INTERNATIONAL TELECOMMUNICATION UNION
COM 12 – D 196 – E
TELECOMMUNICATION
STANDARDIZATION SECTOR
English only
STUDY PERIOD 2001-2004
Original: English
13/12
Question:
Geneva, 24-31 March 2004
STUDY GROUP 12 – DELAYED CONTRIBUTION 196
Source:
Nortel Networks (Canada)
Title:
A Proposed Method for the Accumulation of End-to-End Delay Variation
1.
Introduction
Suppose packets flow through n network segments so that the end-to-end transit delay T has the
form:
n
T   Tk
k 1
with Tk the delay associated with the k-th segment. We seek a method to estimate the variation of
the end-to-end delay T from information about the variation of the component delays Tk .
Delayed Contribution COM 12 D.156 [1] identifies that variances for independent distributions are
additive and, based on this property, suggests using delay variance as a simple measure of (end-toend) delay variation.
Delayed Contribution COM 13 D.529 [2] reviews the difficulties associated with accumulating
component delay measurements into end-to-end quantities when these are all expressed as
quantiles. It also proposes a heuristic method for using component delay percentiles to determine
whether end-to-end delay requirements will be met.
The present Contribution extends these Contributions by proposing a more rigorous approximation
method for combining component quantiles of delay distribution into end-to-end values and has the
properties of relative simplicity of computation and reasonable accuracy.
2.
Definition of Delay Variation
For the purposes of this contribution, delay variation refers to the quantiles t of the random variable
T as defined through the condition
Pr(T  t )  p
where the probability p has a pre-specified value (say p = 0.9, 0.95, or 0.99) that is fixed for the
discussion.
Contact:
Romas Armolavicius
Nortel Networks
Canada
Tel:
+1 613-763-5881
Fax:
Email: romas@nortelnetworks.com
Attention: This is not a publication made available to the public, but an internal ITU-T Document intended only for use by the
Member States of the ITU, by ITU-T Sector Members and Associates, and their respective staff and collaborators in their ITU related
work. It shall not be made available to, and used by, any other persons or entities without the prior written consent of the ITU-T.
-2COM 12 – D 196 – E
3.
Available Information
We assume that we have determined (through measurement):
2
 the mean  k  E Tk  and variance  k2  ETk   k 
for each component Tk . We also assume that we have determined (through measurement) either one
or the other of:
3
 the skewness  k  ETk   k   k3

the quantile t k satisfying Pr(Tk  t k )  p .
In practice, the skewness is unlikely to be measured directly. At best, the third central moment
3
ETk   k  will be measured from which the skewness can be computed as defined above. Most
likely the skewness must be estimated from other data: later in this note we show how skewness can
estimated from measured values of the quantiles t k .
If we also make the assumption that:
 the random variables Tk are all mutually independent
then the mean, variance, and third central moment of T are equal to the sums of the component
means, variances , and third moments respectively. More precisely:

n
n
  E T    E Tk     k
k 1


k 1
n
n
k 1
k 1
 2  ET   2   ETk   k 2   k2
n
E T      E Tk   k 
3
3
k 1
4.
Demonstration of Computational Procedures
The problem under consideration can now be restated as follows: estimate the quantile t of the endto-end delay T as defined by the condition
Pr(T  t )  p .
We consider an example corresponding to packets serially traversing three “Domains” (numbered 1,
2, and 3 in the tables below). There are therefore three components of delay that need to be
accumulated. Three versions of the computations for estimating end-to-end quantiles of delay are
shown depending on whether third moments or quantiles of delay are available. Additional
technical details underlying the methods can be found in the appendix.
4.1
Example 1: Measured Third Moments Available
Step 1
Measure the mean, variance, and third central moment for each delay component. Assuming
independence of the components, the end-to-end mean, variance, and third moment are just the
sums of the corresponding component statistics (add down the middle 3 columns below). The
skewness of the end-to-end delay is computed as shown in the rightmost column (the skewness for
each individual component is not needed for this example and its computation is not shown).
-3COM 12 – D 196 – E
Domain
Mean
Variance
Third
Moment
1
1  1
 12  0.5
1  0.5
2
2  2
 22  1.0
 2  1.0
3
3  3
 32  1.5
 3  1.5
End-to-End
 6
 2  3.0
  3 .0
Skewness
     3 2  0.577
The estimate of a target quantile, say the 99-th percentile ( p  0.99 ) of end-to-end delay proceeds
as follows. If x p is the value satisfying ( x p )  p where  denotes the standard normal (mean 0,
variance 1) distribution function, then x0.99  2.326 is the value satisfying ( x0.99 )  0.99
(tabulated values of this function are readily available) and:



