INTERNATIONAL TELECOMMUNICATION UNION
COM 11-R 205-E
February 2000
Original: English
TELECOMMUNICATION
STANDARDIZATION SECTOR
STUDY PERIOD 1997 - 2000
Question: 16/11
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STUDY GROUP 11 - REPORT R 205
SOURCE*:
STUDY GROUP 11
TITLE:
IMPLEMENTORS' GUIDE (12/99) FOR RECOMMENDATION Q.706 (03/93)
________
Summary
This Q.706 Implementors' Guide shows the MSU queueing delay and its standard deviation for the
extreme error case of signal unit error rate (not MSU error rate) of ~0.004. Graphs are provided for
several MSU sizes: 120 bits (i.e. 15 octets, the Figures 5 and 6/Q.706 value), 25, 50, 100, 150 and
the limiting case of 279 octets.
This Guide provides formulas for the Basic Error Correction (BEC) queueing delay in the presence
of disturbance which yield the correct delay when the link loading is zero.
To cater for long messages and high link loadings, the signalling loop delay is no longer assumed
constant, and is normalized to that of a link with zero loading.
Some typographical errors in the PCR formulas have been corrected, and two technical errors which
affected moderately the standard deviation of queueing delay in the presence of disturbance have
been corrected.
Annex B of Q.706 is expanded to include a brief derivation of the BEC formulas. Annex B.2
contains the calculation for the effect of variable loop delay on the PCR formulas.
This Guide aligns Q.706 with ITU-T Recommendation E.733, used for planning SS No. 7 networks.
* Contact:
TSB
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CONTENTS
Page
1
Introduction .................................................................................................................
3
1.1
References ...................................................................................................................
3
1.2
Acknowledgements .....................................................................................................
3
1.3
Background .................................................................................................................
3
1.4
Scope of the guide.......................................................................................................
4
1.5
Contacts ......................................................................................................................
4
1.6
Document history........................................................................................................
4
2
Changes to Q.706 Section 4.2 .....................................................................................
4
§4.2
§4.2.1
§4.2.2
§4.2.3
§4.2.4
§4.2.5
Queueing delays ..........................................................................................................
General ........................................................................................................................
Basic error correction queueing delays .......................................................................
Preventive cyclic retransmission queueing delays......................................................
Formulas .....................................................................................................................
Examples.....................................................................................................................
4
4
6
7
9
10
3
Replace Figure 5/Q.706 ..............................................................................................
13
4
Replace Figure 6/Q.706 ..............................................................................................
19
5
Replace Figure 8/Q.706 ..............................................................................................
25
6
Replace Table 6/Q.706 ...............................................................................................
26
7
Replace Table 7/Q.706 ...............................................................................................
27
8
Replace Figure 13/Q.706 ............................................................................................
28
9
Replace Figure 15/Q.706 ............................................................................................
29
10
Replace Figure 16/Q.706 ............................................................................................
30
11
In Section 5.2.5, insert a new Table 10bis ..................................................................
31
12
Replacement Annex B of Q.706 .................................................................................
31
B.1
B.1.1
B.1.2
B.1.3
Derivation of basic error correction (BEC) message signal unit (MSU)
queueing delays...........................................................................................................
Preliminary definitions and formulas .........................................................................
BEC queueing delay in the absence of disturbance ....................................................
BEC queueing delay in the presence of disturbance...................................................
31
31
34
36
B.2
PCR Loop delay ..........................................................................................................
38
B.3
Calculation of the kth moments of the MSU emission time for Tod ............................
39
B.4
Approximate calculation of the 95% - values of Tod ..................................................
40
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1
Introduction
1.1
References
[1]
ITU-T Recommendation Q.703 (07/96) - Specifications of Signalling System No. 7 –
Message Transfer Part, Signalling Link.
[2]
J.N. Daigle: Queueing Theory for Telecommunications, published by Addison – Wesley,
1992.
[3]
R.A. Skoog: Performance and Engineering of Common Channel Signaling Networks
supporting ISDN, in Traffic Engineering for ISDN and Planning, published by Elsevier
Science Publishers B.V. (North Holland), 1988.
[4]
L. Kleinrock: Queueing Systems, Volume 1, published by John Wiley & Sons, 1975.
[5]
D.R. Cox and H.D. Miller: The Theory of Stochastic Processes, published by Chapman and
Hall Ltd., 1967.
[6]
ITU-T Recommendation E.733 (11/98) - Methods for dimensioning resources in Signalling
System No. 7 networks.
[7]
ITU-T Recommendation G.821 - Error performance on an international digital connection
forming part of an integrated services digital network, Vol. III.
[8]
Y. Watanabe and Y. Ikeda: Traffic Characteristics of PCR method for CCITT Signalling
System No. 7, in Proceedings of the 10th International Teletraffic Congress 1993, Volume 1.
1.2
Acknowledgements
This Implementors' Guide was prompted by a liaison from ITU-T Study Group 2, which highlighted
the Q.706 BEC inconsistencies and errors. It provided the Laplace Transform of the MSU BEC
queueing delay distribution in the absence of disturbance, and an outline of the BEC virtual service
time derivation for the presence of disturbance. It also contained the mean and variance of the BEC
queueing delays in the absence and in the presence of disturbance, and proposed replacement
graphs for the Pu = 0.004 cases in Figures 5 and 6/Q.706.
1.3
Background
This Q.706 Implementors' Guide modifies Figures 5/Q.706 and 6/Q.706 to show for the extreme
error case the mean and standard deviation of queueing delays at the error rate when link
changeover would occur. Link changeover occurs at a signal unit error rate (not MSU error rate) of
~0.004, and queueing delays with this SU error rate are significantly higher than those shown for an
MSU error rate of 0.004. Graphs are provided for several MSU sizes: 120 bits (i.e. 15 octets, the
Figures 5 and 6/Q.706 value), 25, 50, 100, 150 and the limiting case of 279 octets.
This Guide provides formulas for the Basic Error Correction (BEC) queueing delay in the presence
of disturbance which yield the correct delay when the link loading is zero. Those currently in Q.706
have a term dependent on the MSU error rate even at zero MSU loading. The new formulas are
derived using the concept of "virtual service time" in the presence of disturbance applied to the
BEC formulas for the absence of disturbance.
Annex B of Q.706 is expanded to include a brief derivation of the BEC queueing delay in the
absence of disturbance, it also outlines the derivation and application of the BEC virtual MSU
service time in the presence of disturbance. The Annex now provides the Laplace-Stieltjes
transform of the MSU queueing delay, which is used to derive the mean and variance of the
queueing delay as functions of the first 3 moments of the MSU emission time on to the link. If the
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MSU length distribution is known, the transform of the MSU queueing delay can be inverted to
obtain the queueing time distribution, which would replace the approximate negative exponential
formula given.
The assumption of constant signalling loop delay has been replaced. A part of the loop delay is due
to the emission from the MSU receiving end of an acknowledgement and its absorption at the MSU
sending end. This acknowledgement could be a FISU, or an MSU. The acknowledgement type
depends upon the relative frequency of MSUs on the acknowledgement path, and the
emission/absorption time for an MSU depends upon the MSU length. The time before the
acknowledgement can be transmitted is the residual emission time of the signal unit before it, and
this depends upon the link loading and MSU lengths.
Section 4.2 has been split into separate subsections for BEC and Preventive Cyclic Retransmission
to improve clarity, and some technical and typographical errors have been corrected here and in the
rest of Q.706.
This Guide aligns Q.706 with ITU-T Recommendation E.733, used for planning SS No. 7 networks.
The Guide is intended to be an additional authoritative source of information for implementors, to
be read in conjunction with the Recommendation itself.
1.4
Scope of the guide
This guide records enhancements to the Recommendation in the following categories:
•
corrections of editorial errors;
•
corrections of technical errors;
•
extension to support ISDN User Part (ISUP) and Signalling Connection Control Part
(SCCP) message lengths.
1.5
Contacts
Editor Q.706
1.6
R.A. Adams
Lucent Technologies (UK)
Swindon Road
Malmesbury, Wiltshire SN16 9NA
United Kingdom
Tel:
+44 1666 83 2503
Fax:
+44 1666 83 2712
E-mail: raadams1@lucent.com
Document history
Version
Summary
12/99
New Implementors' Guide
2
Changes to Q.706 Section 4.2
§4.2
Queueing delays
§4.2.1 General
The MTP handles messages from different User Parts on a time-shared basis. With time-sharing,
signalling delay occurs when it is necessary to process more than one message in a given interval of
time. When this occurs, a queue is built up from which messages are transmitted in order of their
times of arrival.
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There are two different types of queueing delays: queueing delay in the absence of disturbances and
total queueing delay.
§4.2.1.1 Assumptions for derivation of the formulas
The queueing delay formulas are basically derived from the M/G/1 queue with priority assignment.
The assumptions for the derivation of the formulas in the absence of disturbances are as follows:
a)
the interarrival time distribution is exponential (M);
b)
the service time distribution is general (G);
c)
the number of servers is one (1);
d)
the service priority refers to the transmission priority within level 2 (see 11.2/Q.703);
however, service of link status signal units and independent flags are not considered;
e)
the signalling link loop delay is the time to receive an acknowledgement to an MSU. It
consists of a constant part, which is the sum of the signal unit MTP level 2 processing time
in signalling terminals and twice the link propagation time, and a variable part. The variable
part consists of the remaining emission time of the signal unit currently being sent in the
return direction (which is the delay to start transmitting the acknowledgement) plus the
emission/absorption time of the acknowledgement. For BEC, this acknowledgement could
be an MSU (with probability ), or a FISU (with probability (1-)) where  is the relative
frequency of MSU arrivals.
 
