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General Information• The network calculus analyses presented in this document were created for the purpose of testing the Disco
Deterministic Network Calculator (DiscoDNC)1 – an open-source deterministic network calculus tool developedby the Distributed Computer Systems (DISCO) Lab at the University of Kaiserslautern.
• Naming of the individual network settings depicts the name of the according functional test for the DiscoDNC.
• The naming scheme used in this document is detailed in NetworkCalculus_NamingScheme.pdf.
• Arrival bounds for PmooArrivalBound.java and analyses using them are listed only if results differ fromPbooArrivalBound_Concatenation.java.
Changelog:Version 1.1 (2014-Dec-30):
• Streamlined the PMOO left-over latency T l.o.f
e2e
computation.
• Adaption to naming scheme version 1.1.
Version 2.0 beta (2015-Jul-11):
• Rework of Arrival Bounds documentation
– Parameters: see DiscoDNC’s computeArrivalBounds( Server server, Set<Flow> flows_to_bound, Flowflow_of_interest ).
– Bounding arrivals moved to the analysis requiring the specific bounds if they differ between flows of interest(may cause duplication).
– The algebraic derivations is included within many tabular bounding procedures. They are adapted toPbooArrivalBound_Concatenation.java, yet, in contrast to the current DiscoDNC code, they may reuseknown results.
• The naming scheme was slightly updated to include sets of servers S and sets of Flows F.
Remark:In this network setting, we have (s3, {f1} , f0) = (s3, {f1} , ;) and (s1, {f2} , f0) = (s1, {f2} , ;) becauseneither (cross-)flow f1 nor f2 interferes with the flow of interest f0 on multiple consecutive hops.
Remark:PmooArrivalBound.java will have the same result as PbooArrivalBound_Concatenation.javabecause f0 does not have cross-traffic interfering on multiple consecutive hops.
Analysis
TFA FIFO Multiplexing Arbitrary Multiplexing
s2
↵s2
= ↵f1s2
+ ↵f2s2
= �5,25 + �5,25
= �10,50
Df1s2
�s2 = b
s2
20 · [t� 20]
+= 50
t = 22
1
2
�s2 = ↵
s2
20 · [t� 20]
+= 10 · t+ 50
t = 45
Bf1s2
↵s2 (Ts2) = 20 · 10 + 50
= 250
s3
↵s3
= ↵f0s3
+ ↵f1s3
= �5,257 1316
+ �5,131 14
= �10,389 116
= ↵f0s3
+ ↵f1s3
= �5,313 89+ �5,166 2
3
= �10,480 59
Df1s3
�s3 = b
s3
20 · [t� 20]
+= 389
1
16
t = 39
29
64
�s3 = ↵
s3
20 · [t� 20]
+= 10 · t+ 480
5
9
t = 88
5
90
Bf1s3
↵s3 (Ts3) = 10 · 20 + 389
1
16
= 589
1
16
↵s3 (Ts3) = 10 · 20 + 480
5
9
= 680
5
9
Df1
=
3Xi=2
Df1si
= 61
61
64
=
3Xi=2
Df1si
= 185
5
9
Bf1
= max
i={2,3}Bf1
si
= 589
1
16
= max
i={2,3}Bf1
si
= 680
5
9
Separate Flow Analysis and PMOO AnalysisArrival Bounds
(s1, {f2} , ;) =: ↵f2s1
FIFO Multiplexing Arbitrary Multiplexing↵f2s1
= ↵f2 ↵ �l.o.f2s2
↵x(f2)s2 = �5,25
�l.o.f2s2
= �s2 ↵x(f2)
s2
= �R
l.o.f2s2 ,T
l.o.f2s2
Rl.o.fns2
hR
s2 � rx(f2)s2
i+= 20� 5
= 15
T l.o.fns2
�s2 = bx(f2)
