in a public telephone booth the arrivals are on an average of 15 per hour.A call on the average takes 15 minutes. If there is just one phone,find (i)expected number of callers in the booth at any time (ii)the proportion of the time the booth is expected to be idle?

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.

In a public telephone booth, the arrivals are on an average of 15 per hour. A call on average takes 15 minutes. If there is just one phone, find (i) the expected number of callers in the booth at any time (ii) the proportion of the time the booth is expected to be idle?

Explore how queueing theory applies to a single telephone booth with an arrival rate of 15 callers per hour and a service rate of 4 calls per hour. Learn about the expected number of callers and the proportion of idle time using the M/M/1 queue model.

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