Magnetic Disk in Computer Architecture
Before you go through this article, make sure that you have gone through the previous article on Magnetic Disk.
We have discussed
 Magnetic disk is divided into platters which are further divided into tracks and sectors.
 Read / write head is a mechanical arm that is used to write to and read from the disk.
 Various important formulas.
In this article, we will discuss practice problems based on magnetic disk.
PRACTICE PROBLEMS BASED ON MAGNETIC DISK
Problem01:
Consider a disk pack with the following specifications 16 surfaces, 128 tracks per surface, 256 sectors per track and 512 bytes per sector.
Answer the following questions
 What is the capacity of disk pack?
 What is the number of bits required to address the sector?
 If the format overhead is 32 bytes per sector, what is the formatted disk space?
 If the format overhead is 64 bytes per sector, how much amount of memory is lost due to formatting?
 If the diameter of innermost track is 21 cm, what is the maximum recording density?
 If the diameter of innermost track is 21 cm with 2 KB/cm, what is the capacity of one track?
 If the disk is rotating at 3600 RPM, what is the data transfer rate?
 If the disk system has rotational speed of 3000 RPM, what is the average access time with a seek time of 11.5 msec?
Solution
Given
 Number of surfaces = 16
 Number of tracks per surface = 128
 Number of sectors per track = 256
 Number of bytes per sector = 512 bytes
Part01: Capacity of Disk Pack
Capacity of disk pack
= Total number of surfaces x Number of tracks per surface x Number of sectors per track x Number of bytes per sector
= 16 x 128 x 256 x 512 bytes
= 2^{28} bytes
= 256 MB
Part02: Number of Bits Required To Address Sector
Total number of sectors
= Total number of surfaces x Number of tracks per surface x Number of sectors per track
= 16 x 128 x 256 sectors
= 2^{19} sectors
Thus, Number of bits required to address the sector = 19 bits
Part03: Formatted Disk Space
Formatting overhead
= Total number of sectors x overhead per sector
= 2^{19} x 32 bytes
= 2^{19} x 2^{5} bytes
= 2^{24} bytes
= 16 MB
Now, Formatted disk space
= Total disk space – Formatting overhead
= 256 MB – 16 MB
= 240 MB
Part04: Formatting Overhead
Amount of memory lost due to formatting
= Formatting overhead
= Total number of sectors x Overhead per sector
= 2^{19} x 64 bytes
= 2^{19} x 2^{6} bytes
= 2^{25} bytes
= 32 MB
Part05: Maximum Recording Density
Storage capacity of a track
= Number of sectors per track x Number of bytes per sector
= 256 x 512 bytes
= 2^{8} x 2^{9} bytes
= 2^{17} bytes
= 128 KB
Circumference of innermost track
= 2 x π x radius
= π x diameter
= 3.14 x 21 cm
= 65.94 cm
Now, Maximum recording density
= Recording density of innermost track
= Capacity of a track / Circumference of innermost track
= 128 KB / 65.94 cm
= 1.94 KB/cm
Part06: Capacity Of Track
Circumference of innermost track
= 2 x π x radius
= π x diameter
= 3.14 x 21 cm
= 65.94 cm
Capacity of a track
= Storage density of the innermost track x Circumference of the innermost track
= 2 KB/cm x 65.94 cm
= 131.88 KB
≅ 132 KB
Part07: Data Transfer Rate
Number of rotations in one second
= (3600 / 60) rotations/sec
= 60 rotations/sec
Now, Data transfer rate
= Number of heads x Capacity of one track x Number of rotations in one second
= 16 x (256 x 512 bytes) x 60
= 2^{4} x 2^{8} x 2^{9} x 60 bytes/sec
= 60 x 2^{21} bytes/sec
= 120 MBps
Part08: Average Access Time
Time taken for one full rotation
= (60 / 3000) sec
= (1 / 50) sec
= 0.02 sec
= 20 msec
Average rotational delay
= 1/2 x Time taken for one full rotation
= 1/2 x 20 msec
= 10 msec
Now, average access time
= Average seek time + Average rotational delay + Other factors
= 11.5 msec + 10 msec + 0
= 21.5 msec
Problem02:
What is the average access time for transferring 512 bytes of data with the following specifications
 Average seek time = 5 msec
 Disk rotation = 6000 RPM
 Data rate = 40 KB/sec
 Controller overhead = 0.1 msec
Solution
Given
 Average seek time = 5 msec
 Disk rotation = 6000 RPM
 Data rate = 40 KB/sec
 Controller overhead = 0.1 msec
Time Taken For One Full Rotation
Time taken for one full rotation
= (60 / 6000) sec
= (1 / 100) sec
= 0.01 sec
= 10 msec
Average Rotational Delay
Average rotational delay
= 1/2 x Time taken for one full rotation
= 1/2 x 10 msec
= 5 msec
Transfer Time
Transfer time
= (512 bytes / 40 KB) sec
= 0.0125 sec
= 12.5 msec
Average Access Time
Average access time
= Average seek time + Average rotational delay + Transfer time + Controller overhead + Queuing delay
= 5 msec + 5 msec + 12.5 msec + 0.1 msec + 0
= 22.6 msec
Problem03:
A certain moving arm disk storage with one head has the following specifications
 Number of tracks per surface = 200
 Disk rotation speed = 2400 RPM
 Track storage capacity = 62500 bits
 Average latency = P msec
 Data transfer rate = Q bits/sec
What is the value of P and Q?
