Convex comparison of service disciplines in real time queues

0.17 MB·English

by  Pascal Moyal

Convex
omparison of servi
e dis
iplines in real
time queues
∗
P. Moyal
8 LMAC - UTC Compiègne
0
0
2

n
Abstra
t
a
J Wepresenta
omparisonoftheservi
edis
iplinesinreal-timequeueing

8 systems (the
ustomers have a deadline before whi
h they should enter
the servi
e booth). We state that giving priority to
ustomers having

] an early deadline minimizes the average stationary lateness. We show
R
thisresultby
omparingadequaterandomve
torswiththeS
hur-Convex
P majorization ordering.
.
h
keywords: Queues with deadline; Earliest Deadline First; Sto
hasti

ompari-
t
a son; S
hur-
onvexordering.
m
[

2
v
1 Introdu
tion
5
8
8 In real-time queuing theory, ea
h
ustomer entering the queueing system is not
2 only identi(cid:28)ed by his arrival time and servi
e duration but also by a deadline.
4. In other words, a given
ustomer has a given period of time (his patien
e, i.e.,
0 theremainingtimebeforehisdeadline)duringwhi
hheshouldentertheservi
e
7 booth. Thispatien
ethende
reasesatunitrateastimegoeson. Ifthedeadline
0
is rea
hed before the
ustomer
ould enter servi
e, the
ustomer is either lost
:
v and the deadline is said hard, or he is kept in the waiting line and the deadline
i is said soft. Hereafter we draw a
omparison of the servi
e dis
iplines in the
X
soft deadline
ase. The performan
e metri
s we fo
us on is the lateness of the
r
a
ustomers with respe
t to their deadlines.
Inbothdeterministi
andrandomenvironments,itappearsthattheEarliest-
Deadline-First (EDF for short) dis
ipline, whi
h
onsists in giving priority to
the
ustomershavingthe nearestdeadline, isoptimal. It isthe morefeasiblein
the stati

ase (see [4℄): if some preemptive dis
ipline
an serve the
ustomers
ofagivens
enariowithoutanylateness,thenpreemptiveEDF alsodoes. Ithas
been shown independently in [6℄ and [10℄ that preemptive EDF is also optimal
for job loss in the hard deadline
ontext. More re
ently, [11℄ have proved that
the soft preemptive EDF minimizes the lateness and tardiness of all the tasks
∗
Laboratoire de Mathématiques Appliquées - Université de Te
hnologie de Compiègne.
Centre de Re
her
hes de Royallieu, BP 20 529, 60 205 COMPIEGNE Cedex, FRANCE.
e-mail: moyalpasdma.ut
.fr
1
in the system at a given time. In the hard deadline
ontext and when random-
ness is assumed, non-preemptive EDF ensures the smallest loss for the strong
sto
hasti
ordering (see [9℄) and the least failure probability, i.e. it minimizes
thestationaryprobabilityofbeinglostbeforeservi
e([8℄). Itisshownin[2℄that
EDFalsominimizessto
hasti
allythenumberoflost
ustomersamongpreemp-
tive dis
iplines. In the soft deadline
ase, non-preemptive EDF also minimizes
the largest lateness among all the
ustomers in the system at a given time (or
at a given (cid:28)nite set of times), for the strong sto
hasti
ordering (see [11℄).
Inthispaper,we
onsiderthesoftdeadline
ase,andstatethatnon-preemptive
EDF also minimizes the mean stationary lateness of the
ustomers. Our main
result (Theorem 1) is in fa
t more general, and establishes that the more the
s
heduler gives priorityto
ustomerswho are
lose to their deadline, the small-
est the mean expe
tation (in the Palm sense) of any
onvex fun
tion of the
stationarylateness. Themainmathemati
altoolusedintheproofistheS
hur-
Convex
omparison of random ve
tors in the Palm spa
e of the arrival pro
ess.
This framework is
onvenient for the
omparison of servi
e dis
iplines in other

ases: the optimality of the SRPT dis
ipline for the residual servi
e ve
tor(see
[5℄)andthatofFIFOforthewaitingtime(seee.g. [3℄)havebeenshowninthat
sense.
This paper is organized as follows. In se
tion 2 we re
all the main proper-
ties of the S
hur-Convex majorization ordering. In se
tion 3 we make pre
ise
the de(cid:28)nitions, assumptions and notation on the G/GI/1 queueing station we