a
a  (1  a ).
f)
Tm
Tf
For an errored MSU in BEC, the variable part contains also the absorption time of the next
(assumed correct) SU after the MSU in error, and the remaining emission time of the SU
being sent in the forward direction before the MSU is retransmitted;
for PCR, the FISU traffic is defined as a3, hence the probability of the acknowledgement
being an MSU is  
(1  a 3 ).t f
a 3  (1  a 3 ).t f
, with a3 calculated assuming the mean value Tm for
TMSU (t).
For both BEC and PCR the link loading is assumed equal in both directions.
For the formulas in the presence of disturbances, the assumptions are as follows:
g)
Bit errors are randomly distributed, this makes transmission errors of message signal units
random;
h)
MSU transmission errors are statistically independent of each other;
i)
the additional delay caused by retransmission(s) of erroneous signal units is considered as a
part of the virtual service time of the concerned signal unit.
Furthermore, the formula of the proportion of messages delayed more than a given time is derived
from the assumption that the probability density function of the queueing delay distribution may be
exponentially decreasing where the delay time is relatively large.
§4.2.1.2 Factors and parameters
a)
The notations and factors required for calculation of the queueing delays are as follows:
Qa
Mean queueing delay in the absence of disturbances
Variance of queueing delay in the absence of disturbances
 a2
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Qt
Mean total queueing delay
 t2
Variance of total queueing delay
Proportion of messages delayed more than T
Traffic loading by message signal units (MSU) (excluding retransmission)
Traffic loading by MSUs including retransmission
Mean emission time of message signal units
Emission time of fill-in signal units (FISUs)
Signalling loop delay including processing time in signalling terminals at zero link
load, excluding FISU absorption and emission times 2Tf and excluding remaining
emission and absorption times Tf
Signalling loop delay (variable)
Error probability of message signal units
Error probability of signal units (FISUs and MSUs)
P(T)
a
aeff
Tm
Tf
K
TL
Pu
Psu
k1 
2nd moment of MSU emission time
k2 
3rd moment of MSU emission time
k3 
4th moment of MSU emission time
Tm2
Tm3
Tm4

T M( 2 )


Tm2
T M( 3)
Tm3
T M( 4 )
Tm4
NOTE - As a consequence of zero insertion at level 2 (see 3.2/Q.703), the length of the emitted
signal unit will be increased by approximately 1.6 per cent on average. However, this increase has
negligible effect on the calculation.
b)
The parameters used in the formulas are as follows:
t f  T f / Tm
t L  TL / Tm
§4.2.2 Basic error correction queueing delays
The functions, factors and parameters used are as follows:

( s) 
e
 st
. dTM (t ) ,
0
(s) is the Laplace-Stieltjes transform of the probability distribution function TM (t) of the
MSU emission time.
 M ( s )(1  a).
Tm
1
 s.T
.(1  e f ).
,
Tf
s.Tm  a.1  ( s )
M (s) is the Laplace-Stieltjes transform of the MSU BEC queueing delay in the absence of
disturbance.
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Pu
,
1  Pu
P (1  Pu )
 u
,
(1  Pu ) 2
N (1) 
N (2)
N ( 3) 
Pu (1  4 Pu  Pu2 )
,
(1  Pu ) 3
a
 
a  (1  a ).
Tm
Tf
TL(1)  K  2(1   )  1  a .T f  2  ak1 .Tm
2  2 k1Tm2  (1   )T f2  2.K ak1Tm  (1  a)T f 
 4.Tm  (1   )T f 
. ak1Tm  (1  a )T f 
 4 K Tm  (1   )T f  23 T f2 (1  a )  ak 2 Tm2  12 .(1  a ).T f  ak1Tm 2
T L(3)  K 3  6.Tm  (1   )T f .k1Tm2  (1   )T f2  2k 2 Tm3  (1   )T f3  2.K ak 2 Tm2  (1  a)T f2 
 4.ak 2 Tm2  (1  a)T f2 .Tm  (1   )T f  6 K 2 Tm  (1   )T f  3.K 2 ak1Tm  (1  a)T f 
 6 K Tm  (1   )T f 2  k1Tm2  (1   )T f2  6.Tm  (1   )T f 2  k1Tm2  (1   )T f2 .ak1Tm  (1  a)T f 
 12 K Tm  (1   )T f 
. ak1Tm  (1  a)T f  12 .(1  a)T f3  a.k 3 .Tm3   32 .K .(1  a)T f  ak1Tm 2
 3.(1   )T f  Tm 
. ak1Tm  (1  a)T f 2  ak1Tm  (1  a)T f .ak 2 Tm2  (1  a)T f2 

T L( 2)  K 2  2 Tm  (1   )T f


(1)
Tvir
 Tm 1  N (1)  N (1) TL(1)

 T N  3N
 3T T  .N


 

2
( 2)
Tvir
 TM(2) N (2)  2N (1)  1  2TmTL(1) N (2)  N (1)  TL(2) N (1)  TL(1) . N (2)  N (1)
(3)
Tvir
a eff  a 



(3)
( 2)
 3N (1)  1  3TM( 2) TL(1) N (3)  2 N ( 2)  N (1)  3TL( 2) Tm N ( 2)  N (1)
(1) 2
L
(3)
 N (1)  TL(3) N (1)  3TL( 2) TL(1) N ( 2)  N (1)  TL(1)
(3)
M
m



   N

3
(3)

 3N ( 2)  2 N (1)

(1)
Tvir
Tm
§4.2.3 Preventive cyclic retransmission queueing delays
Assumptions:
1)
the forced retransmission case is not considered;
2)
after the error occurs, the retransmitted signal units of second priority are accepted at the
receiving end until the sequence number of the last sent new signal unit is caught up by that
of the last retransmitted signal unit.
The functions, factors and parameters used in the formulas are as follows:

a eff .k1  

K 1  a eff
a 3  exp   a.(1   ).t f   

.t f 


Tm
2
2  


: traffic loading caused by fill-in signal
units.

