilibrium cannot be reached.

Step 2: Proportion of the time the booth is idle

The proportion of the time the booth is expected to be idle is given by: $P_0 = 1 - \frac{\lambda}{\mu}$ However, since $\lambda > \mu$, this also means that the booth will never be idle as the traffic intensity exceeds 1.

Thus:

• (i) Expected number of callers in the booth at any time: The system cannot reach equilibrium because the arrival rate exceeds the service rate.
• (ii) Proportion of time the booth is expected to be idle: The booth is never idle, as it is continuously busy under the current parameters.

Would you like more details or clarifications?

Here are five related questions for further exploration:

1. How does the M/M/1 model change if multiple booths are added?
2. What happens if the arrival rate is reduced to match the service rate?
3. How does changing the service rate impact the overall system?
4. What is the probability of having more than one caller waiting in a booth with a lower arrival rate?
5. How does queue length vary with different service distributions?

Tip: When $\lambda > \mu$, queue systems are unstable, and adjustments like adding servers or reducing arrival rates are necessary to prevent overload.