onsider. In se
tion 4 we state and prove our main result. We
on
lude this
paper with some
omments in se
tion 5.
Rn
2 The majorization ordering on
Letusre
allthede(cid:28)nitionandmRaninproΓpe(r1t,ie..s.,onf)themajorization partialsemi-
orderingontheEu
lidianspa
e . Let denotethesetofpermutation
{1,...,n} X = (X ,X ,...,X ) γ ∈ Γ(1,...,n)
1 2 n
of X :=andXfor a,nXy ve
,t.o..r,X Fand anRyn ,
γ γ(1) γ(2) γ(n)
Sdenote F(X) =. FA(fXun
)tion frXom∈ Rnto some spγa
∈e
γ
Γ(i1s,.s.a.,idn)symm(cid:0)etri
whenever (cid:1) X ∈forRnany X and aXny ≤
α α(i)
. The fully ordered version of any is su
h that
X 1 ≤ i < j ≤ n γ ∈ Γ(1,...,n)
α(j)
for any X ∈ Rn X . We say that γ(i)=j is aγ(rje)or=deiring
permutation of if is not fully ordered and and for
i<j X >X
i j
some X, Ysu∈
hRtnhat α, β ∈.Γ(1,...,n) X Y
α β
Let and . Denote and the fully ordered
X Y X ≺Y
versions of and , respe
tively. Then we say that if
n n
X = Y ,
i i
 i=1 i=1
Xn Xn
 Xα(i) ≤ Yβ(i), k =2,...,n.
i=k i=k
≺ X X
The properties of are thoroughly presented in [1℄ and [7℄. Let us quote the
following ones, whi
h will be used in the sequel.
X ≺Y ⇐⇒−X ≺−Y,
(1)
2
−X
wheree.g. denotesthe ve
torwhose
oordinatesaretheoppositeofthatof
X
,
X ≺Y ⇐⇒F(X)≤F(Y) F.
for any
onvex symmetri
real fun
tion (2)
Z ∈Rn γ X
For any fully ordered and any that reorders ,
X −Z ≺X−Z
γ
(3)
(see for instan
e [3℄, Lemma 4.1.1 p.266).
3 The queue with smooth deadlines
(Ω,F,P,θ ) (θ )
Let Pt be a probability spa
e furnished with a bije
tive (cid:29)ow t t∈R,
under whi
h is stationary and ergodi
. Consider a single-server queueing
(N )
station with in(cid:28)nite bu(cid:27)er, fed by a G/GI input: the arrivalpro
ess t t∈R of
...<T <T ≤0<T <T <... (θ )
points −1 0 1ξ :=2T −isT
omnp∈atZible with t t∈R and hen
e
n n+1 n
itsin
rements,orinterarrivals , formastationaryergodi
ξ
sequen
e of random variables (r.v. for shonrt∈).ZTChe genenri
interarrival is
n
assumedto beintegrable. We denoteforall , the -th
ustomerin the
T σ
n n
orderofarrivals(i.e. enteredattime )and ,theservi
edurationrequested
C D
n n
by . We assume furthermore that a smooth deadline is assigned to the
C
n

ustomer , before whi
h he should begin his servi
e. The initial patien
e
C P := D − T
n n n n
of (i.e. the time before his deadline) is hen
e given by .
{σ } {P } (N )
The sequen
es n n∈Z and n n∈Z aresequeΩn
,eFs,oPf0m,θarkso(fN )t t∈R. Under
these settinPg0s[Tone=
0a]n=de1(cid:28)ne the Palmθs:p=a
θe of t t∈R, whi
h is
su
h that 0 a{nσd }where {P T}1(cid:0)is a bije
ti(cid:1)ve stationaryPer0godi
σdis
=retσe(cid:29)◦oθwn. InpaPrti
=ulaPr,◦θnn nn∈Z∈aZnd n n∈Z arestationaryinthat{σ -}.a.s,
n 0 and n 0 , . We assume furthermore that n n∈Z
(N ) {P } σ := σ
is i.i.d., independent of t t∈R and n n∈Z, and that the r.v. 0 and
P :=P
0
areintegrable. Thedeadlinesaresoft,thusany
ustomeragreestowait
in the system as long as his servi
e is not
ompleted, even though he passed
his deadline. Therefore the system is
onservative (any entered
ustomer is
eventually served), and Loynes' stability
ondition for su
h a system,
E0[σ]/E0[ξ]<1,
W C
n n
isassumedtohold. Wedenoteby ,thewaitingtimeof (thetimehemust
B =T +W C
n n n n
waitbeforeenteringservi
e),and theinstantinwhi
h begins
t ≥ T
n
his servi
e. At any time , the residual patien
e (i.e. the remaining time
C t R (t) = D −t
n n n
before the deadline) of at time is given by . The residual
C R := R (B )
n n n n
patien
e of at the time he begins his servi
e thus reads .
C
n
Hen
e the lateness (if any) of with respe
t to his deadline is given by
−
L =(R ) :=−min(R ,0).
n n n
Φ Ψ
Let and denote two admissible non-preemptive servi
e dis
ipline, i.e.
twopoli
iesfollowedbytheserverwhen
hoosingthe
ustomertoserve,depend-
ing only on the state of the system at this instant (not on the already served

