Gompertz Models - Revision history

David J. Groebel at 16:56, 20 June 2014

2014-06-20T16:56:10Z

← Older revision		Revision as of 16:56, 20 June 2014
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	The Gompertz reliability growth model is often used when analyzing reliability data. It is most applicable when the data set follows a smooth curve, as shown in the plot below.		The Gompertz reliability growth model is often used when analyzing reliability data. It is most applicable when the data set follows a smooth curve, as shown in the plot below.

	[[Image:rga7.1.png~~\|thumb~~\|center\|~~400px\|Reliability growth data following a smooth curve.~~]]		[[Image:rga7.1.png\|center\|450px]]

	The Gompertz model is mathematically given by Virene [[RGA_References\|[1]]]:		The Gompertz model is mathematically given by Virene [[RGA_References\|[1]]]:

David J. Groebel: /* Example: Modified Gompertz for Reliability Data */

2014-06-20T16:54:55Z

Example: Modified Gompertz for Reliability Data

← Older revision		Revision as of 16:54, 20 June 2014
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	where <math>p\,\!</math> is the total number of groups (in this case 4) and <math>n\,\!</math> is the total number of items in each group.		where <math>p\,\!</math> is the total number of groups (in this case 4) and <math>n\,\!</math> is the total number of items in each group.

	===Example: Modified Gompertz for Reliability Data===		===Example - Modified Gompertz for Reliability Data===

	{{:Reliability_Data_-_Modified_Gompertz_Model}}		{{:Reliability_Data_-_Modified_Gompertz_Model}}

David J. Groebel: /* Example: Standard Gompertz for Sequential Data */

2014-06-20T16:54:41Z

Example: Standard Gompertz for Sequential Data

← Older revision		Revision as of 16:54, 20 June 2014
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	{{:Reliability_Data_-_Standard_Gompertz_Model}}		{{:Reliability_Data_-_Standard_Gompertz_Model}}

	===Example: Standard Gompertz for Sequential Data===		===Example - Standard Gompertz for Sequential Data===
	{{:Sequential_Data_-_Standard_Gompertz_Model}}		{{:Sequential_Data_-_Standard_Gompertz_Model}}

David J. Groebel: /* Example: Standard Gompertz for Reliability Data */

2014-06-20T16:54:28Z

Example: Standard Gompertz for Reliability Data

← Older revision		Revision as of 16:54, 20 June 2014
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	where <math>p\,\!</math> is the total number of groups (in this case 3) and <math>n\,\!</math> is the total number of items in each group.		where <math>p\,\!</math> is the total number of groups (in this case 3) and <math>n\,\!</math> is the total number of items in each group.

	===Example: Standard Gompertz for Reliability Data===		===Example - Standard Gompertz for Reliability Data===
	{{:Reliability_Data_-_Standard_Gompertz_Model}}		{{:Reliability_Data_-_Standard_Gompertz_Model}}

David J. Groebel at 19:37, 29 May 2014

2014-05-29T19:37:58Z

← Older revision		Revision as of 19:37, 29 May 2014
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	This chapter discusses the two Gompertz models that are used in RGA. The [[Gompertz_Models#The_Standard_Gompertz_Model\|standard Gompertz]] and the [[Gompertz_Models#The_Modified_Gompertz_Model\|modified Gompertz]].		This chapter discusses the two Gompertz models that are used in RGA. The [[Gompertz_Models#The_Standard_Gompertz_Model\|standard Gompertz]] and the [[Gompertz_Models#The_Modified_Gompertz_Model\|modified Gompertz]].

	=The Standard Gompertz Model=		==The Standard Gompertz Model==
	The Gompertz reliability growth model is often used when analyzing reliability data. It is most applicable when the data set follows a smooth curve, as shown in the plot below.		The Gompertz reliability growth model is often used when analyzing reliability data. It is most applicable when the data set follows a smooth curve, as shown in the plot below.

