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TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerging Tel. Tech. 2012; 23:217–226

Published online 26 October 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.1524

RESEARCH ARTICLE

Stability of multiwavelength optical buffers withdelay-oriented schedulingE. Morozov1, W. Rogiest2*,†, K. De Turck2, D. Fiems2,† and H. Bruneel2

1 Institute of Applied Mathematical Research, Karelian Research Centre RAS, Pushkinskaya 11, 185910 Petrozavodsk, Russia2 SMACS Research Group, Department TELIN, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium

ABSTRACT

This contribution presents the first stability analysis of multiwavelength queueing systems. Results are obtained for a wideclass of implementable scheduling disciplines referred to as delay-oriented disciplines, a class to which for example jointhe shortest queue belongs. Scheduling is applied as contention resolution in optical switches accommodated with fibredelay lines as means of buffering. The analysis yields sufficient stability conditions for multiwavelength optical bufferswith a general fibre-delay-line set, renewal input and general service times. Copyright © 2011 John Wiley & Sons, Ltd.

KEY WORDS

multiwavelength optical buffers; stability conditions; regenerative approach; renewal theory; delay-oriented disciplines

*Correspondence

W. Rogiest, SMACS Research Group, Department TELIN, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium.E-mail: [email protected]†W. Rogiest and D. Fiems are postdoctoral fellows with the Research Foundation Flanders (FWO-Vlaanderen).

Received 15 September 2010; Revised 22 August 2011; Accepted 18 September 2011

1. INTRODUCTION

A viable alternative to the current electronic switch-ing in the backbone is optical switching. By avoid-ing the conversion from light to electricity, both opticalburst switching [1, 2] and optical packet switching [3, 4]unleash huge bandwidths but require optical buffering.Specifically, optical buffers mitigate the contention thatarises whenever two or more bursts (or packets) headfor the same destination at the same time. Becauserandom access memory is not available for photonicbroadband applications, buffering is implemented bymeans of fibre delay lines (FDLs), which can be usedto temporarily store light at multiple wavelengths atonce. Whereas the implementation of such a buffer iscompletely different from an electronic buffer (see [5]for a very recent prototype), also the logical operation dif-fers significantly. An ordinary buffer can incur the mini-mal packet delay such that contention is avoided. In FDLbuffering, this is not always possible because only a count-able number of delays can be generated. This accounts forthe creation of the so-called gaps [6] or voids [7, 8].

Callegati’s paper [9] represents the first queueing anal-ysis of an optical buffer with Poisson input, exponentialburst (packet) sizes, a single outgoing wavelength, and

a so-called degenerate FDL set [6]. In a degenerate orequidistant FDL set, the line lengths are equal to multi-ples of a basic value D called the granularity; delay linesare indexed accordingly, with delay line i corresponding toa delay i D. Independent of FDL optical buffers, Lakatos[10] also studied the degenerate waiting time set problemin a completely different context, namely that of multipleairplanes orbiting an airport. The setting relates to the land-ing of airplanes, where arriving airplanes are obliged towait for a discrete number of orbits of fixed length beforelanding (so-called cyclic waiting that is also studied in[11]). In current research on airport scheduling, this typeof waiting is referred to as a holding stack, a holding pat-tern where aircraft are instructed to join a waiting loop atdifferent altitude levels above a feeder fix point [12].

Opposed to a degenerate FDL set, a nondegeneratenumerable FDL set has general line lengths (waitingtimes). Evidently, studies performed for nondegenerate sets[6, 13, 14] are also valid for degenerate sets. In general,optical networks carry data over fibres over multiple wave-lengths at once. Therefore, the case of multiwavelengthoptical buffers is most important from the viewpoint of theapplication. Nevertheless, multiwavelength buffers wereanalysed earlier only in a very specific case [15], validfor a round-robin scheduling discipline and Poisson arrival

Copyright © 2011 John Wiley & Sons, Ltd. 217

E. Morozov et al.

process, and never in a general setting. The schedulingdiscipline is the key for the performance of multiwave-length optical buffers. It consists of a proper assignment ofchannel (wavelength) and delay line and is therefore alsoreferred to as a channel and delay selection algorithm [8].Previous simulation studies [7, 8] identified two relevantcategories of multiwavelength optical buffer schedulingdisciplines, delay-oriented and gap-oriented disciplines,that aim at minimising the delay of and the gap created bythe incoming burst, respectively. Of the two, delay-orienteddisciplines are arguably most likely to be implementedbecause delay-oriented disciplines result in less reorder-ing of the burst sequence. Examples of this category arejoin the shortest queue (JSQ) and minimum length (MINL)(see [7, 15] and Section 4 for more details). An example ofthe other category is minimum gap [7, 8], which realisesless packet/burst loss than delay-oriented disciplines butsuffers increased reordering of burst sequence (and morejitter). Whereas gap-oriented disciplines in fact provide avalid alternative [8], delay-oriented scheduling is assumedin the vast majority of optical switching research [8] and isalso assumed here.

