63
The Max-Plus Algebra: A New Approach To Performance Evaluation of Discrete Event Systems Bernd Heidergott TU Eindhoven Department of Mathematics and Computing Science IFORS, July 8, 2002
The
Max
-Plu
sA
lgeb
ra:
AN
ewA
ppro
ach
ToP
erfo
rman
ceE
valu
atio
nof
Dis
cret
eE
vent
Sys
tem
s
Ber
ndH
eide
rgot
t
TU
Ein
dhov
enD
epar
tmen
tofM
athe
mat
ics
and
Com
putin
gS
cien
ce
IFO
RS
,Jul
y8,
2002
Out
line
OfT
heTu
toria
l
•S
emiri
ngs
•T
he(M
ax,+
)S
emiri
ng
•F
our
Goo
dR
easo
nsfo
rW
orki
ngw
ithth
e(M
ax,+
)S
emiri
ng
•M
axP
lus
atW
ork
–T
hede
term
inis
ticse
tup
[Pub
licTr
ansp
orta
tion
]
–In
term
ezzo
:Max
Plu
sM
odel
s[Q
ueui
ngm
odel
]
–M
axP
lus
for
Sto
chas
ticS
yste
ms
[Hea
psof
piec
es]
•C
oncl
udin
gre
mar
ks
1
Bef
ore
we
star
t,th
eba
sic
refe
renc
e(s
tilli
s)
Fran
cois
Bac
celli
,Guy
Coh
en,G
eert
Jan
Ols
der
and
Jean
–Pie
rre
Qua
drat
Syn
chro
niza
tion
and
Line
arity
:An
Alg
ebra
for
Dis
cret
eE
vent
Sys
tem
sJo
hnW
iley
and
Son
s,N
ew–Y
ork,
1992
Thi
sbo
okis
outo
fprin
t,bu
titc
anbe
dow
nloa
ded
from
the
web
via
http
://w
ww
-roc
q.in
ria.fr
/sci
lab/
cohe
n/S
ED
/boo
k-on
line.
htm
l
2
Sem
irin
gs
3
Sem
iring
:Defi
nitio
n
Ase
miri
ngis
ano
n–em
pty
setR
endo
wed
with
two
bina
ryre
latio
ns⊕
and⊗
such
that
•th
eop
erat
ion⊕
isas
soci
ativ
e,co
mm
utat
ive,
has
zero
elem
ent
�
and⊗
dist
ribut
esov
er⊕
;
•th
eop
erat
ion⊗
isas
soci
ativ
e,ha
sun
itel
emen
t
e
and
�
isab
sorb
ing
for⊗
:
∀a
∈R
:
a
⊗
�
=
�
:
Ase
miri
ngis
deno
ted
byR
=
(R
;
⊕
;
⊗
;
�
;
e
)
.
4
Sem
iring
:Defi
nitio
n(c
ont.)
Ase
miri
ngis
calle
did
empo
tent
if
∀a
∈
R
:
a
⊕
a
=
a
;
and
com
mut
ativ
eif⊗
isco
mm
utat
ive.
Idem
pote
ntse
miri
ngs
are
calle
ddi
oids
inB
CO
Q(1
992)
.
5
Sem
iring
:Exa
mpl
es
R
P
(R
)
I R
∪{−
∞}
I R
∪{∞
}
I R
⊕∩
m
ax
m
in
+
�
R
−∞
∞0
⊗∪
+
+
×
e
∅
0
0
1id
emp.
··
^
··
^
··
^
6
Idem
pote
ncy
Idem
pote
ncy
of⊕
rule
sou
tinv
erta
bilit
yof
⊕.
Pro
of:S
uppo
seth
atfo
r
a
6=�
anu
mbe
r
b
exis
tssu
chth
at
a
⊕
b
=
�
:
Add
ing
a
onbo
thsi
des
yiel
ds
a
⊕
a
⊕
b
=
a
⊕�
:
By
idem
pote
ncy,
this
iseq
uiva
lent
to
a
⊕b
=
a
⊕
�
;
whi
chim
plie
sa
⊕
b
=
a
:
Thi
sco
ntra
dict
s
a
⊕
b
=
�
.
7
Par
ticul
arS
emiri
ngs
The
stru
ctur
e
R
m
a
x
=
(IR
�
=
IR
∪{−
∞};
⊕
=
m
ax;
⊗
=
+
;
�
=
−∞
;
e
=
0)
cons
titut
esan
idem
pote
ntse
miri
ngkn
own
as(m
ax,+
)–al
gebr
a.
The
stru
ctur
e
R
m
i
n
=
(IR
>
=
IR
∪{∞
};
⊕
=
m
in;
⊗=
+
;
>
=
∞
;
e
=
0)
cons
titut
esan
idem
pote
ntse
miri
ngkn
own
as(m
in,+
)–al
gebr
a.N
ote
that
for
the
(min
,+)
alge
bra
the
nota
tion>
for
the
zero
elem
ento
f⊕is
stan
dard
.
8
Oth
erIm
port
antS
emiri
ngs
Inne
twor
kca
lcul
uson
ede
fines
,for
exam
ple,
(f
⊕
g
)(t
)
=
m
ax(f
(t
);
g
(t
))
and
(f
⊗
g
)(t
)
=
sup
0
≤
s
≤
t
�
f
(t
−
s
)
+
g
(s
)�
:
Net
wor
kC
alcu
lus
atIF
OR
S:
Ses
sion
TD
16,v
enue
WR
-11,
”On
prob
abili
stic
netw
ork
calc
ulus
”,M
.Voj
novi
can
dJ.
