QUEUEING NETWORKS
S. Minkevi ius 1;2
1
Institute of Mathemati s and Informati s,Akademijos 4, 08663
2
Vilnius Gediminas Te hni al University, Sauletekio 11, 10223
1;2
Vilnius, Lithuania
ststktl.mii.lt
The paperis devoted tothe analysis ofqueueing systems inthe ontext of the
net-work and ommuni ations theory. We investigate the inequality in an open queueing
network and its appli ations tothe theorems in heavy tra onditions ( uid
approx-imation, fun tional limit theorem, and law of the iterated logarithm) for a queue of
ustomers inan open queueing network.
Keywords: modelsofinformationsystems,queueingtheory,openqueueingnetwork,
heavytra ,queue lengthof ustomers, uid approximation,fun tionallimittheorem,
law ofthe iteratedlogarithm.
1. STATEMENT OF THE PROBLEM
The paperis devoted tothe analysis ofqueueing systems inthe ontext of the
net-work and ommuni ations theory. We investigate the inequality in an open queueing
network and its appli ations tothe theorems in heavy tra onditions ( uid
approx-imation, fun tional limit theorem, and law of the iterated logarithm) for a queue of
ustomers inan open queueing network.
Nowweshallsurvey thepapers onaqueue inheavy tra onditions. Inthe paper
of Chen, Xinyang and Yao [3℄, a semi-martingale re e ting the Brownian motion
ap-proximation is developed for the performan e pro esses su h as workload, queue, and
sojourn time. InthepaperofMasseyandSrinivasan[7℄,thesteady-statedistributionof
the queuepro ess, usingtensorand Krone ker produ ts, shows that itisofthe
matrix-geometri stru ture. J. Daiand W. Daiin[4℄proved that anappropriatelynormalized
queue pro ess onvergesindistributionto ad-dimensionalre e tingthe Brownian
mo-tion under the heavy tra onditions. Puhalskii [9℄ established moderate-deviation
prin iples for the queue, virtual waiting time and sojourn pro esses. In [5℄, Yamada
has showed that the normalized queue pro esses at the nodes onverge in distribution
to a re e ted, multivariate diusion pro ess whose drift and diusion oe ients are
statedependentandnonsingular. Inthe arti leofKushnerandMartins[6℄,the authors
study the problems of the pathwise average ost per unit time for ontrolled and
un- ontrolledopen queueing networks inheavy tra . In the paperof ZhangHanqin and
XuGuang-hui[13℄strongapproximationsforanopenqueueingnetworkinheavytra
pro-motion. Inthearti leofReimanandSimon[11℄,the authors onsideranopenqueueing
network with multiple lasses, priorities, "arbitrary" routing,and general servi e time
distribution. Usinga heavy tra limittheorem for open queueing networks, Reiman
[10℄ found the orre t diusion approximation for sojourn times in Ja kson networks
with a single-server station. As one an see, there are only several works designed to
explore a queue in a more ompli ated than the lassi al single-server queue: tandem,
multiphasequeue, open queueingnetwork (seethe arti lesof Boxma[1,2℄,Zhang
Han-qinandXuGuang-hui[13℄,MasseyandSrinivasan[7℄,andSakalauskasandMinkevi ius
[12℄).
In this paper, we investigate an open queueing network model in heavy tra .
The servi e dis ipline is rst ome, rst served (FCFS). We onsider open queueing
networks with the FCFS servi e dis iplineat ea h station and general distributions of
interarrival and servi e times. We study the queueing network with k single server
stations, ea h of whi h has an asso iated in nite apa ity waiting room. Ea h station
hasanarrivalstreamfromoutsidethenetwork,and the arrivalstreamsare assumedto
bemutuallyindependentrenewalpro esses. Customersareservedintheorderofarrival
and after servi e they are randomly routed to either another station in the network,
or out of the network entirely. Servi e times and routing de isions form mutually
independent sequen es of independent identi allydistributed randomvariables.
The basi omponents of the queueing network are arrival pro esses, servi e
pro- esses, and routingpro esses. In parti ular, there are mutually independent sequen es
of independent identi ally distributed random variables n z (j) n ;n1 o , n S (j) n ;n1 o and n (j) n ;n1 o
for j = 1;2;:::;k. The random variables z (j) n and S (j) n are stri tly positive, and (j) n
has support in f0;1;2;:::;kg. We de ne
j = M h S (j) n i 1 > 0; j =D S (j) n > 0, j = M h z (j) n i 1 > 0; and a j =D z (j) n > 0; j = 1;2;:::;k;
allofthesesequen es areassumedtobe nite. Wedenotep
ij =P (i) n =j >0; i;j =
1;2;:::;k:Inthe ontextofthequeueingnetwork,therandomvariablesz (j)
n
fun tionas
interarrivaltimes(fromoutsidethenetwork)atthestationj,whileS (j)
n
isthenthservi e
timeatthe stationj,and (j)
n
isaroutingindi atorfor thenth ustomerserved atthe
stationj. If (i)
n
=j(whi ho urswithprobabilityp
ij
),thenthenth ustomerservedat
thestationiisroutedtothestationj. When (i)
n
=0;theasso iated ustomerleavesthe
network. To onstru trenewalpro essesgeneratedbytheinterarrivalandservi etimes,
we assume z j (0) =0;z j (l) = l P m=1 z (j) m ;S j (0) = 0;S j (l) = l P m=1 S (j) m ;l 1;j = 1;2;:::;k: We now de ne a j (t) = max(l0:z j (l)t ), x j (t) = max(l 0:S j (l)t), ~ j (t) as
the total number of ustomers routed to the jth station of the network until time t,
j
(t) as the total number of ustomers after servi e departure from the jth station of
thenetwork untiltime t,
ij
(t)asthetotal numberof ustomers afterservi e departure
timet, p t ij = ij i (t)
asapart ofthe totalnumberof ustomers whi h,afterservi e atthe
ithstationofthenetwork,areroutedtothejthstationofthenetwork, i;j =1;2;:::;k
and t > 0. Note that this system is quite general, en ompassing the tandem system,
a y li networksof GI=G=1queues,networks ofGI=G=1queues withfeedba k, and an
open queueing network.