In addition, scheduling disciplines may or may not applyvoid filling [6]. Void filling optimises the use of the FDLsby allowing for inserting bursts into voids created bypreceding bursts but requires complex logic and deliversbursts out of sequence.

For the case of a single wavelength (server), exploit-ing the regenerative property of the waiting time processallowed to establish general conditions for stability forgeneral independent and identically distributed (iid) inter-arrival times and burst sizes (or service times) and general(nondegenerate) FDL sets [14]. Other less general contri-butions even yielded exact stability bounds (albeit withoutformal proofs) in several specific cases [16–19]. Stabilityconditions of multiwavelength FDL buffers are the subjectof the present paper. In particular, we investigate stabil-ity conditions for the multiwavelength optical buffer withnondegenerate FDL sets and general iid interarrival timesand burst sizes. In Kendall notation, this corresponds toa GI/GI/m queueing model. As such, results obtained inthis paper apply to a broad class of traffic environments,including common settings like M/M/m but also allowingfor a power law probability distribution (e.g. Pareto distri-bution) for interarrival times as well as service times. Theexact conditions are listed in Section 3, Equations 9–11.

The buffer setting largely coincides with the oneassumed in earlier studies on multiwavelength opticalbuffers [7, 15, 20]. We assume the optical buffer locatedat the output port of an optical switch, and dedicated toa single optical outlet, as is also discussed and illustratedthrough figures in [20]. The optical fibre carries m dif-ferent wavelengths. The buffer consists of buffer controllogic, FDLs and a switching matrix at both the input andoutput of the fibre lines, allowing to switch a burst to anyline within the FDL set. Further, full wavelength conver-sion is available so that a burst can be converted to anyof different wavelengths. Further, we assume that each

burst traverses a given delay line only once (feed-forward),traverses exactly one delay line (single-stage) before beingtransmitted on the outgoing fibre [21]. The case of opti-cal buffers with feedback, as an alternative to feed-forwarddesign, is studied in [22]. Buffer control exercises a delay-oriented scheduling discipline—such as JSQ or MINL—and there is no void filling. We provide stability condi-tions for all delay-oriented disciplines. In this manner,limits on the wavelength conversion capability (as stud-ied for instance in [21, 23]) can also be accounted for(see Section 2). By the absence of void filling, the state ofthe buffer is described by a scheduling horizon (SH) pro-cess per wavelength. Moreover, for each wavelength, burstsare scheduled according to a first-come-first-served disci-pline; however, no such discipline is enforced when consid-ering the complete buffer (see [24] for discussion). Further,note that in the classical multiserver system with identicalservers and a JSQ discipline, the distribution of the work-load process does not depend on the assignment rule ofservers, whereas for optical systems, the assignment cor-responds to a whole class of delayed-oriented disciplines,each associated with another workload process.

The stability analysis presented in this work extendsour results on single-wavelength optical systems [14]. Ituses renewal theory and a characterisation of the limitingbehaviour of the forward renewal (regeneration) time ofthe SH process. This characterisation provides a straight-forward way to establish positive recurrence (finiteness ofthe mean regeneration period) of the embedded renewalprocess of regenerations in terms of given distributions.This approach turns out to be effective for the stabilityanalysis of many other queueing systems [25, 26]. In [26],this approach is presented in a general form, whereas [27]contains a detailed description of the method with applica-tions to various known and new models. As this paper isa natural extension of previous work [14], we follow thesame methodology, which allows us to reduce correspond-ing proofs, in particular in what concerns verification ofnegative drift of the basic process.

Introducing the Markovian SH process allows to basethe stability analysis on the well-known Kiefer–Wolfowitzrepresentation of the workload process in general multi-server queues [28]. We here show that the forward renewaltime (of the regenerations of the SH process) does notgo to infinity in probability. Such an approach does notrequire to prove finiteness of the mean regeneration perioddirectly, which is typically a difficult problem. In partic-ular, our approach requires to verify the following twosteps: (i) using a negative drift assumption, one mustshow that the SH does not go to infinity in probabil-ity; and (ii) using a regeneration assumption, one showsthat the forward regeneration time also does not go toinfinity. Then the characterisation mentioned previouslyimmediately implies positive recurrence of the SH process.When compared with the single-server case, an addi-tional requirement is to establish tightness of the incre-ments of the SH; in this regard, Lemma 2 constitutes avital new part in this contribution. Note that the negative

218 Trans. Emerging Tel. Tech. 23:217–226 (2012) © 2011 John Wiley & Sons, Ltd.DOI: 10.1002/ett

E. Morozov et al.

drift assumption in our work has a natural form arisingin stability analysis of general Markov chains and state-dependent queues [29, 30]. Finally, note that although thebasic process considered in the paper is Markovian (andthe Markovity is useful in intermediate steps of analy-sis), the present method also works successfully outsideof a Markovian framework [25–27]. Further, note that thepresent approach is different from but compatible withthe conventional approach to stability analysis of classicalmultiserver system using regeneration of Harris Markovchains (for more detail, see Chapter VII in [31].)

