Mathematical Derivation of Reliability


Due to the technological development, there is an addition a figure of sophisticated and complex system. During the past 50 old ages, due to the addition of these complex systems,the thought of Reliability Engineering has developed.Due to their alteration in nature, these types of jobs have been theextremely attracted the attending by the assorted writer from different subjects. So in overall scenario, a theory which deals with the assorted method and assorted techniques those are increase and maximise the effectivity of systems is called dependability theory.

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Dependability can be explained in many different legion plants.In simple signifier, “Reliability means the chance of the unit in which the particular map perform without any failure in a given clip t” .But in a more strict manner, the dependability can be define as:

Dependability of a Component/System/Device is the step in the signifier of chance under a specific period of clip and in a given operating conditions to execute its map adequate without failure.

This definition of dependability chiefly contain the four types of the of import factors

Function: A map is a series of operations. If a system/device whose dependability is to be determined is in map when, it is working or runing. e.g. In a defense mechanism system, a big figure of Stationss are set up to direct a signal for establishing missiles to assail enemy war planes based on the sensing of the attack of the planes. While the dependability of the launching of missiles get increased with the addition of a station, at the same clip.

OPERATING Conditions:There device or system whose dependability is to be determined must execute its map under given specific conditions. It has been shown that the temperature, clime, storage, installing, pollution, transit, force per unit area daze, quiver, electromotive force, acceleration, torsion, gravitation etc.This conditions are chiefly named as environment of a system.e.g. An air conditioner which performs satisfactorily in temperature zones may non hold the same public presentation features for hot and cool clime conditions. Similarly, an car daze absorber may be satisfactory if the vehicle is used chiefly on main road. The life span may be really short if the vehicle is by and large used on uneven or unsmooth roads.

Time:It is the negatively correlated with the dependability.The impairment of the parts of units is natural. So the public presentation degree of the unit will besides travel down with clip. Most of the units or system fails when they are operated over long periods. So it is really necessary to take a clip bound within which the public presentation of a unit is satisfactory or non.

Probability:Probability is the ratio of the nu. Of favorable instance and the entire nu. of thorough instances. Probability means a certainty. Its value lies in between the 0 and 1.e.g. picking a one of nines card at random from a deck incorporating 52 cards is 1/52.


Let “T” be the figure of constituents which are repeatedly tested. But after a clip ‘t’ , merely ‘p’ constituent are survive the trial and staying ‘q’components fail. This implies that ( T=p +q ) is changeless throughout the trial because as the trial returns, the figure of failed constituents q additions precisely as the figure of lasting constituent P lessenings.

So at any clip ‘t’ the chance of endurance or dependability is defined as

R ( T ) = p/ ( p+q ) =p/T

Here P or Q are counted at a specific clip, ’t’ .

Now at any clip T, the undependability can be expressed as

F ( T ) =q/ ( p+q ) =q/T

So at any clip ‘t’

R ( T ) + F ( T ) =1

Because R ( T ) and F ( T ) are reciprocally sole events.

Now R ( T ) =p/ ( p+q ) = ( T-q ) /T

R ( T ) = ( 1-q/T )

Diff. w.r.t ‘t’

=( 1-q/T ) = – ( 1/T )

or= -T…………….. ( 1 )

which is the rate at which the constituent fail.

Dividing equ. ( 1 ) by P in both sides, we get

== –

Here ? ( T ) = failure rate =

Then ? ( T ) = –

Or ? ( T ) dt = –

Integrating in both sides with taking proper bound,

== ln R ( T )

ln R ( T ) =

This is the mathematical expression to find the dependability of a system.

Properities:The dependability R ( T ) or the chance of endurance at any clip T, can be expressed as

  1. – 0? R ( T ) ? 1
  2. – Roentgen ( 0 ) = 1 and R ( ? )
  3. – In general, R ( T ) is non a increasing map of clip.


Failure rate is besides called the Hazard rate. It is the ratio of the entire figure of failures in a peculiar interval to the entire figure Of points in that intervalAlso, Mathematically, the Hazard rate or the instantaneous failure rate R ( T ) can expressed in footings of the conditional chance as

K ( T ) =[ Probability of the unit which will neglect between T and ( t+T ) but the unit has non failed in the interval ( 0, T ) ]

The Hazard map is defined as the bound of the failure rate as the bound tends to zero. Thus the Hazard map is besides defined as

K ( T ) =



Here K ( T ) dt represents the chance that the system at age T will neglect in the little interval T to t+. The Hazard map is of import because it describes the alteration in the failure rate over the entire life of the constituents.

In statistical theory, the decease rate is similar to the failure rate like the force of mortality is correspondent to the Hazard map. Then the Hazard map or failure rate map is equal to the ratio of the chance denseness map and the dependability map.