ustomers,noron the future
ustomersto enter). We re
all, that a dis
iplineis
3
said non-preemptivewhen theserveralways
ompletesan initiated servi
eeven
when a priority
ustomer enters the system in the meanwhile. The
omplian
e
oftheservi
edis
iplinewith respe
ttothe real-time
onstraintis
hara
terized
Φ ≪ Ψ Φ
as follows. We write , whenever always
hooses a
ustomer having a
deadline earlier (and maybe already passed!) than that of the
ustomer
ho-
Ψ
sen by , in the
ase where both
ustomers are present in the system and the
t Φ C Ψ C
i j
servi
e booth is free (i.e. at this instant , if
hooses and
hooses ,
R (t) ≤ R (t)
i j
then ). In parti
ular the Earliest Deadline First (EDF) servi
e
dis
iplinealwaysgivesprioritytothe
ustomerhavingtheearliestdeadline,and
the Latest Deadline First (LDF) one gives priority to the
ustomer having the
latest deadline. Then, by de(cid:28)nition
EDF ≪Φ ≪LDF Φ.
for any admissible dis
ipline
4 Comparison of servi
e dis
iplines
Letusemphasizeforagivensystemthedependen
eontheservi
edis
iplineby
Φ Φ
adding when ne
essary the supers
ript whenever the servi
e dis
ipline is .
Sin
e the stability
ondition is assumed to hold, their exists a stationary (i.e.
θ WΦ Φ