The remainder of this paper is organised as follows.In Section 2, we describe the system with the JSQ dis-cipline. The stability analysis of this system is presentedin Section 3. The proof of stability conditions consists ofthe following steps. We introduce the remaining workloadsassigned to different wavelengths, constituting the (vector-valued) SH process. Then we establish a negative drift ofthis Markov process and prove the tightness of the differ-ence between the components of SH process. Finally, weuse the characterisation of the renewal process of regener-ations to show positive recurrence. In Section 4, we intro-duce a class of delay-oriented disciplines and show that thestability analysis developed for JSQ discipline carries overto the complete class of delay-oriented disciplines, whichin particular includes JSQ and MINL scheduling.

2. DESCRIPTION OF THE MODEL

We consider an optical buffer with m wavelengths. Forease of exposition, we assume that m> 1 although theanalysis also holds for a single wavelength mD 1 (the sta-bility analysis of the single-wavelength system is presentedin [14].)

Let tk be the arrival instant of burst k, Bk be the burstsize (transmission time) of burst k, Tk D tkC1 � tk be theinterarrival time between the kth and .kC1/th arrivals, andUk D Bk � Tk ; k � 1. It is assumed that fTkg and fBkgare independent sequences of the iid random variables.

The system consists of an optical buffer that is charac-terised by a denumerable set AD fa0; a1; : : :g of availableFDL lengths or delays, ai 2 RC; i 2 N, a0 D 0. Theset contains nonidentical elements, that is ai ¤ aj fori ¤ j , and the line lengths are sorted in ascending order,a0 < a1 < : : :. Moreover, we assume limi!1 ai D 1such that there is always a sufficiently long delay line.Obviously, an actual buffer always contains a limited num-ber of lines; however, from a modelling point of view,assuming infinite buffer size is quite natural. Indeed, in sev-eral performance studies of finite optical buffers, a modelwith infinite buffer size is used as reference model, fromwhich results for finite buffers are extracted by means ofheuristics (see for example ([15, 16]). Note that we stickto the usual convention of referring to ai as FDL lengths,although they in fact represent the time needed to traversethe FDL.

The main feature of the optical buffer system is the con-straint imposed by the FDLs: optical buffers in generalcannot assign the exact required delay but only a delayai 2A. Therefore, on a given wavelength, if a burst needs adelay x so as not to overlap with previous bursts, the actualdelay ai is chosen such that ai�1 < x � ai or

dxeA WDminfai 2A W ai � xg; x 2RC (1)

The difference �.x/ D dxeA � x accounts for the timeduring which the outgoing wavelength remains unused,even though the scheduled burst (and possibly later-arrivedbursts) is still present in the buffer and awaiting transmis-sion on that wavelength. The quantity�.x/, called void (orgap), constitutes capacity loss in the state x. This variableplays an important role in stability analysis.

Any delay-oriented discipline, including JSQ andMINL, takes the scheduling decision on the basis of the SHof the m different wavelengths. At any instant t , the SH ofa wavelength is defined as the amount of time since instantt until all bursts assigned to this wavelength have left thatwavelength. Under the JSQ discipline, arriving bursts arerouted to the wavelength with the smallest SH, and this isregardless of the exact delay that will be assigned. Opposedto this, MINL first identifies the j different wavelengthsthat result in minimal delay and then routes the arrivingburst to the wavelength with minimal void �.x/, with xbeing the SH of the chosen wavelength. A more formaldescription of MINL is provided in Section 4.

Let H .i/k

denote the i th smallest SH upon arrival of thekth burst, and introduce the m-dimensional SH processH D fHkg, where

Hk D .H.1/k; : : : ;H

.m/k

/; H.1/k� � � � �H

.m/k

; k � 1

The SH process satisfies a Kiefer–Wolfowitz-type recursion,

HkC1 D R��l

H.1/k

mACUk

�C;�H.2/k� Tk

�C; : : : ;

�H.m/k� Tk

�C�(2)

where the operator R puts the components in increasingorder. We give an example of evolution of the workloadprocess for initially empty two-server system, that isH1 D

.0; 0/. Then H2 D R.d0eA; UC1 / D .H

.1/2 ; H

.2/2 / (note

that H .1/2 D 0), H3 D R..dH

.1/2 eA C U2/

C; .H.2/2 �

T2/C/, and so on.