(1  a 3 ).t f
a 3  (1  a 3 ).t f
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t L(1) 
t L( 2)
1  a eff
K 
 1   
Tm 
2
 K
 
 Tm

2
a k 


.t f     eff 1 


2 


 

 

K
  2.
.   (1   ).T f   k1  (1   )t 2f  K . a eff k1  (1  a eff )t f

T
m

 

   (1   )t f . a eff k1  (1  a eff )t f  13 . a eff k 2  (1  a eff )t 2f

3






 K 
K
   k 2  (1   )t 3f 
t L(3)  
. a eff k 2  (1  a eff )t 2f    (1   )t f . a eff k 2  (1  a eff )t 2f

T
T
m
 m
2
2
 K 
 K 
 .   (1   )t f  32 

 3.

 T  . (1  a eff ).t f  a eff .k1
 Tm 
 m
K
 3.
.  k1  (1   )t 2f  32 .  k1  (1   )t 2f . (1  a eff ).t f  a eff .k1
Tm



 3.


 







K
.   (1   )t f . (1  a eff ).t f  a eff .k1  14 . (1  a eff )t 3f  a eff .k 3
Tm

a eff  1  a 3
a z  1  a  a3
H 1  a.t L(1)
H 2  a.k1 .t L(1)  a 2 .t L( 2)
H 3  a.k 2 .t L(1)  3.a 2 .k1 .t L( 2)  a 3 .t L(3)
F1 
F2 
F3 
qa 
a.t L(1)
2
a.k1 .t L(1)
2
a.k 2 .t L(1)
2

a 2 .t L( 2)
a
3
2
.k1 .t L( 2)

a 3 .t L(3)
4
k1 ( a  az )  a3t f
2 (1  a )
k2 ( a  az )  a3t 2f
ak1
sa 
qa 
1  a
3(1  a )
( a  az ) k3  a3t 3f
3 ak1sa  2 ak2 qa
ta 

2 (1  a )
4 (1  a )
Z1 = 2  Pu(1  H1)
Z2 = 4k1  Pu(5k1  6H1  H2)
Z3 = 8k2  Pu(19k2  27k1H1  9H2  H3)
Y2 = sa  4k1  F2  2{qa(2  F1)  2F1}
Y3 = ta  8k2  F3  3{sa(2  F1)  qa(4k1  F2)  2F2  4k1F1}  12qaF1

1  a{2  Pu (1  at L(1) )}
qd 
2  q a  at L(1) / 2
aZ 2  Y2
2(1  aZ1 )
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sd 
aZ 2
aZ  Y3
qd  3
1  aZ1
3(1  aZ1 )
qb 
q a  1  F1
1 a
sb 
qc 
sc 
s a  k1  F2
(1  a)
2

2{q a (1  F1 )  F1 }
(1  a )
2

ak1 q b
(1  a) 2
q d  1  Pu (1  H1 )
1 a
s d  k1  Pu (3k1  H 2 )
(1  a)
Pv  Pu .a.
2
 2.
q d  Pu {q d (1  H 1 )  2 H 1 }
(1  a)
q a  2  at L(1) / 2 
a  a 2 t L(1)
1  Pu

1  2a
1  2a

2

ak1 q c
(1  a) 2




§4.2.4 Formulas
The formulas of the mean and the variance of the queueing delays are described in Table 2. The
proportion of messages delayed more than a given time Tx is:
 T  Qx   x 
P(Tx )  exp   x

x


where Qx and x denote the mean and the standard deviation of queueing delay, respectively. This
approximation is better suited in absence of disturbances. In the presence of disturbances the actual
distribution may be deviated further. Relation between P(Tx) and Tx is shown in Figure 2.
TABLE 2/Q.706
Queueing delay formula
Error
correction
method
Disturbance
Absent
BEC
Present
Absent
PCR
Present
Qa 
T ( 2) 
1 
a
.  T f 
 M 
2 
1  a Tm 
 2a
T ( 2) 
T ( 3)
1  a
1
a

