Analysis of Queues with Bath Arrivals
G.Wolfner 1
and M. Telek 2
1
HungarianTeleommuniationCompany,TeleommuniationsDevelopmentInstitute
H-1456Budapest,PoB2{Hungary
2
Departmentof Teleommuniations,TehnialUniversityofBudapest
Sztozek2,H-1111Budapest {Hungary
Email: fwolfner,telekghit.bme.hu
Abstrat
The steady state distributionof quasi-birth-deathproessesanbe eÆiently obtained
by matrix-geometri (MG) methods. Sine a number of teleommuniation problems are
modelledbyproesseswithbatharrivals,theextensionofMGmethodsfortheseproesses
haspratialimportane. ThispaperpresentsanextensionofMGmethodswhihiseetive
fortheanalysisofquasi-birth-deathproesseswithbatharrivals. Theproposed methodis
omparedwithoneof thewell-knownmethods.
1 Introdution
Withtheextremelyrapidevolutionofommuniationandomputersystemsandwiththeinten-
tionoftheirintegration,whosemostwell-knownexampleistheintrodutionoftheasynhronous
transfermode (ATM),thepresentandfutureommuniationnetworksareharaterizedbythe
oexisteneof dierent transmission/servierequirements, ommuniationprotools andtrans-
missionspeeds. WithverysimpleassumptionsonthestohastibehaviourofthenetworktraÆ
(memoryless orMarkov modulated arrivaland servie) the transfer of data from one part of a
network to an other resultsin omplexqueue behaviourat thetransfer point. For example, in
onepartofthenetworkpaketsofsize1500bytearetransmitted(IPpaketsizeusedinEthernet
LANs)whileinanother partellsofsize48+5 byte (thesizeofATMellpayload+header)are
thebase dataunits. At theborder of theseommuniationprotools thearrivalof a1500 byte
paketon theone siderequires thetransmissionof 32 ells on theother side. Thisphenomena
isommonly referred to asbath arrival.
A queueing system withmemoryless arrival and servie an be analysedby theunderlying
Markov hain. When in addition the arrival and servie is queue length independent and the
bath sizeis boundedtheunderlyingMarkov hainhasa nieblokstrutureandis referredto
asaquasi-birth-death(QBD)Markovhain[8 ℄. Thereareseveralnumerialmethodstoevaluate
thesteadystatebehaviourofQBDproesses. Themostwell-knownistheoneproposedbyNeuts
whih is often referred to as Matrix Geometri (MG) method [8 ℄ and is based on an iterative
proedure alled SimpleSubstitution (SS) method. Mitrani and Chakka proposed a one step
methodbasedonthespetralexpansionofsubmatries[2,6 ℄. WhileLatouhe andRamaswami
proposed an other iterative proedure with better numerial properties [5 ℄. Naoumov et al.
enhanedthismethodbyreduingtheomplexityoftheiterationsteps[7 ℄withahighermemory
requirement.
To analyze real ommuniation networks an eetive extension of these methods for bath
arrivals is neessary. An obvious solution of proesses with bath arrivals is to enlarge of the
ThisworkwassupportedbytheEuropeanCommunitythroughCOPERNICUS'94ontratn. 1463, bythe
HungarianOTKAontratn.F-23971andbytheHungarianMinistryofCultureandEduationFKFP-0420/1997
projet.
ment is the signiant inrease of omputation omplexityand omputer storage requirement.
Spetralexpansionisoneofthemethodshasbeendevelopedforbatharrivalsand hasshowna
good performaneomparedto theothers[4 ℄. There isan extensionoftheSS algorithm forx
sizebatharrivalsthatalsoshowssomeadvantagesinperformaneandstoragerequirement[10 ℄.
Inthispaperweproposeanalternativemethodthatgenerallyhasabetter performanebothin
omputationomplexityandomputerstoragerequirementthantheabovementionedones. Re-
entlyapaperthatusesasimilairapproahfornitstatespaeametotheauthers'knowledge.
[11 ℄.