To better grasp the difference between JSQ and MINL,the two delay-oriented disciplines, we consider an instruc-tive comparison illustrated in Figure 1. Assume the firstvalues of an (infinite) FDL set are chosen (arbitrarily)as follows:

AD fa0; a1; a2; a3; a4; : : :g D f0; 8; 14; 16; 22; : : :g (3)

with ai ; i � 5 arbitrary. Now, consider the simple arrivaltrace of burst sizes and arrival instants given in Table I. For

Trans. Emerging Tel. Tech. 23:217–226 (2012) © 2011 John Wiley & Sons, Ltd.DOI: 10.1002/ett

219

E. Morozov et al.

t

a1

a2

a3

a4

10

20

00 10

B1

...

: Hk

B2

B3 B4

20

B5

B6

B7

30

B8

B9...

JSQ

(a) JSQ

t

a1

a2

a3

a4

10

20

00 10

B1

...

B2

B3

B4

20

B5

B6B7

30

B8B9

...

MINL

: Hk

(b) MINL

Figure 1. The evolution of both scheduling horizon components (marked ı), in case of two servers .m D 2/. For the same arrivaltrace (see Table I), JSQ and MINL lead to a different assignment. The burst sizes Bk (marked ") and arrival instants tk (marked �) of

the arrival trace .1� k � 10/ are given in Table I.

Table I. Values of traffic trace considered in Figure 1.

k 1 2 3 4 5 6 7 8 9

Bk 8:8 7:0 5:0 4:5 12:5 2:4 4:4 7:6 4:8tk 0:0 4:0 7:0 10:5 14:0 16:5 21:0 24:0 30:0

this trace, Figure 1 compares the evolution of the horizonprocessH D fHkg, in case of JSQ (Figure 1(a)) and MINL(Figure 1(b)), respectively. Clearly, the disciplines lead to acompletely different system evolution, with different burstsassigned to different delay lines and wavelengths. Note thatthis is still completely different from what would happen ina classic random access memory buffer. In such a system,the waiting room is a continuum, ADRC, and the waitingtime is simply identical to the horizon, which consistentlyleads to smaller waiting times.

In the following, we adhere a regenerative approachto stability [26, 27]. Therefore, we construct regenerationinstants for the process H as follows. We put ˇ0 D 0 and

ˇnC1 D inf�k > ˇn WH

.m/kD 0

�; n� 0 (4)

where inf; WD1. Notice that H .m/kD 0 for a given k,

impliesH .i/kD 0, i D 1; : : : ; m (by definition) or an empty

system, accounting for a regeneration at time tk .Our main purpose is to establish conditions that imply

positive recurrence of the renewal process ˇDfˇng, whichmeans

ˇ1 <1 with probability 1 (w.p.1) and

E.ˇ2 � ˇ1/ WD ˛ <1: (5)

The Markov chain H is called positive recurrent if theprocess ˇ is positive recurrent [32]. We focus on stabilityconditions for the zero-delayed case when the first burstarrives in an empty system at instant t D 0, which isthus a regeneration epoch. For this case, ˇ1DˇnC1 � ˇn

stochastically (provided ˇnC1 <1), n � 1 and Eˇ1 D ˛.The forward regeneration time at instant n is defined as

ˇ.n/D infk.ˇk � n W ˇk � n > 0/; n� 1 : (6)

Note that by counting arrivals, we work effectively indiscrete time. We characterise the recurrence property ofthe renewal process ˇ via the limiting behaviour of itsforward regeneration time. Indeed, it is known [33] that

˛ D1 implies ˇ.n/)1 (in probability) as n!1 :

(7)Obviously, in the zero-delayed case, ˛ <1 implies posi-tive recurrence (5).

3. STABILITY ANALYSIS

In this section, we study stability of the JSQ discipline.Extending the analysis to delay-oriented disciplines is thesubject of Section 4. Our approach is based on the charac-terisation (7) of the renewal process ˇ associated with theregenerations of the SH process. A key idea of the proofis to show that under the assumptions given in the follow-ing, the forward regeneration time ˇ.n/ 6)1. By property(7), this implies ˛ <1 and thus positive recurrence of therenewal process ˇ and the SH process H .

We first introduce some notation:

gn D anC1 � an; n� 0 I �� D supn�0

gn I

�0 D lim supn!1

gn I ı� D infn�0

gn (8)

and recall the definition dxeA � x D �.x/, x � 0. Notethat �.x/ � 0 for x � 0 and �.ak/ D 0 for any k � 0.We now formulate some assumptions that hold through-out the paper. Here, and in the remainder of the paper, wewill omit subscripts to denote corresponding generic ran-dom variables. First, finiteness of the mean burst size and


E. Morozov et al.

mean interarrival times, EB <1 and ET <1, is assumedthat implies

EU 2 .�1; C1/ (9)

In addition, the following FDL assumptions

ı� > 0; �� <1 (10)

are imposed as well as the following regeneration assump-tion

P.T > ��CB/ > 0 (11)

Before proceeding to the main stability theorem, we firstprove some lemmas. Denote

OHk D

mXiD1

H.i/k; �H .k/D OHkC1 � OHk ; k � 1 (12)

The following lemma provides an estimate of the meanincrement E.�H .k/jHk D y/ for large values of y WD.y1; : : : ; ym/.

