Question

Oxnard Casualty wants to ensure that their e-mail server has 99.98 percent reliability. They will use several independent servers in parallel, each of which is 95 percent reliable. What is the smallest number of independent file servers that will accomplish the goal?

Answer 1

The smallest number of servers required is 3 servers

Step-by-step explanation:

Given

`Reliability = 99.98\%`

`Individual Servers = 95\%`

Required

Minimum number of servers needed

Let p represent the probability that a server is reliable and the probability that it wont be reliable be represented with q

Such that

`p = 95\%\\`

It should be noted that probabilities always add up to 1;

So,

`p + q = 1`'

Subtract p from both sides

`p - p + q = 1 - p`

`q = 1 - p`

Substitute
`p = 95\%\\`

`q = 1 - 95\%`

Convert % to fraction

`q = 1 - (95)/(100)`

Convert fraction to decimal

`q = 1 - 0.95`

`q = 0.05`

------------------------------------------------------------------------------------------------

To get an expression for one server

The probabilities of 1 servers having 99.98% reliability is as follows;

`p = 99.98\%`

Recall that probabilities always add up to 1;

So,

`p + q = 1`

Subtract q from both sides

`p + q - q = 1 - q`

`p = 1 - q`

So,

`p = 1 - q = 99.98\%`

`1 - q = 99.98\%`

Let the number of servers be represented with n

The above expression becomes

`1 - q^n = 99.98\%`

Convert percent to fraction

`1 - q^n = (9998)/(10000)`

Convert fraction to decimal

`1 - q^n = 0.9998`

Add
`q^n` to both sides

`1 - q^n + q^n= 0.9998 + q^n`

`1 = 0.9998 + q^n`

Subtract 0.9998 from both sides

`1 - 0.9998 = 0.9998 - 0.9998 + q^n`

`1 - 0.9998 = q^n`

`0.0002 = q^n`

Recall that
`q = 0.05`

So, the expression becomes

`0.0002 = 0.05^n`

Take Log of both sides

`Log(0.0002) = Log(0.05^n)`

From laws of logarithm
`Loga^b = bLoga`

So,

`Log(0.0002) = Log(0.05^n)` becomes

`Log(0.0002) = nLog(0.05)`

Divide both sides by
`Log0.05`

`(Log(0.0002))/(Log(0.05)) = (nLog(0.05))/(Log(0.05))`

`(Log(0.0002))/(Log(0.05)) = n`

`n = (Log(0.0002))/(Log(0.05))`

`n = (-3.69897000434)/(-1.30102999566)`

`n = 2.84310893421`

`n = 3 (Approximated)`

Hence, the smallest number of servers required is 3 servers