t       x p  1  x 2p 
6





0.577

t  6  3.0  2.326 
1  2.3262 
6




t  10.77
Therefore, with probability 0.99, the end-to-end delay T is less than (approximately) t  10.77 .
4.2
Example 2: Measured Quantiles of Delay Available
Step 1
Measure the mean and variance for each delay component. Estimate the mean and variance of the
end-to-end delay by summing the columns as shown.
Domain
Mean
Variance
1
1  1.0
 12  0.5
2
 2  2.0
 22  1.0
-4COM 12 – D 196 – E
3
 3  3.0
 32  1.5
End-to-End
  6 .0
 2  3.0
Step 2
Measure the quantiles for each delay component at a given probability, say p  0.99 . Estimate the
corresponding skewness and third moment using the formula shown at the top of the table below.
As for example 1, x0.99  2.326 is the value satisfying ( x0.99 )  0.99 where  denotes the
standard normal (mean 0, variance 1) distribution function.
Domain
Skewness
xp 
 k  6
tk   k
k
1 x
2.326 
1
1  6
 2  6
3
3  6
End-to-End   
3.32  1
1   1   13 2  0.459
 0.947
 2   2   23 2  0.947
 0.783
 3   3   33 2  1.438
6.55  3
1.5
2
1  2.326
32
 1.297
5.02  2
1. 0
2
1  2.326 
2.326 
 k   k   k3 2
2
p
0.5
2
1  2.326
2.326 
2
Third Moment
 0.547
  1  1  1  2.844
-5COM 12 – D 196 – E
Assuming independence of the delay components, the third moment of the end-to-end delay is just
the sum of the component third moments. The end-to-end skewness is computed by dividing by
 3 2 as shown in the table.
Step 3
The estimate of the 99-th percentile ( p  0.99 ) of end-to-end delay t as follows.




2 
t      x p  1  x p 
6



0.547

t  6  3.0  2.326 
1  2.326 2 
6




t  10.73
Therefore, with probability 0.99, the end-to-end delay T is less than (approximately) t  10.73 .
4.3
Example 3: Third Central Moments and Quantiles of Delay Available
This demonstrates a method essentially equivalent to those shown in the previous two examples but
not requiring explicit use of the standard normal probabilities. It is independent of the specific
probability p (except that the value of p must be sufficiently high that the approximations used are
valid). It is demonstrated here for the case where both quantiles of delay and third central moments
are available.
Step 1
Measure the mean, variance, and third central moment for each delay component. Assuming
independence of the components, the end-to-end mean, variance, and third moment are just the
sums of the corresponding component statistics (add down the middle 3 columns below). The
skewness for each component and end-to-end is computed from the third moments as shown in the
rightmost column.
Domain
Mean
Variance
Third
Moment
Skewness
1
1  1
 12  0.5
1  0.5
 1  1   13 2  1.414
2
2  2
 22  1.0
 2  1.0
 2   2   23 2  1.0
3
3  3
 32  1.5
 3  1.5
 3   3   33 2  0.817
End-to-End
 6
 2  3.0
  3 .0
     3 2  0.577
-6COM 12 – D 196 – E
Step 2
Measure the delay for a common quantile for each of the components. For example, suppose that
we select the 99-th percentile, and that we measure the following quantities.
Domain
Delay t
1
t1  3.32
2
t 2  5.02
3
t 3  6.55
Step 3
Estimate the 99-th percentile of end-to-end delay t as follows. For each component of delay,
compute the expression shown in the table and sum the results. The top row shows the general
formula used for the table entries: the definitions  k   k 3 and    3 are used to simplify the
expressions.
Domain
 k   k2 
 t  k
1  2 k   k
 k

   k2  1

k
1
 1.297   3.32  1   1.297 
  
1  2
  
 1
 3   0.5   3 
1  0.5 
 1.143
1.297 3
2
2
 0.947   5.02  2   0.947 
  
1  2
  
 1
 3   1.0   3 
 2  1.0 
 2.304
0.947 3
2
3
 0.783   6.55  3   0.783 
  
1  2
  
 1
 3   1.5   3 
 3  1.5 
 3.4674
0.783 3
2
End-to-End   1   2   3  6.914
-7COM 12 – D 196 – E
Then compute the 99-th percentile of end-to-end delay t as follows.
t