Lemma 1. Under assumptions (9) and (10), the follow-ing limit inequality holds:

lim supy!1

E.�H .k/jHk D y/��0C EB �mET (13)

Proof . Because the SH process is Markovian, the follow-ing relation holds:

E.�H .k/jHk D y/D E

.dy1eACUk/

C � y1

C

mXiD2

.yi � Tk/C � yi

!(14)

By straightforward calculations, we find

E..dy1eA CUk/C � y1/

D

ZR

�.�.y1/C y1C z/

C � y1

�FU .dz/

D�y1P .U ��y1 ��.y1//

C�.y1/P .U > �y1 ��.y1//

C

Zz��y1��.y1/

zFU .dz/ (15)

From EU > �1, we have �xP.U � �x/ " 0 asx!1. Note that convergence in Equation (13) is compo-nentwise, and in particular, y1 !1. Hence, we have thefollowing limit for the first term in Equation (15):

limy1!1

�y1P.U � �y1 ��.y1//

D limy1!1

�y1P.U � �y1/D 0 (16)

Moreover, the following limits for the second and thirdterms hold:

lim supy1!1

�.y1/P.U > �y1 ��.y1//� lim supy1!1

�.y1/

��0 ; (17)

limy1!1

Zz��y1��.y1/

zFU .dz/D EU (18)

such that

lim supy1!1

E..dy1eACUk/C � y1/��0C EU (19)

Finally, we have

limyi!1

E..yi � Tk/C � yi /D�ET ; i D 2; : : : ; m (20)

Combining Equations (14), (19) and (20) then yields thestated result. �

As mentioned earlier, unlike in the single-wavelengthcase, the present system setting requires establishing tight-ness of the components of the SH. This is the aim of the

following lemma. Introduce the differences dn D H.m/n �

H.1/n , n� 1, and prove the following.

Lemma 2. The sequence fdng is tight under assumptions(9) to (11).

Proof . Our approach follows Kiefer and Wolfowitz [28].(More details can also be found in [34].) Denote

Cn D

m�1XiD1

.H.m/n �H

.i/n /D .m�1/H

.m/n �

m�1XiD1

H.i/n � 0

(21)and prove that the sequence fCng is tight. Because dn �Cn, the tightness of fCng implies the statement of Lemma2. By the basic recursion (2) and as in [28], we find thefollowing bound for CnC1:

CnC1 �max�Cn ��.H

.1/n /�Bn;

.m� 1/�BnC�.H

.1/n /

��max

�Cn �Bn; .m� 1/.BnC�

�/�

(22)

Repeatedly invoking this inequality further yields

CnC1 �max

[email protected]� 1/.Bk C��/� nX

iDkC1

Bi ;

k D 1; : : : ; n

1A WD Yn; n� 1 (23)


221

E. Morozov et al.

whereP; WD 0. Note that we here implicitly used the

condition C1 D 0. Because the Bn are iid, we can redefineYn keeping stochastic equivalence in such a way that

Yn Dmax

[email protected]� 1/.Bk C��/� k�1X

iD1

Bi ; k D 1; : : : ; n

1A

(24)By the strong law of large numbers, we have, with

probability 1,

1

k

k�1XiD1

Bi ! EB > 0;Bk C�

�

k! 0; k!1 (25)

This implies that lim supk!1�.m� 1/.Bk C�

�/ �Pk�1iD1 Bk

�< 0. Hence, supYn < 1 and the series fYng

is tight. Now the tightness of fCng follows. �

Remark 1. The tightness of fdng also holds for anyinitial state H1 [34].

Remark 2. It is tempting to simplify the aforementionedproof. For instance, one may reason along the followingline. Consider an instant k when the minimal componentof the SH vector becomes the maximal one. For this k,we have

dH.1/keA CBk DH

.1/kC�.H

.1/k/CBk >H

.m/k

(26)

or equivalently, Bk C �.H.1/k/ > dk . However, the

latter does not yield a useful upper bound for dk becausethe burst size Bk follows an unknown conditional distribu-

tion (on the event fBk > dk ��.H.1/k/g).

Now we establish the main stability result under theassumption of the zero initial stateH1D 0, with a first burstarriving at a totally empty system at instant t D 0.

Theorem 1. Assume conditions (9)–(11) and the follow-ing negative drift condition

�0C EB <mET (27)

hold. Then the zero-delayed regenerative process H ispositive recurrent.

