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Network Calculus Tests – Feed Forward Network Settings
Version 2.0 beta (2015-Jul-11)
© Steffen Bondorf 2013 - 2015, Some Rights Reserved.
Except where otherwise noted, this work is licensed under Creative Commons Attribution-ShareAlike 4.0.See http://creativecommons.org/licenses/by-sa/4.0/
1
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.
• Minor consistency fixes for variable names.
1http://disco.cs.uni-kl.de/index.php/projects/disco-dnc
FeedForward_1SC_3Flows_1AC_3Paths
s0 s1
s2 s3f1
f0
f2
S = {s0, s1, s2, s3} with
�s0 = �
s1 = �s2 = �
s3 = �Rsi ,Tsi
= �20,20, i 2 {0, 1, 2}
F = {f0, f1, f2} with
↵f0= ↵f1
= ↵f2= �
r
fn,b
fn = �5,25, n 2 {0, 1, 2}
Flow f0
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
= �R
l.o.f0s0 ,T
l.o.f0s0
= �s0 = �20,20
↵f0s1
= ↵f0 ↵ �l.o.f0s0
= �r
f0s1 ,b
f0s1
rf0s1
= 5
bf0s1
= ↵f0�T l.o.
�= 5 · 20 + 25
= 125
= = �5,125
(s3, {f1} , ;) =: ↵f1s3
FIFO Multiplexing Arbitrary Multiplexing(s1, {f2} , ;) =: ↵f2s1
=
: ↵fnsi
with (n, i) 2 {(1, 3) , (2, 1)}↵fnsi
= ↵fn ↵ �l.o.fns2
↵x(fn)s2 = �5,25
�l.o.fns2
= �s2 ↵x(fn)
s2
= �R
l.o.fns2 ,T
l.o.fns2
Rl.o.fns2
hR
s2 � rx(fn)s2
i+= 20� 5
= 15
T l.o.fns2
�s2 = bx(fn)
s2
20 · [t� 20]
+= 25
t = 21
1
4
�s2 = ↵x(fn)
s2
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
= = �15,21 14
= �15,28 13
↵fnsi
= ↵fn ↵ �l.o.fns2
= �r
fnsi
,b
fnsi
rfnsi
= 5
bfnsi
= ↵fn(T l.o.fn
s2)
= 5 · 2114
+ 25
= 131
1
4
= ↵fn(T l.o.fn
s2)
= 5 · 2813
+ 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
�(reuse of previous result)
= ↵f0 ↵ �l.o.f0s0
↵ �l.o.f0s1
= ↵f0s1↵ �l.o.f0
s1
↵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
↵f0s1
= �5,125
↵f0s3
= ↵f0 ↵ �l.o.f0s1
= �r
f0s3 ,b
f0s3
rf0s3
= 5
bf0s3
= ↵f0s1
�T l.o.f0s1
�= 5 · 26 9
16
+ 125
= 257
13
16
= ↵f0s1
�T l.o.f0s1
�= 5 · 377
8
+ 125
= 313
8
9
= = �5,257 1316
= �5,313 89
Analysis
TFA FIFO Multiplexing Arbitrary Multiplexing
s0
↵s0 = ↵f0
s0= �5,25
Df0s0
�s0 = b
s0
20 · [t� 20]
+= 25
t = 21
1
4
FIFO per micro flow�s0 = b
s0
20 · [t� 20]
+= 25
t = 21
1
4
Bf0s0
↵s0 (Ts0) = 20 · 5 + 25
= 125
s1
↵s1
= ↵f0s1
+ ↵f2s1
= �5,125 + �5,131 14
= �10,256 14
= ↵f0s1
+ ↵f2s1
= �5,125 + �5,166 23
= �10,291 23
Df0s1
�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
Bf0s1
↵s1 (Ts1) = 10 · 20 + 256
1
4
= 456
1
4
↵s1 (Ts1) = 10 · 20 + 291
2
3
= 491
2
3
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
Df0s3
�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
Bf0s3
↵s3 (Ts3) = 10 · 20 + 389
1
16
= 589
1
16
↵s3 (Ts3) = 10 · 20 + 480
5
9
= 680
5
9
Df0
=
Xi={0,1,3}
Df0si
= 93
33
64
=
Xi={0,1,3}
Df0si
= 178
17
36
Bf0
= max
i={0,1,3}Bf0
si
= 589
1
16
= max
i={0,1,3}Bf0
si
= 680
5
9
Separate Flow Analysis and PMOO AnalysisArrival Bounds
(s3, {f1} , f0) =: ↵f1s3
FIFO Multiplexing Arbitrary Multiplexing(s1, {f2} , f0) =: ↵f2s1
=
: ↵fnsi
with (n, i) 2 {(1, 3) , (2, 1)}↵fnsi
= ↵fn ↵ �l.o.fns2
↵x(fn)s2 = �5,25
�l.o.fns2
= �s2 ↵x(fn)
s2
= �R
l.o.fns2 ,T
l.o.fns2
Rl.o.fns2
hR
s2 � rx(fn)s2
i+= 20� 5
= 15
T l.o.fns2
�s2 = bx(fn)
s2
20 · [t� 20]
+= 25
t = 21
1
4
�s2 = ↵x(fn)
s2
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
= = �15,21 14
= �15,28 13
↵fnsi
= ↵fn ↵ �l.o.fns2
= �r
fnsi
,b
fnsi
rfnsi
= 5
bfnsi
= ↵fn�T l.o.fns2
�= 5 · 211
4
+ 25
= 131
1
4
= ↵fn�T l.o.fns2
�= 5 · 281
3
+ 25
= 166
2
3
= = �5,131 14
= �5,166 23
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.
Analyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s0↵x(f0)s0 = �0,0
�l.o.f0s0
= �s0 ↵x(f0)
s0 = �20,20
s1
↵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
9
= = �15,26 916
= �15,37 79
s3
↵x(f0)s3 = ↵f1
s3= �5,131 1
4= ↵f1
s3= �5,166 2
3
�l.o.f0s3
= �s3 ↵x(f0)
s3
= �R
l.o.f0s3 ,T
l.o.f0s3
Rl.o.f0s3
hR
s3 � rx(f0)s3
i+= 20� 5
= 15
T l.o.f0s3
�s3 = bx(f0)
s3
20 · [t� 20]
+= 131
1
4
t = 26
9
16
�s3 = ↵x(f0)
s3
20 · [t� 20]
+= 5 · t+ 166
2
3
t = 37
7
9
= = �15,26 916
= �15,37 79
�l.o.f0
hs0,s1,s3i = �R
l.o.f0hs0,s1,s3i,T
l.o.f0hs0,s1,s3i
=
Oi={0,1,3}
�l.o.f0si
= �15,73 18
=
Oi={0,1,3}
�l.o.f0si
= �15,95 59
Df0
�l.o.f0
hs0,s1,s3i = bf0
15 ·t� 73
1
8
�+= 25
t = 74
19
24
�l.o.f0
hs0,s1,s3i = bf0
15 ·t� 95
5
9
�+= 25
t = 97
2
9
Bf0
↵f0
⇣T l.o.f0
hs0,s1,s3i
⌘= 5 · 731
8
+ 25
= 390
5
8
↵f0
⇣T l.o.f0
hs0,s1,s3i
⌘= 5 · 955
9
+ 25
= 502
7
9
PMOO Arbitrary Multiplexing
s0↵x(f0)s0
= �0,0↵x̄(f0)s0
s1↵x(f0)s1
= ↵f2s1
= �5,166 23↵x̄(f0)
s1
s3↵x(f0)s3
= ↵f1s3
= �5,166 23↵x̄(f0)
s3
�l.o.f0
hs0,s1,s3i = �R
l.o.f0hs0,s1,s3i,T
l.o.f0hs0,s1,s3i
Rl.o.f0
hs0,s1,s3i
=
^i2{0,1,3}
⇣R
si � rx(f0)si
⌘= (20� 5) ^ (20� 5) ^ (20� 5)
= 15
T l.o.f0
hs0,s1,s3i
=
Xi2{0,1,3}
Tsi +
bx̄(f0)si + rx(f0)
si · Tsi
Rl.o.f0e2e
!