–Y.L
eB
oude
c.
Ses
sion
TD
16,v
enue
WR
-11,
”Som
ere
sults
ofde
term
inis
ticne
twor
kca
lcul
usap
plie
dto
com
mun
icat
ion
netw
orks
”,P.
Thi
ran
and
J.–Y
.Le
Bou
dec.
Mor
eon
sem
iring
s:S
teph
ane
Gau
bert
.Met
hods
and
appl
icat
ions
of(m
ax,+
)–lin
ear
alge
bra
InP
rocc
edin
gsof
the
STA
CS
’199
7,Le
ctur
eN
otes
inC
ompu
ter
Sci
ence
,
vol1
200,
Spr
inge
r,19
97(t
his
repo
rtca
nbe
acce
ssed
via
the
WE
Bat
http
://w
ww
.inria
.fr/R
RR
T/R
R-3
088.
htm
l)
9
Mat
rices
and
Vec
tors
inR
=
(R
;
⊕
;
⊗
;
�
;
e
)
For
mat
rices
A
∈
R
I
×
K
�
and
B
∈
R
K
×
J
�
we
defin
eth
em
atrix
prod
uct
A
⊗
B
inth
eus
ualw
ay:
(A
⊗
B
)i
j
=
K
M
k
=
1
A
i
k
⊗
B
k
j
:
Spe
cific
ally
,we
intr
oduc
eth
e
i
t
h
pow
erof
A
by
A
i
=
A
⊗
:
:
:
⊗A
|
{
z
}
i
times
;
whe
re
A
0
=
E
.
Add
ition
ofm
atric
es
A
∈
R
J
×
I
�
and
B
∈R
J
×
I
�
,den
oted
by
A
⊕
B
,is
give
nby
(A
⊕B
)i
j
=
A
i
j
⊕
B
i
j
:
10
Th
e(M
ax,+
)S
emir
ing
11
The
(Max
,+)
Sem
iring
:Bas
icC
alcu
lus
[Rec
allt
hatR
m
a
x
=
(IR
∪{−
∞};
⊕
=
m
ax;
⊗
=
+
;
�
=
−∞
;
e
=
0)
]
5
⊕
3
=
5
⊕
�
=
e
⊕
3
=
43
=
�
4
⊗
3
=
�
3
√
9
=
√−
1
=
12
The
(Max
,+)
Sem
iring
:Mat
rix-V
ecto
rC
alcu
lus
0 B @
e
�
3
2
1 C A
⊗
0 B @
5 1
1 C A
=
13
The
(Max
,+)
Sem
iring
:Pol
ynom
ials
Con
side
rth
epo
lyno
mia
l
�
x
⊕
1�
2
By
alge
brai
cco
mpu
tatio
ns,
�
x
⊕
1�
2
=
�
x
⊕
1�
⊗
�
x
⊕
1�
=
�
x
⊗
x
�
⊕
�
x
⊗
1�
⊕�
x
⊗
1�
⊕
(1
⊗
1�
=
x
2
⊕
�
x
⊗
1�
⊕2
Num
eric
ally
,
=
m
ax
�
2x
;
x
+
1
;
2�
=
m
ax
�
2x
;
2�
=
x
2
⊕
2
:
14
The
(Max
,+)
Sem
iring
:Pol
ynom
ials
(con
t.)
2x
2
x
+
1
������������������
��
��
��
��
��
��
��
��
-
6
��
��
��
��
��
��
�
x
⊕
1�
2
=
x
2
⊕�
x
⊗
1�
⊕
2
=
x
2
⊕
2
:
Gen
eral
ly,
�
a
⊕
b
�
n
=
a
n
⊕
a
n
−
1
b
⊕···⊕
a
b
n
−
1
⊕
b
n
:
15
War
ning
!H
ybrid
For
mul
as
LetA
∈
IR
J
×
J
�
and
x
0
∈
IR
J �
and
cons
ider
x
(k
+
1)=
A
⊗
x
(k
)
;
k
≥
0
x
(0)=
x
0
:
How
dow
ein
terp
rett
hefo
llow
ing
form
ula:
lim
k
→∞
1 k
x
(k
)
=
lim
k
→∞
1 k
A
k
⊗x
0
?
16
Fo
ur
Go
od
Rea
son
sfo
rW
ork
ing
wit
hth
e(M
ax,+
)S
emir
ing
17
AF
irst
Goo
dR
easo
nfo
r(M
ax,+
):C
ompu
ting
Max
imal
Wei
ghts
inG
raph
s
LetA
∈
IR
J
×
J
�
.The
com
mun
icat
ion
grap
hof
A
,den
oted
byG
(A
)
,is
defin
edas
follo
ws.
G
(A
)
has
node
s{1
;
:
:
:
;
J
},an
da
pair
(i
;
j
)
∈
J
×
J
isan
arc
ofth
egr
aph
if
A
j
i
6=�
.
For
any
arc
(i
;
j
)
inG
(A
)
,we
call
A
j
i
the
wei
ghto
farc
(i
;
j
)
and
the
wei
ghto
fapa
thin
G
(A
)
isde
fined
byth
esu
mof
the
wei
ghts
ofal
larc
sco
nstit
utin
gth
epa
th.