Proof . Obviously, the following uniform upper boundholds:

maxy�0

E.�H .k/jHk D y/��C EB �

mXiD2

E.min.yi ; T //

��C EB WD C : (28)

Denote "DmET ��0 � EB > 0. By Lemma 1 and con-dition (27), there exists a value x0 � 0 such that

supy2A0

E.�H .k/jHk D y/� �"

2;

where A0 WD Œx0; 1/m. Now we obtain the following

upper bound for E�H .k/,

E�H .k/� CP.Hk 62A0/�"

2P.Hk 2A0/; k � 1:

(29)

Assume for a moment that H.1/k)1; k!1.

Because H .1/k

is dominated by H .i/k

, i D 2; : : : ; m, wealso have Hk ) 1 and therefore P.Hk 2 A0/ ! 1.By Equation (29), this implies lim supk!1 E�H .k/ < 0.Hence, there exist a k0 such that

EH .1/k�1

mE OHk �

1

mE OHk0 <1; k � k0 (30)

Hence, we obtain a contradiction and H .1/k»1. This

implies the existence of constants "0 > 0 and C0 <1 anda nonrandom (sub)sequence nk !1 such that

infk

P.H .1/nk � C0/� "0 (31)

In the following second step of the proof, we show thatstarting in a compact set, the SH process hits a regenera-tion within a finite interval with a positive probability thatis lower bounded by a positive constant (uniformly overthe states of the set). Notice that the proof of the negativedrift of the SH process is similar to that has been appliedin single-server case in [14]. This is enough to deduce

that H .1/k6)1, but because now the SH process is mul-

tidimensional, it is required that the maximal component

H.m/k6)1. By virtue of Lemma 2, the increments dk are

tight. Hence, one can find a constant R0 (� C0) such that

infk

P�Hnk 2 Œ0; R0�

m��"0

2(32)

We now fix any nk satisfying Equation (31) and introducethe following events:

E.nk/D˚Hnk 2 Œ0; R0�

m�I

B.nk/D˚mTnk > Bnk C�

�C ı0�: (33)

Here, ı0 > 0 is such that P.B.nk// � ı1 for someı1 > 0. Existence of such ı0; ı1 is guaranteed byEquation (11). Then, on the event B.nk/ and provided thatall wavelengths are still busy at instant nk C 1, we find

OHnkC1 DH.1/nk C�.H

.1/nk /CBnk � Tnk

C

mXiD2

.H.i/nk � Tnk /

D OHnk � .mTnk �Bnk ��.H.1/nk //�

OHnk � ı0(34)

From Equation (34), we find that on the event B.nk/ andprovided that all wavelengths are busy before the arrivalof burst nk C 1, the workload in the system decreases


E. Morozov et al.

by ı0. This holds for each interarrival time as long asall wavelengths are busy. Because (on the event E.nk/)OHnk � mR0, we need at most D0 WD dmR0=ı0e eventsB.nk C i/; i D 0; : : : ;D0 � 1, until a burst �.nk/ findsan empty wavelength upon arrival. In other words, withprobability

P

[email protected]/\

D0�1\iD0

B.nk C i/

1A� "0

2ıD01 > 0 (35)

burst �.nk/ arrives in interval Œnk ; nk CD0/ and finds an

empty wavelength. Moreover, by Equation (34),H .i/�.nk/

�

OH�.nk/ �OHnk �mR0 for i D 2; : : : ; m. We now generate

the following events:

D�.nk/Ci D fT�.nk/Ci > B�.nk/Ci C ı0g;i D 0; : : : ;D0 � 1 (36)

to have a burst arriving in totally empty buffer. Indeed (onthese events), each burst �.n/Ci can be routed to the sameempty wavelength i D 0; : : : ;D0�1. Therefore, workloaddecreases continuously for the other m � 1 wavelengths.By Equation (36), the workload decreases by at least ı0 perburst arrival. Hence, there is a burst that arrives in intervalŒ�.nk/; �.nk/CD0� in a completely empty optical buffer.Note that P.Dn/ � P.Bn/ � ı1 for each n. Thus, therequired number of the events Dn has probability

P

0@D0\iD0

Dk.n/Ci

1A� ı D0�11 > 0 (37)

Summarising, for each burst nk satisfying (31), thereis a burst that enters an empty optical buffer in interval

Œnk ; nk C 2D0� with a probability � "0=2 ı2D01 . In other

words, the forward regeneration time ˇ.nk/ is (uniformly)limited by the constant 2D0 with a positive probability, andpositive recurrence follows. �

Corollary 1. Under assumptions of Theorem 1, the sta-tionary distribution P.Hk 2 �/! P.H1 2 �/ as k !1exists.

Proof . By assumption (11), aperiodicity of the renewalprocess ˇ follows because

P.ˇ1 D 1/D P.H2 D 0jH1 D 0/

D P.T � B/� P.T > B C��/ > 0: (38)

The statement of the Corollary 1 then immediatelyfollows from regenerative theory. �

Corollary 2. For the degenerate case �n�D, the zero-delayed process H is positive recurrent if conditions D CEB <mET and P.T > DCB/ > 0 hold.