The rest of the paper is organized as follows. The next Setion desribes the analyses of
the QBD proesses. In Setion 3 the problem of bath arrivals and the proposed algorithm is
introdued. that isan extensionof MGsolution. Setion4 isdevotedto theomparison of the
proposed algorithmand thealgorithmproposedbyNaoumov etal. whihseems tobe thebest
methodsfora generalQBDproess. Thepaperis onludedinSetion 5.
2 Analyses of quasi-birth-death proesses
Consider a Disrete Time Markov Chain (DTMC) where the state of system is desribed by
two randomvariables: Z
n
=fI
n
;J
n g;I
n
istaking its value from f1;2;:::;Ng and J
n
istaking
its value from f0;1;:::g. This DTMC is alled a quasi-birth-death proess if only the state
transitionswhere J
n+1 J
n
2f 1;0;1g have a positiveprobability. The states withthe same
value of J
n (J
n
= j) denes a level (j). By the above denitionstate transitions are possible
insidethe levelsand betweenthe neighbouringlevels.
The nonzero transition probabilitiesaregiven bythesubmatries
A (j)
(lateral transitions): A (j)
(i;k) = Pr(I
n+1
= k;J
n+1
= jjI
n
= i;J
n
= j) 1
(i;k 2
f1;2:::;Ng, j2f0;1;:::g);
B (j)
(upward transitions): B (j)
(i;k) = Pr(I
n+1
= k;J
n+1
= j+1jI
n
= i;J
n
=j) (i;k 2
f1;2;:::;Ng,j2f0;1;:::g);
C (j)
(downward transitions): C (j)
(i;k) = Pr(I
n+1
= k;J
n+1
= j 1jI
n
= i;J
n
= j)
(i;k 2f1;2;:::;Ng,j 2f1;2;:::g);
and all theothertransition probabilitiesequalto 0.
As itis seenfrom these denitions A (j)
, B (j)
and C (j)
arematries of sizeN N. Assume
thatan m (m1) thresholdexistssuh that
A (j)
=A; 8j m,
B (j)
=B; 8j m 1,
C (j)
=C ; 8jm+1,
whihmeansthatthetransitionprobabilitiesarelevelindependentifjm. Theblokstruture
of the transition probability matrix of a QBD proess is shown in Figure 1. The upper-left
submatrixofsize(Nm)(Nm)is referredto astheirregularpartof thetransitionprobability
matrixand therest asits regular part.
The methods developed for the analysis of level independent QBD proesses, suh as the
spetral expansionortheMG methods, protfrom the strutureof theregular partand allow
anykindofbehaviourintheirregularpartinludingtransitionsbetweennon-neighbouringlevels.
Denote thesteady state distributionby
p
i;j
= lim
n!1 Pr(I
n
=i;J
n
=j)
1
(i;k)denotes thekthelementoftheithrowofamatrix
= 6
6
6
6
6
6
6
4 A
(0)
B (0)
C (1)
A (1)
B (1)
C (2)
A (2)
B
C (3)
A B
C A B
.
.
. .
.
. .
.
. 7
7
7
7
7
7
7
5
Figure1: The nonzero bloksof thetransition probabilitymatrixof aQBDproess (m=3)
and introduevetorsv
j
(j 0) as
v
j
=[p
0;j
;:::;p
N;j
℄:
As a onsequene of the blok struture of the transition probability matrix the steady
state distributionof a QBD proess an be obtainedby solvingthe following systemof vetor
equations:
(a) v
0
= v
0 A
(0)
+v
1 C
(1)
(b) v
j
= v
j 1 B
(j 1)
+v
j A
(j)
+v
j+1 C
(j+1)
m>j >0
() v
j
= v
j 1 B+v
j A+v
j+1
C jm
(d) 1 =
P
1
j=0 v
j e
T
N
(1)
wheree T
N
istheolumnvetor of1sof sizeN.
Themethodsdevelopedforthesteady-stateanalysisarebasedonthefat[8 ℄that8j m 1
v
j+1
=v
j
R ; (2)
where R is a matrix of size N N, and it is theminimal non-negative solutionof the matrix
equation
B+R A+R 2
C=R : (3)
A simple way to evaluateR is to applythe SS algorithm whose iteration step is shown in
Figure 2 [8℄. The sequene of R
n
(n=0;1;:::) isentry-wise nondereasing and itonverges to
matrixR [8℄.