= 20 +
0 + 0 · 2015
+ 20 +
166
23 + 5 · 2015
+ 20 +
166
23 + 5 · 2015
= 60 +
533
13
15
= 95
5
9
= = �15,95 59
Df0
�l.o.f0
hs0,s1,s3i = bf0
15 ·t� 95
5
9
�+= 25
t = 97
2
9
Bf0
↵f0
⇣T l.o.f0
hs0,s1,s3i
⌘= 5 · 955
9
+ 25
= 502
7
9
Flow f1
Total Flow AnalysisArrival Bounds
(s3, {f1} , ;) =: ↵f1s3
FIFO Multiplexing Arbitrary Multiplexing(s1, {f2} , ;) =: ↵f2s1
=
: ↵fnsi
with (n, i) 2 {(1, 3) , (2, 1)}↵fnsi
= ↵fn ↵ �l.o.fns2
↵x(fn)s2 = �5,25
�l.o.fns2
= �s2 ↵x(fn)
s2
= �R
l.o.fns2 ,T
l.o.fns2
Rl.o.fns2
hR
s2 � rx(fn)s2
i+= 20� 5
= 15
T l.o.fns2
�s2 = bx(fn)
s2
20 · [t� 20]
+= 25
t = 21
1
4
�s2 = ↵x(fn)
s2
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
= = �15,21 14
= �15,28 13
↵fnsi
= ↵fn ↵ �l.o.fns2
= �r
fnsi
,b
fnsi
rfnsi
= 5
bfnsi
= ↵fn�T l.o.fns2
�= 5 · 211
4
+ 25
= 131
1
4
= ↵fn�T l.o.fns2
�= 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.
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
(s1, {f0} , f2) =: ↵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
Analyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s2↵x(f2)s2 = ↵f1
= �5,25�l.o.f2s2
= �s2 ↵x(f1)
s2 = �15,21 14
= �15,28 13
s1
↵x(f2)s1 = ↵f0
s1= �5,125
�l.o.f2s1
= �s1 ↵x(f2)
s1
= �R
l.o.f2s1 ,T
l.o.f2s1
Rl.o.f2s1
hR
s1 � rx(f2)s1
i+= 20� 5
= 15
T l.o.f2s1
�s1 = b
s1
20 · [t� 20]
+= 125
t = 26
1
4
�s1 = ↵x(f2)
s1
20 · [t� 20]
+= 5 · t+ 125
t = 35
= = �15,26 14
= �15,35
�l.o.f2
hs2,s1i = �R
l.o.f2hs2,s1i,T
l.o.f2hs2,s1i
=
2Oi=1
�l.o.f2si
= �15,47 12
=
2Oi=1
�l.o.f2si
= �15,63 13
Df2
�l.o.f2e2e
= bf2
15 ·t� 47
1
2
�+= 25
t = 49
1
6
�l.o.f2e2e
= bf2
15 ·t� 63
1
3
�+= 25
t = 65
Bf2
↵f2
⇣T l.o.f2
e2e
⌘= 5 · 471
2
+ 25
= 262
1
2
↵f2
⇣T l.o.f2
e2e
⌘= 5 · 631
3
+ 25
= 341
2
3
PMOO Arbitrary Multiplexing
s2↵x(f2)s2
= ↵f1= �5,25
↵x̄(f2)s2
s1↵x(f2)s1
= ↵f0s1
= �5,125↵x̄(f2)s1
�l.o.f2
hs2,s1i = �R
l.o.f2hs2,s1i,T
l.o.f2hs2,s1i
Rl.o.f2
hs2,s1i
=
^i2{2,1}
⇣R
si � rx(f2)si
⌘= (20� 5) ^ (20� 5)
= 15
T l.o.f2
hs2,s1i
=
Xi2{2,1}
Tsi +
bx̄(f2)si + rx(f2)
si · Tsi
Rl.o.f2e2e
!
= 20 +
25 + 5 · 2015
+ 20 +
125 + 5 · 2015
= 40 +
350
15
= 63
1
3
= = �15,63 13
Df2
�l.o.f2
hs2,s1i = bf2
15 ·t� 63
1
3
�+= 25
t = 65
Bf2
↵f2
⇣T l.o.f2
hs2,s1i
⌘= 5 · 631
3
+ 25
= 341
2
3
FeedForward_1SC_4Flows_1AC_4Paths
s0 s1
s2 s3
f1
f0
f2
f3
S = {s0, s1, s2, s3} with
�s0 = �
s1 = �s2 = �
s3 = �Rsi ,Tsi
= �20,20, i 2 {0, 1, 2, 3}
F = {f0, f1, f2, f3} with
↵fn= �
r
fn,b
fn = �5,25, n 2 {0, 1, 2, 3}
Arrival Bounds
(s0, {f3} , ;) =: ↵f3s0
FIFO Multiplexing Arbitrary Multiplexing(s1, {f2} , ;) =: ↵f2s1
(s3, {f1} , ;) =: ↵f1s3
=
: ↵fnsi
with (n, i) 2 {(3, 0) , (2, 1) , (1, 3)}↵fnsi
= ↵fn ↵ �l.o.fns2
↵x(fn)s2 = �5,25 + �5,25 = �10,50
�l.o.fns2
= �s2 ↵x(fn)
= �R
l.o.fns2 ,T
l.o.fns2
Rl.o.fns2
hR
s2 � rx(fn)s2
i+= 20� 5
= 15
T l.o.fns2
�s2 = bx(fn)
s2
20 · [t� 20]
+= 50
t = 22
1
2
�s2 = ↵x(fn)
s2
20 · [t� 20]
+= 10 · t+ 50
t = 45
= = �10,22 12
= �10,45
↵fnsi
= ↵fn ↵ �l.o.fns2
= �r
fnsi
,b
fnsi
rfnsi
= 5
bfnsi
↵fn�T l.o.fns2
�= 5 · 22 1
2 + 25 = 137
12 ↵fn
�T l.o.fns2
�= 5 · 45 + 25 = 250
= = �5,137 12
= �5,250
(s3, {f0} , ;) = ↵f0s1
FIFO Multiplexing Arbitrary Multiplexing↵f0s1
= ↵f0 ↵ �l.o.f0s0
↵f0s0
= ↵f0= �5,25
↵x(f0)s0 = ↵f3
s0= �5,137 1
2= ↵f3
s0= �5,250
�l.o.f0s0
= �s0 ↵x(f0)
s0
= �R
l.o.f0s0 ,T
l.o.f0s0
Rl.o.f0s0
hR
s0 � rx(f0)s0
i+= 20� 5
= 15
T l.o.f0s0
�s0 = bx(f0)
s0
20 · [t� 20]
+= 137
1
2
t = 26
7
8
�s0 = ↵x(f0)
s0
20 · [t� 20]
+= 5 · t+ 250
t = 43
1
3
= = �15,26 78
= �15,43 13
↵f0s1