The
n,
(A
n
)j
i
yiel
dsth
em
axim
alw
eigh
tofa
path
ofle
ngth
n
(tha
tis,
cons
istin
gof
n
arcs
)fr
omno
de
i
tono
de
j
,and
(A
n
)j
i
=
�
refe
rsto
the
fact
that
ther
eis
nopa
thof
leng
th
n
from
i
to
j
,inG
(A
)
.
18
AS
econ
dG
ood
Rea
son
for
(Max
,+):
Sol
ving
Line
arE
quat
ions
Let
x
;
b
∈
IR
J �
and
A
∈
IR
J
×
J
�
,sol
ve
x
=
A
⊗
x
⊕
b
:
(1)
We
defin
eth
epo
wer
serie
sof
A
by
A
∗
=
∞
M
i
=
0
A
i
;
whi
chis
finite
if
A
isa
low
ertr
iang
ular
mat
rix.
The
n,
x
=
A
∗⊗
b
solv
es(1
).
19
Inte
rmez
zo:
Irre
duci
bilit
y
Am
atrix
A
∈
IR
J
×
J
�
isca
lled
irred
ucib
leif
itsco
mm
unic
atio
ngr
aph
isst
rong
lyco
nnec
ted.
Inw
ords
:for
any
two
node
s
i
;
j
ther
eex
itsa
path
inG
(A
)
.
20
AT
hird
Goo
dR
easo
nfo
r(M
ax,+
):E
igen
valu
esan
dE
igen
vect
ors
For
any
irred
ucib
lem
atrix
A
∈
IR
J
×
J
�
,uni
quel
yde
fined
inte
gers
c
(A
)
,�(A
)
and
aun
ique
lyde
fined
real
num
ber
�
=
�
(A
)
exis
tsuc
hth
at,f
oral
ln
≥c
(A
)
:
A
n
+
�
(
A
)
=
�
⊗
�
(
A
)
⊗
A
n
:
The
num
ber
c
(A
)
isca
lled
the
coup
ling
time
of
A
,�(A
)
isca
lled
the
cycl
icity
of
A
and
�
⊗
�
(
A
)
isth
eun
ique
eige
nval
ueof
A
⊗
�
(
A
)
.
Thi
sis
also
calle
dth
e”P
erro
n-Fr
oben
ius
The
orem
of(m
ax,+
)al
gebr
a”.
�
isal
soca
lled
the
Lyap
unov
expo
nent
ofA
.
21
AT
hird
Goo
dR
easo
nfo
r(M
ax,+
):E
igen
valu
esan
dE
igen
vect
ors
(con
t.)
LetA
∈
IR
J
×
J
�
and
x
0
∈
IR
J �
and
cons
ider
x
(k
+
1)=
A
⊗
x
(k
)
;
k
≥
0
x
(0)=
x
0
:
For
allk
≥
c
(A
)
:
x
(k
+
�
(A
))=
A
k
+
�
(
A
)
⊗
x
0
=
�
⊗
�
(
A
)
⊗A
k
⊗x
0
=
�
⊗
�
(
A
)
⊗x
(k
)
:
We
say
that
{x(k
)
}en
ters
itspe
riodi
cre
gim
eaf
ter
(atm
ost)
c
(A
)
tran
sitio
ns.
For
any
initi
alve
ctor
x
(0)
,the
limiti
ngbe
havi
our
ofth
ese
quen
ce{x
(k
)
}is
lim
k
→∞
x
j
(k
)
k
=
�
;
1
≤
j
≤
J
:
22
AT
hird
Goo
dR
easo
nfo
r(M
ax,+
):E
igen
valu
esan
dth
eC
omm
un.G
raph
LetC
deno
teth
ese
tofc
ircui
tsin
G
(A
)
.Itt
hen
hold
sth
at
�
=
m
ax
p
∈C
wei
ghto
fp
leng
thof
p
:
Aci
rcui
twho
seav
erag
ew
eigh
tis
max
imal
(equ
als
�
)is
calle
dci
rtic
al.
The
criti
calg
raph
isth
esu
bgra
phof
G
(A
)
that
cont
ains
the
criti
calc
ircui
tson
ly.
The
criti
calg
raph
dete
rmin
esth
ecy
clic
ityof
A
:Ift
hecr
itica
lgra
phis
stro
ngly
conn
ecte
d,th
enth
ecy
clic
ityof
A
isgi
ven
byth
egr
eate
stco
mm
ondi
viso
rof
the
leng
ths
ofal
lcirc
uits
inth
ecr
itica
lgra
ph.
The
criti
calg
raph
char
acte
rizes
the
eige
nspa
ceof
A
⊗
�
(
A
)
.