R
0
=0
n=0
DO
R
n+1
=B+R
n A+R
2
n C
n=n+1
WHILE (jjR
n R
n 1 jj)
Figure2: The simplesubstitution(SS)algorithm
The disadvantage of the method is the slow onvergene, espeially when theutilization of
themodelledsystemapproahes1. Latouhe etRamaswamiproposedanotheriterativemethod,
a LogarithmiRedution(LR) algorithm,thatonverges muhfaster[5 ℄.
Reently Naoumov et al. proposed an enhanement of the LR algorithm (Figure 3), that
requireslessoperationsperiteration[7℄. AniterationstepoftheLRmethodismoreompliated
than a step of the SS algorithm,but the fewer iterations makes theLR algorithm faster. The
experienesso far have shown that usually the number of needed iteration steps are less than
20 [4 ,5 , 10 ℄.
Mitrani and Chakka proposed a diret method, alled spetral expansion [2 , 6 ℄. In this
method theleastN eigenvaluesand theassoiatedeigenvetorsmustbeobtained from
=
B+A+ 2
C
;
L=B
M=C
W =A I
DO
X = N 1
L
Y = N 1
M
Z =LY
W =W +Z
N =N+Z+MX
L=LX
M =MY
WHILE (jjZjj)
R= BW 1
Figure 3: The logarithmiredution(LR)algorithm ofNaoumov etal.
whereis aomplexnumberand is avetor of N omplexelements.
The advantage of this method is the diret solution and the easy alulation of the state
probabilities based on the eigenvalues and the eigenvetors, but numerial problems an arise
bytheloseeigenvaluesandeigenvetors.
The detailedomparison of this method and the algorithm of Naoumov et alhas not pub-
lishedyet. Sinethetheoretial omplexityofthealgorithmsaresimilar(O(N 3
))thedierene
betweenthe algorithmsan be shown byexperienes. Thepreviously publishedresultsshow a
better performaneof spetralexpansion,espeiallywhenthe utilisationisnear1,butspetral
expansionmaylead inaurateresultsbeause ofits numerialproblems.
3 The extension of MG approah for bath arrivals
3.1 Proesses with bath arrivals
Consider the DTMC Z
n
= fI
n
;J
n
g, where I
n
is taking its value from f1;2;:::;Ng and J
n
is taking its value from f0;1;:::g, as in the previous setion. Models with bath arrivals and
single serverdiersfrom QBDproesses onlybytheallowed upwardtransitions. Now, upward
transitionsareallowedfromleveljtolevelj+l(l =1;2;:::;y),wherey isthemaximumbath
size.
In thisasethe nonzero transitionprobabilitiesaregiven bythesubmatries
A (j)
(lateral transitions): A (j)
(i;k) = Pr(I
n+1
= k;J
n+1
= jjI
n
= i;J
n
= j) (i;k 2
f1;2;:::;Ng,j2f0;1;:::g);
B (j)
l
(upward transitions): B (j)
(i;k) = Pr(I
n+1
= k;J
n+1
= j+1jI
n
= i;J
n
=j) (i;k 2
f1;2;:::;Ng,j2f0;1;:::g and l2f0;1;:::;yg);
C (j)
(downward transitions): C (j)
(i;k) = Pr(I
n+1
= k;J
n+1
= j 1jI
n
= i;J
n
= j)
(i;k 2f1;2;:::;Ng,j 2f1;2;:::g);
and all theothertransition probabilitiesequalto 0.
Assume that anm (my) thresholdexistssuh that
A (j)
=A; 8j m;
B (j)
=B; 8j m y;
C (j)
=C ; 8jm+1;
ofthetransitionprobabilitymatrixofaproess withbatharrivalsisshowninFigure4(where
y=3 and m=6).