Remark 3. Condition (12) takes the classical form,EB <mET ; if �0 D 0.

Remark 4. In general, the queueing system underconsideration belongs to a class of state-dependent queue-ing systems because a change of the SH upon arrival ofa burst depends not only on the burst size but also on thecurrent state of the SH process. This dependency compli-cates the stability analysis, and as a result, the extensionto arbitrary initial conditions cannot be carried out as inthe single-server case [14] and is not considered further inthis work.

4. DELAY-ORIENTED DISCIPLINES

In this section, retaining the main notation, we intro-duce a class of delay-oriented disciplines that in particularincludes the JSQ discipline and then extend the precedingstability analysis to this class.

Recall Equation (2) for the JSQ discipline. Upon arrivalof a burst, the wavelength with the shortest SH is selected,and the delay of the burst is obtained by ceiling the SH(with respect to A). Define the set

Gk D fj W dH.j /keA D dH

.1/keAg; (39)

of the wavelengths with minimal SH upon arrival of theburst k � 1. Although for JSQ case the buffer always incursminimal delay by selecting the wavelength with the short-est SH, it is possible, provided #fj WGkg WD jGk j> 1, thatselection of other wavelengths also yields minimal delay.

This motivates the introduction of the delay-orienteddisciplines, which formally can be introduced as follows.If jGk j> 1, then each discipline belonging to this class ischaracterised by the specification of a rule that assigns awavelength �k 2 Gk to burst k � 1. It is assumed that theselector �k is a stopping time with respect to the � -field(filtration) Ck WD fBj ; Tj I j � kg. Moreover, the selec-tor is assumed to regenerate each time the system becomesempty.

For delay-oriented disciplines, Equation (2) is modifiedas follows:

HkC1 DR�.H

.1/k� Tk/

C; : : : ; .H.�k�1/k

� Tk/C;

.dH.�k/

keACBk � Tk/

C; .H.�kC1/k

� Tk/C;

: : : ; .H.m/k� Tk/

C�

(40)

Before proceeding to the stability theorem, we note thatfor JSQ and MINL disciplines, the selectors are defined,respectively, as follows:

�k D 1 ; �k Dmaxfj W j 2Gkg; k � 1 (41)

Whereas JSQ selects the wavelength with the shortestSH, MINL selects the one with the wavelength with thelongest SH that incurs the same delay. As a consequence,


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MINL creates smaller voids. Therefore, performance ofMINL (both in terms of loss and waiting time) is typicallybetter than that of JSQ because minimising void sizes alsominimises the capacity loss due to these voids [7].

Delay-oriented disciplines are not limited to those previ-ously mentioned. For instance, one can consider a schedulethat minimises the need for wavelength conversion, whichis a critical factor in practical implementations [21, 23].Namely, let f�kg be an iid sequence and �k 2 f1; : : : ; mg.Then the considered discipline assigns wavelength �k D�k if �k 2 Gk ; otherwise, it selects in accordance with,say, JSQ or MINL. Another example is the random min-imal delay discipline for which �k is uniformly selectedfrom the set Gk .

Although it is intuitively clear that the stability conditionfor JSQ discipline implies stability for any delay-orienteddiscipline, we give a formal proof of this fact. Therefore,we introduce the functions

fi .h; b; t/D ..dh.i/eAC b � t /

C; .h.1/ � t /C; : : : ;

.h.i�1/ � t /C; .h.iC1/ � t /C; : : : ;

.h.m/ � t /C/ (42)

for i D 2; : : : ; m� 1 (with evident modification for i D 1;m). Here, h is an m-dimensional vector with componentsh.1/ � h.2/ � � � � � h.m/, and b and t are non-negativeconstants. Also, let � denote componentwise inequality.We now prove the following theorem.

Theorem 2. Let fHkg and f QHkg be the SH process underJSQ discipline and a delay-oriented discipline with selec-tor f�kg, respectively. For given fBkg; fTkg, if H1 D QH1,then QHk �Hk for all k � 1.

Proof . Note that for the JSQ discipline, HkC1 D

R.f1.Hk ; Bk ; Tk//, and in view of Equations (40) and(42), QHk satisfies QHkC1 D R.f�k .

QHk ; Bk ; Tk//; k � 1:

We now prove the theorem by induction on k. We triv-ially have QH1 � H1. Assuming QHn � Hn for all n � kand some k � 1, we establish QHkC1 � HkC1. Indeed, byEquation (42),

f�k .QHk ; Bk ; Tk/� f1. QHk ; Bk ; Tk/� f1.Hk ; Bk ; Tk/

(43)

It is easy to establish that the Kiefer–Wolfowitz operatorpreserves componentwise inequality, that is g � h impliesR.g/ � R.h/ (also see, for instance, [35]). Hence, fromEquation (43), we have

QHkC1 D R.f�k .QHk ; Bk ; Tk//� R.f1.Hk ; Bk ; Tk//

DHkC1 (44)�

By the dominance property (44), a regeneration pointfor fHkg is a regeneration point for f QHkg as well. Now,it follows from the analysis that the stability conditions

for JSQ discipline (Theorem 1) imply the stability of anydelay-oriented discipline.