s2
20 · [t� 20]
+= 25
t = 21
1
4
�s2 = ↵x(f2)
s2
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
= = �15,21 14
= �15,28 13
↵f2s1
= ↵f2 ↵ �l.o.f2s2
= �r
f2s1 ,b
f2s1
rf2s1
= 5
bf2s1
= ↵f2�T l.o.f2s2
�= 5 · 211
4
+ 25
= 131
1
4
= ↵f2�T l.o.f2s2
�= 5 · 281
3
+ 25
= 166
2
3
= = �5,131 14
= �5,166 23
(s3, {f0} , ;) =: ↵f0s3
FIFO Multiplexing Arbitrary Multiplexing↵f0s3
= ↵f0 ↵��l.o.f0s0
⌦ �l.o.f0s1
�↵x(f0)s0 = �0,0
�l.o.f0s0
= �s0 ↵x(f0)
s0
= �R
l.o.f0s0 ,T
l.o.f0s0
= �s0 = �20,20
↵x(f0)s1 = ↵f2
s1= �5,131 1
4= ↵f2
s1= �5,166 2
3
�l.o.f0s1
= �s1 ↵x(f0)
s1
= �R
l.o.f0s1 ,T
l.o.f0s1
Rl.o.f0s1
hR
s1 � rx(f0)s1
i+= 20� 5
= 15
T l.o.f0s1
�s1 = bx(f0)
s1
20 · [t� 20]
+= 131
1
4
t = 26
9
16
�s1 = ↵x(f0)
s1
20 · [t� 20]
+= 5 · t+ 166
2
3
t = 37
7
8
= = �15,26 916
= �15,37 78
�l.o.f0s0
⌦ �l.o.f0s1
= �l.o.f0
hs0,s1i
= �20,20 ⌦ �15,26 916
= �15,46 916
= �20,20 ⌦ �15,37 78
= �15,57 78
↵f0s3
= ↵f0 ↵��l.o.f0s0
⌦ �l.o.f0s1
� rf0s3
= 5
bf0s3
= ↵f0s1
⇣T l.o.f0
hs0,s1i
⌘= 5 · 46 9
16
+ 25
= 257
13
16
= ↵f0s1
⇣T l.o.f0
hs0,s1i
⌘= 5 · 577
8
+ 25
= 313
8
9
= = �5,257 1316
= �5,313 89
Remark:PmooArrivalBound.java will have the same result as PbooArrivalBound_Concatenation.javabecause f0 does not have cross-traffic interfering on multiple consecutive hops.
Analyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s2↵x(f1)s2 = ↵f2
= �5,25�l.o.f1s2
= �s2 ↵x(f1)
s2
= �R
l.o.f1s2 ,T
l.o.f1s2
= �15,21 14
= �15,28 13
s3
↵x(f1)s3 = ↵f0
s3= �5,257 13
16= �5,313 8
9
�l.o.f1s3
= �s3 ↵x(f1)
s3
= �R
l.o.f1s3 ,T
l.o.f1s3
Rl.o.f1s3
hR
s3 � rx(f1)s3
i+= 20� 5
= 15
T l.o.f1s3
�s3 = bx(f1)
s3
20 · [t� 20]
+= 257
13
16
t = 32
57
64
�s3 = ↵x(f1)
s3
20 · [t� 20]
+= 5 · t+ 313
8
9
t = 47
16
27
= = �15,32 5764
= �15,47 1627
�l.o.f1
hs2,s3i = �R
l.o.f1hs2,s3i,T
l.o.f1hs2,s3i
=
3Oi=2
�l.o.f1si
= �15,54 964
=
3Oi=2
�l.o.f1si
= �15,75 2527
Df1
�l.o.f1
hs2,s3i = bf1
15 ·t� 54
9
64
�+= 25
t = 55
155
192
�l.o.f1
hs2,s3i = bf1
15 ·t� 75
25
27
�+= 25
t = 77
16
27
Bf1
↵f1
⇣T l.o.f1
hs2,s3i
⌘= 5 · 54 9
64
+ 25
= 295
45
64
↵f1
⇣T l.o.f1
hs2,s3i
⌘= 5 · 7525
27
+ 25
= 404
17
27
PMOO Arbitrary Multiplexing
s2↵x(f1)s2
= ↵f2s2
= �5,25↵x̄(f1)s2
s3↵x(f1)s3
= ↵f0s3
= �5,313 89↵x̄(f1)
s3
�l.o.f1
hs2,s3i = �R
l.o.f1hs2,s3i,T
l.o.f1hs2,s3i
Rl.o.f1
hs2,s3i
=
^i2{2,3}
⇣R
si � rx(f1)si
⌘= (20� 5) ^ (20� 5)
= 15
T l.o.f1
hs2,s3i
=
Xi2{2,3}
Tsi +
bx̄(f1)si + rx(f1)
si · Tsi
Rl.o.f1e2e
!
= 20 +
25 + 5 · 2015
+ 20 +
313
89 + 5 · 2015
= 40 +
538
89
15
= 75
25
27
= = �15,75 2527
Df1
�l.o.f1
hs2,s3i = bf1
15 ·t� 75
25
27
�+= 25
t = 77
16
27
Bf1
↵f1
⇣T l.o.f1
hs2,s3i
⌘= 5 · 7525
27
+ 25
= 404
17
27
Flow f2
Total Flow AnalysisArrival Bounds
(s1, {f0} , ;) =: ↵f0s1
FIFO Multiplexing Arbitrary Multiplexing↵f0s1
= ↵f0 ↵ �l.o.f0s0
↵x(f0)s0 = �0,0
�l.o.f0s0
= �s0 ↵x(f0)
s0= �
s0 = �20,20
↵f0s1
= ↵f0 ↵ �l.o.f0s0
= �r
f0s1 ,b
f0s1
rf0s1
= 5
bf0s1
= ↵f0�T l.o.