= 2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4 A
(0)
B (0)
1 B
(0)
2 B
(0)
3
C (1)
A (1)
B (1)
1 B
(1)
2 B
(1)
3
C (2)
A (2)
B (2)
1 B
(2)
2 B
(2)
3
C (3)
A (3)
B (3)
1 B
(3)
2 B
3
C (4)
A (4)
B (4)
1 B
2 B
3
C (5)
A (5)
B
1 B
2 B
3
C (6)
A B
1 B
2 B
3
C A B
1 B
2 B
3
.
.
. .
.
. .
.
. .
.
. .
.
. 3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
Figure4: Theblokstrutureofthetransitionprobabilitymatrixofaproesswithbatharrivals
(y=3,m=6)
Toobtainthe steadystate distributionthefollowingsystemof equationsmustbe solved:
(a) v
j
= P
j 1
i=0 v
i B
(i)
j i +v
j A
(j)
+v
j+1 C
(j+1)
j<y
(b) v
j
= P
y
i=1 v
j i B
(j i)
i
+v
j A
(j)
+v
j+1 C
(j+1)
m>jy
() v
j
= P
y
i=1 v
j i B
i +v
j A+v
j+1
C jm
(d) 1 =
P
1
j=0 v
j e
T
N
(4)
3.2 Analysis by blok size enlargement
A possiblesolutionto analyse systems of thiskindis thetransformation of the probleminto a
QBDproessbyintroduingbloksofsizeyNyN (Figure4) [4℄. Theadvantageofthisblok
size enlargement is that the standard QBD methods an be used, but its disadvantage is the
inreaseinomputation omplexity(O(y 3
N 3
))and storage requirement (O(y 2
N 2
)).
Thespetralexpansionmethodhasbeenextendedtoanalysesystemswithmulti-leveljumps
without blok size enlargement [2 , 6 ℄. Although the omputation omplexity of solving the
spetral deomposition is similar to the blok enlargement (O(y 3
N 3
)) numerial experienes
shows a better performane of this method [4℄. But the aforementioned numerial instability
remainsits mainproblem.
3.3 A level-blok-size method
Here we propose a method, whih is more eetive than the blok size enlargement both in
omputationomplexityandinstorage requirement and doesnothave the numerial problems
of spetral expansion. The proposed method is an extension of the SS method for proesses
withbath arrivals.
If the MG methods(either the simplesubstitution orthe logarithmi redution algorithm)
withbloksizeenlargement areused thenmatriesof sizeyNyN aretreated. Asa resultof
theMG methodsan Rmatrix ofsizeyN yN is obtained. Denote theN N submatriesof
thisRas
R= 2
6
4 R
1;1
R
1;y
.
.
.
.
.
.
R
y;1
R
y;y 3
7
5:
The extensionof theSS methodis basedon thefollowingimportant theorem:
v
j
= y 1
X
i=0 v
j y+i T
i
; 8jm;
whereT
i
=R
i+1;1 .
Proof: Consider the proess as a QBD proess with blok size enlargement. Equation (2)
holdsfortherst yN sizevetor ofthe regularpart:
[v
m
;:::;v
m+y 1
℄=[v
m y
;:::;v
m 1
℄R;
i.e., thetheorem holdsforj =m.
Now, assumingthatthe sizeofthe irregularpart ism 0
=m+l; l0 theregular partof
theproess remainsthesame, and sodoesR. Inthisase Equation(2) gives:
[v
m+l
;:::;v
m+l +y 1
℄=[v
m+l y
;:::;v
m+l 1
℄R:
Sine thisequationis satised8j =l0the theoremis proved. 2
The mainonsequeneof thetheorem isthat therst N olumnsofR (T
i
; i=0;:::y 1)
ontains suÆient information to determine the steady state distribution of the proess. The
followingtwo theoremsallowto obtainthe T
i
(i=0;:::y 1) matries.