23
AT
hird
Goo
dR
easo
nfo
r(M
ax,+
):C
ompu
tatio
nalI
ssue
s
•P
ower
algo
rithm
(eig
enva
lue
and
eige
nvec
tor)
•K
arp’
sal
gorit
hm(e
igen
valu
e)
•T
heH
owar
dal
gorit
hmal
low
sfo
rco
mpu
ting
the
eige
nval
uean
dan
eige
nvec
tor
inal
mos
tlin
ear
time
(eig
enva
lue
and
eige
nvec
tor)
•C
ompu
ting
the
coup
ling
time
isN
Pha
rdin
the
num
ber
ofcr
itica
lcirc
uits
,no
effic
ient
algo
rithm
sex
ista
ndon
ly(c
rude
)up
per
boun
dsar
ekn
own
24
AF
ourt
hG
ood
Rea
son
for
(Max
,+):
Sub
addi
tivity
For
A
∈
IR
J
×
J
�
,set
|| A
|| ∨
=
J
M
i
=
1
J
M
j
=
1
A
i
j
=
m
ax
{
A
i
j
:1
≤
i
;
j
≤J
}:
Itth
enho
lds
for
A
;
B
∈
I R
J
×
J
�
that
|| A
⊗
B
|| ∨≤
||A
|| ∨
+
||B
|| ∨:
LetA
(k
)
∈
I R
J
×
J
�
,for
k
≥
0
,and
set
�
l
k
=
� � � � � �
� � � � � �
k
−
1
O i
=
l
A
(i
)� � � � � �
� � � � � �
∨
;
then
{�l
k
:
k
≥
1
;0
≤
l
<
k
}is
suba
dditi
ve,t
hati
s,
�
l
k
≤
�
l
m
+
�
m
k
;
for
allm
with
l
<
m
<
k
.
25
Max
Plu
sat
Wo
rk
26
(Max
,+)-
Line
arD
iscr
ete
Eve
ntS
yste
ms
LetA
∈
IR
J
×
J
�
and
cons
ider
the
hom
ogen
eous
recu
rsio
n
x
(k
+
1)=
A
⊗
x
(k
)
;
k
≥
0
x
(0)=
x
0
:
Alte
rnat
ivel
y,co
nsid
erth
ein
hom
ogen
eous
recu
rsio
n
x
(k
+
1)=
A
⊗
x
(k
)
⊕
b
(k
+
1)
;
k
≥
0
x
(0)=
x
0
;
with
b
(k
+
1)
∈
I R
J �
.
Asy
stem
who
sest
ate–
dyna
mic
follo
ws
eith
erof
the
abov
ere
curs
ion
isca
lled
(max
,+)–
linea
r.
(Max
,+)–
linea
rsy
stem
sar
ise
natu
rally
inth
epr
esen
ceof
sync
hron
izat
ion.
27
(Max
,+)–
Line
arS
yste
ms:
AP
ublic
Tran
spor
tatio
nE
xam
ple
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
2
3
-
��
-•• ••
Fou
rtr
ains
circ
ulat
eon
sepa
rate
dlin
es.A
tthe
cent
erst
atio
n,th
ede
part
ure
oftr
ains
issy
nchr
oniz
edin
orde
rto
letp
asse
nger
sch
ange
trai
ns.
28
AP
ublic
Tran
spor
tatio
nE
xam
ple:
The
(Max
,+)
Mod
el
LetA
i
j
deno
teth
etr
avel
time
from
stat
ion
j
to
i
(incl
udin
gdw
ellt
imes
)an
dle
txj
(k
)
deno
teth
e
k
t
h
depa
rtur
etim
efr
omst
atio
n
j
,the
n
x
1
(k
+
1)=
A
1
2
⊗
x
2
(k
)
x
2
(k
+
1)=
�
A
2
1
⊗
x
1
(k
)�
⊕
�
A
2
3
⊗
x
3
(k
)�
x
3
(k
+
1)=
A
3
2
⊗
x
2
(k
)
:
Inm
atrix
–vec
tor
natio
n:
x
(k
+
1)
=
0 B @
�
A
1
2
�
A
2
1
�
A
2
3
�
A
3
2
�
1 C A
⊗
x
(k
)
:
29
AP
ublic
Tran
spor
tatio
n:C
ompu
ting
the
Eig
enva
lue
1
2
3
A
2
1
A
1
2
A
3
2
A
2
3
••
•..........................
....................................................
...................................................
..................................................
................................................
................. .... .... .... .... ... .... .... ... . .. .
. .. . .. .. .. . .. .. .. . .. .. . . . .. .. . . . . . . .. . . . . . . .. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . .
. . . . . . . . . . .. . . . . . .. . .. . .. .. . .. . .. . .. .. . .. . .. ... .... .
.. ... ... ... ... ... .................................
................................................
..................................................
...................................................
....................................................
.............................................
....................................................
..................................................
..................................................
.................................................
................ .... .... .... .... ... .... .... ... . .. ..
.. . .. .. .. . .. .. .. . .. .. . . . . ... . . . . . . .. . . . . . . .. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . .
. . . . . . . . . .. . . . . . .. . .. . .. ... .. . .. . .. .. . .. . .. ... .... ..
. ... ... ... ... ... ..................................
.................................................
..................................................
...................................................
....................................................
................
�
�
If
A
1
2
+
A
2
1
>
A
2
3
+
A
3
2
,the
n
1
→2
→
1
isth
ecr
itica
lcirc
uit
�
=
A
1
2
+
A
2
1
2an
d
�
(A
)
=
2
:
30
Tim
etab
leD
esig
n,I
LetA
bea
mat
rixof
cycl
icity
1
mod
elin
gth
eac
tual
trav
elin
gtim
esof
trai
nson
the
trac
ks.T
hen,
the
vect
orof
(k
+
1)s
t
depa
rtur
etim
espe
rtr
ack
follo
ws
x
(k
+
1)=
A
⊗
x
(k
)
;
k
∈
I N
;
x
(0)=
x
0
:
Ass
ume
that
,in
addi
tion
toth
at,
A
isirr
educ
ible
.By
the
(max
,+)
Per
ron-
Frob
eniu
sth
eore
m,f
orsu
ffici
ently
larg
e
k
,
x
(k
+
1)
=
�
⊗x
(k
)
;
whe
re
�
deno
tes
the
eige
nval
ueof
A
.