s0
�= 5 · 20 + 25
= 125
= = �5,125
(s1, {f2} , ;) =: ↵f2s1
FIFO Multiplexing Arbitrary Multiplexing↵f2s1
= ↵f2 ↵ �l.o.f2s2
↵x(f2)s2 ↵f1
= �5,25
�l.o.f2s2
= �s2 ↵x(f2)
s2
= �R
l.o.f2s2 ,T
l.o.f2s2
Rl.o.f2s2
hR
s2 � rx(f2)s2
i+= 20� 5
= 15
T l.o.f2s2
�s2 = bx(f2)
s2
20 · [t� 20]
+= 25
t = 21
1
4
�s2 = ↵x(f2)
s2
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
= = �15,21 14
= �15,28 13
↵f2s1
= ↵f2 ↵ �l.o.f2s2
= �r
f2s1 ,b
f2s1
rf2s1
= 5
bf2s1
= ↵f2�T l.o.f2s2
�= 5 · 211
4
+ 25
= 131
1
4
= ↵f2�T l.o.f2s2
�= 5 · 281
3
+ 25
= 166
2
3
= = �5,131 14
= �5,166 23
Analysis
TFA FIFO Multiplexing Arbitrary Multiplexing
s2
↵s2
= ↵f1+ ↵f2
= �5,25 + �5,25
= �10,50
Df2s2
�s2 = b
s2
20 · [t� 20]
+= 50
t = 22
1
2
�s2 = ↵
s2
20 · [t� 20]
+= 10 · t+ 50
t = 45
Bf2s2
↵s2 (Ts2) = 20 · 10 + 50
= 250
s1
↵s3
= ↵f0s3
+ ↵f1s3
= �5,125 + �5,131 14
= �10,256 14
= ↵f0s3
+ ↵f1s3
= �5,125 + �5,166 23
= �10,291 23
Df2s1
�s1 = b
s1
20 · [t� 20]
+= 256
1
4
t = 32
13
16
�s1 = ↵
s1
20 · [t� 20]
+= 10 · t+ 291
2
3
t = 69
1
6
Bf2s1
↵s1 (Ts1) = 10 · 20 + 256
1
4
= 456
1
4
↵s1 (Ts1) = 10 · 20 + 291
2
3
= 491
2
3
Df2
=
2Xi=1
Df2si
= 55
5
16
=
2Xi=1
Df2si
= 114
1
6
Bf2
= max
i={1,2}Bf2
si
= 456
1
4
= max
i={1,2}Bf2
si
= 491
2
3
Separate Flow Analysis and PMOO AnalysisArrival Bounds
Remark:PmooArrivalBound.java will have the same result as PbooArrivalBound_Concatenation.javabecause f0 does not have cross-traffic interfering on multiple consecutive hops.
Analysis
TFA FIFO Multiplexing Arbitrary Multiplexing
s0
↵s0
= ↵f0+ ↵f1
= �5,25 + �5,25
= �10,50
Df0s0
�s0 = b
s0
20 · [t� 20]
+= 50
t = 22
1
2
�s0 = ↵
s0
20 · [t� 20]
+= 10 · t+ 50
t = 45
Bf0s0
↵s0 (Ts0) = 10 · 20 + 50
= 250
s2
↵s2
= ↵f0s2
+ ↵f1s2
= �5,231 14+ �5,131 1
4
= �10,362 12
= ↵f0s2
+ ↵f1s2
= �5,266 23+ �5,166 2
3
= �10,433 13
Df0s2
�s2 = b
s2
20 · [t� 20]
+= 362
1
2
t = 38
1
8
�s2 = ↵
s2
20 · [t� 20]
+= 10 · t+ 433
1
3
t = 83
1
3
Bf0s2
↵s2 (Ts2) = 10 · 20 + 362
1
2
= 562
1
2
↵s2 (Ts2) = 10 · 20 + 433
1
3
= 633
1
3
Df0P
i={0,2} Df0si
= 60
58
Pi={0,2} D
f0si
= 128
13
Bf0max
i={0,2} Bf0si
= 562
12 max
i={0,2} Bf0si
= 633
13
Separate Flow Analysis and PMOO AnalysisArrival Bounds
(s2, {f1} , f0) =: ↵f1s2
FIFO Multiplexing Arbitrary Multiplexing↵f1s2
= ↵f1 ↵��l.o.f1s0
⌦ �l.o.f1s1
�↵x(f0)s0 = �5,25
�l.o.f1s0
= �s0 ↵x(f1)
s0
= �R
l.o.f1s0 ,T
l.o.f1s0
Rl.o.f1s0
= 15
T l.o.f1s0
�s0 = bx(f1)
s0
20 · [t� 20]
+= 25
t = 21
1
4
�s0 = ↵x(f1)
s0