ompatible with ) waiting time under dis
ipline . Hen
e there exists
a stationary residual patien
e when entering servi
e and a stationary lateness,
whi
h are respe
tively given by
RΦ :=P −WΦ
and LΦ = RΦ −.
(cid:0) (cid:1)
We now state our main result.
g :R→R
Theorem 1 For any
onvex fun
tion ,
E0 g RΦ ≤E0 g RΨ Φ≪Ψ.
whenever (4)
(cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:0) (cid:1)(cid:3)
C
0
Proof. Let us assume that
ustomer (cid:28)nds an empty queue upon arrival.
k ≥ 0 φ(k) ψ(k)
We denote for any , (resp. ) the rank, in the in
reasing order
k Φ Ψ
of deadlines, of the -th
ustomer served under (resp. ). Let for any
j ≥ 0 C j
α(j)
, be the -th
ustomer in the in
reasing order of deadlines (i.e.
D ≤D i<j n≥0 γ(n) n
α(i) α(j)
for ),and forall , besu
hthatthe -th
ustomer
Ψ γ(n) Φ k ≥ 0 C
φ◦α(k)
served by is the -th
ustomer served by . Sin
e for all ,
C k Φ Ψ
ψ◦α(k)
(resp. ) is the -th
ustomer served under (resp. ), we have
γ =α◦ψ◦φ−1◦α−1.
N Nγ
Let and be the
ounting measures representing respe
tively the original
double-markedinputpointpro
ess,andtheinputpointpro
esswhentheservi
e
γ
times of the
ustomers are rearrangedfollowing , i.e.
N := δ ,
Tn,σn,Pn
n≥0
X
4
Nγ := δ .
Tn,σγ(n),Pn
n≥0
X
Then, it is easily seen that
N Nγ
and have the same distribution (5)
(this is the "inter
hange argumenNt" for theNγG/GI input, see [3℄, p.26N7). We
Nadγd whenTne
essary a supers
ript (resp. ) whenever the input is (resp.
τ
). Let be the (cid:28)rst time in whi
ha
ustomerentersan empty queuewhen
T
0
starting with an empty queue just before time . Note, that sin
e the system
T
τ
is
onservative, does not depend on the servi
e dis
ipline, nor on the order
n≤τ −1
of servi
es. We have for any
φ−1(n)−1 φ−1(n)−1
BαN(γn,)Φ = σγ◦α◦φ(i) = σα◦ψ◦φ−1◦α−1◦α◦φ(i)
i=1 i=1
X X
φ−1(n)−1
= σ =BN,Ψ ,
α◦ψ(i) γ◦α(n)
i=1
X
C Ψ
γ◦α(n)
that is the instant in whi
h
ustomer begins his servi
e under . Thus
we have
RNγ,Φ =DNγ −BNγ,Φ =DNγ −BN,Ψ =DN −BN,Ψ.
α α α α γ◦α α γ◦α (6)
Let us moreoverremark that
γ BN,Ψ
α
Lemma 1 is a
omposition of reordering permutations of .
n γ(α(n))6=α(n)
Proof of Lemma 1. The (cid:28)rst integer , if any, su
hthat issu
h
α(n) Ψ α(n)
that at the -th end of servi
e under (whi
h is as well the -th end
Φ γ(k) = k k = 0,...,α(n)−1
of servi
e under sin
e for ), there are in both
Φ Ψ C C D <D
systems (under and ) two
ustomers, say i1 and i2 su
h that i1 i2,
C Φ C Ψ
whereas the server talk=es1
,a2rejof= iα1−u1n(ide)r and ofBNi,2Ψun<deBrN,Ψ. In other
words, denoting for , l l , we have α(j2) α(j1) whereas
i = α(j ) > α(j ) = i Φ C C
φ2−1(j )<2 φ−1(j 1). 1. Now, sin
e gives priψority to i1 over i2, we have
1 2
Hen
e from the de(cid:28)nition of ,
BN,Ψ =BN,Ψ <BN,Ψ =BN,Ψ .
γ◦α(j1) α◦ψ◦φ−1(j1) α◦ψ◦φ−1(j2) γ◦α(j2)
γ i j BN,Ψ γ
1 α
Thus the permutation inter
hanging and reorders , and reads
γ =γ ◦...◦γ , γ
p 1 i
where the are su
h permutations.
It is thus a
onsequen
e of Lemma 1, together with (3) and (1) that
DN −BN,Φ ≺DN −BN,Ψ =RN,Ψ,
α γ◦α α α α
that is with (6), RNγ,Φ ≺RN,Ψ.
α α
5
F :Rτ →R
Hen
e from (2), for any
onvex symmetri
,
F RNγ,Φ ≤F RN,Ψ
α α
(cid:16) (cid:17) g :R(cid:0) →R(cid:1)
and in parti
ular for all
onvex fun
tion ,
τ−1 τ−1
g RNγ,Φ ≤ g RN,Ψ .
α(i) α(i)
Xi=1 (cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
Finally, taking expe
tations in this last inequaliy and in view of (5),
τ−1 τ−1
E g RN,Φ =E g RN,Φ
i α(i)
" # " #
Xi=1 (cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
τ−1 τ−1
≤E g RN,Ψ =E g RN,Ψ .
α(i) i
" # " #
Xi=1 (cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
This leads to (4) using the
y
le formula of Palm probability.
g(x):=x−
Sin
e is a
onvex fun
tion, we have
Corollary 1 The mean lateness is minimized by EDF and maximized by LDF,
Φ
i.e. for any admissible dis
ipline ,
E0[L ]≤E0 LΦ ≤E0[L ].
EDF LDF
(cid:2) (cid:3)
5 Comments
Let us (cid:28)rst emphasizethe di(cid:27)eren
esbetween ourresultsand that of[11℄, The-
orem 4 p.229. Corollary 1 states in parti
ular that in the steady state, the
lateness an arriving
ustomer will undergo is (in average) minimized by EDF
among all non-preemptive dis
iplines. Theorem 4 of [11℄ shows a
omplemen-
N
tary, but not dire
tly related result: for a given and a given set of instants
t < t < ... < t
1 2 N
, any in
reasing fun
tion (for the
oordinatewise ordering)
(Z(t ),Z(t ),....,Z(t ))
1 2 N
of the random ve
tor is sto
hasti
ally minimized by
t ≥ 0 Q(t) t
EDF, denoting for all , the number of
ustomers in the systems at
Z(t)=maxQ(t)(−R (t)). t≥0
and i=1 i In parti
ular, this implies that forany , the
maximallateness(termedtardiness in [11℄) amongthe
ustomersin the system
t
at is sto
hasti
ally minimized by EDF. The performan
e metri
s there is the
maximallatenessratherthantheaveragelateness,asitisthe
aseinourstudy.
On another hand, remark that Corollary 1 is
onsistant with the optimal-
ity of the FIFO servi
e dis
ipline for the average waiting time (see e.g. [3℄,
Property 4.1.3 p. 266P), in=tpheP
0ase where the initial patien
e of the
ustomers
n
is deterministi
(say , -a.s.). Then, EDF redu
es to FIFO (the (cid:28)rst

ustomer in the system has the earliest deadline), and the waiting time ve
tor
W =p−R p:=(p,p...,p)
onabusy
y
lereads ,where . Hen
etheS
hur
onvex
W R R
FIFO FIFO EDF
minimality of amounts to that of , i.e. .
6
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7