= ↵f0 ↵ �l.o.f0s0
= �r
f0s1 ,b
f0s1
rf0s1
= 5
bf0s1
↵f0�T l.o.f0s0
�= 5 · 26 7
8 + 25 = 159
38 ↵f0
�T l.o.f0s0
�= 5 · 43 1
3 + 25 = 241
23
= = �5,159 38
= �5,241 23
(s3, {f0} , ;) = ↵f0s3
FIFO Multiplexing Arbitrary Multiplexing↵f0s3
= ↵f0 ↵��l.o.f0s0
⌦ �l.o.f0s1
�(reuse of previous result)
= ↵f0 ↵ �l.o.f0s0
↵ �l.o.f0s1
= ↵f0s1↵ �l.o.f0
s1
↵f0s1
= �5,159 38
= �5,241 23
↵x(f0)s1 = ↵f2
s1= �5,137 1
2= ↵f2
s1= �5,250
�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]
+= 137
1
2
t = 26
7
8
�s1 = ↵x(f0)
s1
20 · [t� 20]
+= 5 · t+ 250
t = 43
1
3
= = �15,26 78
= �15,43 13
↵f0s3
= ↵f0s1↵ �l.o.f0
s1
= �r
f0s3 ,b
f0s3
rf0s3
= 5
bf0s3
↵f0s1
�T l.o.f0s1
�= 5 · 26 7
8 + 159
38 = 293
34 ↵f0
s1
�T l.o.f0s1
�= 5 · 43 1
3 + 241
23 = 458
13
= = �5,293 34
= �5,458 13
(s3, {f3} , ;) = ↵f3s3
FIFO Multiplexing Arbitrary MultiplexingPbooArrivalBound_Concatenation.java
↵f3s3
= ↵f3 ↵��l.o.f3s2
⌦ �l.o.f3s0
�(reuse of previous result)
= ↵f3 ↵ �l.o.f3s2
↵ �l.o.f3s0
= ↵f3s0↵ �l.o.f3
s0
↵x(f3)s0 = ↵f0
= �5,25
�l.o.f3s0
= �s0 ↵x(f3)
s0
= �R
l.o.f3s0 ,T
l.o.f3s0
Rl.o.f3s0
hR
s0 � rx(f3)s0
i+= 20� 5
= 15
T l.o.f3s0
�s0 = bf0
20 · [t� 20]
+= 25
t = 21
1
4
�s0 = ↵f0
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
= = �15,21 14
= �15,28 13
↵f3s0
= �5,137 12
= �5,250
↵f3s3
= ↵f3s0↵ �l.o.f3
s0
= �r
f3s3 ,b
f3s3
rf3s3
= 5
bf3s3
= ↵f3s0
�T l.o.f3s0
�= 5 · 211
4
+ 137
1
2
= 243
3
4
= ↵f3s0
�T l.o.f3s0
�= 5 · 281
3
+ 250
= 391
2
3
= = �5,243 34
= �5,391 23
PmooArrivalBound.java
↵f3s3
= ↵f3 ↵ �l.o.f3
hs2,s0i
s2↵x(f3)s2 = ↵f1
+ ↵f2
= �5,25 + �5,25
= �10,50
↵x̄(f3)s2
s0↵x(f3)s
0
= ↵f0= �5,25
↵x̄(f3)s
0
�l.o.f3
hs2,s0i = �R
l.o.f3hs2,s0i,T
l.o.f3hs2,s0i
Rl.o.f3
hs2,s0i
=
^i2{2,0}
⇣R
si � rx(f3)si
⌘= (20� 10) ^ (20� 5)
= 10
T l.o.f3
hs2,s0i
=
Xi2{2,0}
Tsi +
bx̄(f3)si + rx(f3)
si · Tsi
Rl.o.f3
hs2,s0i
!
= 20 +
50 + 10 · 2010
+ 20 +
25 + 5 · 2010
= 77
1
2
= = �10,77 12
↵f3s3
= ↵f3 ↵ �l.o.f3
hs2,s0i
rf3s3
= 5
bf3s3
= ↵f3(T l.o.f3
hs2,s0i)
= 5 · 7712
+ 25
= 412
1
2
= = �5,412 12
Flow f0
Total Flow AnalysisAnalysis
TFA FIFO Multiplexing Arbitrary MultiplexingPbooArrivalBound_Concatenation.java PmooArrivalBound.java
s0
↵s0
= ↵f0s0
+ ↵f3s0
= �5,25 + �5,137 12
= �10,162 12
= ↵f0s0
+ ↵f3s0
= �5,25 + �5,250
= �10,275
Df0s0
�s0 = b
s0
20 · [t� 20]
+= 162
1
2
t = 28
1
8
�s0 = ↵
s0
20 · [t� 20]
+= 10 · t+ 275
t = 67
1
2
Bf0s0
↵s0 (Ts0) = 10 · 20 + 162
1
2
= 262
1
2
↵s0 (Ts0) = 10 · 20 + 275
= 475
s1
↵s1
= ↵f0s1
+ ↵f2s1
= �5,159 38+ �5,137 1
2
= �10,296 78
= ↵f0s1
+ ↵f2s1
= �5,241 23+ �5,250
= �10,491 23
Df0s1
�s1 = b
s1
20 · [t� 20]
+= 296
7
8
t = 34
27
32
�s1 = ↵
s1
20 · [t� 20]
+= 10 · t+ 491
2
3
t = 89
1
6
Bf0s1
↵s1 (Ts1) = 10 · 20 + 296
7
8
= 496
7
8
↵s1 (Ts1) = 10 · 20 + 491
2
3
= 691
2
3
s3
↵s3
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,293 34+ �5,137 1
2+ �5,243 3
4
= �15,675
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,458 13+ �5,250 + �5,391 2
3
= �15,1100
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,458 13+ �5,250 + �5,412 1
2
= �15,1120 56
Df0s3
�s3 = b
s3
20 · [t� 20]
+= 675
t = 53
3
4
�s3 = ↵
s3
20 · [t� 20]
+= 15 · t+ 1100
t = 300
�s3 = ↵
s3
20 · [t� 20]
+= 15 · t+ 1120
5
6
t = 304
1
6
Bf0s3
↵s3 (Ts3) = 15 · 20 + 675
= 975
↵s3 (Ts3) = 15 · 20 + 1100
= 1400
↵s3 (Ts3) = 15 · 20 + 1120
5
6
= 1420
5
6
Df0 Df0s0
+Df0s1
+Df0s3
= 116
2332 Df0
s0+Df0
s1+Df0
s3= 456
23 Df0
s0+Df0
s1+Df0
s3= 460
56
Bf0max
�Bf0
s0, Bf0
s1, Bf0
s3
= 975 max
�Bf0