Failure:If a map do non execute undertaking under the given status in clip T is called failure. A system is normally acceptable if the frequence of failure is really low. This types of system is considered as dependable. There are the many factors those consequence the dependability of system or those are the cause of failures in systems.

  1. – Lacks in design
  2. – Lacks in stuff
  3. – Lacks in processing
  4. – Improper service conditions
  5. – Complexity of unit used
  6. – Poor care policies

In pattern, there are three types of failures occurred during the complete life rhythm of the operation of the system: –

  1. – Early Failures
  2. – Random failures
  3. – Wear-out Failures

In these above, three types of failures wear-out failures are normally decrease the effectivity of the system and made the system undependable. Wear-out failures are carried due to the have oning out of constituents. This types of failures occur if the system is non maintained decently. This types of failure additions quickly with clip. In pattern, these failures can non be wholly removed but can be decrease for some clip by transporting out preventative care of the system at regular interval of clip. The preventative care is planned in such a manner that the care period between two care is shorter than the average wear-out life of the system. The life characteristic curve consists of three different types of failure.


Any device do non execute in the similar mode in all the conditions. Due to the ripening of the constituents and other assorted types of failures, no device can non execute for a boundlessly long clip. So it is really necessary to mensurate the life clip of the system. This step is considered as the Mean Time to System Failures ( MTSF ) .

Then MTSF is the mean clip between the consecutive failure of the system or it is the expected clip for which the device is in operation before it compeletely fails.


The handiness of a unit/device is the chance that it is performed satisfactorily in the given clip T, under the given conditions at any point. The system handiness means the system effectivity as a whole because it is the combination of both dependability ( which is the system’s capableness of endurance ) and maintenability ( which is the system’s capableness of fix ) .

In other words, the handiness is a step that license for a system to mend when any failure occurred.


For non –repairable systems, dependability is ever equal to handiness But For repairable system, dependability is less than of equal to handiness.

Difference between Reliability and Availability

Reliability is an interval map.

Availability is a point map which describes the behavior of the system at a peculiar clip.

Reliability map prevents the failure of the system during the interval consideration.

Availability do non better any limitation on the behavior of system.

There are many types of handiness


See if the system is working at clip T so S ( T ) and allow when it is non working so T ( T ) =1 the handiness of system is defined as

Availability ( PA( T ) ) = P [ S ( T ) =1 / T ( T ) =0 ] , at any clip T.


If the system will be perform with in tolerance so it the supposed fraction of a given interval of time.tThen

IA( T ) =from the clip interval [ 0, T ]


It is the supposed fraction of clip in the long tally that the system perform satisfactory. So mathematically Restricting Interval Availability is defined as

ALT() =A( T )


The dependability of any system depends on the figure of failure occurred in the system. So for the sweetening of system dependability, it is necessary that the design applied scientist understands the cause of failure those are due to the designs lacks.

In pattern, most of the complex systems consist of constituents and subsystems.

There are some undermentioned basic nomenclatures to find the system dependability.

PARTS/ELEMENT:A really little subdivision of a system which can non farther divided without being destroyed is called portion.

Circuit:The aggregation of component which holding some specific map is called circuit.

Unit of measurement:A unit or system is the aggregation component/element or circuit/assembles which represents a self-contained component of a complete operating system to execute a specific map. The measures and qualities of the different constituent /subsystem is straight consequence the system’s dependability.

If these units of a system are connected in different ways, it gives the undermentioned system constellations.


Those systems in which a big figure of assemblies are connected in the series are called series constellations. In this type of system if one of the constituent fail in the series so the whole system dislocation or fails or if all the constituents are connected in a series so each constituents should ever be in working or run. e.g. In the festival of Diwali, if the one bulb in the freshness bulb fails so the whole lead fails. A system which incorporating ‘n’ constituents in series can be represented by

Series – System

Fig –

The dependability ( RSecond) of the ‘n’ constituent connected in series is equal to

RoentgenSecond= R­1.R2.R3­— — — — — — — — —RoentgenN


By and large RIis the dependability of a the IThursdayconstituent connected in series in the system.

If R1=R2=R3­— — — — — — — — — — — — –RoentgenN= E’ , the mean dependability of each constituent so the dependability of the system is given by

RoentgenSecond= ( E’ )N

[ In this constellation, if the figure of constituents is additions, so the system’s dependability decreases. ]

If a system consists of merely one constituent holding dependability 85 % . Then the dependability of the whole system is 85 % . If two constituents are connected in series of dependability 85 % so the system dependability is equal to 72 % and so on.

The following tabular array shows that the system dependability lessening when the figure of constituents in the series additions.

Number of Components

System Reliability














Parallel Configuration is besides referred to as redundancy. It is a technique in which more than one constituent are connected in analogue to execute a specific map and to acquire a higher reliability.e.g.