  
 M   
 M
12 4  1  a Tm 
3 1  a Tm
Qt 
a eff
T ( 2) 
1 

.  T f 
 vir
(1) 
2 
1  a eff Tvir

 2t
( 3)
a eff
T ( 2) 
Tvir
1  a eff
1
  

  
 vir

(1) 
(1)
12 4  1  a eff Tvir
3 1  a eff Tvir

2
T f2
2
T f2
Qa
 qa
Tm
Qt
 (1  Pu  Pv ) qa
Tm
+ Puqb + Pvqc
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Variance 2
Mean Q
 a2
Tm2
t2
Tm2
 s a  q a2
 (1  Pu  Pv ) sa
 Pusb  Pv sc 
Qt2
Tm2
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P (TX ), probability of MSU delayed more than Tx
10 –1
10 –2
10 –3
10 –4
10 –5
10 –6
Qx
Q x + 8 x
Q x + 4 x
Qx + 12 x
Queueing delay time (Tx )
Qx
x
s
T1156380-93/d02
Mean queueing delay (see Figure 3)
Standard deviation (see Figure 4)
FIGURE 2/Q.706
Probability of message signal unit delayed more than Tx
FIGURE 2/Q.706...[D02] = 10.5 CM
§4.2.5 Examples
Assuming the traffic models given in Table 3, examples of queueing delays are calculated as listed
in Table 4.
In many cases, the bit error probability of signalling data links (terrestrial and satellite) is better than
10–7, at which rate the effects on queueing delays are minimal.
NOTE - The values for models A and B in Table 3 were determined based on TUP messages. With
the increase of the effective message length, using ISUP and TC, these values may be expected to
be increased. Sheets 2 to 6 of Figures 5/Q.706 and 6/Q.706 show the effect of increased MSU
length on BEC queueing delay.
TABLE 3/Q.706
Traffic model
Model
A
Message length (bits)
120
104
304
constant
Percent
100
92
8
100
Mean message length
(octets)
15
15
25-279
k1
1.0
1.2
1.0
k2
1.0
1.9
1.0
k3
1.0
3.8
1.0
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B
C
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TABLE 4/Q.706
List of examples
Figure
Error control
Queueing delay
Disturbance
Model
3
BEC/PCR
Mean
Absent
A and B
4
BEC/PCR
Standard
deviation
Absent
A and B
5 sheet 1
BEC
Mean
Present
A
6 sheet 1
BEC
Standard
deviation
Present
A
7
PCR
Mean
Present
A
8
PCR
Standard
deviation
Present
A
5 sheets 2-6
BEC
Mean
Present
C
6 sheets 2-6
BEC
Standard
deviation
Present
C
ms
2.5
2.0
1.2
R
PC
Mean queueing delay (Qa )
1.0
sic
Ba
1.5
TL
0 ms
= 60
0.8
0.6
1.0
30
ms
0.4
s
30 m
0.5
0.2
0
0
0.1
0.2
0.3
0.4
a, traffic loading of MSU
0.5
Q a /Tm , mean queueing delay in multiples of T m
Tm = 1.875 ms (120 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
0 Erlangs
0.6
T1156390-93/d03
Model A
Model B
FIGURE 3/Q.706
Mean queueing delay of each channel of traffic
in absence of disturbance
FIGURE 3/Q.7...[D03] = 12.5 CM
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ms
2.5
T m = 1.875 ms (120 bits and 64 kbit/s)
T f = 0.75 ms (48 bits and 64 kbit/s)
Standard deviation ( )
Ba
sic
2.0
1.5
TL = 600 ms
1.0
PCR
0.5
T L = 30 ms
0
0
0.1
0.2
0.3
0.4
a, traffic loading of MSU
0.5
0.6 Erlangs
T1156400-93/d04
Model A
Model B
FIGURE 4/Q.706
Standard deviation of queueing delay of each channel of traffic
in absence of disturbance
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3
Replace Figure 5/Q.706 with:
18
16
14
12
Mean total queueing delay (Qt) ms
10
8
Psu =0.004
6
Pu = 0.001
4
2
Pu = 0
0
0
0.2
0.4
0.6
0.8
MSU channel load, Erlangs
FIGURE 5/Q.706 (sheet 1 of 6)
Mean total queueing delay of each channel of traffic Basic error correction method, 15 octet MSUs
Tm= 1.875 ms (120 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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20
Mean total queueing delay (Qt) ms
15
10
Psu =0.004
Pu = 0.001
5
Pu = 0
0
0
0.2
0.4
0.6
0.8
MSU channel load, Erlangs
FIGURE 5/Q.706 (sheet 2 of 6)
Mean total queueing delay of each channel of traffic Basic error correction method, 25 octet MSUs
Tm= 3.125 ms (200 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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30
25
Mean total queueing delay (Qt) ms
20
15
Psu =0.004
10
Pu = 0.001
5
Pu = 0
0
0
0.2
0.4
0.6
0.8
MSU channel load, Erlangs
FIGURE 5/Q.706 (sheet 3 of 6)
Mean total queueing delay of each channel of traffic Basic error correction method, 50 octet MSUs
Tm= 6.25ms (400 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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50
40
Mean total queueing delay (Qt) ms
30
20
Psu =0.004
Pu = 0.001
10
Pu = 0
0
0
0.2
0.4
0.6
0.8
MSU channel load, Erlangs
FIGURE 5/Q.706 (sheet 4 of 6)
Mean total queueing delay of each channel of traffic Basic error correction method, 100 octet MSUs
Tm= 12.5 ms (800 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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80
70
60
Mean total queueing delay (Qt) ms
50
40
30
Psu =0.004
20
Pu = 0.001
10
Pu = 0
0
0
0.2
0.4
0.6
0.8
MSU channel load, Erlangs
FIGURE 5/Q.706 (sheet 5 of 6)
Mean total queueing delay of each channel of traffic Basic error correction method, 150 octet MSUs
Tm= 18.75 ms (1 200 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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160
140
120
Mean total queueing delay (Qt) ms
100
80
60
Psu =0.004
40
Pu = 0.001
20
Pu = 0
0
0
0.2
0.4
0.6
0.8
MSU channel load, Erlangs
FIGURE 5/Q.706 (sheet 6 of 6)
Mean total queueing delay of each channel of traffic Basic error correction method, 279 octet MSUs
Tm= 34.875 ms (2 232 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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4
Replace Figure 6/Q.706 with:
Standard deviation of queueing delay () ms
25
20
15
10
Psu = 0.004
Pu = 0.001
5
Pu = 0
0
0
0.2
0.4
0.6
0.8
a, channel load in Erlangs
FIGURE 6/Q.706 (sheet 1 of 6)
Standard deviation of queueing delay of each channel of traffic Basic error correction method, 15 octet MSUs
Tm= 1.875 ms (120 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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25
Standard deviation of queueing delay () ms
20
15
Psu = 0.004
10
Pu = 0.001
5
Pu = 0
0
0
0.2
0.4
0.6
0.8
a, channel load in Erlangs
FIGURE 6/Q.706 (sheet 2 of 6)
Standard deviation of queueing delay of each channel of traffic Basic error correction method, 25 octet MSUs
Tm = 3.125 ms (200 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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35
Standard deviation of queueing delay () ms
30
25
20
15
Psu = 0.004
Pu = 0.001
10
Pu = 0
5
0
0
0.2
0.4
0.6
0.8
a, channel load in Erlangs
FIGURE 6/Q.706 (sheet 3 of 6)
Standard deviation of queueing delay of each channel of traffic Basic error correction method, 50 octet MSUs
Tm = 6.25 ms (400 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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60
Standard deviation of queueing delay () ms
50
40
30
Psu = 0.004
20
Pu = 0.001
10
Pu = 0
0
0
0.2
0.4
0.6
0.8
a, channel load in Erlangs
FIGURE 6/Q.706 (sheet 4 of 6)
Standard deviation of queueing delay of each channel of traffic Basic error correction method, 100 octet MSUs
Tm = 12.5 ms (800 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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100
Standard deviation of queueing delay () ms
80
60
Psu = 0.004
40
Pu = 0.001
20
Pu = 0
0
0
0.2
0.4
0.6
0.8
a, channel load in Erlangs
FIGURE 6/Q.706 (sheet 5 of 6)
Standard deviation of queueing delay of each channel of traffic Basic error correction method, 150 octet MSUs
Tm = 18.75 ms (1 200 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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Standard deviation of queueing delay () ms
200
150
100
Psu = 0.004
Pu = 0.001
50
Pu = 0
0
0
0.2
0.4
0.6
0.8
a, channel load in Erlangs
FIGURE 6/Q.706 (sheet 6 of 6)
Standard deviation of queueing delay of each channel of traffic Basic error correction method, 279 octet MSUs
Tm = 34.875 ms (2 232 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
TL = 30 ms loop delay with zero link loading
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5
Replace Figure 8/Q.706 with:
14
Tm = 1.875 ms (120 bits and 64 kbits/s)
Tf = 0.75 ms (48 bits and 64 Kbits/s)
12
Pu=0.004
0.004
0.001
Standard deviation (sigma) ms
10
0.001
8
TL = 600 ms
6
TL = 30 ms
4
2
Pu = 0
0
0
0.1
0.2
0.3
0.4
0.5
a, traffic loading of MSU Erlangs
FIGURE 8/Q.706
Standard deviation of queueing delay of each channel of traffic Preventive cyclic retransmission error correction method, 15 octet MSUs
Tm = 1.875 ms (120 bits and 64 kbit/s)
Tf = 0.75 ms (48 bits and 64 kbit/s)
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6
Replace Table 6/Q.706 with:
TABLE 6/Q.706
Outgoing link delay Tod with basic error correction method
Outgoing link delay (ms)
a [Erl]
0.2
TL [ms]
30
Disturbance
No
Yes
600
No
Yes
0.4
30
No
Yes
600
No
Yes
MSU – Length (Bytes)
Value
15
23
50
140
279
Mean
2.7
4.0
8.2
21.5
39.6
95%
9.3
14.0
30.1
66.0
61.6
Mean
2.8
4.1
8.4
22.0
40.5
95%
10.8
15.5
31.7
69.9
69.0
Mean
2.7
4.0
8.2
21.5
39.6
95%
9.3
14.0
30.1
66.0
61.6
Mean
35.2
36.9
42.3
59.7
83.6
95%
277.7
283.6
303.5
357.5
391.1
Mean
3.5
5.2
10.8
27.6
46.9
95%
12.3
18.6
40.0
88.7
87.1
Mean
3.8
5.5
11.4
31.0
50.1
95%
15.0
21.4
43.6
98.2
104.8
Mean
3.5
5.2
10.8
27.6
46.9
95%
12.3
18.6
40.0
88.7
87.1
Mean
109.7
113.0
124.0
158.2
199.7
95%
580.0
591.3
629.3
738.7
834.6
a Traffic loading by MSUs (excluding retransmission).
TL Signalling link loop delay at zero link load, with mean bit error probability < 10-5 in the presence of
disturbance.
In this table, the mean values and 95% points for all MSU lengths for 30 ms and 600 ms TL, for
both a = 0.2 and 0.4, with disturbance present, have been replaced where appropriate.
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Replace Table 7/Q.706 with:
TABLE 7/Q.706
Outgoing link delay Tod with preventive cyclic retransmission method
Outgoing link delay (ms)
a [Erl]
0.2
TL [ms]
30
Disturbance
No
Yes
600
No
Yes
0.4
30
No
Yes
600
No
Yes
MSU – Length (Bytes)
Value
15
23
50
140
279
Mean
4.2
6.1
11.4
23.9
39.1
95%
12.6
18.9
38.3
73.4
60.0
Mean
4.2
6.2
11.9
25.0
42.1
95%
12.9
19.6
40.8
84.4
87.9
Mean
4.2
6.5
14.1
35.7
56.0
95%
12.7
19.4
42.2
93.2
86.2
Mean
5.2
7.6
(a)
(a)
(a)
95%
29.7
38.8
(a)
(a)
(a)
Mean
5.0
7.6
15.6
32.4
45.5
95%
15.0
22.9
48.7
98.9
84.0
Mean
5.7
9.0
20.7
56.1
92.9
95%
25.8
42.1
107.2
318.2
444.5
Mean
5.0
7.7
16.7
41.8
63.9
95%
15.0
23.0
50.0
111.5
108.9
Mean
63.7
74.7
(a)
(a)
(a)
95%
386.6
443.1
(a)
(a)
(a)
NOTE (a) - The signal unit error rate for the effective message traffic link load would cause the link to
fail.
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Replace Figure 13/Q.706 with:
100
90
80
Mean outgoing link delay (ms)
70
60
BEC:
50
TL = 30 ms
a = 0.4 Erl
Pb = 0/1 e – 5
PCR: TL = 30 ms
a = 0.4 Erl
Pb = 0
40
PCR:
30
TL = 30 ms
a = 0.4 Erl
PCR: TL = 30 ms
a = 0.2 Erl
Pb = 0
Pb = 1.0 e – 5
PCR: TL = 30 ms
a = 0.2 Erl
P b = 1.0 e – 5
20
10
BEC: TL = 30 ms
a = 0.2 Erl
Pb = 0/1 e – 5
0
15 23
50
140
MSU mean length (byte)
BEC
TL
a
Pb
279
T1133000-91/d13
Basic error correction method / PCR = preventive cyclic retransmission method
Signalling loop propagation time including processing time in signalling terminal
Traffic load by MSUs excluding retransmission
Bit error probability
FIGURE 13/Q.706
Mean outgoing link delay – TL = 30 ms
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Replace Figure 15/Q.706 with:
ERROR! OBJECTS CANNOT BE CREATED FROM EDITING FIELD CODES.
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10
Replace Figure 16/Q.706 with:
1500
1400
1300
1200
1100
95% level values of outgoing link delay (ms)
1000
900
BEC: TL = 600 ms
a = 0.4 Erl
Pb = 1.0 e-5
PCR: TL = 600 ms
a = 0.4 Erl
Pb = 1.0 e-5
800
700
BEC: TL = 600 ms
a = 0.2 Erl
Pb = 1.0 e-5
PCR: TL = 600 ms
a = 0.2 Erl
Pb = 0
600
500
BEC: TL = 600 ms
a = 0.4 Erl
Pb = 0
400
PCR: TL = 600 ms
a = 0.2 Erl
Pb = 1.0 e-5
300
BEC: TL = 600 ms
a = 0.2 Erl
Pb = 0
200
PCR: TL = 600 ms
a = 0.4 Erl
Pb = 0
100
0
0
50
100
150
200
250
300
MSU mean length (octets)
FIGURE 16/Q.706
95% level value of outgoing link delay – TL = 600 ms
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11
In Section 5.2.5, insert a new Table 10bis
TABLE 10bis/Q.706
Maximum PCR link loading and 95% level value for Tod for long MSUs with TL = 600 ms
Bit error rate
Maximum link load (Erlangs)
95% level value for Tod (ms)
279 octet MSUs
140 octet MSUs
279 octet MSUs
140 octet MSUs
1.0  10
-6
0.45
0.45
1 100
500
1.5  10-6
0.45
0.45
1 400
660
2.0  10
-6
0.32
0.45
300
750
3.0  10-6
0.23
0.45
210
1 150
3.5  10
0.21
0.45
200
1 300
~3.8  10-6
< 0.2
< 0.15
~200
~100
~5.0  10
< 0.17
< 0.1
~180
~90
-6
-6
Note that this table is calculated assuming constant length MSUs. For MSUs of variable length, the
maximum link loading would be reduced.
12
Replacement Annex B of Q.706
(This annex forms an integral part of this Recommendation).
B.1
Derivation of basic error correction (BEC) message signal unit (MSU) queueing
delays
B.1.1
Preliminary definitions and formulas
This section is included to summarize the concepts used in deriving the BEC queueing delay.
In the following, the construct Prob{event} is used to denote the probability of the event happening.
Thus Prob{ X  x} denotes the probability that the random variable X is less than the value x, etc.
Laplace-Stieltjes transform and expectation of a continuous random variable (references [2]
and [4])
The Laplace-Stieltjes transform of a probability distribution is defined as follows:
if a continuous non-negative random variable X has the probability distribution function F(x), such
that F ( x)  Prob{X  x} , then the Laplace-Stieltjes transform of F(x), ( s), is defined as:

