Question

A simple telephone system consists of two end offices and a single toll office to which each end office is connected by a 1-MHz full-duplex trunk. The average telephone is used to make four calls per 8-hour workday. The mean call duration is \(6 \mathrm{~min}\). Ten percent of the calls are long distance (i.e., pass through the toll office). What is the maximum number of telephones an end office can support? (Assume \(4 \mathrm{kHz}\) per circuit.) Explain why a telephone company may decide to support a lesser number of telephones than this maximum number at the end office.

Short answer

Maximum theoretical support is 50,000 telephones, but practical factors reduce this.

Step-by-step solution

Understand the Problem

We are to determine the maximum number of telephones that can be supported by an end office for a given full-duplex trunk with a bandwidth of 1 MHz. Each telephone conducts calls with a mean duration and frequency, and a certain percentage of these are long-distance, requiring use of the toll office trunk.

Calculate Frequency Utilization per Telephone

Each telephone makes 4 calls per 8-hour day. The mean call duration is 6 minutes. Therefore, the total call duration per telephone per day is \(4 \times 6 = 24\) minutes (or 0.4 hours). The fraction of the day a telephone is in use is \(\frac{0.4}{8} = 0.05\).

Determine Long Distance Calls Usage

Since 10% of the calls are long distance, the usage for long distance calls per telephone is \(0.05 \times 0.10 = 0.005\) or 0.5% of the time in a day.

Determine Trunk Capacity in Terms of Phone Circuits

The trunk between both end offices is 1 MHz, and each telephone call uses \(4 \text{kHz}\). Hence, the trunk can support \(\frac{1\, \text{MHz}}{4\, \text{kHz}} = 250\) call circuits simultaneously.

Calculate Total Potential Telephone Support

Given the total trunk utilization for long-distance calls is 0.5%, the effective number of circuits utilized long-distance is \(250 \times 0.005 = 1.25\). Therefore, non-integers suggest handling a single long-distance call per time unit when fully used. The total number of telephones managed by each end office can be \(250 / 0.005 = 50,000\), requiring revisiting an average of offered processing based on total circuit use.

Discuss Practical Limitations

While the theoretical maximum is 50,000 telephones, in reality, companies opt for lower numbers due to peak call times, maintenance, call quality concerns, and unanticipated system demands.

Key concepts

Bandwidth Allocation

The concept of bandwidth allocation is crucial in the design of telephone networks. Bandwidth refers to the range of frequencies available for transmitting data. In this exercise, we have a total of 1 MHz bandwidth available on the full-duplex trunk connecting the end offices to the toll office. This bandwidth must be shared among all active calls.
Each call requires a certain portion of this total bandwidth. For our telephone system, each circuit uses 4 kHz. Therefore, with 1 MHz available, the trunk can support up to 250 phone circuits simultaneously ( \( \frac{1000 \text{kHz}}{4 \text{kHz}} = 250 \) ).
Bandwidth allocation ensures that multiple calls can occur simultaneously without exceeding the available capacity, preventing data loss or interruption.

Call Duration

Call duration impacts network capacity and planning. It is the amount of time a call occupies a network circuit, affecting how many calls can be handled concurrently. In the given exercise, the average call duration is 6 minutes.
On a typical 8-hour workday, each telephone will make 4 calls, resulting in a cumulative call duration of 24 minutes per telephone per day ( \( 4 \times 6 = 24 \) minutes). This converts to 0.4 hours.
The fraction of time a telephone vehicle is in operation translates to how occupied the communication circuits become. Understanding and calculating the impact of call duration is vital in network design to ensure efficient use of resources.

Trunk Capacity

Trunk capacity is an essential element in telephone network design. It determines the number of simultaneous calls a trunk can handle. Here, we have a trunk with 1 MHz available to accommodate calls from multiple telephones.
Given that each call consumes a 4 kHz bandwidth, the trunk can support up to 250 calls at the same time. This capacity is critical for determining how many telephones can effectively be connected without compromising service quality. The trunk's ability to manage calls directly affects service efficiency, influencing how many users can participate in calls simultaneously, especially during peak hours when call volume is at its highest.

Long-Distance Calls

Long-distance calls involve higher costs and more extensive network involvement compared to local calls. They must pass through the toll office, utilizing additional resources and potentially affecting the overall capacity.
In this exercise, 10% of all calls are long-distance. This affects trunk utilization as these calls consume bandwidth differently, requiring more coordination and toll fees.
Understanding the proportion and demand of long-distance calls helps in network planning to ensure adequate allocation of resources. Calculating usage shows that long-distance calls utilize the trunk's resources by 0.5%, impacting how available capacity is divided among all potential calls.