First,letusdenotebyQ
j
(t)thequeueof ustomersatthejthstationofthequeueing
networkat time t; j = j + k P i=1 i p ij j >0; ^ 2 j =( j ) 3 a j + k P i=1 ( i ) 3 i (p ij ) 2 + ( j ) 3 j >0;j =1;2;:::;k:
Supposethatthequeueof ustomersinea hstationoftheopenqueueingnetworkis
unlimited. Allrandomvariablesarede nedonthe ommonprobabilityspa e(;F;P).
Weassume that the following onditions are ful lled:
j + k X i=1 i p ij > j ; j =1;2;:::;k: (1)
Note that these onditions quarantee that there exists a queue of ustomers and it
is onstantly growing.
2. MAIN RESULTS
At rst we will prove a lemma in whi h the queue length of ustomers in an open
queueingnetwork is estimated by a ompositionof renewal pro esses
Lemma 1. If Q(0) = 0, then jQ j (t) x^ j (t)j k P i=1 w i (t)+ k P i=1 i (t); where w j (t) = k P i=1 x i (t)jp t ij p ij j and j (t)= sup 0st (x j (s) j (s)), j =1;2;:::;k and t >0:
Proof. Byde nitionof the queue lengthof ustomersatthe stationof the network,we
get that Q j (t)= ~ j (t) j (t)= ~ j (t) x j (t)+x j (t) j (t) ~ j (t) x j (t)+ + sup 0st (x j (s) j (s))= k X i=1 i (t)p t ij +a j (t) x j (t)+ + sup 0st (x j (s) j (s)) k X i=1 x i (t)p t ij +a j (t) x j (t)+ + sup 0st (x j (s) j (s)) k X i=1 x i (t)(p t ij p ij +p ij )+a j (t) x j (t)+ sup 0st (x j (s) j (s)) k X i=1 x i (t)p ij +a j (t)
x j (t)+ X i=1 x i (t)jp t ij p ij j+ sup 0st (x j (s) j (s))= =x^ j (t)+ k X i=1 w i (t)+ j (t)x^ j (t)+ ( k X i=1 w i (t)+ k X i=1 i (t) ) ; (2) wherej =1;2;:::;k and t>0:
This implies that
Q j (t)x^ j (t)+ ( k X i=1 w i (t)+ k X i=1 i (t) ) ; (3) j =1;2;:::;k and t >0 (see (2)).
On the otherside, weobtain that
Q j (t) ~ j (t) x j (t)= k X i=1 i (t)p t ij +a j (t) x j (t) k X i=1 ( i (t) x i (t)+x i (t))p t ij +a j (t) x j (t) = k X i=1 x i (t)p t ij +a j (t) x j (t)+ k X i=1 ( i (t) x i (t))p t ij = k X i=1 x i (t)(p t ij p ij +p ij )+a j (t) x j (t) k X i=1 ( i (t) x i (t))p t ij k X i=1 x i (t)(p t ij p ij +p ij )+a j (t) x j (t) k X i=1 x i (t)jp ij p t ij j k X i=1 (x i (t) i (t)) x^ j (t) k X i=1 w i (t) sup 0st k X i=1 (x i (s) i (s)) ! ; (4) j =1;2;:::;k and t >0:
This implies that
Q j (t)x^ j (t) k X i=1 w i (t) k X i=1 i (t); (5) j =1;2;:::;k and t >0: Denote q(t)= k P i=1 w i (t)+ k P i=1 i
(t); t>0:So, weobtain that
jQ
j
(t) x^
j
The proof of the lemma is ompleted.
3. APPLICATION OF THE INEQUALITY
Note that inequality (6) is the key inequality to prove several laws ( uid
approx-imation, fun tional limit theorem, and law of the iterated logarithm) for a queue of
ustomers inopen queueing networks in heavy tra onditions.
At rst we present a theorem on the uid approximation for a queue of ustomers
inopen queueingnetworks in heavy tra onditions.
Theorem 1. If onditions (1) are ful lled, then
Q 1 (t) t ; Q 2 (t) t ;:::; Q k (t) t )( 1 ; 2 ;:::; k ); 0t1:
Next,wepresentatheoremonthefun tionallimittheoremforaqueueof ustomers
inopen queueingnetworks in heavy tra onditions.
Theorem 2. If onditions (1) are ful lled, then
Q 1 (nt) 1 nt ^ 1 p n ; Q 2 (nt) 2 nt ^ 2 p n ;:::; Q k (nt) k nt ^ k p n ) (z 1 (t);z 2 (t);:::;z k (t)); where z j
(t); j = 1;2;:::;k; 0 t 1 are independent
stan-dard Winer pro esses.
One of the results of the paper is the following theorem onthe lawof the iterated
logarithmfor aqueue of ustomers inanopen queueing network.
Theorem 3. If onditions (1) are ful lled, then
P lim t!1 Q j (t) j t ^ j a(t) =1 =1; j =1;2;:::;k and a(t)= p 2tlnlnt:
Proof. The proof of thesetheorems is similartothat in[12℄, and weomit theseproofs.
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