2
2

  
 1  2   1   2

  





t  10.71
Therefore, with probability 0.99, the end-to-end delay T is less than (approximately) t  10.71 .
Note:
The estimates of the 99-th percentile of end-to-end delay produced by the computations in example
1 (t = 10.77), example 2 (t = 10.73), and example 3 (t = 10.71) differ slightly but are all close to the
exact value computed from convolution of the underlying distributions (t = 10.75) .
5.
Conclusions and Recommendations
This Contribution extends earlier Contributions by proposing a more rigorous method for
combining component quantiles of delay distribution into end-to-end values, and has the properties
of relative simplicity of computation and reasonable accuracy. It is recommended that this approach
be adopted in a new Recommendation for estimating end to end performance in IP networks from a
user perspective, as discussed during the Q.13/12 meeting of September 2003.
6.
References
1. COM 12-D.156-E “Estimating end to end performance in IP networks”, Nortel Networks, Sept.
2003
2. COM 13-D.529-E “Proposed New Appendix XI/Y.1541 on Concatenating QoS Values”, USA,
Feb. 2004
3. “A Note on the Normal Power Approximation”, Colin B. Ramsey, ASTIN Bulletin, Vol 2. No
1, April 1991
-8COM 12 – D 196 – E
Appendix 1
Mathematical Background
If the distributions of each of the components Tk were known in detail the distribution of the endto-end delay T could be computed using convolutions. Convolutions are challenging in practice:
most implementations will rely on Laplace transform techniques including methods to invert
transforms numerically to recover the underlying probability distributions. To use this method we
would also have to make assumptions about the exact nature of the component distributions.
Instead, we employ an alternative method that uses the available information without requiring
additional assumptions or complex methods.
The basic idea is to transform a random variable T with known mean  , variance  2 , and
skewness  into a symmetric random variable Z which is standard normal (mean 0, variance 1) or
nearly so. One such method, called the Normal Power approximation (see [3] ) works as follows:
T 
 Define the standardized variable X 


 The Normal Power approximation states that X  Z  ( Z 2  1) where Z is a standard
6
normal (mean 0, variance 1) random variable
Once the details are worked through we get the approximation:
1
3
t   
Pr(T  t )  
9  6 
 2  


 
  

where  is the cumulative standard normal distribution function
x
( x) 


e
 x2
2
2
dx .
Although the values of this function are readily available, a more transparent relationship can be
derived that eliminates all reference to  and allows the quantile t to be directly computed from the
component quantiles t k .
In fact, since the probabilities in all the quantile definitions Pr(Tk  t k )  p , Pr(T  t )  p have the
common value p, if we define x p to be the unique value satisfying x p   p , then we have:
1
k
1

 tk  
9  6 k 
 k

3
  k2 
 x p and


k

3
t   
2
9  6 
     xp.

  
If we multiply the above by  k2 and  2 respectively and add over all the components we deduce
from the additivity of variances of independent distributions that:
-9COM 12 – D 196 – E
2 
t  
2
1  2  
   1




where we have set   
n
  k2 
 t  k
1  2 k   k
 k
k 1
and  k 

   k2  1

k
k
. Although this looks complex, it requires only simple
3
3
algebra to compute the end-to-end quantile t from the components t k and the available measured
quantities.
Special Cases
In the approximation
1
3
t  
Pr(T  t )  
9  6 
 2  


 
  

if we let   0 we produce the result
t  
Pr(T  t )  

  
corresponding to the case where T has a normal distribution with mean  , variance  2 . If we let all
the skewness terms   0 ,  k  0 the algebraic expression of the previous section reduces to:
n
  t       k  t k   k  .
k 1
Some further manipulation removes the variances to produce
n
t   2   t k   k 2 .
k 1
This shows that when the component delays Tk are normally distributed with mean  k , and
variance  k2 , then the corresponding quantiles follow a composition law similar to that for
variances.
This composition law for normal variates can also be derived directly. The algebraic expression of
the previous section can be viewed as a generalization of this particular composition law.
Estimating Skewness from Quantiles
Consider a random variable T whose mean  and variance  2 are known and where the quantile t
in Pr(T  t )  p is known but where the skewness  is not known. Using the Normal Power
approximation we have
1
3
t  
Pr(T  t )  
9  6 
 2    p


 
  

- 10 COM 12 – D 196 – E
From tabulated values of the standard normal distribution function  we can find the unique value
x p satisfying x p   p . Therefore
1

t   2 3
9  6 
     xp

  
This can be solved for  producing
  6
xp 
t

1  x 2p
___________________