31
Tim
etab
leD
esig
n,II
LetX
deno
tean
eige
nvec
tor
of
A
and
take
x
0
=
X
,the
n
x
(k
+
1)
=
A
k
⊗
X
=
(k
·�)
⊗
X
:
Hen
ce,X
repr
esen
tsa
timet
able
and
�
the
freq
uenc
y,or
,spe
edof
the
timet
able
.
X
isop
timal
inth
ese
nse
that
itre
pres
ents
the
timet
able
with
the
high
estf
requ
ency
oftr
ains
per
trac
kph
ysic
ally
poss
ible
.
For
�
≥
�
,let
d
(k
)
=
(k
·�)
⊗
X
deno
teth
eve
ctor
ofpl
anne
d
k
t
h
depa
rtur
etim
espe
rtr
ack
acco
rdin
gto
timet
able
X
,the
nth
eac
tual
depa
rtur
etim
es
x
(k
)
are
give
nby
x
(k
+
1)
=
A
⊗
x
(k
)
⊕
d
(k
+
1)
:
The
diffe
renc
e
�
−
�
isan
indi
cato
rfo
rth
ero
bust
ness
ofth
etim
etab
le.
32
Pro
paga
tion
ofD
elay
s,I
By
sim
ple
alge
bra,
x
(k
+
1)=
A
⊗
x
(k
)
⊕
d
(k
+
1)
=
A
⊗
(A
⊗
x
(k
−
1)
⊕
d
(k
))
⊕
d
(k
+
1)
=
A
2
⊗
x
(k
−
1)
⊕
A
⊗
d
(k
)
⊕d
(k
+
1)
=
A
2
⊗
x
(k
−
1)
⊕
d
(k
+
1). . .
=
A
k
⊗
x
(1)
⊕
d
(k
+
1)
:
Lett
hefir
sttr
ain
whi
chde
part
son
trac
kj
bede
laye
d,so
that
x
j
(1)
>
d
j
(1)
,and
assu
me
that
this
isth
eon
lytr
ain
that
isde
laye
d.
33
Pro
paga
tion
ofD
elay
s,I
I
The
initi
alde
lay
ontr
ack
j
caus
esa
dela
yfo
rth
e
(k
+
1)s
t
trai
nde
part
ing
ontr
ack
i
if
M
l
(A
k
)i
l
⊗
x
l
(1)
>
d
i
(k
+
1)
:
Bec
ause
ofou
ras
sum
ptio
nth
at
x
i
(1)
≤
d
i
(1)
,for
i
6=j
,itf
ollo
ws
that
the
(k
+
1)s
t
depa
rtur
eon
trac
k
i
isde
laye
dbe
caus
eof
ade
lay
inth
ein
itial
depa
rtur
eon
trac
k
j
if
(A
k
)i
j
⊗
x
j
(1)
>
d
i
(k
+
1)
:
Inth
isw
ayw
eob
tain
the
seto
fall
dela
yed
trai
ns.
34
Pro
paga
tion
ofD
elay
s,I
II
Obs
erve
that
the
mat
rices
A
k
(k
=
1;
2;
:
:
:
)
can
beca
lcul
ated
inad
vanc
ean
dth
atto
dete
rmin
eth
epr
opag
atio
nof
anin
itial
dela
yon
trac
k
j
,we
only
need
the
j
t
h
colu
mn
ofth
ese
mat
rices
.
Afte
r
k
∗st
eps,
whe
re
k
∗is
give
nby
k
∗
=
m
in
8 < :
k
� � � � � �
(A
k
)i
j
⊗
x
j
(1)
≤d
i
(k
+
1)
∀i9 = ;
;
the
initi
alde
lay
on
j
isou
toft
hesy
stem
.
35
Pro
paga
tion
ofD
elay
san
dth
eC
oupl
ing
Tim
e
Con
side
rth
esy
stem
with
initi
alve
ctor
X
′ :
x
′ (k
+
1)=
A
⊗
x
′ (k
)
;
k
∈
I N
;
x
′ (0)=
X
′ :
Rec
allt
hatt
heco
uplin
gtim
eof
A
isde
note
dby
c
(A
)
.Hen
ce,i
ndep
ende
ntof
X
′ ,it
hold
sth
at
x
(k
)
=
a
⊗
x
′ (k
)
;
k
≥c
(A
)
;
for
som
efin
itenu
mbe
r
a
.(H
ere,
we
assu
me
that
A
has
anun
ique
eige
nvec
tor.)
Inw
ords
,ade
lay
eith
erdi
esou
tafte
rat
mos
t
c
(A
)
tran
sitio
nsor
resu
ltsin
aun
iform
dela
yof
a
time
units
onal
ltra
cks:
a
isth
epa
rtof
the
dela
yth
atre
ache
sth
ecr
itica
lci
rcui
t.
36
App
licat
ions
of(m
ax,+
)to
Rai
lway
Sys
tem
sat
IFO
RS
Sem
i-ple
nary
,Sec
tion
TA16
,ven
ueW
R-1
1”M
ax–p
lus
alge
bra
and
itsap
plic
atio
nto
railw
aysy
stem
s”,G
.J.O
lsde
r,(T
oolP
ET
ER
)
Ses
sion
TC
16,v
enue
WR
-11,
”Lon
g-te
rmca
paci
tyan
alys
isof
tunn
els
ona
railw
aylin
e”,A
.de
Kor
t,B
.Hei
derg
otta
ndH
.Ayh
an.