Solution
Given
 Number of tracks per surface = 200
 Disk rotation speed = 2400 RPM
 Track storage capacity = 62500 bits
Time Taken For One Full Rotation
Time taken for one full rotation
= (60 / 2400) sec
= (1 / 40) sec
= 0.025 sec
= 25 msec
Average Latency
Average latency or Average rotational latency
= 1/2 x Time taken for one full rotation
= 1/2 x 25 msec
= 12.5 msec
Data Transfer Rate
Data transfer rate
= Number of heads x Capacity of one track x Number of rotations in one second
= 1 x 62500 bits x (2400 / 60)
= 2500000 bits/sec
= 2.5 x 10^{6} bits/sec
Thus, P = 12.5 and Q = 2.5 x 10^{6}
Problem04:
A disk pack has 19 surfaces and storage area on each surface has an outer diameter of 33 cm and inner diameter of 22 cm. The maximum recording storage density on any track is 200 bits/cm and minimum spacing between tracks is 0.25 mm. Calculate the capacity of disk pack.
Solution
Given
 Number of surfaces = 19
 Outer diameter = 33 cm
 Inner diameter = 22 cm
 Maximum recording density = 200 bits/cm
 Inter track gap = 0.25 mm
Number Of Tracks On Each Surface
Number of tracks on each surface
= (Outer radius – Inner radius) / Inter track gap
= (16.5 cm – 11 cm) / 0.25 mm
= 5.5 cm / 0.25 mm
= 55 mm / 0.25 mm
= 220 tracks
Capacity Of Each Track
Capacity of each track
= Maximum recording density x Circumference of innermost track
= 200 bits/cm x (3.14 x 22 cm)
= 200 x 69.08 bits
= 13816 bits
= 1727 bytes
Capacity Of Disk Pack
Capacity of disk pack
= Total number of surfaces x Number of tracks per surface x Capacity of one track
= 19 x 220 x 1727 bytes
= 7218860 bytes
= 6.88 MB
Problem05:
Consider a typical disk that rotates at 15000 RPM and has a transfer rate of 50 x 10^{6} bytes/sec. If the average seek time of the disk is twice the average rotational delay and the controller’s transfer time is 10 times the disk transfer time. What is the average time (in milliseconds) to read or write a 512 byte sector of the disk?
Solution
Given
 Rotation speed of the disk = 15000 RPM
 Transfer rate = 50 x 10^{6} bytes/sec
 Average seek time = 2 x Average rotational delay
 Controller’s transfer time = 10 x Disk transfer time
Time Taken For One Full Rotation
Time taken for one full rotation
= (60 / 15000) sec
= 0.004 sec
= 4 msec
Average Rotational Delay
Average rotational delay
= 1/2 x Time taken for one full rotation
= 1/2 x 4 msec
= 2 msec
Average Seek Time
Average seek time
= 2 x Average rotational delay
= 2 x 2 msec
= 4 msec
Disk Transfer Time
Disk transfer time
= Time taken to read or write 512 bytes
= 512 bytes / (50 x 10^{6} bytes/sec)
= 10.24 x 10^{6} sec
= 0.01024 msec
Controller’s Transfer Time
Controller’s transfer time
= 10 x Disk transfer time
= 10 x 0.01024 msec
= 0.1024 msec
Average Time To Read Or Write 512 Bytes
Average time to read or write 512 bytes
= Average seek time + Average rotational delay + Disk transfer time + Controller’s transfer time + Queuing delay
= 4 msec + 2 msec + 0.01024 msec + 0.1024 msec + 0
= 6.11 msec
Problem06:
A hard disk system has the following parameters
 Number of tracks = 500
 Number of sectors per track = 100
 Number of bytes per sector = 500
 Time taken by the head to move from one track to another adjacent track = 1 msec
 Rotation speed = 600 RPM
What is the average time taken for transferring 250 bytes from the disk?