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	As it can be seen from the mathematical definition, the Gompertz model is a 3-parameter model with the parameters <math>a\,\!</math>, <math>b\,\!</math> and <math>c\,\!</math>. The solution for the parameters, given <math>{{T}_{i}}\,\!</math> and <math>{{R}_{i}}\,\!</math>, is accomplished by fitting the best possible line through the data points. Many methods are available; all of which tend to be numerically intensive. When analyzing reliability data in the RGA software, you have the option to enter the reliability values in percent or in decimal format. However, <math>a\,\!</math> will always be returned in decimal format and not in percent. The estimated parameters in the RGA software are unitless. The next section presents an overview and background on some of the most commonly used algorithms/methods for obtaining these parameters.		As it can be seen from the mathematical definition, the Gompertz model is a 3-parameter model with the parameters <math>a\,\!</math>, <math>b\,\!</math> and <math>c\,\!</math>. The solution for the parameters, given <math>{{T}_{i}}\,\!</math> and <math>{{R}_{i}}\,\!</math>, is accomplished by fitting the best possible line through the data points. Many methods are available; all of which tend to be numerically intensive. When analyzing reliability data in the RGA software, you have the option to enter the reliability values in percent or in decimal format. However, <math>a\,\!</math> will always be returned in decimal format and not in percent. The estimated parameters in the RGA software are unitless. The next section presents an overview and background on some of the most commonly used algorithms/methods for obtaining these parameters.

	==Parameter Estimation==		===Parameter Estimation===
	===Linear Regression===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		====Linear Regression====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on <math>Y\,\!</math>, then the sum of the squares of the vertical deviations from the points to the line is minimized. If the regression is on <math>X\,\!</math>, the line is fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized. To illustrate the method, this section presents a regression on <math>Y\,\!</math>. Consider the linear model given by Seber and Wild [[RGA_References\|[2]]]:		The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on <math>Y\,\!</math>, then the sum of the squares of the vertical deviations from the points to the line is minimized. If the regression is on <math>X\,\!</math>, the line is fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized. To illustrate the method, this section presents a regression on <math>Y\,\!</math>. Consider the linear model given by Seber and Wild [[RGA_References\|[2]]]:

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	:<math>\widehat{\beta }=({{X}^{T}}X)^{-1}{{X}^{T}}Y\,\!</math>		:<math>\widehat{\beta }=({{X}^{T}}X)^{-1}{{X}^{T}}Y\,\!</math>

	===Non-linear Regression===		====Non-linear Regression====
	Non-linear regression is similar to linear regression, except that a curve is fitted to the data set instead of a straight line. Just as in the linear scenario, the sum of the squares of the horizontal and vertical distances between the line and the points are to be minimized. In the case of the non-linear Gompertz model <math>R=a{{b}^{{{c}^{T}}}}\,\!</math>, let:		Non-linear regression is similar to linear regression, except that a curve is fitted to the data set instead of a straight line. Just as in the linear scenario, the sum of the squares of the horizontal and vertical distances between the line and the points are to be minimized. In the case of the non-linear Gompertz model <math>R=a{{b}^{{{c}^{T}}}}\,\!</math>, let:

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	When using the Gauss-Newton method or some other estimation procedure, it is advisable to try several sets of starting values to make sure that the solution gives relatively consistent results.		When using the Gauss-Newton method or some other estimation procedure, it is advisable to try several sets of starting values to make sure that the solution gives relatively consistent results.

	===Choice of Initial Values===		====Choice of Initial Values====
	The choice of the starting values for the nonlinear regression is not an easy task. A poor choice may result in a lengthy computation with many iterations. It may also lead to divergence, or to a convergence due to a local minimum. Therefore, good initial values will result in fast computations with few iterations, and if multiple minima exist, it will lead to a solution that is a minimum.		The choice of the starting values for the nonlinear regression is not an easy task. A poor choice may result in a lengthy computation with many iterations. It may also lead to divergence, or to a convergence due to a local minimum. Therefore, good initial values will result in fast computations with few iterations, and if multiple minima exist, it will lead to a solution that is a minimum.