( s) 
e
 sx
. dF ( x) , where the lower limit of integration includes the origin of x.
0

The expectation of the random variable X is defined as E[ X ]   x.dF ( x) , which is the mean value
0
of X. Note that:
(s)  E[e sX ]
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Moments of a continuous distribution (reference [4])
The nth moment of a probability distribution function F ( x)  Prob{X  x} is defined as:

F
( n)


d n( s)
dsn
x n . dF ( x )  ( 1)n
0
 E[ X n ]
s0

The mean value of X is thus F(1), and its variance is
  x  F  . dF ( x)  F
(1) 2
(2)
 
 F (1)
2
.
0
Moments of a discrete distribution (reference [4])
If Y is a non-negative integer random variable, with probability function p( y)  Prob{Y  y}, the kth
moment of Y is defined as:
Y (k ) 

 p( y). y k  E[Y k ]
y 0
Generating function (references [2] and [4])
If Y is a non-negative integer random variable, with probability function p( y)  Prob{Y  y}, the
generating function of Y, G(z), is defined as:
G( z) 

 p( y).z y  E[ z Y ], where the expectation of zY is
E[ z Y ] .
y 0
If T denotes a random time variable, with probability distribution function F(t) such that
F (t )  Prob{T  t}, and if there is a Poisson process of rate , with Y denoting the number of events of
this process to occur during time T, the generating function of Y, G(z), is:

G( z )  E[ z ] 
Y
 E[ z
Y
| T  t ].dF (t ) , where the E expression in the integral is the expectation of zY
0
conditional upon the time interval T having the value t. For a Poisson process of rate , the number
of events in a time t is y with probability Prob{Y  y| T  t} 
E[ z Y | T  t ] 
e t .(t ) y
, hence:
y!