New
!Ses
sion
TC
16,v
enue
WR
-11,
”Per
form
ance
eval
uatio
nof
trai
nne
twor
ktim
etab
les”
,R.G
over
de.
37
Inte
rmez
zo:
Max
Plu
sM
od
els
38
A(M
ax,+
)-Li
near
Que
uing
Sys
tem
Con
side
ran
open
syst
emof
J
sing
le–s
erve
rqu
eues
inta
ndem
,with
infin
itebu
ffers
.W
eas
sum
eth
atth
esy
stem
star
tsem
pty.
�
�
�
�
�
�
�
�
�
�
�
�
--
--
-
12
···
J
�
0
�
1
�
2
�
J
The
sequ
ence
ofde
part
ure
times
then
follo
ws
x
(k
+
1)
=
A
⊗x
(k
)
;
with
A
=
2 6 6 6 6 4
�
0
�
�
:
:
:
�
�
0
⊗
�
1
�
1
�
:
:
:
�
�
0
⊗
�
1
⊗
�
2
�
1
⊗
�
2
�
2
:
:
:
�
. . .. . .
. . .. . .
�
0
⊗
�
1
⊗
�
2
⊗···⊗
�
J
�
1
⊗
�
2
⊗···⊗
�
J
�
2
⊗···⊗
�
J
:
:
:
�
J
3 7 7 7 7 5
:
39
A(M
ax,+
)-Li
near
Que
uing
Sys
tem
We
now
cons
ider
the
open
tand
emqu
euin
gsy
stem
with
one
item
initi
ally
resi
ding
atea
chqu
eue.
�
�
�
�
�
�
�
�
�
�
�
�
---
--
-•
••
12
···
J1
2J
�
0
�
1
�
2
�
J
The
sequ
ence
ofde
part
ure
times
then
follo
ws
x
(k
+
1)
=
A
⊗x
(k
)
;
whe
re
A
=
2 6 6 6 6 4
�
0
�
:
:
:
�
�
�
0
�
1
�
:
:
:
. . .
:
:
:
�
J
−
2
�
J
−
1
�
:
:
:
�
�
J
−
1
�
J
3 7 7 7 7 5
:
40
(Max
,+)-
Line
arM
odel
s
(Max
,+)
mod
els
desc
ribe
poin
tsin
time,
e.g.
,w
hen
ace
rtai
nev
ento
ccur
sfo
rth
e
k
t
h
time.
We
have
noin
form
atio
nab
outt
heph
ysic
alst
ate
ofth
esy
stem
.
Asy
stem
is(m
ax,+
)–lin
ear
ifan
don
lyif
itca
nbe
mod
eled
bya
FIF
Oev
entg
raph
(i.e,
aP
etri–
nets
uch
that
each
plac
eha
sex
actly
one
up-s
trea
man
don
edo
wns
trea
mtr
ansi
tion)
.
Inte
rms
ofqu
euin
g:(-
)cu
stom
ercl
asse
s,(-
)ov
erta
king
ofcu
stom
ers,
(-)
rout
ing;
(+)
sync
hron
izat
ion,
(+)
fork
and
join
,(+
)bl
ocki
ng
41
Max
Plu
sfo
rS
toch
asti
cS
yste
ms
42
Hea
psof
Pie
ces
J
=
{1;
2;
:
:
:
;
5
}a
seto
fres
ourc
es.
(a)
1
2
3
4
5
(b)
1
2
3
4
5
(c)
1
2
3
4
5
43
Hea
psof
Pie
ces
The
heap
a
b
a
c
b
:
1
2
3
4
5
44
Hea
psof
Pie
ces
:The
(Max
,+)
Mod
el
Den
ote
by
x
j
(k
)
the
heig
htof
the
pile
ofpi
eces
over
reso
urce
j
afte
r
k
bloc
ks.
Itca
nbe
show
nth
ata
piec
e
a
can
bem
odel
edby
a
J
×
J
dim
ensi
onal
mat
rix
M
(a
)
.
Pili
nghe
aps
ina
stoc
hast
icw
ayle
ads
toa
sequ
ence
{ M(k
)
}of
mat
rices
and
the
heig
htve
ctor
follo
ws
x
(k
+
1)
=
M
(k
)
⊗
x
(k
)
;
k
≥
0
;
whe
re
x
(0)
deno
tes
the
initi
alco
ntou
r.
The
heig
htve
ctor
ofth
ehe
ap
(a
b
a
c
b
)
isgi
ven
by:
x
(5)
=
x
(a
b
a
c
b
)
=
M
(b
)
⊗M
(c
)
⊗
M
(a
)
⊗
M
(b
)
⊗
M
(a
)
⊗
x
(0)
:
45
Type
sof
Lim
itsfo
rH
eaps
ofP
iece
s
The
vect
or
^x
(k
)
=
�
x
2
(k
)
−
x
1
(k
);
x
3
(k
)
−
x
1
(k
);
:
:
:
;
x
J
(k
)
−x
1
(k
)�
desc
ribes
the
cont
our
ofth
ehe
ap.