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	Then:		Then:
	::<math>\frac{S_3-S_2}{S_2-S_1}=\frac{\sum_{i=2n}{m-1} c^{T_i}-\sum_{i=n}^{2n-1} c^T_i}{\sum_{i=0}^{n-1} c^{T_i}}\,\!</math>		:<math>\frac{S_3-S_2}{S_2-S_1}=\frac{\sum_{i=2n}{m-1} c^{T_i}-\sum_{i=n}^{2n-1} c^T_i}{\sum_{i=0}^{n-1} c^{T_i}}\,\!</math>
	::<math>\frac{S_3-S_2}{S_2-S_1}=c^T_{2n}\frac{\sum_{i=0}{n-1} c^{T_i}-c^{T_n}\sum_{i=0}^{n-1} c^T_i}{c^{T_n}\sum_{i=0}^{n-1} c^{T_i}}\,\!</math>		:<math>\frac{S_3-S_2}{S_2-S_1}=c^T_{2n}\frac{\sum_{i=0}{n-1} c^{T_i}-c^{T_n}\sum_{i=0}^{n-1} c^T_i}{c^{T_n}\sum_{i=0}^{n-1} c^{T_i}}\,\!</math>
	::<math>\frac{S_3 - S_2}{S_2-S_1}=\frac{c^{T_2n}-c^{T_n}}{c^{T_n}-1}=c^{T_{a_n}}=c^{n\cdot I+T_0}\,\!</math>		:<math>\frac{S_3 - S_2}{S_2-S_1}=\frac{c^{T_2n}-c^{T_n}}{c^{T_n}-1}=c^{T_{a_n}}=c^{n\cdot I+T_0}\,\!</math>

	Without loss of generality, take <math>{{T}_{{{a}_{0}}}}=0\,\!</math> ; then:		Without loss of generality, take <math>{{T}_{{{a}_{0}}}}=0\,\!</math> ; then:
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	</li>		</li>
	</ol>		</ol>
	==Confidence Bounds==		===Confidence Bounds===
	The approximate reliability confidence bounds under the Gompertz model can be obtained with non-linear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation is used to obtain the confidence bounds on reliability.		The approximate reliability confidence bounds under the Gompertz model can be obtained with non-linear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation is used to obtain the confidence bounds on reliability.

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	where <math>p\,\!</math> is the total number of groups (in this case 3) and <math>n\,\!</math> is the total number of items in each group.		where <math>p\,\!</math> is the total number of groups (in this case 3) and <math>n\,\!</math> is the total number of items in each group.

	==Example: Standard Gompertz for Reliability Data==		===Example: Standard Gompertz for Reliability Data===
	{{:Reliability_Data_-_Standard_Gompertz_Model}}		{{:Reliability_Data_-_Standard_Gompertz_Model}}

	==Example: Standard Gompertz for Sequential Data==		===Example: Standard Gompertz for Sequential Data===
	{{:Sequential_Data_-_Standard_Gompertz_Model}}		{{:Sequential_Data_-_Standard_Gompertz_Model}}

	==Cumulative Reliability==		===Cumulative Reliability===
	For many kinds of equipment, especially missiles and space systems, only success/failure data (also called discrete or attribute data) is obtained. Conservatively, the cumulative reliability can be used to estimate the trend of reliability growth. The cumulative reliability is given by Kececioglu [[RGA_References\|[3]]]:		For many kinds of equipment, especially missiles and space systems, only success/failure data (also called discrete or attribute data) is obtained. Conservatively, the cumulative reliability can be used to estimate the trend of reliability growth. The cumulative reliability is given by Kececioglu [[RGA_References\|[3]]]:

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	The modified Gompertz model is more flexible than the original, especially when fitting growth data with S-shaped trends.		The modified Gompertz model is more flexible than the original, especially when fitting growth data with S-shaped trends.

	==Parameter Estimation==		===Parameter Estimation===
	To implement the modified Gompertz growth model, initial values of the parameters <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>d\,\!</math> must be determined. When analyzing reliability data in RGA, you have the option to enter the reliability values in percent or in decimal format. However, <math>a\,\!</math> and <math>d\,\!</math> will always be returned in decimal format and not in percent. The estimated parameters in RGA are unitless.		To implement the modified Gompertz growth model, initial values of the parameters <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>d\,\!</math> must be determined. When analyzing reliability data in RGA, you have the option to enter the reliability values in percent or in decimal format. However, <math>a\,\!</math> and <math>d\,\!</math> will always be returned in decimal format and not in percent. The estimated parameters in RGA are unitless.