 E[ z Y | T  t , Y  y]. Prob{Y  y| T  t}
y 0


zy.
y 0
e  t .(t ) y
,
y!
 e   .(1 z ).t

Hence G( z)   e .(1 z ).t .dF (t ) (.[1  z]) , where (s) is the Laplace-Stieltjes transform of F(t).
0
Derivation of ordinary M/G/1 system queue size generating function (references [2],
[4] and [5])
For an M/G/1 system (i.e. a system with Poisson arrivals, general service time and one server) in
which each arriving item's service time is an independent identically distributed random variable,
the generating function for the queue size is calculated by considering the number in the queue left
behind by a departing item (i.e. by an item completing service). The set of departure instants form
an embedded Markov chain - that is, the state of the system can be determined at one departure
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instant from the state at the previous departure instant and the number of arrivals in the meantime.
For any system that changes state by one step (up or down), the limiting distribution for the number
of items in the system left behind by a departure is the same as the limiting distribution of the
number found in the system by an arriving item. Since in our case the arrival process is Poissonian,
and a Poisson arrival observes the system as a random observer, the statistics of the queue length
measured at the set of departure instants are the same as those for arrival instants, and hence provide
the solution for all points in time.
If we denote the number of items left behind by the nth departing item as qn, the number of arriving
items during the nth service by vn, and putting (q – 1)+ = max(q – 1,0), we have:
q n1  (q n  1)   v n1 , where n+1 and (qn – 1)+ are independent random variables. As n  ,
q n 1 q n  q, v n 1  v n  v, and the generating function Q(z) for the number in the queue is given
by:
Q( z )  E[ z q ]  E[ z ( q 1)

v

]  E[ z ( q 1) ]  E[ z v ]  P( z )  V ( z ) ,
with P(z) and V(z) the generating functions

for (q  1) and v respectively.
v represents the number of arrivals during a service time, hence from the formula for G(z) given
previously, V ( z)  ( .[1  z]), where ( s) is the Laplace-Stieltjes transform of the service time. Now,

P( z )  E[ z ( q 1) ] 

 E[ z (q1)
x 0

1 
| q  x].Prob{q  x} Prob{q  0}   E[ z q | q  x].Prob{q  x}
z x 1

1 
1
 Prob{q  0}   E[ z q | q  x].Prob{q  x}  Prob{q  0}
z x 0
z

Hence:
1
1
P( z )  (1  )  Prob{q  0}  Q( z ) , so
z
z
Q( z ) 
(1  z ).(1  a ).  (  .[1  z ])
 (  .[1  z ])  z
Here Prob{q  0}  1  a , where a is the server utilization, and a    .
d ( s)
ds
s0
  . Tm .
Tm is the mean service time.
Probability distribution function and transform of the residual time to the next event
(references [5], [2] and [4])
Consider the following diagram, where an observation is made at an arbitrary time t. The time x to
the next event (the residual time) has a probability distribution function Rt(x). The probability
density function of Rt(x) is rt(x), which gives the probability that the next event occurs in the
interval ( x, x  x] as r t ( x). x .
t
t+x
time axis
y
event
Rt = PDF of time to next event
observation
event
Suppose the kth event occurs at a time X1  X 2 ... X k , with { X1 , X 2 ,..., X k } independent positive
random variables. Let X2, X3, ... be identically distributed with probability density functions f(x)
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and probability distribution functions F(x) such that f ( x) 
dF ( x)
, and let X1 have the probability
dx
density function f1 (x) which might be the same as f(x). Let the probability density function of events
be h(t), then:
h(t ) 
lim Prob{event in (t , t  t ]}
t  0
t
From the diagram, the next event is assumed to occur in the interval ( x, x  x] after t, this might be
the first event with probability f1 (t  x). x , or there could have been an event at a time interval y
before t, and the next event occurs in the interval ( x  y, x  y  x] . Hence:
t

r t ( x). x  f 1 (t  x). x  h(t  y). f ( y  x). dy. x
0
As t  , h(t ) 
1

, r t ( x)  r ( x) ,
R t ( x )  R( x ) ,
f 1 (t  x )  0
, where  is the mean event
interarrival time. Hence, in equilibrium,
r ( x) 
1



 f ( y  x). dy

1  F ( x)

0
.
The Laplace-Stieltjes transform of R(x) is:

 r ( s) 

0
e  s.x dR( x)



e  s.x r ( x). dx
0


e
 s. x
0

1  F ( x)

. dx

1  ( s)
. s
where (s) is the Laplace-Stieltjes transform of F(x).
B.1.2
BEC queueing delay in the absence of disturbance
The basis of this calculation is the M/G/1 system with server multiple vacations, described in
reference [2].
In the BEC method (reference [1]), fill - in signal units (FISUs) are transmitted continually if there
are no MSUs to transmit. The first MSU to arrive waits until the FISU has finished emission, and
then starts being emitted itself (i.e. it starts "service"). Denote as "type 1" MSUs those MSUs
arriving whilst a FISU is being emitted, and "type 2" MSUs as the rest (type 2 MSUs arrive whilst
an MSU is being emitted). In order to calculate the probability generating function for the number
of queued MSUs, it is helpful to modify (temporarily) the order of MSU service: any type 2 MSU
arriving whilst a type 1 MSU is being emitted is emitted before any waiting type 1 MSUs. This
trick enables the calculation of the queue length, which is independent of the order of service (i.e.
the order of MSU transmission).
Thus, the start of emission of a type 1 MSU starts a "busy period", which lasts for the time of
emission of this MSU and emission of all its associated type 2 MSU arrivals. The transmitter will
transmit FISUs when the busy period associated with the last type 1 MSU to arrive during the
previous FISU emission ends. Just one type 1 MSU is emitted in a busy period.
Consider the queue size when an MSU has just been emitted. Since the MSU arrival process is
Poissonian, and the emission times of successive MSUs are independent, the future evolution of the
system depends only upon the MSU queue size immediately following an MSU departure and
future MSU arrivals. In addition, the number of type 1 MSUs present is independent of the number
of type 2 MSUs present, and if we write the total number of MSUs left behind by the departure of
an arbitrary MSU as M, with M1 as the number of type 1 MSUs, and M2 the number of type 2
MSUs, we have:
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M = M1 + M2, and generating function N MSU ( z)  E[ z M ]  E[ z M1 M 2 ]  E[ z M1 ]  E[ z M 2 ]  N1 ( z)  N 2 ( z) .
N2 (z) is analogous to Q(z) in the previous section.
N1 (z) is given by the number of MSUs arriving during the residual emission time of a FISU, since
the distribution of the number of type 1 MSUs arriving from the Poisson arrival process is the same
as the distribution of the number of type 1 MSUs left behind by the departure of an arbitrary MSU.
N 1 ( z) 
1   F (  .[1  z])
 .[1  z]. T f
, where F ( s) is the Laplace-Stieltjes transform of the FISU emission time

probability distribution, with Tf the FISU emission time.  F ( s)   e st .  (t  Tf ). dt  e sTf , so:
0
N 1 ( z) 
  .[1 z ].T f
1 e
 .[1  z ]. T f
Hence:
  .[1 z ].T
N MSU ( z) 
f
(1  z).(1  a ).  (  .[1  z]) 1  e