Solution
Given
 Number of tracks = 500
 Number of sectors per track = 100
 Number of bytes per sector = 500
 Time taken by the head to move from one track to another adjacent track = 1 msec
 Rotation speed = 600 RPM
Average Seek Time
Average seek time
= (Time taken by the head to move from track1 to track1 + Time taken by the head to move from track1 to track500) / 2
= (0 + 499 x 1 msec) / 2
= 249.5 msec
Time Taken For One Full Rotation
Time taken for one full rotation
= (60 / 600) sec
= 0.1 sec
= 100 msec
Average Rotational Delay
Average rotational delay
= 1/2 x Time taken for one full rotation
= 1/2 x 100 msec
= 50 msec
Capacity Of One Track
Capacity of one track
= Number of sectors per track x Number of bytes per sector
= 100 x 500 bytes
= 50000 bytes
Data Transfer Rate
Data transfer rate
= Number of heads x Capacity of one track x Number of rotations in one second
= 1 x 50000 bytes x (600 / 60)
= 50000 x 10 bytes/sec
= 5 x 10^{5} bytes/sec
Transfer Time
Transfer time
= (250 bytes / 5 x 10^{5} bytes) sec
= 50 x 10^{5} sec
= 0.5 msec
Average Time Taken To Transfer 250 Bytes
Average time taken to transfer 250 bytes
= Average seek time + Average rotational delay + Transfer time + Controller overhead + Queuing delay
= 249.5 msec + 50 msec + 0.5 msec + 0 + 0
= 300 msec
Problem07:
A hard disk has 63 sectors per track, 10 platters each with 2 recording surfaces and 1000 cylinders.
The address of a sector is given as a triple (c, h, s) where c is the cylinder number, h is the surface number and s is the sector number. Thus, the 0th sector is addressed as (0,0,0), the 1st sector as (0,0,1) and so on.
Part01:
The address <400, 16, 29> corresponds to sector number
 505035
 505036
 505037
 505038
Part02:
The address of 1039 sector is
 <0, 15, 31>
 <0, 16, 30>
 <0, 16, 31>
 <0, 17, 31>
Solution
Know this Concept? In general, when counting of items is started from 0, then  For any itemn, number ‘n’ specifies the number of items that must be crossed in order to reach that item.
Example If counting is started from 0, then  To reach cylinder5, the number of cylinders that must be crossed = 5 cylinders
 To reach surface5, the number of surfaces that must be crossed = 5 surfaces
 To reach sector5, the number of sectors that must be crossed = 5 sectors

To solve this question, we assume there is only one track on each surface.
Part01:
We have to calculate the sector number for the address <400, 16, 29>
Step01:
To reach our desired cylinder, we have to cross 400 cylinders.
Total number of sectors that are crossed in 400 cylinders
= Number of cylinders x Number of surfaces per cylinder x Number of tracks per surface x Number of sectors per track
= 400 x (10 x 2) x 1 x 63
= 504000
Now, after crossing 400 cylinders (cylinder0 to cylinder399), we are at cylinder400.
Step02:
To reach our desired surface, we have to cross 16 surfaces.
Total number of sectors that are crossed in 16 surfaces
= Number of surfaces x Number of tracks per surface x Number of sectors per track
= 16 x 1 x 63
= 1008
Now, after crossing 16 surfaces (surface0 to surface15) in cylinder400, we are at surface16.
Step03:
To reach our desired sector, we have to cross 29 sectors.
Now, after crossing 29 sectors on surface16 of cylinder400, we are at sector29.
Thus
Total number of sectors that are crossed
= 504000 + 1008 + 29
= 505037
Thus,
 After crossing 505037 sectors, we are at sector505037.
 So, required address of the sector is 505037.
 Option (C) is correct.
Part02:
We have to find the address of the sector2039.
Let us check all the options one by one.
OptionA:
For the address <0, 15, 31>, the sector number is
Sector number = 0 + (15 x 1 x 63) + 31 = 976
OptionB:
For the address <0, 16, 30>, the sector number is
Sector number = 0 + (16 x 1 x 63) + 30 = 1038
OptionC:
For the address <0, 16, 31>, the sector number is
Sector number = 0 + (16 x 1 x 63) + 31 = 1039
OptionD:
For the address <0, 17, 31>, the sector number is
Sector number = 0 + (17 x 1 x 63) + 31 = 1102
Thus, Option (C) is correct.
Next Article Addressing Modes
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