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	\end{align}\,\!</math>		\end{align}\,\!</math>

			:<math>{{\nu }^{(0)}}=\left[ \begin{matrix}
	::<math>{{\nu }^{(0)}}=\left[ \begin{matrix}
	g_{1}^{(0)} \\		g_{1}^{(0)} \\
	g_{2}^{(0)} \\		g_{2}^{(0)} \\
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	As mentioned previously, when using the Gauss-Newton method or some other estimation procedure, it is advisable to try several sets of starting values to make sure that the solution gives relatively consistent results. Note that RGA uses a different analysis method called the Levenberg-Marquardt. This method utilizes the best features of the Gauss-Newton method and the method of the steepest descent, and occupies a middle ground between these two methods.		As mentioned previously, when using the Gauss-Newton method or some other estimation procedure, it is advisable to try several sets of starting values to make sure that the solution gives relatively consistent results. Note that RGA uses a different analysis method called the Levenberg-Marquardt. This method utilizes the best features of the Gauss-Newton method and the method of the steepest descent, and occupies a middle ground between these two methods.

	==Confidence Bounds==		===Confidence Bounds===

	The approximate reliability confidence bounds under the modified Gompertz model can be obtained using non-linear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation can be used to obtain the confidence bounds on reliability.		The approximate reliability confidence bounds under the modified Gompertz model can be obtained using non-linear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation can be used to obtain the confidence bounds on reliability.
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	where <math>p\,\!</math> is the total number of groups (in this case 4) and <math>n\,\!</math> is the total number of items in each group.		where <math>p\,\!</math> is the total number of groups (in this case 4) and <math>n\,\!</math> is the total number of items in each group.

	==Example: Modified Gompertz for Reliability Data==		===Example: Modified Gompertz for Reliability Data===

	{{:Reliability_Data_-_Modified_Gompertz_Model}}		{{:Reliability_Data_-_Modified_Gompertz_Model}}

	=More Examples=		==More Examples==
	==Standard Gompertz for Grouped per Configuration Data==		===Standard Gompertz for Grouped per Configuration Data===
	{{:Grouped_per_Configuration_-_Standard_Gompertz_Model}}		{{:Grouped_per_Configuration_-_Standard_Gompertz_Model}}

	==Comparing Standard and Modified Gompertz==		===Comparing Standard and Modified Gompertz===
	{{:Comparing_the_Standard_and_Modified_Gompertz_Models}}		{{:Comparing_the_Standard_and_Modified_Gompertz_Models}}

	~~<br>~~

David J. Groebel at 18:48, 29 May 2014

2014-05-29T18:48:01Z

Show changes

Chris Kahn at 20:33, 15 May 2014

2014-05-15T20:33:58Z

Chris Kahn at 18:18, 14 May 2014

2014-05-14T18:18:31Z

← Older revision		Revision as of 18:18, 14 May 2014
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	To estimate the values of the parameters <math>a,b\,\!</math> and <math>c\,\!</math>, do the following:		To estimate the values of the parameters <math>a,b\,\!</math> and <math>c\,\!</math>, do the following:

	~~:1)~~ Arrange the currently available data in terms of <math>T\,\!</math> and <math>R\,\!</math> as in the table below. The <math>T\,\!</math> values should be chosen at equal intervals and increasing in value by 1, such as one month, one hour, etc.		<ol>
			<li>Arrange the currently available data in terms of <math>T\,\!</math> and <math>R\,\!</math> as in the table below. The <math>T\,\!</math> values should be chosen at equal intervals and increasing in value by 1, such as one month, one hour, etc.