Theorem 3.2 The T
i
;i = 0;:::;y 1 matries are the minimal nonnegative solutions of the
following system of matrix equations:
T
0
=B
y +T
0 (A+T
y 1 C)
T
i
=B
y i +T
i (A+T
y 1
C)+T
i 1
C i=1;:::;y 1 (5)
Proof: ApplyingTheorem3.1fortheleft handsideofEquations (4)wehave:
v
j
= P
y 1
i=0 v
j y+i T
i
(6)
and applyingit fortheright handsideof Equations(4) we have:
P
y
i=1 v
j i B
i +v
j A+v
j+1 C=
P
y
i=1 v
j i B
i +v
j A+
P
y 1
i=0 v
j+1 y+i T
i
C =
= P
y
i=1 v
j i B
i +v
j A+
P
y 2
i=0 v
j+1 y+i T
i
C+v
j T
y 1 C=
= P
y
i=1 v
j i B
i +v
j
(A+T
y 1 C)+
P
y 2
i=0 v
j+1 y+i T
i
C=
= P
y 1
i=0 v
j y+i B
y i +
P
y 1
i=0 v
j y+i T
i
(A+T
y 1 C)+
P
y 1
i=1 v
j y+i T
i 1
C=
=v
j y (B
y +T
0 (A+T
y 1 C))+
P
y 1
i=1 v
j y+i (B
y i +T
i (A+T
y 1 C)+T
i 1 C)
(7)
The theorem omes from the equality of the oeÆients of v
j y+i
; i =0;1;:::;y 1 in
Equation (6)and (7). 2
Theorem 3.3 If X (0)
i
=0; i=0;1;:::;y 1 then the iteration
X (n+1)
0
=B
y +X
(n)
0
(A+X (n)
y 1 C)
X (n+1)
i
=B
y i +X
(n)
i
(A+X (n)
y 1
C)+X (n)
i 1
C i=1;:::;y 1
onverges on the minimal non-negative solutions of Equation(5).
thealgorithm an be establishedinthesame wayasit isin[8 ℄: rst itis proved that the
sequenesof X n
i
areentry-wisenondereasing thentheonvergene isveried.
Sine the B
y
, A and C matries onsist of nonnegative elements, X (1)
i
X
(0)
i
= 0; i =
0;:::;y 1 entry-wise. The inreaseof X (n+1)
i
; i=0;:::;y 1; n1 an beproved by
indution:
X (n+1)
0
=B
y +X
(n)
0
(A+X (n)
y 1
C) B
y +X
(n 1)
0
(A+X (n 1)
y 1
C)=X (n)
0
and
X (n+1)
i
=B
y i +X
(n)
i
(A+X (n)
y 1
C)+X (n)
i 1
C
B
y i +X
(n 1)
i
(A+X (n 1)
y 1
C)+X (n 1)
i 1
C=X (n)
i
i=1;:::;y 1
X (0)
i T
i
; i=0;:::;y 1 entry-wiseand theX (n)
i T
i
; i=0;:::;y 1; n0 also an
beveriedbyindution:
X (n+1)
0
=B
y +X
(n)
0
(A+X (n)
y 1
C) B
y +T
0 (A+T
y 1
C)=T
0
and
X (n+1)
i
=B
y i +X
(n)
i
(A+X (n)
y 1
C)+X (n)
i 1
C
B
y i +T
i (A+T
y 1 C)+T
i 1 C =T
i
i=1;:::;y 1
Sine an upper-bounded monotone inreasing sequene must onverge, the sequenes
X (n)
i
; i = 0;:::;y 1 onverge entry-wise. The limit matries satisfy Equation (5)
and they are not greater than the minimal nonnegative solutions, thus sequenes of
X (n)
i
; i = 0;:::;y 1 onverge on the minimal non-negative solutions of Equation (5).
2
As the result of Theorem 3.3 the algorithm in Figure 5 an be used to obtain the T
i
; i =
0;:::;y 1 matries. The omplexity of one iteration step of the algorithm is O(y N 3
) that is
signiantlybetter thantheomplexityof theother mentionedmethods.