Remark 5. In the classical multiserver system with iden-tical servers and a JSQ discipline, the distribution of theworkload process does not depend on which server �k 2Gk the kth burst is assigned to. Therefore, in the classi-cal setting, there is no need to specify the selection rule.Any such selection can be treated as a JSQ discipline. Incontrast, for optical systems, the class of delayed-orienteddisciplines has meaning and is well recognised.

5. CONCLUSION

In this paper, stability of multiwavelength optical buffers isstudied. The set of FDLs is general, and both the interar-rival times and burst sizes are assumed to be iid sequences.The stability proof is based on the characterisation of theforward regeneration time in the renewal process generatedby the regenerations of the SH process and allows to estab-lish minimal stability conditions in a straightforward way.Further, we prove the tightness of the increments of theSH process, as vital new part in this contribution. We firstestablish stability of multiwavelength optical buffers with aJSQ discipline. Then, the analysis is extended to the entireclass of delay-oriented disciplines.

ACKNOWLEDGEMENTS

E. Morozov is supported by the Russian Foundation forBasic Research under grant 10-07-00017.

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AUTHORS’ BIOGRAPHIES

Evsey V. Morozov obtained the PhD from Kiev Insti-tute of Cybernetics in1979 and the Dr Scientist Degree inMathematics from Moscow Institute of Control Sciencesin 1996. He is the leading researcher at the Institutefor Applied Mathematical Research of Karelian Research


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Centre, Russian Academy of Sciences and also full pro-fessor in the Faculty of Mathematics of the PetrozavodskState University. His research interests include stabilityanalysis of queues and networks, rare event and regenera-tive simulation, Gaussian queues, fluid analysis of queues.He has published over 120 papers on these and relatedtopics. He had numerous invitation research visits andgave intensive lecture courses on stochastic modelling oftelecommunication systems at the Universities of Helsinki,Oulu, Kuopio, Pisa, Zaragoza and Navarra.

Wouter Rogiest was born in Ghent, Belgium, in 1980. Heobtained the master’s degree in Electrical Engineering in2004 and the PhD degree in Engineering in 2008, bothfrom Ghent University, Belgium. Since October 2004,he has been with the Department of Telecommunica-tions and Information Processing of the same univer-sity as a researcher and also 11 months with Bell Labs,Alcatel-Lucent Bell (October 2008–August 2009) as a net-work architect. His research interests include stochasticmodels and queueing models for performance evaluationand optimisation of communication systems, in particularoptical backbone networks. In fall 2011 (26 September–23 December), he is a visiting researcher at the Universityof Kyoto, Japan (Takahashi Laboratory).

Koen De Turck was born in Zottegem, Belgium, in 1981.He obtained the MS degree in Engineering in 2004 fromGhent University. He subsequently joined the SMACSResearch Group, Department of Telecommunications andInformation Processing, also at Ghent University, wherehe obtained his PhD degree in 2011. His research interestsinclude among others the application of probability theoryto telecommunication networks.

Dieter Fiems received his engineering degree from KAHOSt-Lieven in 1997, a post-graduate degree in Computer Sci-ence from Ghent University in 1998 and the PhD degreein engineering from Ghent University in 2004. Since1998, he is a researcher at the Department of Telecom-munications and Information Processing of Ghent Univer-sity, as a member of the SMACS Research Group. Since2009, he is also part-time professor at this department.His research interests include among others queueingtheory, branching processes, traffic characterisation andstochastic modelling of telecommunication networks.

Herwig Bruneel was born in Zottegem, Belgium, in 1954.He received the master’s degree in Electrical Engineer-ing, the master’s degree in Computer Science and thePhD degree in Computer Science in 1978, 1979 and 1984,respectively, all from Ghent University, Belgium. He is fullprofessor in the Faculty of Engineering and head of theDepartment of Telecommunications and Information Pro-cessing at the same university. He also leads the SMACSResearch Group within this department. His main per-sonal research interests include stochastic modelling andanalysis of communication systems, discrete-time queueingtheory and the study of automatic repeat request proto-cols. He has published more than 400 papers on thesesubjects and is coauthor of the book H. Bruneel and B. G.Kim, ‘Discrete-Time Models for Communication SystemsIncluding ATM’ (Kluwer Academic Publishers, Boston,1993). From October 2001 to September 2003, he servedas the Academic Director for Research Affairs at GhentUniversity. Since 2009, he holds a career-long Methusalemgrant from the Flemish Government at Ghent Univer-sity, specifically on stochastic modelling and analysis ofcommunication systems.