	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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			</li>
	~~:2)~~ Calculate the natural log <math>R\,\!</math>.		<li>Calculate the natural log <math>R\,\!</math>.
	~~:3)~~ Divide the column of values for log <math>R\,\!</math> into three groups of equal size, each containing <math>n\,\!</math> items. There should always be three groups. Each group should always have the same number, <math>n\,\!</math>, of items, measurements or values.		</li>
	~~:4)~~ Add the values of the natural log <math>R\,\!</math> in each group, obtaining the sums identified as <math>{{S}_{1}}\,\!</math>, <math>{{S}_{2}}\,\!</math> and <math>{{S}_{3}}\,\!</math>, starting with the lowest values of the natural log <math>R\,\!</math>.		<li>Divide the column of values for log <math>R\,\!</math> into three groups of equal size, each containing <math>n\,\!</math> items. There should always be three groups. Each group should always have the same number, <math>n\,\!</math>, of items, measurements or values.
	~~:5)~~ Calculate <math>c\,\!</math> using the following equation:		</li>
			<li>Add the values of the natural log <math>R\,\!</math> in each group, obtaining the sums identified as <math>{{S}_{1}}\,\!</math>, <math>{{S}_{2}}\,\!</math> and <math>{{S}_{3}}\,\!</math>, starting with the lowest values of the natural log <math>R\,\!</math>.
			</li>
			<li>Calculate <math>c\,\!</math> using the following equation:
	::<math>c={{\left( \frac{{{S}_{3}}-{{S}_{2}}}{{{S}_{2}}-{{S}_{1}}} \right)}^{\tfrac{1}{n}}}\,\!</math>		::<math>c={{\left( \frac{{{S}_{3}}-{{S}_{2}}}{{{S}_{2}}-{{S}_{1}}} \right)}^{\tfrac{1}{n}}}\,\!</math>
	~~:6)~~ Calculate <math>a\,\!</math> using the following equation:		</li>
			<li>Calculate <math>a\,\!</math> using the following equation:
	::<math>a={{e}^{\left[ \tfrac{1}{n}\left( {{S}_{1}}+\tfrac{{{S}_{2}}-{{S}_{1}}}{1-{{c}^{n}}} \right) \right]}}\,\!</math>		::<math>a={{e}^{\left[ \tfrac{1}{n}\left( {{S}_{1}}+\tfrac{{{S}_{2}}-{{S}_{1}}}{1-{{c}^{n}}} \right) \right]}}\,\!</math>
	~~:7)~~ Calculate <math>b\,\!</math> using the following equation:		</li>
			<li>Calculate <math>b\,\!</math> using the following equation:
	::<math>b={{e}^{\left[ \tfrac{({{S}_{2}}-{{S}_{1}})(c-1)}{{{\left( 1-{{c}^{n}} \right)}^{2}}} \right]}}\,\!</math>		::<math>b={{e}^{\left[ \tfrac{({{S}_{2}}-{{S}_{1}})(c-1)}{{{\left( 1-{{c}^{n}} \right)}^{2}}} \right]}}\,\!</math>
	~~:8)~~ Write the Gompertz reliability growth equation.		</li>
	~~:9)~~ Substitute the value of <math>T\,\!</math>, the time at which the reliability goal is to be achieved, to see if the reliability is indeed to be attained or exceeded by <math>T\,\!</math>.		<li>Write the Gompertz reliability growth equation.
			</li>
			<li>Substitute the value of <math>T\,\!</math>, the time at which the reliability goal is to be achieved, to see if the reliability is indeed to be attained or exceeded by <math>T\,\!</math>.
			</li>
			</ol>
	==Confidence Bounds==		==Confidence Bounds==
	The approximate reliability confidence bounds under the Gompertz model can be obtained with non-linear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation is used to obtain the confidence bounds on reliability.		The approximate reliability confidence bounds under the Gompertz model can be obtained with non-linear regression. Additionally, the reliability is always between 0 and 1. In order to keep the endpoints of the confidence interval, the logit transformation is used to obtain the confidence bounds on reliability.

Kate Racaza: /* Comparing Standard and Modified Gompertz */

2014-01-21T20:37:19Z

Comparing Standard and Modified Gompertz

@@ Line 570: / Line 570: @@
 ==Comparing Standard and Modified Gompertz==
-Using the data in the following table, determine whether the standard Gompertz or modified Gompertz would be better suited for analyzing the given data.
+{{:Comparing_the_Standard_and_Modified_Gompertz_Models}}
 <br>

Kate Racaza: /* Standard Gompertz for Grouped per Configuration Data */

2014-01-21T20:35:41Z

Standard Gompertz for Grouped per Configuration Data

@@ Line 567: / Line 567: @@
 =More Examples=
 ==Standard Gompertz for Grouped per Configuration Data==
-A new design is put through a reliability growth test. The requirement is that after the ninth stage the design will exhibit an 85% reliability with a 90% confidence level. Given the data in the following table, do the following:
+{{:Grouped_per_Configuration_-_Standard_Gompertz_Model}}
 ==Comparing Standard and Modified Gompertz==

