Question

We are using a private branch exchange (PBX), a telephone switch, to connect our office building to the telephone company's central office. There are 50 telephones in the building, and each telephone is used on average 5% of the workday to make calls to/from phones outside the building. The PBX enables us to connect telephones inside the building without having to use the services of the telephone company. Telephone calls to/from phones outside the building are connected by the central office. We want to determine the required capacity of the line between the PBX and the telephone company's central office, expressed as a multiple N of the bandwidth required for each phone call. The probability that a call will be blocked due to not enough capacity between the PBX and the central office is B = [(y^N)/N!] / [1 + y + (y^2)/2! + ... + (y^N)/N!], where y = the average number of active phone calls at any particular time to/from phones outside the building, "^" means "raised to the power" and "!" means factorial. (See information about the Erlang B formulae for background, if desired.) What is the minimum size for N that ensures that B < 2%?

Answer 1

There are 50 telephones in the building, and each telephone is used on average 5% of the workday to make calls to/from phones outside the building. So the average number of calls in a particular time is . The probability

We have to find the least such that .

We cannot use any calculus methods for finding the required minimum integer value.

I used Maple to find the minimum value of  .

For ,   and , and we see is decreasing as increases. So the minimum value of .