s0, Bf0
s1, Bf0
s3
= 1400 max
�Bf0
s0, Bf0
s1, Bf0
s3
= 1420
56
Separate Flow Analysis and PMOO AnalysisAnalyses
PbooArrivalBound_Concatenation.java
SFA FIFO Multiplexing Arbitrary Multiplexing
s0
↵x(f0)s0 = ↵f3
s0= �5,137 1
2= ↵f3
s0= �5,250
�l.o.f0s0
= �s0 ↵x(f3)
s0
= �R
l.o.f0s0 ,T
l.o.f0s0
Rl.o.f0s0
hR
s0 � rx(f0)s0
i+= 15
T l.o.f0s0
�s0 = bx(f0)
s0
20 · [t� 20]
+= 137
1
2
t = 26
7
8
�s0 = ↵x(f0)
s0
20 · [t� 20]
+= 5 · t+ 250
t = 43
1
3
= = �15,26 78
= �15,43 13
s1
↵x(f0)s1 = ↵f1
s1= �5,137 1
2= ↵f1
s1= �5,250
�l.o.f0s1
= �s1 ↵x(f0)
s1
= �R
l.o.f0s1 ,T
l.o.f0s1
Rl.o.f0s1
hR
s1 � rx(f0)s1
i+= 15
T l.o.f0s1
�s1 = bx(f0)
s1
20 · [t� 20]
+= 137
1
2
t = 26
7
8
�s1 = ↵x(f0)
s1
20 · [t� 20]
+= 5 · t+ 250
t = 43
1
3
= = �15,26 78
= �15,43 13
s3
↵x(f0)s3
= ↵f2s3
+ ↵f3s3
= �5,137 12+ �5,243 3
4
= �10,381 14
= ↵f2s3
+ ↵f3s3
= �5,250 + �5,391 23
= �10,641 23
�l.o.f0s3
= �s3 ↵x(f0)
s3
= �R
l.o.f0s3 ,T
l.o.f0s3
Rl.o.f0s3
hR
s3 � rx(f0)s3
i+= 10
T l.o.f0s3
�s3 = bx(f0)
s3
20 · [t� 20]
+= 381
1
4
t = 39
1
16
�s3 = ↵x(f0)
s3
20 · [t� 20]
+= 10 · t+ 641
2
3
t = 104
1
6
= = �10,39 116
= �10,104 16
�l.o.f0e2e
= �R
l.o.f0e2e
,T
l.o.f0e2e
Ni={0,1,3} �
l.o.f0si
= �10,92 1316
Ni={0,1,3} �
l.o.f0si
= �10,190 56
Df0
�l.o.f0e2e
= bf0
10 ·t� 92
13
16
�+= 25
t = 95
5
16
�l.o.f0e2e
= bf0
10 ·t� 190
5
6
�+= 25
t = 193
1
3
Bf0
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 9213
16
+ 25
= 489
1
16
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 1905
6
+ 25
= 979
1
6
PmooArrivalBound.java
SFA Arbitrary Multiplexing
s0
↵x(f0)s0 = ↵f3
s0= �5,250
�l.o.f0s0
= �s0 ↵x(f3)
s0
= �R
l.o.f0s0 ,T
l.o.f0s0
Rl.o.f0s0
hR
s0 � rx(f0)s0
i+= 15
T l.o.f0s0
�s0 = ↵x(f0)
s0
20 · [t� 20]
+= 5 · t+ 250
t = 43
1
3
= = �15,43 13
s1
↵x(f0)s1 = ↵f1
s1= �5,250
�l.o.f0s1
= �s1 ↵x(f0)
s1
= �R
l.o.f0s1 ,T
l.o.f0s1
Rl.o.f0s1
hR
s1 � rx(f0)s1
i+= 15
T l.o.f0s1
�s1 = ↵x(f0)
s1
20 · [t� 20]
+= 5 · t+ 250
t = 43
1
3
= = �15,43 13
s3
↵x(f0)s3 = ↵f2
s3+ ↵f3
s3= �5,250 + �5,412 1
2= �10,662 1
2
�l.o.f0s3
= �s3 ↵x(f0)
s3
= �R
l.o.f0s3 ,T
l.o.f0s3
Rl.o.f0s3
hR
s3 � rx(f0)s3
i+= 10
T l.o.f0s3
�s3 = ↵x(f0)
s3
20 · [t� 20]
+= 10 · t+ 662
1
2
t = 106
1
4
= = �10,106 14
�l.o.f0e2e
= �R
l.o.f0e2e
,T
l.o.f0e2e
Ni={0,1,3} �
l.o.f0si
= �10,192 1112
Df0
�l.o.f0e2e
= bf0
10 ·t� 192
11
12
�+= 25
t = 195
5
12
Bf0
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 19211
12
+ 25
= 989
7
12
PbooArrivalBound_Concatenation.java
PMOO Arbitrary Multiplexing
s0↵x(f0)s0
= ↵f3s0
= �5,250↵x̄(f0)s0
s1↵x(f0)s1
= ↵f2s1
= �5,250↵x̄(f0)s1
s3↵x(f0)s3
= ↵f2s3
+ ↵f3s3
= �5,250 + �5,391 23
= �5,641 23
↵x̄(f0)s3
�l.o.f0e2e
= �R
l.o.f0e2e
,T
l.o.f0e2e
Rl.o.f0e2e
=
^i2{0,1,3}
⇣R
si � rx(f0)si
⌘= (20� 5) ^ (20� 5) ^ (20� 10)= 10
T l.o.f0e2e
=
Xi2{0,1,3}
Tsi +
bx̄(f0)si + rx(f0)
si · Tsi
Rl.o.f0e2e
!
= 20 +
250 + 5 · 2010
+ 20 +
250 + 5 · 2010
+ 20 +
641
23 + 10 · 2010
= 60 +
1541
23
10
= 214
1
6
= = �10,214 16
Df0
�l.o.f0e2e
= bf0
10 ·t� 214
1
6
�+= 25
t = 216
2
3
Bf0
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 2141
6
+ 25
= 1095
5
6
PmooArrivalBound.java
PMOO Arbitrary Multiplexing
s0↵x(f0)s0
= ↵f3s0
= �5,250↵x̄(f0)s0
s1↵x(f0)s1
= ↵f2s1
= �5,250↵x̄(f0)s1
s3↵x(f0)s3
= ↵f2s3
+ ↵f3s3
= �5,250 + �5,412 12
= �10,662 12
↵x̄(f0)s3
�l.o.f0e2e
= �R
l.o.f0e2e
,T
l.o.f0e2e
Rl.o.f0e2e
=
^i2{0,1,3}
⇣R
si � rx(f0)si
⌘= (20� 5) ^ (20� 5) ^ (20� 10)= 10
T l.o.f0e2e
=
Xi2{0,1,3}
Tsi +
bx̄(f0)si + rx(f0)
si · Tsi
Rl.o.f0e2e
!