FOR i=0TO y 1
T (0)
i
=0
ENDFOR
n=0
DO
T (n+1)
0
=B
y +T
(n)
0
(A+T (n)
y 1 C)
FOR i=1TO y 1
T (n+1)
i
=B
y i +T
(n)
i
(A+T (n)
y 1 C)+T
(n+1)
i 1 C
ENDFOR
n=n+1
WHILE (max
i (jjT
(n)
i T
(n 1)
i
jj)
Figure5: The proposednumerial methodto obtaintheT
i
matries
3.4 The steady state distribution
When the T
i
; i = 0;:::;y 1 matries are known only the vetors v
0
;v
1
;:::;v
m 1
miss to
determine the steady state distribution of the proess. Equations (4a), (4b) and (4d) an be
used to obtain these unknowns. The number of unknowns is mN and the number of linearly
independentequationsisthesame. Sinev
m
= P
y 1
i=0 v
m y+i T
i
(seeTheorem3.1)theunknowns
inEquation(4a)and(4b)arev
0
;v
1
;:::;v
m 1
. TheinnitesuminEquation(4d)anberesolved
bythefollowingtheorem:
1
X
j=m v
j
= y 1
X
i=0 v
m y+i 0
B
i
X
n=0 T
n 0
I y 1
X
l =0 T
l 1
A 1
1
C
A
Proof: Lets= P
1
j=m v
j
and makethe followingtransformations:
s= P
1
j=m v
j
= P
y 1
i=0 P
1
j=0 v
m+jy+i
= P
y 1
i=0 P
1
j=0 P
y 1
n=0 v
m+(j 1)y+i+n T
n
=
P
y 1
n=0
P
y 1
i=0 P
1
j=0 v
m+(j 1)y+i+n
T
n
= P
y 1
n=0
P
y 1
i=n v
m y+i +s
T
n
Asof onsequeneof these theTheoremomesas:
s = P
y 1
n=0
P
y 1
i=n v
m y+i +s
T
n
s
I P
y 1
l =0 T
l
= P
y 1
i=0 v
m y+i
P
i
n=0 T
n
s = P
y 1
i=0 v
m y+i
P
i
n=0 T
n
I P
y 1
l =0 T
l
1
2
By thistheorem we have a systemof equations withthesame numberunknownsand inde-
pendent equations. The result of the system of equations is the v
0
;v
1
;:::;v
m 1
vetors, thus
thesteady state distributionisobtained.
3.5 Continuous time proesses
So far the disrete time Markov hains has been disussed, but the results an be applied to
ontinuoustimeMarkovhains(CTMC)aswell. Asimplewaytodosoistheappliationofthe
method of randomization, that produes a DTMC from a CTMC with the same steady state
distribution. LetQbethegeneratormatrixoftheCTMC,and q=max
i;j
jQ(i;j)j. TheDTMC
withtransition probabilitymatrix
=Q=q+I;
where the divisionmeans division of all entry of the matrix and I is the identity matrix with
theappropriatedimension hasthesame steady state distribution[3℄.
4 Performane omparison
4.1 The system model
A simple queueing system has been evaluated to investigate the performane of the proposed
method. AsystemwithaMarkovmodulatedsoureisonsidered. Thesouretransmitspakets
toanoutputlink. Theoutputlinkworksinaslottedmanner: therearexsizetimeslotsandin
every timeslotat mostone dataunitan betransmitted. Thetransmissionofa databeginsat
thebeginningofa timeslot. Werefer todataunitsasellsbelow. An innitebuerisassumed
at theoutput link.
The souresubmitsat most onepaketat theendof thetimeslots and allof these pakets
have the same size. The probability of a paket arrival in a time slot depends on phase of
the Markov modulated soure. The soure may hange its phase at the end of the time slots
independent ofpaket arrivals.
These assumptions are realisti onsidering a le server where TCP/IP over ATM is used.
TheslottedoutputlinkhasthepropertiesofATMandpaketsonsistingofaxnumberofells
isa possiblemodel forlargeletransferssinemostoftheIP paketshasthesizeofmaximum
transferunit(MTU)duringbulktransfer[9 ℄. ForexampleinEthernet-basednetworkstheMTU
of an IP datagram is 1500 bytes, therefore the maximum paket size is 32 ells. The default
MTU value in IP over ATM environment is hosen to be 9180 byte, and thus the MTU size
is 192 ells [1℄. The Markov modulated soure represents a phase dependent arrival, e.g., the
renewal proess withphase-typedistributedinterarrivaltimes.