 (  .[1  z])  z
 .[1  z]. T f
The size of the queue, and hence the generating function for it, does not depend on the order of
service, so it is the same for first come, first transmitted MSUs. Hence, denoting the transform of
the total time spent in the system by an MSU by (s) (this is the transform of the queueing plus
emission time - i.e. the transform of the sojourn time of the MSU),
N MSU ( z)  ( .[1  z])
Hence:
( s)  (1  a ).
Tm
 ( s)
 s .T
.(1  e f ).
Tf
s. Tm  a .1   ( s)
, since a   . Tm
Now, the sojourn time S equals the queueing time Q of the MSU plus its emission time TMSU on to
the link, and its queueing time is independent of its emission time. Hence:
( s)  E[e  s.S ] E[e  s (Q TMSU ) ] E[e  sQ ]  E[e  sTMSU ] M (s)  (s), where  M (s)
is the Laplace-Stieltjes
transform of the MSU queueing delay time. Hence:
 M ( s )(1  a).
Writing  M (s) 
g ( s)
,
h( s )
Tm
1
 s.T
.(1  e f ).
Tf
s.Tm  a.1  ( s )
 M ( s).h( s)  g ( s) , the various moments of  M (s) can be determined by
differentiating the latter equation sufficiently and taking limits as s  0 . The first moment is the
mean queueing delay in the absence of disturbance,
Qa  
d M ( s )
ds
T ( 2)
1 
a
 . T f 
 M

1  a Tm
s 0 2 




, where TM(2)  k1  Tm2 is the second moment of the MSU
emission time.
The variance of the queueing delay in the absence of disturbance can be derived from the mean and
the second moment of  M (s) to be:
 2a
2
T ( 2) 
T ( 3)
1  a
1
a

  
 M   
 M ,
12 4  1  a Tm 
3 1  a Tm
T f2
where TM( 3)  k 2  Tm3 is the third moment of the MSU
emission time.
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B.1.3
B.1.3.1
BEC queueing delay in the presence of disturbance
Virtual emission time of an MSU, and its moments (reference [3])
In the BEC method, MSUs that have been transmitted are held in the retransmission buffer until
acknowledged. An error in a received MSU causes the receiver to request retransmission, and this
MSU plus any subsequently transmitted MSUs are then re-sent in order. Hence the delay caused by
the error in the MSU can be viewed as an increase in its emission (service) time.
If the MSU error rate is Pu, the signalling loop delay is TL and the number of retransmissions of the
MSU is N, then the virtual service (emission) time of the MSU is:
Tvir  TMSU  N .TMSU  TL (1)  ... TL ( N ) , with N different and independent values of TL.
For randomly distributed MSU errors, the probability that the number N of retransmissions is n is:
Prob{N  n}  Pun  (1  Pu ) . The kth moment of N is thus
N (1) 
N ( 2) 
N (3) 
N (k ) 

 nk  Pun  (1  Pu )
, and hence:
n 0
Pu
,
1  Pu
Pu (1  Pu )
(1  Pu ) 2
,
Pu (1  4 Pu  Pu2 )
(1  Pu ) 3
The Laplace-Stieltjes transform of the distribution of TL can be derived as follows:
The acknowledgement to an MSU is carried in the backward sequence number of either a FISU or
an MSU, the choice depends upon the relative frequency of MSUs and FISUs on the link in the
return direction. A decision to send a negative acknowledgement (NACK) is made after receiving
the next SU with correct cyclic redundancy check (CRC) bits after receiving an errored SU. Before
the NACK is sent (in the next SU to be emitted in the return direction), the SU's backward sequence
number must be inserted, and its CRC bits computed and inserted. The CRC bits can be computed
and inserted during SU emission. After the current SU has been emitted, the SU carrying the NACK
is sent. At the NACK receiving end, the CRC bits must be recomputed and the SU absorbed
completely before it is accepted as a NACK.
Hence, the overall signalling loop delay for an errored SU consists of the propagation time of the
MSU, plus the absorption time of the next (assumed error-free) SU, plus its processing time at level
2 at the receiving end, plus the time to finish emission of the SU before the NACK, plus the NACK
SU emission time, plus the propagation delay on the link, plus the MTP level 2 processing time of
the NACK, plus the remaining emission time of the SU currently being transmitted in the forward
direction. The erroneous MSU can start retransmission after the now currently emitting SU.
In the above, errors in the next SU after the errored SU in the forward direction, and errors in the
return direction, extend the loop delay time, but this is a second order effect and is ignored here.
From the above,

 
T L* ( s )  E e  sTL  E e
f

E e
 sSU r 

 f  m
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
 s K  2(1 )T f TMSU (1) TMSU ( 2)  2.SU r

. 1 e
 sT f
   
. 1  ( s )
m
f
m
  f T f   m Tm 
s.

  f   m 
,
   
 

  E e  sK . E 2 e  sSU  .E 2 e  sSU r  ,


E e  sSU   (1   ).e
 sT f
  .( s )
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hence:
 sK

 (1   )e
TL* ( s )
e
f 
1 a
a
, m 
Tf
Tm
 sT f
  

  1  e  sT f    1  ( s )
f
m
  .( s )  
s


2

2
,


SUr is the residual time to transmit an SU before the NACK, which can be a FISU or MSU. TMSU is
the time to emit and absorb the NACK if it is in an MSU. Now, the residual time for SU emission is
independent of the time to absorb the NACK, so each component of the expectation is independent,
and hence the total expectation is the product of the expectations of the components.
The first 3 moments of TL are:
TL(1)  K  2(1   )  1  a .T f  2  ak1 .Tm
2  2 k1Tm2  (1   )T f2  2.K ak1Tm  (1  a)T f 
 4.Tm  (1   )T f 
. ak1Tm  (1  a )T f 
 4 K Tm  (1   )T f  23 T f2 (1  a )  ak 2 Tm2  12 .(1  a ).T f  ak1Tm 2

T L( 2)  K 2  2 Tm  (1   )T f


 


T L(3)  K 3  6. Tm  (1   )T f . k1Tm2  (1   )T f2  2 k 2 Tm3  (1   )T f3  2.K ak 2 Tm2  (1  a)T f2






 4. ak 2 Tm2  (1  a)T f2 . Tm  (1   )T f  6 K 2 Tm  (1   )T f  3.K 2 ak1Tm  (1  a)T f




 6.Tm  (1   )T f 
.ak1Tm  (1  a)T f 
3
3
 12 K Tm  (1   )T f 
. ak1Tm  (1  a)T f  12 .(1  a)T f  a.k 3 .Tm   32 .K .(1  a)T f  ak1Tm 2
 3.(1   )T f  Tm 
. ak1Tm  (1  a)T f 2  ak1Tm  (1  a)T f .ak 2 Tm2  (1  a)T f2 
 6 K Tm  (1   )T f
2
a
where  
a  (1  a ).
 k1Tm2
 (1   )T f2
2
 k1Tm2
 (1   )T f2
is the relative frequency of MSUs.
Tm
Tf
The kth moment of the distribution of Tvir is given by:
(k )
k
Tvir
 E[Tvir
]  (1) k
*
d k Tvir
( s)
ds k
*
, where Tvir
( s) is the Laplace-Stieltjes transform of the distribution of
s 0
Tvir. The transform is obtained as follows:

 



*
Tvir
( s | N  n)  E e  sTvir (t| N n)  E e  s ( n 1)TMSU .E e  sTL (t1 ) ... TL (t n )  [n  1]s . TL* ( s)
*
Tvir
( s) 

n
 Tvir* (s | N  n).Prob{N  n}   Prob{N  n}.[n  1]s .TL* (s)


n 0
n 0
n
The first 3 moments of the distribution of Tvir are thus:


