A set of telephone lines is to be installed so as to connect...

A set of telephone lines is to be installed so as to connect town $A$ to town $B$. The town A has 2000 telephones. If esch of tbe telephone users of $A$ were to be guarabteed instsnt secess to make calls to $B, 2000$ telephone lines would be needed. This would be rather extravagant. Suppose that during the busicst hour of the day each subseriber in $A$ requires, on the average, a telephone connection to $B$ for two minutes, and that theae telephone cslis are made st random. Find the rrinimum nunber If of telephone lines to $\boldsymbol{B}$ which must be installed so that at most only 1 jereent of the callers of town $A$ will fail to hsve immediate becess to a telephone line to $B$. (Suggestion: spproximate the distribution by A Gausian distribution to facilitste the arithrpetic.)

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) establishes that the sum (or average) of a large number of independent, identically distributed random variables tends toward a normal distribution, regardless of the original distribution. This theorem underpins the Gaussian approximation, making it a powerful tool for analyzing telephone traffic and ensuring that capacity planning meets specified quality of service targets.

Gaussian (Normal) Approximation

The Gaussian approximation is a statistical method used to approximate the distribution of a sum of many independent random variables, even if the original distribution is not normal. In telecommunications, when dealing with a large number of independent call attempts, this approximation simplifies the analysis of call blocking probabilities and system performance.

Telecommunications Traffic Engineering

This concept deals with the study and design of systems that handle the transmission of data and voice, particularly focusing on how resources like telephone lines are allocated relative to user demand. It involves analyzing user behavior patterns, call durations, and overall system load to ensure that adequate capacity is provided without excessive redundancy.

Blocking Probability

Blocking probability refers to the likelihood that a user will be denied service because all available communication channels are in use. In telecommunications, it is essential to limit this probability to maintain a certain level of service quality, ensuring that only a small percentage of call attempts fail due to unavailability of a line.

Random Call Arrival Processes

This concept concerns the understanding that telephone call attempts occur at random times rather than in a predictable pattern. The randomness in call arrivals requires statistical models to predict the number of simultaneous calls and to design systems that effectively handle peak loads, balancing between overcapacity and service denial.

Transcript

00:03 Okay, in this problem, we have a cell phone company that is trying to make sure that they have coverage.

00:14 So what they've done is they have blocked off an area that's 10 feet, 10 miles by 10 miles to give them 100 square mile area of service.

00:27 And they've decided to place four towers.

00:29 And i've marked each of the centers where they want the towers.

00:33 So the first thing we want to do is write the equation of each of those towers, and each of those towers have a radius of 3 .5 miles.

00:52 So the four equations would be found by using the formula for a circle in standard form.

01:07 Okay, so i'm going to do these.

01:09 Let's color code them.

01:11 The first one i'll do blue.

01:13 And that circle would be in the form x minus 2 .5 squared plus y that's a y sorry y minus 2 .5 squared equals and if they all have a radius of 3 .5 then we know that 3 .5 squared is equal to 12 .25 so the radius would be 12 .25.

01:45 That's the equation of the circle that i plotted there with a blue center.

01:51 And then the one with the green center, its equation would be x minus 7 .5 squared plus y plus 2 .5.

02:02 I'm sorry, not plus, y minus 2 .5 squared equals 12 .25.

02:12 And then let's do a yellow circle.

02:14 And that yellow circle, its equation would be x minus 7 .5 squared plus y minus 7 .5 squared equals 12 .25...

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