The calculation of how many servers can be installed in a data centre was made easier by the introduction of a standardised floor tile, which is a square that is 2 ft by 2 ft, occupying 4 square feet of floor area.

Two facing rows of servers can either fit in a 7-tile pitch (if a 3 ft by 2 ft rack is used) or an 8-tile pitch (if a 4 ft by 2 ft rack is used). In either case, there must be one unoccupied tile width around the outside of the racks and two tiles between the rows.

Here is an illustration of the ground plan for a cluster of servers in a 7-tile pitch and for a cluster of servers in an 8-tile pitch. In each case, there are four racks of servers in each row.

Figure 1 7-tile pitch (for 3 ft by 2 ft racks in rows of four)

Figure 2 8-tile pitch (for 4 ft by 2 ft servers in rows of four)

a is answered!

• a.The ratio of occupied tiles to the total number of tiles in the 8-tile pitch shown in Figure 2 is : 1/3

  • i.Find the equivalent ratio for the cluster of racks shown in Figure 1.

  • ii.The general expression for the ratio of occupied tiles to the total tiles in an 8-tile pitch, if each row contains n racks is:

    n/2(n+2)

    Find the general expression for the ratio of occupied tiles to the total tiles in a 7-tile cluster if there are n racks in each row check that when n = 4, your expression gives the same answer as you wrote down in (a)(i). Answer N/2(N+3)

  (3 marks)

• b.

  • i.Write the ratio of the general expression you were given for the 8-tile cluster to the one you found for the 7-tile cluster in (a)(ii) and simplify your result. Show your working carefully.

  • ii.Explain what your result from (b)(i) means in terms of how space efficient an 8-tile pitch is compared to a 7-tile pitch.

  (5 marks)

• c.

  • i.The machine room of a data centre has an area, A, in square feet, that will hold an exact number of 8-tile clusters, each with a row size, n. Find a formula for the number of racks, N, the machine room can accommodate in terms of A and n. Describe your reasoning at each step.

  • ii.Use the formula you developed in (c)(i) to find the number of racks in a rectangular machine room if it has a length of 1200 ft and a width of 800 ft.

  (7 marks)