The systembehaviourat theendof thenth timeslotis haraterized by
thenumberofells inthebuerof theoutputlink (J
n ) and
thephase ofthesoure (I
n ).
The systemhas thefollowingparameters:
C: thenumberof phasesof thesoure;
r: the numberof ells inapakets;
D
0
(i;k)=Pr(I
n+1
=k; no message arrivesjI
n
=i), D
0
is amatrixof sizeCC
D
1
(i;k)=Pr(I
n+1
=k; message arrivesjI
n
=i), D
1
is amatrix ofsizeCC
ThestohastiproessfI
n
;J
n
gisaDTMC.FromthesteadystatedistributionofthisDTMC
thethequeue lengthdistributionand the paketdelaydistributionan beobtained. The state
transition ofthesystem areasfollows:
If no paket arrives then a ellleaves thebuer, if it wasnot empty, at the beginning of
thetimeslot, and thesourehasa phasetransition from phaseito k:
(i;j)!(k;max (j 1;0)) (8)
The probabilityof thisstate transitionis D
0 (i;k).
If paket arrives then it is stored in the buer and a ell leaves the buer, if it was not
empty,atthebeginningofthetimeslot,andthesourehasaphasetransitionfromphase
ito k:
(i;j)!(k;max (j 1;0)+r) (9)
The probabilityof thisstate transitionis D
1 (i;k).
As a onsequene of (8) and (9) the blokstruture of the transition probabilitymatrix is
asinFigure 6. Thisstrutureorrespondsto theproblem presentedinSetion 3.
2
6
6
6
4 D
0
0 0 0 D
1
D
0
0 0 0 D
1
D
0
0 0 0 D
1
.
.
.
.
.
. 3
7
7
7
5
Figure 6: The blokstrutureof transitionprobabilitymatrix(r=4)
4.2 Numerial results
We haveomparedtheproposedmethod(referredto asLevel-Blok-Size (LBS)method) to the
method proposed by Naoumov et al. (see Figure 3, referred to as LR method), sine the LR
methodisone of thebestamong thepublishedgeneralmethodsforthesteadystate analysisof
levelindependentQBDproesses.
BothmethodshavebeenimplementedinCusingtheMeshahlibrary 2
formatrixoperation.
TheCPUtimemeasurementshavebeenperformedonaPCwithIntelPentiumproessor,using
2
MeshahlibraryformatrixomputationisdevelopedatShoolofMathematialSienes,AustralianNational
UniversitybyDavidE.StewartandZbigniewLeykanditisavailablevianetlib(ftp.netlib.org//meshah).
0.1 1 10 100
8 16 32 64 100
Computationtime
[se℄
Paketlength(r)
3 3
3
3 3
3
3
3
3
3
3
3
3
3
3
3
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ LBS
LR
=40%
3
=80%
+
Figure 7: Computationtimeversusthe paketlength(C =5 and =40; 80%)
Numberofiterationstep Numberof iterationstep
Utilization (C=5; r=16) (C=5; r=32)
LR MG LR MG
20% 10 440 9 241
40% 11 1203 10 642
60% 13 2601 12 1374
80% 14 6341 13 3345
90% 15 13087 14 6917
95% 16 25370 15 13445
97% 17 40471 16 21499
Table 1: The numberof iterationstep
MSDOSandGNUCompiler. ThereportedCPUtimeinludesonlythetimeneededtoobtain
the R matrix in the LR algorithm and the T
i
matries in the proposed LBS method. In the
experimentstherequiredrelative auray() wasset to10 10
inbothalgorithms. The results
show thatthedierenes intheobtained steadystate distributionaremarginalat thisrelative
auray.
First the inuene of the bath size, that is related to the paket length (y = r 1), to
the omputation time has been investigated (Figure 7). In these experiments the number of
phases of the soure (C) was 5. We have ompared the performane of these methods with
two dierentsystemutilization parameters(40% and 80%). Thepaket lengthhasonlya little
inuene on the omputation time with the proposed method, butit has a signiant impat
withtheLRmethodwhihusesbloksizeenlargement. Thisdiereneanbeexplainedbythe
omplexityofthealgorithms,theomplexityofLRalgorithmisO(r 3
N 3
),whiletheomplexity
of theproposedmethod isO(rN 3
).