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
=
=�15,21 14
=�15,28 13
↵x(f1)s1 = �0,0
�l.o.f1s1
= �s1 ↵x(f1)
s1 = �s1 =�20,20
�l.o.f1s0
⌦ �l.o.f1s1
= �l.o.f1
hs0,s1i
= �R
l.o.f1hs0,s1i,T
l.o.f1hs0,s1i
= �l.o.f1s0
⌦ �s1
= �15,21 14⌦ �20,20
= �15,41 14
= �l.o.f1s0
⌦ �s1
= �15,28 13⌦ �20,20
= �15,48 13
↵f1s2
= ↵f1 ↵��l.o.f1s0
⌦ �l.o.f1s1
�= �
r
f1hs0,s1i,b
f1hs0,s1i
rf1hs0,s1i = 5
bf1hs0,s1i
↵f1
⇣T l.o.f1
hs0,s1i
⌘= 5 · 411
4
+ 25
= 231
1
4
↵f1
⇣T l.o.f1
hs0,s1i
⌘= 5 · 481
3
+ 25
= 266
2
3
= = �5,231 14
= �5,266 23
Remark:PmooArrivalBound.java will have the same result as PbooArrivalBound_Concatenation.javabecause f0 does not have cross-traffic interfering on multiple consecutive hops.
Remark:PmooArrivalBound.java will have the same result as PbooArrivalBound_Concatenation.javabecause f1 does not have cross-traffic interfering on multiple consecutive hops.
Analysis
TFA FIFO Multiplexing Arbitrary Multiplexing
s0
↵s0 = ↵f0
+ ↵f1= �5,25 + �5,25 = �10,50
Df0s0
�s0 = b
s0
20 · [t� 20]
+= 50
t = 22
1
2
�s0 = ↵
s0
20 · [t� 20]
+= 10 · t+ 50
t = 45
Bf0s0
↵s0 (Ts0) = 10 · 20 + 50
= 250
s1
↵s1 = ↵f1
s1= �5,131 1
4= ↵f1
s1= �5,166 2
3
Df1s1
�s1 = b
s1
20 · [t� 20]
+= 131
1
4
t = 26
9
16
FIFO per micro flow�s1 = b
s1
20 · [t� 20]
+= 166
2
3
t = 28
1
3
Bf1s1
↵s1 (Ts1) = 5 · 20 + 131
1
4
= 231
1
4
↵s1 (Ts1) = 5 · 20 + 166
2
3
= 266
2
3
s2
↵s2
= ↵f0s2
+ ↵f1s2
= �5,231 14+ �5,131 1
4
= �10,362 12
= ↵f0s2
+ ↵f1s2
= �5,266 23+ �5,166 2
3
= �10,433 13
Df1s2
�s2 = b
s2
20 · [t� 20]
+= 362
1
2
t = 38
1
8
�s2 = ↵
s2
20 · [t� 20]
+= 10 · t+ 433
1
3
t = 83
1
3
Bf1s2
↵s2 (Ts2) = 10 · 20 + 362
1
2
= 562
1
2
↵s2 (Ts2) = 10 · 20 + 433
1
3
= 633
1
3
Df1P2
i=0 �f1si
= 87
316
P2i=0 �
f1si
= 156
23
Bf1max
2i=0 B
f1si
= 562
12 max
2i=0 B
f1si
= 633
13
Separate Flow Analysis and PMOO AnalysisArrival Bounds
(s2, {f0} , ;) =: ↵f0s2
FIFO Multiplexing Arbitrary Multiplexing↵f0s2
= ↵f0 ↵ �l.o.f0s0
↵x(f0)s0 = �5,25
�l.o.f0s0
= �s0 ↵x(f0)
s0
= �R
l.o.f0s0 ,T
l.o.f0s0
Rl.o.f0s0
= 15
T l.o.f0s0
�s0 = bx(f0)
s0
20 · [t� 20]
+= 25
t = 21
1
4
�s0 = ↵x(f0)
s0
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
=
=�15,21 14
=�15,28 13
↵f0s2
= ↵f0 ↵ �l.o.f0s2
= ↵f0 ↵ �l.o.f0s2
rf0s2
= 5
bf0s2
↵f0�T l.o.f0s0
�= 131
14 ↵f0
�T l.o.f0s0
�= 166
23
= = �5,131 14
= �5,166 23
Remark:PmooArrivalBound.java will have the same result as PbooArrivalBound_Concatenation.javabecause f1 does not have cross-traffic interfering on multiple consecutive hops.