← Older revision		Revision as of 20:33, 15 May 2014
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	*<math>0<c<1\,\!</math>		*<math>0<c<1\,\!</math>
	*<math>T>0\,\!</math>		*<math>T>0\,\!</math>
	*<math>R=\,\!</math> the system's reliability at development time, launch number or stage number, <math>T\,\!</math>.		*<math>R=\,\!</math> the system's reliability at development time, launch number or stage number, <math>T\,\!</math>
	*<math>a=\,\!</math> the upper limit that the reliability approaches asymptotically as <math>T\to \infty \,\!</math>, or the maximum reliability that can be attained.		*<math>a=\,\!</math> the upper limit that the reliability approaches asymptotically as <math>T\to \infty \,\!</math>, or the maximum reliability that can be attained
	*<math>ab=\,\!</math> initial reliability at <math>T=0.\,\!</math>		*<math>ab=\,\!</math> initial reliability at <math>T=0\,\!</math>
	*<math>c=\,\!</math> the growth pattern indicator (small values of <math>c\,\!</math> indicate rapid early reliability growth and large values of <math>c\,\!</math> indicate slow reliability growth).		*<math>c=\,\!</math> the growth pattern indicator (small values of <math>c\,\!</math> indicate rapid early reliability growth and large values of <math>c\,\!</math> indicate slow reliability growth)

	As it can be seen from the mathematical definition, the Gompertz model is a 3-parameter model with the parameters <math>a\,\!</math>, <math>b\,\!</math> and <math>c\,\!</math>. The solution for the parameters, given <math>{{T}_{i}}\,\!</math> and <math>{{R}_{i}}\,\!</math>, is accomplished by fitting the best possible line through the data points. Many methods are available; all of which tend to be numerically intensive. When analyzing reliability data in the RGA software, you have the option to enter the reliability values in percent or in decimal format. However, <math>a\,\!</math> will always be returned in decimal format and not in percent. The estimated parameters in the RGA software are unitless. The next section presents an overview and background on some of the most commonly used algorithms/methods for obtaining these parameters.		As it can be seen from the mathematical definition, the Gompertz model is a 3-parameter model with the parameters <math>a\,\!</math>, <math>b\,\!</math> and <math>c\,\!</math>. The solution for the parameters, given <math>{{T}_{i}}\,\!</math> and <math>{{R}_{i}}\,\!</math>, is accomplished by fitting the best possible line through the data points. Many methods are available; all of which tend to be numerically intensive. When analyzing reliability data in the RGA software, you have the option to enter the reliability values in percent or in decimal format. However, <math>a\,\!</math> will always be returned in decimal format and not in percent. The estimated parameters in the RGA software are unitless. The next section presents an overview and background on some of the most commonly used algorithms/methods for obtaining these parameters.
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	:*<math>0<b<1,0<c<1,\text{and}T\ge 0\,\!</math>		:*<math>0<b<1,0<c<1,\text{and}T\ge 0\,\!</math>

	:*<math>R\,\!</math> is the system's reliability at development time <math>T\,\!</math> or at launch number <math>T\,\!</math>, or stage number <math>T\,\!</math>.		:*<math>R\,\!</math> is the system's reliability at development time <math>T\,\!</math> or at launch number <math>T\,\!</math>, or stage number <math>T\,\!</math>
	:*<math>d\,\!</math> is the shift parameter.		:*<math>d\,\!</math> is the shift parameter
	:*<math>d+a\,\!</math> is the upper limit that the reliability approaches asymptotically as <math>T\to\infty\,\!</math>		:*<math>d+a\,\!</math> is the upper limit that the reliability approaches asymptotically as <math>T\to\infty\,\!</math>
	:*<math>d+ab\,\!</math> is the initial reliability at <math>T=0\,\!</math>		:*<math>d+ab\,\!</math> is the initial reliability at <math>T=0\,\!</math>
	:*<math>c\,\!</math> is the growth pattern indicator (small values of <math>c\,\!</math> indicate rapid early reliability growth and large values of <math>c\,\!</math> indicate slow reliability growth).		:*<math>c\,\!</math> is the growth pattern indicator (small values of <math>c\,\!</math> indicate rapid early reliability growth and large values of <math>c\,\!</math> indicate slow reliability growth)

	The modified Gompertz model is more flexible than the original, especially when fitting growth data with S-shaped trends.		The modified Gompertz model is more flexible than the original, especially when fitting growth data with S-shaped trends.