In Figure 7 it an be observed that the systemutilization inuenedthe omputation time
aswell,thus we investigated theeet of system utilization. We have foundthat theproposed
algorithm isvery sensitive to theutilizationand beomes ineÆient omparingto theLR algo-
rithmwhen utilizationonverges on 1(Figure8). Thisan beexplainedbythehighnumberof
iterationstepsof theLBSalgorithm(Table 1) andthisfatorrespondsto thepreviousresults
withtheSSalgorithm[2,6 ℄. TheutilizationlevelwhentheLRalgorithmbeomesmoreeÆient
stronglydepends onthemaximumbath size.
We then investigated The impat of the number of phases of the soure (C) is depited
0.1 1 10
20 30 40 50 60 70 80 90 100
Computationtime
[se℄
Utilization() 3
3
3
3
3 3
3
3
3
3
3
3 3
3
+
+
+
+
+ +
+
+
+
+
+
+ +
+
2
2
2
2
2 2
2
2
2
2
2
2 2
2
LBS
LR
C=5;r =16
3
C =10;r =16 +
C=5;r =32
2
Figure8: Computation timeversustheutilization(C =5and r=16; 32)
in Figure 9. For this experiment the message length (r) was 16, i.e., the bath size was 15.
Thebehaviourofthealgorithmsarequitesimilar,althoughtheproposedalgorithmseemsmore
eÆientwhen the numberof phasesof thesoureis higher. Thelarger memoryrequirement of
theLR algorithmmayause thisphenomenon.
Ourlastinvestigationwasthesensitivityofthealgorithmsontherequiredauray(stoping
riteria,). Figure 10showsresultswith anetwork utilisationof60%. The urvesinthegure
show that the required auray has only a little inuene of the omputation time of the
LR algorithm, while omputation time of the LBS method is strongly depends on it. In the
previously presented results we used a strit required auray, so the results would show a
better performaneof theLBS algorithminthease of lessstrit requiredauray.
5 Conlusions
In this paper an extension of the MG approah for proesses with bath arrivals has been
presented and a numerial method has been proposed to obtain the steady state behaviour.
A proess of this kind an be analysed as a QBD proess with a larger blok size, thus the
algorithms available for the analysis of QBD proesses are ineÆient. We proposed a method
thatperformstheomputationwithoutbloksizeenlargement. Theproposedapproahredues
theomputation omplexityand thememoryrequirement of thenumerial analysis.
A performaneomparison of the proposed method with an eÆient general method (LR)
hasalso beenpresented. Theresultsshowthat theproposedmethodis eÆient inaseof large
bath size,but itbeomesineÆientifthesystem utilizationonverges on 1.
Referenes
[1℄ R.Atkinson.DefaultIPMTUforuseoverATMAAL5,InternetRFC1626.NavalResearh
Laboratory, 1994.
[2℄ RamChakka. PerformaneandReliability Modellingof ComputingSystemsUsingSpetral
Expansion. PhD thesis,UniversityofNewastle uponTyne,1995.
[3℄ W.Feller.AnIntrodutiontoProbabilityTheoryandItsAppliation,volume2.JohnWiley
&Sons, 1971. 2nd edition.
0.1 1 10 100
4 8 12 16 20 24
Computationtime
[se℄
Numberofphasesofthesoure(C) 3
3
3
3
3
3
3
3
3
3
3
3
+
+
+
+
+
+
+
+
+
+
+ LBS
LR
=40%
3
=80%
+
Figure 9: Computationtimeversusthenumberof phasesofthe soure
[4℄ B.HaverkortandA.Ost.Steadystateanalysesofinnitestohastipetrinets: Aomparing
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muniations, pages265{276, Lund,Sweden,September1996.
1 10
6 8 10 12 14 16
Computation time
[se℄
Requiredauray( lg ()) 3
3
3
3
3
3
3
3 3 3
3
3
+
+
+
+
+
+
+ +
LBS
LR
C =5; r =16
3
C =5; r =32 +
Figure10: Computationtime versustheauray