Firs
tord
erlim
it:T
hegr
owth
rate
may
conv
erge
with
prob
abili
tyon
eto
war
dsa
num
ber
lim
k
→∞
1 k
x
(k
)
=
a
:
Sec
ond
orde
rlim
it:T
heup
per
cont
our
may
conv
erge
wea
kly
tow
ards
alim
iting
rand
omva
riabl
e
^x
:lim
k
→∞
^x
(k
)
:
46
Fir
stO
rder
Lim
its
47
Firs
tOrd
erE
rgod
icT
heor
yfo
rS
toch
astic
(Max
,+)–
Line
arS
yste
ms
Bas
icas
sum
ptio
n:{A
(k
)
}is
ara
ndom
sequ
ence
such
that
A
(k
)
has
a.s.
atle
ast
one
entr
ydi
ffere
ntfr
om
�
onea
chro
w.
We
stud
yth
ese
quen
ce
x
(k
+
1)=
A
(k
)
⊗
x
(k
)
;
k
∈IN
;
x
(0)=
x
0
;
with
x
0
finite
.
Let
|| x(k
)
|| ∧=
m
in
{
x
j
(k
)
}
and
||x(k
)
|| ∨
=
m
ax
{
x
j
(k
)
}
:
48
The
Firs
tOrd
erE
rgod
icT
heor
em
Let{
A
(k
)
}be
ani.i
.d.s
eque
nce
ofin
tegr
able
mat
rices
(with
the
”bas
icas
sum
ptio
n”in
forc
e).
By
Kin
gman
’ssu
badd
itive
ergo
dic
theo
rem
,fini
teco
nsta
nts
�
top
and
�
bot
exis
tsu
chth
atfo
ral
lfini
tein
itial
cond
ition
s
x
0
lim
k
→∞
||x(k
)
|| ∧
k
=
lim
k
→∞
E
"
||x(k
)
|| ∧
k
#
=
�
bot
and
lim
k
→∞
||x(k
)
|| ∨
k
=
lim
k
→∞
E
"
||x(k
)
|| ∨k
#
=
�
top
with
prob
abili
tyon
e.
�
top
isca
lled
the
max
imal
orto
pLy
apun
ovex
pone
ntan
d
�
bot
the
min
imal
orbo
ttom
Lyap
unov
expo
nent
.
49
Refi
ning
the
Firs
tOrd
erE
rgod
icT
heor
em
We
say
that
ara
ndom
mat
rix
A
has
afix
edsu
ppor
tifa
nel
emen
tofA
isei
ther
with
prob
abili
tyon
efin
ite,o
rw
ithpr
obab
ility
one
equa
lto
�
.
Fix
edsu
ppor
tim
plie
sth
atth
eva
lues
ofth
eel
emen
tsar
era
ndom
butn
otth
eir
posi
tion.
Our
queu
ing
exam
ple
has
fixed
supp
ort,
the
heap
sof
piec
esm
odel
sfa
ilsto
have
afix
edsu
ppor
t.
50
The
Firs
tOrd
erE
rgod
icT
heor
em(
2n
d
Ver
sion
)
Let{
A
(k
)
}be
ani.i
.d.s
eque
nce
ofm
atric
essu
chth
at
•
A
(k
)
isin
tegr
able
,
•
A
(k
)
isirr
educ
ible
(pre
supp
oses
fixed
supp
ort)
,
•an
yfin
iteel
emen
tofA
(k
)
ispo
sitiv
ean
dal
ldia
gona
lele
men
tsof
A
(k
)
are
finite
,
then
,for
1
≤
j
≤
J
, lim
k
→∞
x
j
(k
)
k
=
lim
k
→∞
E
"
x
j
(k
)
k
#
=
�
a.s.
�
=
�
top
=
�
bot
isca
lled
the
Lyap
unov
expo
nent
of{ A(k
)
}.
51
The
Firs
tOrd
erE
rgod
icT
heor
em:
Ext
ensi
ons
The
first
orde
rer
godi
cth
eore
mca
nbe
exte
nded
tore
duci
ble
mat
rices
.
The
”fixe
dsu
ppor
t”as
sum
ptio
nca
nbe
rela
xed
for
mat
rices
with
disc
rete
stat
e-sp
ace
via
patte
rns.
The
basi
cde
finiti
onis
:The
reis
am
atrix
C
inth
est
ate–
spac
eof
{A(k
)
}su
chth
at(i)
C
isirr
educ
ible
,has
aun
ique
eige
nvec
tor,
and
(ii)
an
N
∈
I N
exis
tsw
ith
P
�
A
(N
)
⊗
A
(N
−
1)
⊗···A
(1)
=
C
�
>
0
:
52
Sec
on
dO
rder
Lim
its
53
Wai
ting
Tim
es
Bas
iceq
uatio
n:
x
(k
+
1)
=
A
(k
)
⊗
x
(k
)
⊕
�
(k
+
1)
⊗
B
(k
)
;
whe
re
�
(k
)
deno
tes
the
time
ofth
e
k
t
h
arriv
alto
the
syst
em.W
ede
note
by
�
0
(k
)
the
k
t
h
inte
rarr
ival
time
toth
esy
stem
,whi
chim
plie
s
�
(k
)
=
k
X
i
=
1
�
0
(i
)
;
k
≥1
;
with
�
(0)
=
0
.