= 20 +
250 + 5 · 2010
+ 20 +
250 + 5 · 2010
+ 20 +
662
12 + 10 · 2010
= 60 +
1462
12
10
= 216
1
4
= = �10,216 14
Df0
�l.o.f0e2e
= bf0
10 ·t� 216
1
4
�+= 25
t = 218
3
4
Bf0
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 2161
4
+ 25
= 1106
1
4
Flow f1
Total Flow AnalysisAnalysis
PbooArrivalBound_Concatenation.java
TFA FIFO Multiplexing Arbitrary Multiplexing
s2
↵s2
↵f3s3
= ↵f1s2
+ ↵f2s2
+ ↵f3s2
= �5,25 + �5,25 + �5,25
= �15,75
Df1s2
�s2 = b
s2
20 · [t� 20]
+= 75
t = 23
3
4
�s2 = ↵
s2
20 · [t� 20]
+= 15 · t+ 75
t = 95
Bf1s2
↵s2 (Ts2) = 15 · 20 + 75
= 375
s3
↵s3
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,137 12+ �5,293 3
4+ �5,243 3
4
= �15,675
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,250 + �5,458 13+ �5,391 2
3
= �15,1100
Df1s3
�s3 = b
s3
20 · [t� 20]
+= 675
t = 53
3
4
�s3 = ↵
s3
20 · [t� 20]
+= 15 · t+ 1100
t = 300
Bf1s3
↵s3 (Ts3) = 15 · 20 + 675
= 975
↵s3 (Ts3) = 15 · 20 + 1100
= 1400
Df1 Df1s2
+Df1s3
= 77
12 Df1
s2+Df1
s3= 395
Bf1max
�Bf1
s2, Bf1
s3
= 975 max
�Bf1
s2, Bf1
s3
= 1400
PmooArrivalBound.java
TFA Arbitrary Multiplexing
s2
↵s2
↵f3s3
= ↵f1s2
+ ↵f2s2
+ ↵f3s2
= �5,25 + �5,25 + �5,25
= �15,75
Df1s2
�s2 = ↵
s2
20 · [t� 20]
+= 15 · t+ 75
t = 95
Bf1s2
↵s2 (Ts2) = 15 · 20 + 75
= 375
s3
↵s3
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,250 + �5,458 13+ �5,412 1
2
= �15,1120 56
Df1s3
�s3 = ↵
s3
20 · [t� 20]
+= 15 · t+ 1120
5
6
t = 304
1
6
Bf1s3
↵s3 (Ts3) = 15 · 20 + 1120
5
6
= 1420
5
6
Df1 Df1s2
+Df1s3
= 399
16
Bf1max
�Bf1
s2, Bf1
s3
= 1420
56
Separate Flow Analysis and PMOO AnalysisAnalyses
PbooArrivalBound_Concatenation.java
SFA FIFO Multiplexing Arbitrary Multiplexing
s2↵x(f1)s2
= ↵f2s2
+ ↵f3s2
= �5,25 + �5,25
= �10,50�l.o.f1s2
= �s2 ↵x(f1)
s2 = �10,22 12
= �10,45
s3
↵x(f1)s3
= ↵f0s3
+ ↵f3s3
= �5,293 34+ �5,243 3
4
= �10,537 13
= ↵f0s3
+ ↵f3s3
= �5,458 13+ �5,391 2
3
= �10,850
�l.o.f1s3
= �s3 ↵x(f1)
s3
= �R
l.o.f1s3 ,T
l.o.f1s3
Rl.o.f1s3
hR
s3 � rx(f1)s3
i+= 10
T l.o.f1s3
�s3 = bx(f1)
s3
20 · [t� 20]
+= 537
1
2
t = 46
7
8
�s3 = ↵x(f1)
s3
20 · [t� 20]
+= 10 · t+ 850
t = 125
= = �10,46 78
= �10,125
�l.o.f1e2e
= �R
l.o.f1e2e
,T
l.o.f1e2e
N3i=2 �
l.o.f1si
= �10,69 38
N3i=2 �
l.o.f1si
= �10,170
Df1
�l.o.f1e2e
= bf1
10 ·t� 69
3
8
�+= 25
t = 71
7
8
�l.o.f1e2e
= bf1
10 · [t� 170]
+= 25
t = 172
1
2
Bf1
↵f1
⇣T l.o.f1
e2e
⌘= 5 · 693
8
+ 25
= 371
7
8
↵f1
⇣T l.o.f1
e2e
⌘= 5 · 170 + 25
= 875
PmooArrivalBound.java
SFA Arbitrary Multiplexing
s2↵x(f1)s2
= ↵f2s2
+ ↵f3s2
= �5,25 + �5,25
= �10,50�l.o.f1s2
= �s2 ↵x(f1)
s2 = �10,45
s3
↵x(f1)s3
= ↵f0s3
+ ↵f3s3
= �5,458 13+ �5,412 1
2
= �10,870 56
�l.o.f1s3
= �s3 ↵x(f1)
s3
= �R
l.o.f1s3 ,T
l.o.f1s3
Rl.o.f1s3
hR
s3 � rx(f1)s3
i+= 10
T l.o.f1s3
�s3 = ↵x(f1)
s3
20 · [t� 20]
+= 10 · t+ 870
5
6
t = 127
1
12
= = �10,127 112
�l.o.f1e2e
= �R
l.o.f1e2e
,T
l.o.f1e2e
N3i=2 �
l.o.f1si
= �10,172 112
Df1
�l.o.f1e2e
= bf1
10 ·t� 172
1
12
�+= 25
t = 174
7
12
Bf1
↵f1
⇣T l.o.f1e2e
⌘= 5 · 172 1
12
+ 25
= 885
5
12
PbooArrivalBound_Concatenation.java
PMOO Arbitrary Multiplexing
s2↵x(f1)s2
= ↵f2s2
+ ↵f3s2
= �5,25 + �5,25
= �10,50↵x̄(f1)s2
s3↵x(f1)s3
= ↵f0s3
+ ↵f3s3
= �5,458 13+ �5,391 2
3
= �10,850↵x̄(f1)s3
�l.o.f1e2e
= �R
l.o.f1e2e
,T
l.o.f1e2e
Rl.o.f1e2e
=
^i2{2,3}
⇣R
si � rx(f1)si
⌘= (20� 10) ^ (20� 10)
= 10
T l.o.f1e2e
=
Xi2{2,3}
Tsi +
bx̄(f1)si + rx(f1)
si · Tsi
Rl.o.f1e2e
!