A system in which the two constituent are connected in analogue is represented by




I-P-O and I-Q-O are the two possible waies to execute any operation on the system. The parallel system can neglect if all the constituents are fail.If one of the constituent is working so system is besides working.

In pattern, Parallel constellation is used in the system when

  1. – to diminish the consequence of opportunity failure.
  2. – where the mistake location in the system can ne’er the determined or the failure of a constituent is non detected.
  3. – when a constituent is non sufficiently dependable to acquire required consequence.

Here system dependability ( RSecond) in which two constituents are connected in analogue is

RoentgenSecond= 1- ( 1-R1) ( 1-R2)

Here R1and Roentgen2is the separate dependability of each. Similarly dependability RSecondin which n constituents are connected in analogue is equal to

RoentgenSecond=1 –

In this constellation if the figure of component /subsystem additions, the system’s dependability additions.

If a system consists of merely one constituent holding dependability 25 % .Then the dependability of the whole system is 25 % . If two constituents are connected in analogue, each holding dependability 25 % so the dependability of the whole system is 97 % . And so on. The following tabular array shows that the system dependability increases when the figure of constituent in parallel additions.

Number of Component

System Reliability














This type of constellation consists of many phases connected in series and each phases contains a figure of excess constituents. Here dependability of system is

equal to the merchandise of the dependability of each phases.

1stphase 2neodymiumphase IThursdayphase mThursdayphase

Here the dependability ( RSecond) of this types of constellation is equal to

RoentgenSecond= [ 1- ( 1-R )N]

Where R =

( Here RIis the dependability of the IThursdayconstituent in series )


This type of constellation consists of many phases connected in parallel and each phases contains many constituent connected in series constellation.





The dependability ( RSecond) of this type of system is equal to

RoentgenSecond= [ 1- ( 1-R )Nitrogen]Meter

Standby System: –

In a standby system, merely one constituent is runing and one or more constituent are standby to take over the operation when the first fail occurs. The operation of standby constituents is consecutive. The system constellation holding ability to execute its map when at least one unit out of n unit is working.


In this figure, system consists of two constituents. The constituent 1 is successfully runing upto clip t. If this constituent 1 fails at clip T1& A ; lt ; T and the the constituent 2 operates from clip T1to t.

p-OUT-of –n CONFIGURATION ( P, N ) :

In this type of system merely p constituents successfully operate out – of – n constituents e.g. A battery holding n cells, in which a lower limit of P cells should be in operation for the needed electromotive force. See that all the system are statistically indistinguishable. Then the dependability of the system is

Roentgenp/n=RoentgenI( 1-R )n-i

Where R is the dependability of a individual constituent of system.


There are following many technique to better the system dependability.

Partss betterment method

Effective and Creative design

Structural redundancy





In this method, either the dependability of all the constituents are improved or at least the most critical constituents are identified and their dependability improved. Due to the mechanization, it is really dearly-won technique, but it is rather effectual upto a certain point.


Reliability ever depends how the system is to be design.Whether it is design in series constellation or parallel constellation or series-parallel constellation or parallel –series constellation.

( 3 ) – Structural Redundancy:

Redundancy is besides a really of import technique which is used to better the system dependability.Interconnected power systems, protective systems for atomic reactors, temperature control systems for infinite vehicle, data- processing systems, there are the few illustrations of systems, where redundancy is extremely used. There are two types of redundancy depending upon the demand in the system.

( I ) – Active Redundancy:

In this type of redundancy, all the subsystems are arranged in parallel signifier. Here all the units perform ab initio and the system continues to execute adequately until all the fractional monetary units fail. In this instance the system, holding a positive chance of failure when it is perform map accurately or non. This is because of the environmental conditions and temperature.

( two ) – Passive voice or STANDBY REDUNDANCY:

Standby redundancy applies to a constituents where there is an alternate agencies of taking topographic point the undertaking that is switched on by a detection and shift over device when the primary constituents fails.

( a ) – Cold Standby:

It is a imperfect detection cold standby system. It is wholly inactive.Then its dependability will non alter when it is put into operating province. In this instance, primary unit operates and one or more secondary constituents in taken as standby.

( B ) – Warm Standby:

It is partly energized unit which is in reduced in size, of import or strength so the chief online unit. So if the warm standby fail so the chance of its weakness is smaller than the chance of the weakness of online unit.

( degree Celsius ) – Hot Standby:

It is to the full active in the system. In this instance the standby constituent fails without being operated because it has shelf life. e.g. any type of batteries.


The chance that a system will be restored to its operational effectivity within a given period of clip, when the care action is performed in conformity with prescribed processs and resources.

( a ) -MAINTENANCE Action:

The act of keeping a system in its operating status. It may consists of a series of events such as fix, accommodation, trial or preventative care, as required.

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