(1)
Tvir
 Tm 1  N (1)  N (1) TL(1)


 1 3T

 

2
( 2)
Tvir
 TM(2) N (2)  2N (1)  1  2TmTL(1) N (2)  N (1)  TL(2) N (1)  TL(1) . N (2)  N (1)

(3)
Tvir
 TM(3) N (3)  3N ( 2)  3N (1)
 
2

( 2) (1)
M TL
N
(3)

 2 N ( 2)  N (1)  3TL( 2) Tm N ( 2)  N (1)

   N
 3Tm TL(1) . N (3)  N (1)  TL(3) N (1)  3TL( 2) TL(1) N ( 2)  N (1)  TL(1)
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


3
(3)
 3N ( 2)  2 N (1)

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In addition, aeff   .Tvir(1)   .Tm 
(1)
Tvir
T (1)
 a  vir
Tm
Tm
aeff and these moments of Tvir replace a and the corresponding moments of TM in the formulas for Qa
and a2 to give:
Qt 
 2t 
B.1.3.2
Tf2
12
( 2) 
a eff
Tvir
1
 Tf 


(1) 
2 
1  a eff Tvir

2

( 3)
1  aeff
T ( 2) 
1
aeff
Tvir


 vir



(1)
(1)
4  1  aeff Tvir
3 1  aeff Tvir

Relationship between SU error rate and MSU error rate (references [3] and [6])
From the previous section, a eff
  eff
PSU  
  eff   f


 f
.Pu  

  eff   f


 Pu T L(1)
1 

Tm

 a
1  Pu




. The SU error rate is given by:

a eff Pu
6 Pu / m
.PFISU 
 (1  a eff ).
, where

Tm
T

a eff  (1  a eff )
a eff  (1  a eff ) m
Tf
Tf
eff Tm   f T f  1, eff Tm  aeff and PFISU is the FISU error rate.
Here, the mean MSU length is m octets. If the bit error rate is Pb, then, since a FISU is 6 octets long,
Pb 


m 1  a eff
1  1  a eff a eff 


  PSU and Pu  8m  Pb   a eff 

8  6
m 
6

   P


SU
and at changeover PSU = ~0.004. These equations furnish a quadratic equation for Pu which can be
solved for the particular MSU length and link loading.
B.2
PCR Loop delay
As in the BEC case, the loop delay is influenced by the link loading and MSU size. The FISU
loading on the link is calculated as a function of the loop delay, but the loop delay is a function of
the FISU loading. The equation for the FISU loading can be solved by iteration.
The first three moments of the loop delay replace its corresponding powers in the formulae for H
and F, this is shown below. The Laplace-Stieltjes transforms for H*(s) and F*(s) in the original PCR
paper contained TL as a constant. H*(s) is the Laplace-Stieltjes transform of the distribution of the
time for retransmitting all the MSUs in the retransmission buffer (RTB), F*(s) is the LaplaceStieltjes transform of the distribution of the time for transmitting MSUs in the RTB ahead of an
MSU previously transmitted with errors.
If we write H*(s | TL) and F*(s | TL) for the original paper's H*(s) and F*(s) respectively, we can cater
for TL being a variable by taking the expectations of H*(s | TL) and F*(s | TL) with respect to TL. Thus
we have:
TL (t )  K  (1   ).T f   .TMSU (t )  RSU (t ) , where  is the relative frequency of MSUs, TMSU (t) is the
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emission/absorption time of an MSU and RSU(t) is the remaining emission time of the SU before the
SU containing an acknowledgement is sent. Now, denoting the Laplace-Stieltjes transform of TL(t)
by T*L(s), we have:



TL* ( s )  E e  s.TL (t )  e  sK .  .( s )  (1   ).e
 sT f
. .1  (s) s .1  e  since the remaining emission
 s.T f
f
time of the SU being emitted is independent of the absorption/emission time of the next SU to be
sent.  is given by:
.Tm   f .T f  1 ,  

,
 f
with f the FISU generation rate.
The various moments of TL can be obtained from this transform using TL( k )  (1) k
1
H * (s | TL )  e mTL 1( s ) ,
1
F * (s | TL )  H * ( s |  .TL ) d  e mTL 1( s ) d 


0
0
d k TL* ( s )
ds k
. Now,
s 0
1  e mTL 1( s )
 m .TL 1  (s)
*
Hence the new H (s) is given by:
  1  ( s ).K  (1  ).T f   .TMSU (t )  R SU (t )
  1  ( s ). .TMSU (t )  (1  ).T f 
H * (s)  E e m
 e   m 1  ( s ).K .E e m
.E  e   m 1  ( s ).RSU 



H * ( s )  e  Kz .  .( z )  (1   ).e
 zT f

. . 1  ( z) z .1  e  , where z  
 z .T f
f

 1  (s) ,
m.
a   m .Tm
The first three moments of H*(s) can be derived from this, and they are equal to the values in the
original paper with TL replaced by TL(1), TL2 replaced by TL(2), etc. To obtain the moments of the new
1
F*(s), replace TL in the original same order moment for H by TL .  . d (i.e.
1

2
TL), TL2 by
0
1

TL2 .  2 .d (i.e.
1 .T 2
3 L
), etc., and then replace TL by TL(1), TL2 by TL(2) etc.
0
B.3
Calculation of the kth moments of the MSU emission time for Tod
The kth moment of the MSU emission time is required for the factors, k1, k2 and k3 in 4.2:

TM( k )

 t .t
k
M
(t )dt
tM (t) = distribution density function
0
t M (t ) 
dTM (t )
, TM (t) is the
dt
probability distribution function.
T constant (= Tm):
tM (t) =  (t – Tm)
T negative exponential distributed:
t M (t ) 
t
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

With the Laplace-Stieltjes transformation ( s)  e  st . t M (t )dt :
0


t k t M (t )dt ( 1) k .
0
d k  ( s)
ds k
s 0
T constant:
(s) = exp(–s  Tm)
so:
2nd moment = TM(2)  Tm2
3rd moment = TM(3)  Tm3
4th moment = TM(4)  Tm4
1
1  s  Tm
T negative exponential distributed:
( s) 
so:
2nd moment = 2. Tm2
3rd moment
= 6. Tm3
4th moment = 24. Tm4
This leads to the following values for the factors k1, k2 and k3 (see 4.2):
general:
T constant:
k1 =
TM( 2 )
Tm2
k2 =
TM( 3)
Tm3
k3 =
TM( 4 )
Tm4
k1 = 1
k2 = 1
k3 = 1
T negative exponential distributed:
k1 = 2
k2 = 6
k3 = 24
B.4
Approximate calculation of the 95% - values of Tod
Tod.95% = Qt.95% + Tm.95%
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Qt.95% and Tm.95% are calculated with the help of the approximation in 4.2.4:
 Tmean  
 T
P(T95% )  exp   95%





P(T T95%) =
1 – P(T95%) with
P(T T95%) =
0.95

Tmean + 2  with
Tmean
mean value of T
=
standard deviation of T
where T95%

=
This gives:
Qt.95%

Qt + 2 
Tm.95%

3Tm
for mean MSU length 15, 23 or 50 bytes
(MSU length distribution as negative exponential)
Tm.95%

2/3  3Tm + 1/3  Tm
(2/3 negative exponential distribution + 1/3
constant)
and
Tm.95%
= 7/3  Tm
for mean MSU length 140 bytes
= Tm
for mean MSU length 279 bytes
(MSU length distribution as constant).
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