5. Open-ended quiz

Checking criteria: (1) application of the course materials discussed after midterm, (2) concreteness, (3) originality, (4) [no] jumps in the protocol procedure, (5) possibility of performance calculation of the designed scheme, (6) meeting the design requirements

 

Q. It is early 20th century. There is a large city with population P where the citizens are very willing to subscribe to the telephone service if the telephone has no call blocking possibility. As a telephone engineer, you are given X crosspoints (X≪P) by the company to design a switching hardware for the city. You could make either a single huge crossbar switch or a multi-stage switch. What is the maximum number of telephone subscribers that you can get by making a wise design decision?

 

 

The single stage crossbar switch can eliminate the blocking phenomenon, but the cross point waste is too much. Therefore, it is better to select a multi-stage switch. 2kN + k(N/n)^2 Since it has the number of cross points, it can be seen that less than N^ is used. However, due to a kind of bottleneck, blocking occurs even with different outputs. To lower this, you can make it conform to the Clos criteria principle. We learned in class that k >= 2n-1 is the most well-known condition and is a realistically strong condition. In order to fix N and minimize the number of intersections, we can obtain the value of n that minimizes the number of intersections by differentiating n in the equation (using the first derivative theorem)’

 

Calculation gives the number of intersections >= 4N((2N)^(1/2)-1). Now, possible N (number of subscribers, number of lines) can be calculated as follows through the inverse function.

In fact, if you look at the graph, the number of lines does not increase proportionally, but increases slowly even when a lot of crosspoints are assigned in a log-like manner. Therefore, unless the company is stupid and assigns me a lot of cross points, there is a limit to joining many citizens with the space-division switch. At this time, it is impossible to use the time-division switch that was not learned in class because of the condition of the early 20th century (digital technology not developed). A conceivable method at this time is to consider the introduction of the communication cost differentiation policy for each communication quality of the current telecommunication companies. In the past, an important institution installed a separate switchboard so that it does not go down in an emergency even if other lines are down, so for users who use a lot of telephones, a separate switchboard is added to ensure that the blocking phenomenon is removed.

 

In fact, it is statistically important to set (n, k) so that blocking does not occur, so it is up to the competitor to decide how much blocking the user will tolerate in this problem.