= 20 +
50 + 10 · 2010
+ 20 +
850 + 10 · 2010
= 40 +
1300
10
= 170
= = �10,170
Df1
�l.o.f1e2e
= bf1
10 · [t� 170]
+= 25
t = 172
1
2
Bf1↵f1
⇣T l.o.f1
e2e
⌘= 5 · 170 + 25
= 875
PmooArrivalBound.java
PMOO Arbitrary Multiplexing
s2↵x(f1)s2
= ↵f2s2
+ ↵f3s2
= �5,25 + �5,25
= �10,50↵x̄(f1)s2
s3↵x(f1)s3
= ↵f0s3
+ ↵f3s3
= �5,458 13+ �5,412 1
2
= �10,870 56
↵x̄(f1)s3
�l.o.f1e2e
= �R
l.o.f1e2e
,T
l.o.f1e2e
Rl.o.f1e2e
=
^i2{2,3}
⇣R
si � rx(f1)si
⌘= (20� 10) ^ (20� 10)
= 10
T l.o.f1e2e
=
Xi2{2,3}
Tsi +
bx̄(f1)si + rx(f1)
si · Tsi
Rl.o.f1e2e
!
= 20 +
50 + 10 · 2010
+ 20 +
870
56 + 10 · 2010
= 40 +
1320
56
10
= 172
1
12
= = �10,172 112
Df1
�l.o.f1e2e
= bf1
10 ·t� 172
1
12
�+= 25
t = 174
7
12
Bf1
↵f1
⇣T l.o.f1
e2e
⌘= 5 · 172 1
12
+ 25
= 885
5
12
Flow f2
Total Flow AnalysisAnalysis
TFA FIFO Multiplexing Arbitrary Multiplexing
s2
↵s2
= ↵f2s2
+ ↵f1s2
+ ↵f3s2
= �5,25 + �5,25 + �5,25
= �15,75
Df2s2
�s2 = b
s2
20 · [t� 20]
+= 75
t = 23
3
4
�s2 = ↵
s2
20 · [t� 20]
+= 15 · t+ 75
t = 95
Bf2s2
↵s2 (Ts2) = 15 · 20 + 75
= 375
s1
↵s1
= ↵f2s1
+ ↵f0s1
= �5,137 12+ �5,159 3
8
= �10,296 78
= ↵f2s1
+ ↵f0s1
= �5,250 + �5,241 23
= �10,491 23
Df2s1
�s1 = b
s1
20 · [t� 20]
+= 296
7
8
t = 34
27
32
�s1 = ↵
s1
20 · [t� 20]
+= 10 · t+ 491
2
3
t = 89
1
6
Bf2s1
↵s1 (Ts1) = 10 · 20 + 296
7
8
= 496
7
8
↵s1 (Ts1) = 10 · 20 + 491
2
3
= 691
2
3
Df2 Df2s2
+Df2s1
= 58
1932 Df2
s2+Df2
s1= 184
16
Bf2max
�Bf2
s2, Bf2
s1
= 496
78 max
�Bf2
s2, Bf2
s1
= 691
23
Separate Flow Analysis and PMOO AnalysisAnalyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s2↵x(f2)s2
= ↵f1s2
+ ↵f3s2
= �5,25 + �5,25
= �10,50�l.o.f2s2
= �s2 ↵x(f2)
s2 = �10,22 12
= �10,45
s1
↵x(f2)s1 = ↵f0
s1= �5,159 3
8= ↵f0
s1= �5,241 2
3
�l.o.f2s1
= �s1 ↵x(f2)
s1
= �R
l.o.f2s1 ,T
l.o.f2s1
Rl.o.f2s1
hR
s2 � rx(f2)s2
i+= 15
T l.o.f2s1
�s1 = bx(f2)
s1
20 · [t� 20]
+= 159
3
8
t = 27
31
32
�s1 = ↵x(f2)
s1
20 · [t� 20]
+= 5 · t+ 241
2
3
t = 42
7
9
= = �10,27 3132
= �10,42 79
�l.o.f2e2e
= �R
l.o.f2e2e
,T
l.o.f2e2e
N2i=1 �
l.o.f2si
= �10,50 1532
N2i=1 �
l.o.f2si
= �10,87 79
Df2
�l.o.f2e2e
= bf2
10 ·t� 50
15
32
�+= 25
t = 52
31
32
�l.o.f2e2e
= bf2
10 ·t� 87
7
9
�+= 25
t = 90
5
18
Bf2
↵f2
⇣T l.o.f2
e2e
⌘= 5 · 5015
32
+ 25
= 277
11
32
↵f2
⇣T l.o.f2
e2e
⌘= 5 · 877
9
+ 25
= 463
8
9
PMOO Arbitrary Multiplexing
s2↵x(f2)s2
= ↵f1s2
+ ↵f3s2
= �5,25 + �5,25
= �10,50↵x̄(f2)s2
s1↵x(f2)s1
= ↵f0s1
= �5,241 23↵x̄(f2)
s1
�l.o.f2e2e
= �R
l.o.f2e2e
,T
l.o.f2e2e
Rl.o.f2e2e
=
^i2{2,1}
⇣R
si � rx(f2)si
⌘= (20� 10) ^ (20� 5)
= 10
T l.o.f2e2e
=
Xi2{2,1}
Tsi +
bx̄(f2)si + rx(f2)
si · Tsi
Rl.o.f2e2e
!