The
n,
W
j
(k
)
=
x
j
(k
)
−
�
(k
)
deno
tes
the
time
the
k
t
h
cust
omer
arriv
ing
toth
esy
stem
spen
dsin
the
syst
emun
tilco
mpl
etio
nof
serv
ice
atse
rver
j
.
54
Wai
ting
Tim
es(c
ont.)
The
vect
orof
k
t
h
wai
ting
times
,den
oted
by
W
(k
)
=
(W
1
(k
);
:
:
:
;
W
J
(k
))
follo
ws
the
recu
rsio
n
W
(k
+
1)
=
A
(k
)
⊗
C
(�
0
(k
+
1))
⊗
W
(k
)
⊕B
(k
)
;
k
≥
0
;
whe
re
C
(h
)
deno
tes
adi
agon
alm
atrix
with
−
h
onth
edi
agon
alan
d
�
else
whe
re. 55
The
Sec
ond
Ord
erT
heor
em:
Wai
ting
Tim
es
•Le
t{
A
(k
)
}be
inte
grab
le,a
llfin
iteel
emen
tsar
ea.
s.no
n–ne
gativ
e,an
dth
edi
agon
alel
emen
tsar
ea.
s.no
n–ne
gativ
e.
•T
hese
quen
ce{ (
A
(k
);
B
(k
))
}is
stat
iona
ryan
der
godi
c,an
din
depe
nden
tof
{�(k
)
}.
•T
hem
axim
alLy
apun
ovex
pone
ntof
{ A(k
)}
issm
alle
rth
an
E
[�
0
(k
)]
.
The
n,{ W
(k
)
}co
nver
ges
inst
rong
coup
ling
toan
uniq
uest
atio
nary
regi
me
W
,with
W
=
M
j
≥
0
C
(�
(
−j
))
⊗
j
O
i
=
1
A
(
−
i
)
⊗
B
(
−
(j
+
1))
:
56
Fu
rth
erTo
pic
s
57
Asy
mpt
otic
Ana
lysi
s
Sta
bilit
yth
eory
for
(max
,+)–
linea
rsy
stem
sfo
rpa
rtic
ular
type
sof
inpu
tdis
trib
utio
ns.
Ong
oing
rese
arch
ison
sube
xpon
entia
ldis
trib
utio
ns.
Asy
mpt
otic
sfo
rsu
bexp
onen
tialn
etw
orks
atIF
OR
S:
Ses
sion
TB
16,v
enue
WR
-11,
”Asy
mpt
otic
sof
clos
edqu
euei
ngne
twor
ksw
ithsu
bexp
onen
tials
ervi
cetim
es”,
H.A
yhan
,Z.P
alm
owsk
iand
S.S
chle
gel.
Ses
sion
TB
16,v
enue
WR
-11,
”Sub
expo
nent
iala
sym
ptot
ics
for
stat
iona
ryop
en(m
ax,p
lus)
syst
ems”
,F.B
acce
lli,S
.Fos
san
dM
.Lel
arge
.
58
Com
puta
tiona
lIss
ues
Inco
ntra
stto
the
dete
rmin
istic
setti
ngth
ere
exis
tno
effic
ient
algo
rithm
for
com
putin
g
�
;
�
top
;
�
bot
orst
atio
nary
wai
ting
times
.
Ong
oing
rese
arch
ison
com
puta
tiona
lapp
roac
hes
via
Tayl
orse
ries
expa
nsio
ns.
The
Tayl
orse
ries
appr
oach
atIF
OR
S:
New
!Ses
sion
TB
16,v
enue
WR
-11,
”Pol
ynom
iala
lgor
ithm
sfo
rTa
ylor
expa
nsio
nsof
max
–plu
ssy
stem
s”,A
.Jea
n–M
arie
and
M.H
eusc
h.
Ses
sion
TC
16,v
enue
WR
-11,
”Tai
lpro
babi
lity
ofw
aitin
gtim
esin
max
–plu
s–lin
ear
syst
ems”
,H.A
yhan
and
D.–
W.S
on.
Ses
sion
TD
16,v
enue
WR
-11,
”Num
eric
alev
alua
tion
ofm
ax–p
lus–
linea
rsy
stem
s”,
B.H
eide
rgot
t.59
Co
ncl
ud
ing
Rem
arks
60
Bey
ond
(Max
,+)
•(m
in,m
ax,+
)sy
stem
s
•M
Mca
lcul
us
•to
pica
lmap
ping
s
•m
onot
one
sepa
rabl
efr
amew
ork
•et
c....
61
Upc
omin
g(M
ax,+
)ev
ents
•
6t
h
Inte
rnat
iona
lWor
ksho
pon
Dis
cret
eE
vent
Sys
tem
s(W
OD
ES
’02)
,Z
arag
oza,
Spa
in,2
–4O
ctob
er,2
002.
•In
tern
atio
nalW
orks
hop
onM
ax–P
lus
Alg
ebra
,Bir
min
gham
,UK
,Jul
ior
Aug
ust,
2003
(inpl
anni
ng)
•IE
EE
Con
trol
Sys
tem
sS
ocie
ty:F
irstM
ultid
isci
plin
ary
Inte
rnat
iona
lSym
posi
umon
Pos
itive
Sys
tem
s:T
heor
yan
dA
pplic
atio
ns,F
acul
tyof
Eng
inee
ring,
Uni
vers
ityof
Rom
e”L
aS
apie
nza”
Rom
a,Ita
ly,2
8-30
Aug
ust,
2003
.
62