= 20 +
50 + 10 · 2010
+ 20 +
241
23 + 5 · 2010
= 40 +
591
23
10
= 99
1
6
= = �10,99 16
Df2
�l.o.f2e2e
= bf2
10 ·t� 99
1
6
�= 25
t = 101
2
3
Bf2
↵f2
⇣T l.o.f2
e2e
⌘= 5 · 991
6
+ 25
= 520
5
6
Flow f3
Total Flow AnalysisAnalysis
PbooArrivalBound_Concatenation.java
TFA FIFO Multiplexing Arbitrary MultiplexingPbooArrivalBound_Concatenation.java PmooArrivalBound.java
s2
↵s2
= ↵f3s2
+ ↵f1s2
+ ↵f2s2
= �5,25 + �5,25 + �5,25
= �15,75
Df3s2
�s2 = b
s2
20 · [t� 20]
+= 75
t = 23
3
4
�s2 = ↵
s2
20 · [t� 20]
+= 15 · t+ 75
t = 95
Bf3s2
↵s2 (Ts2) = 15 · 20 + 75
= 375
s0
↵s0
= ↵f0s0
+ ↵f3s0
= �5,25 + �5,137 12
= �10,162 12
= ↵f0s0
+ ↵f3s0
= �5,25 + �5,250
= �10,275
Df3s0
�s0 = b
s0
20 · [t� 20]
+= 162
1
2
t = 28
1
8
�s0 = ↵
s0
20 · [t� 20]
+= 10 · t+ 275
t = 67
1
2
Bf3s0
↵s0 (Ts0) = 10 · 20 + 162
1
2
= 262
1
2
↵s0 (Ts0) = 10 · 20 + 275
= 475
s3
↵s3
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,293 34+ �5,137 1
2+ �5,243 3
4
= �15,675
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,458 13+ �5,250 + �5,391 2
3
= �15,1100
= ↵f0s3
+ ↵f1s3
+ ↵f3s3
= �5,458 13+ �5,250 + �5,412 1
2
= �15,1120 56
Df3s3
�s3 = b
s3
20 · [t� 20]
+= 675
t = 53
3
4
�s3 = ↵
s3
20 · [t� 20]
+= 15 · t+ 1100
t = 300
�s3 = ↵
s3
20 · [t� 20]
+= 15 · t+ 1120
5
6
t = 304
1
6
Bf3s3
↵s3 (Ts3) = 15 · 20 + 675
= 975
↵s3 (Ts3) = 15 · 20 + 1100
= 1400
↵s3 (Ts3) = 15 · 20 + 1120
5
6
= 1420
5
6
Df3 Df3s2
+Df3s0
+Df3s3
= 105
58 Df3
s2+Df3
s0+Df3
s3= 462
12 Df3
s2+Df3
s0+Df3
s3= 466
23
Bf3max
�Bf3
s2, Bf3
s0, Bf3
s3
= 975 max
�Bf3
s2, Bf3
s0, Bf3
s3
= 1400 max
�Bf3
s2, Bf3
s0, Bf3
s3
= 1420
56
Separate Flow Analysis and PMOO AnalysisAnalyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s2↵x(f3)s2
= ↵f1s2
+ ↵f2s2
= �5,25 + �5,25
= �10,50�l.o.f3s2
= �s2 ↵x(f3)
s2 = �10,22 12
= �10,45
s0↵x(f3)s0 = ↵f0
s0= �5,25
�l.o.f3s0
= �s0 ↵x(f3)
s0 = �15,21 14
= �15,28 13
s3
↵x(f3)s3
= ↵f0s3
+ ↵f1s3
= �5,293 34+ �5,137 1
2
= �10,431 14
= ↵f0s3
+ ↵f1s3
= �5,458 13+ �5,250
= �10,708 13
�l.o.f3s3
= �s3 ↵x(f3)
s3
= �R
l.o.f3s3 ,T
l.o.f3s3
Rl.o.f3s3
hR
s3 � rx(f3)s3
i+= 10
T l.o.f3s3
�s3 = bx(f3)
s3
20 · [t� 20]
+= 431
1
4
t = 41
9
16
�s3 = ↵x(f3)
s3
20 · [t� 20]
+= 10 · t+ 708
1
3
t = 110
5
6
= = �10,41 916
= �10,110 56
�l.o.f3e2e
= �R
l.o.f3e2e
,T
l.o.f3e2e
Ni={2,0,3} �
l.o.f3si
= �10,85 516
Ni={2,0,3} �
l.o.f3si
= �10,184 16
Df3
�l.o.f3e2e
= bf3
10 ·t� 85
5
16
�+= 25
t = 87
13
16
�l.o.f3e2e
= bf3
10 ·t� 184
1
6
�+= 25
t = 186
2
3
Bf3
↵f3
⇣T l.o.f3
e2e
⌘= 5 · 85 5
16
+ 25
= 451
9
16
↵f3
⇣T l.o.f3
e2e
⌘= 5 · 1841
6
+ 25
= 945
5
6
PMOO Arbitrary Multiplexing
s2↵x(f3)s2
= ↵f1s2
+ ↵f2s2
= �5,25 + �5,25
= �10,50↵x̄(f3)s2
s0↵x(f3)s0
= ↵f0s0
= �5,25↵x̄(f3)s0
s3↵x(f3)s3
= ↵f0s3
+ ↵f1s3
= �5,458 13+ �5,250
= �10,708 13
↵x̄(f3)s3
�l.o.f3e2e
= �R
l.o.f3e2e
,T
l.o.f3e2e
Rl.o.f3e2e
=
^i2{2,0,3}
⇣R
si � rx(f3)si
⌘= (20� 10) ^ (20� 5) ^ (20� 10)
= 10
T l.o.f3e2e
=
Xi2{2,0,3}
Tsi +
bx̄(f3)si + rx(f3)
si · Tsi
Rl.o.f3e2e
!
= 20 +
50 + 10 · 2010
+ 20 +
25 + 5 · 2010
+ 20 +
708
13 + 10 · 2010
= 60 +
1283
13
10
= 188
1
3
= = �10,188 13
Df3
�l.o.f3e2e
= bf3
10 ·t� 188
1
3
�+= 25
t = 190
5
6
Bf3
↵f3
⇣T l.o.f3
e2e
⌘= 5 · 1881
3
+ 25
= 966
2
3
FeedForward_1SC_2Flows_1AC_2Paths
s2s0
s1
f0
f1
S = {s0, s1, s2} with
�s0 = �
s1 = �s2 = �
Rsi ,Tsi= �20,20, i 2 {0, 1, 2}
F = {f0, f1} with
↵f0= ↵f1
= �r
fn,b
fn = �5,25, n 2 {0, 1}
Flow f0
Total Flow AnalysisArrival Bounds
(s1, {f1} , ;) =: ↵f1s1
FIFO Multiplexing Arbitrary Multiplexing(s2, {f0} , ;) =: ↵f0s2
=
: ↵fnsi
with (n, i) 2 {(1, 1) , (0, 2)}↵fnsi
= ↵fn ↵ �l.o.fns0
↵x(fn)s0 = �5,25
�l.o.fns0
= �s0 ↵x(fn)
s0
= �R
l.o.fns0 ,T
l.o.fns0
Rl.o.fns0
= 15
T l.o.fns0
�s0 = bx(fn)
s0
20 · [t� 20]
+= 25
t = 21
1
4
�s0 = ↵x(fn)
s0
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
=
=�15,21 14
=�15,28 13
↵fnsi
= ↵fn ↵ �l.o.fns0
= �r
fnsi
,b
fnsi
rfnsi
= 5
bfnsi
↵f0�T l.o.fns0
�= 5 · 211
4
+ 25
= 131
1
4
↵f0�T l.o.fns0
�= 5 · 281
3
+ 25
= 166
2
3
= = �5,131 14
= �5,166 23
(s2, {f1} , ;) =: ↵f1s2
FIFO Multiplexing Arbitrary Multiplexing↵f1s2
= ↵f1 ↵��l.o.f1s0
⌦ �l.o.f1s1
�(reuse of previous result)
= ↵f1 ↵ �l.o.f1s0
↵ �l.o.f1s1
= ↵f1s1↵ �l.o.f1
s1
↵x(f1)s1 = �0,0
�l.o.f1s1
= �s1 ↵x(f1)
s1 = �s1 =�20,20
↵f1s1
= �5,131 14
= �5,166 23
↵f1s2
= ↵f1 ↵ �l.o.f1s1
= �r
f1s2 ,b
f1s2
rf1s2
= 5
bf1s2
↵f0�T l.o.f0s0
�= 5 · 20 + 131
1
4
= 231
1
4
↵f0�T l.o.f0s0
�= 5 · 20 + 166
2
3
= 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.
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.
Analyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s0
↵x(f0)s0 = ↵f1
= �5,25
�l.o.f0s0
= �s0 ↵x(f0)
s0
= �R
l.o.f0s0 ,T
l.o.f0s0
Rl.o.f0s0
hR
s0 � rx(f0)s0
i+= 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
s2
↵x(f0)s2 = �5,231 1
4= �5,266 2
3
�l.o.f0s2
= �s2 ↵x(f0)
s2
= �R
l.o.f0s2 ,T
l.o.f0s2
Rl.o.f0s2
hR
s2 � rx(f0)s2
i+= 15
T l.o.f0s2
�s2 = bx(f0)
s2
20 · [t� 20]
+= 231
1
4
t = 31
9
16
�s2 = ↵x(f0)
s2
20 · [t� 20]
+= 5 · t+ 266
2
3
t = 44
4
9
= = �15,31 916
= �15,44 49
�l.o.f0e2e
= �R
l.o.f0e2e
,T
l.o.f0e2e
Ni={0,2} �
l.o.f0si
= �15,52 1316
Ni={0,2} �
l.o.f0si
= �15,72 79
Df0
�l.o.f0e2e
= bf0
15 ·t� 52
13
16
�+= 25
t = 54
23
48
�l.o.f0e2e
= bf0
15 ·t� 72
7
9
�+= 25
t = 74
4
9
Bf0
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 5213
16
+ 25
= 289
1
16
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 727
9
+ 25
= 388
8
9
PMOO Arbitrary Multiplexing
s0↵x(f0)s
0
= �5,25↵x̄(f0)s
0
s2↵x(f0)s2
= �5,266 23↵x̄(f0)
s2
�l.o.f0e2e
= �R
l.o.f0e2e
,T
l.o.f0e2e
Rl.o.f0e2e
=
^i2{0,2}
⇣R
si � rx(f0)si
⌘= (20� 5) ^ (20� 5)
= 15
T l.o.f0e2e
=
Xi2{0,2}
Tsi +
bx̄(f0)si + rx(f0)
si · Tsi
Rl.o.f0e2e
!
= 20 +
25 + 5 · 2015
+ 20 +
266
23 + 5 · 2015
= 40 +
491
23
15
= 72
7
9
= = �15,72 79
Df0
�l.o.f0e2e
= bf0
15 ·t� 72
7
9
�+= 25
t = 74
4
9
Bf0
↵f0
⇣T l.o.f0
e2e
⌘= 5 · 727
9
+ 25
= 388
8
9
Flow f1
Total Flow AnalysisArrival Bounds
(s1, {f1} , ;) =: ↵f1s1
FIFO Multiplexing Arbitrary Multiplexing(s2, {f0} , ;) =: ↵f0s2
=
: ↵fnsi
with (n, i) 2 {(1, 1) , (0, 2)}↵fnsi
= ↵fn ↵ �l.o.fns0
↵x(fn)s0 = �5,25
�l.o.fns0
= �s0 ↵x(fn)
s0
= �R
l.o.fns0 ,T
l.o.fns0
Rl.o.fns0
= 15
T l.o.fns0
�s0 = bx(fn)
s0
20 · [t� 20]
+= 25
t = 21
1
4
�s0 = ↵x(fn)
s0
20 · [t� 20]
+= 5 · t+ 25
t = 28
1
3
=
=�15,21 14
=�15,28 13
↵fnsi
= ↵fn ↵ �l.o.fns0
rfnsi
= 5
bfnsi
↵f0�T l.o.fns0
�= 131
14 ↵f0
�T l.o.fns0
�= 166
23
= = �5,131 14
= �5,166 23
(s2, {f1} , ;) =: ↵f1s2
FIFO Multiplexing Arbitrary Multiplexing↵f1s2
= ↵f1 ↵��l.o.f1s0
⌦ �l.o.f1s1
�= ↵f1 ↵ �l.o.f1
s0↵ �l.o.f1
s1= ↵f1
s1↵ �l.o.f1
s1
↵x(f1)s1 = �0,0
�l.o.f1s1
= �s1 ↵x(f1)
s1 = �s1 =�20,20
↵f1s1
= �5,131 14
= �5,166 23
↵f1s2
= ↵f1s1↵ �l.o.f1
s1
= ↵f1 ↵ �l.o.f1s1
rf1s2
= 5
bf1s2
↵f0�T l.o.f0s0
�= 231
14 ↵f0
�T l.o.f0s0
�= 266
23
= = �5,231 14
= �5,266 23
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.
Analyses
SFA FIFO Multiplexing Arbitrary Multiplexing
s0
↵x(f1)s0 = ↵f0
= �5,25
�l.o.f1s0
= �s0 ↵x(f1)
s0
Rl.o.f1s0
hR
s0 � rx(f1)s0
i+= 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
s1↵x(f1)s1 = �0,0
�l.o.f1s1
= �s1 ↵x(f1)
s1 = �s1 = �20,20
s2
↵x(f1)s2 = �5,131 1
4= �5,166 2
3
�l.o.f1s2
= �s2 ↵x(f1)
s2
Rl.o.f1s2
hR
s2 � rx(f1)s2
i+= 15
T l.o.f1s2
�s2 = bx(f1)
s2
20 · [t� 20]
+= 131
1
4
t = 26
9
16
�s2 = ↵x(f1)
s2
20 · [t� 20]
+= 5 · t+ 166
2
3
t = 37
7
9
= = �15,26 916
= �15,37 79
�l.o.f1e2e
= �R
l.o.f1e2e
,T
l.o.f1e2e
N2i=0 �
l.o.f1si
= �15,67 1316
N2i=0 �
l.o.f1si
= �15,86 19
Df1
�l.o.f1e2e
= bf1
15 ·t� 67
13
16
�+= 25
t = 69
23
48
�l.o.f1e2e
= bf1
15 ·t� 86
1
9
�+= 25
t = 87
7
9
Bf1
↵f1
⇣T l.o.f1
e2e
⌘= 5 · 6713
16
+ 25
= 364
1
16
↵f1
⇣T l.o.f1
e2e
⌘= 5 · 861
9
+ 25
= 455
5
9
PMOO Arbitrary Multiplexing
s0↵x(f1)s0
= �5,25↵x̄(f1)s0
s1↵x(f1)s0
= �0,0↵x̄(f1)s0
s2↵x(f1)s2
= �5,166 23↵x̄(f1)
s2
�l.o.f1e2e
= �R
l.o.f1e2e
,T
l.o.f1e2e
Rl.o.f1e2e
=
^i2{0,1,2}
⇣R
si � rx(f1)si
⌘= (20� 5) ^ (20� 0) ^ (20� 5)
= 15
T l.o.f3e2e
=
Xi2{0,1,2}
Tsi +
bx̄(f1)si + rx(f1)
si · Tsi
Rl.o.f1e2e
!
= 20 +
25 + 5 · 2015
+ 20 +
0 + 0 · 2015
+ 20 +
166
23 + 5 · 2015
= 60 +
391
23
15
= 86
1
9
= = �15,86 19
Df1
�l.o.f1e2e
= bf1
15 ·t� 86
1
9
�+= 25
t = 87
7
9
Bf1
↵f1
⇣T l.o.f1
e2e
⌘= 5 · 861
9
+ 25
= 455
5
9
