Burstiness in Broadband Integrated Networks

Fabrice GUILLEMIN, Jacqueline BOYER and Alain DUPUIS

Centre National d'Etudes des T�el�ecommunicationsRoute de Tr�egastel, 22301 Lannion, FRANCE

Abstract

Since broadband integrated networks have to cope with a wide range of bit rates,the notion of burstiness which expresses the irregularity of a ow, has been recognizedas a vital question for such networks. In burstiness characterization encountered inthe literature, special attention is given to the squared coe�cient of variation of in-terarrival time (Cv2) in a cell arrival process. In order to observe the impact of abursty ow on a queue, we introduce in this paper a new class of arrival process, then-stage Markov Modulated Bernoulli Process, MMBPn , for short, and its peculiarcase, the n-stage Hyper-Bernoulli process, denoted by HBPn. We numerically solve theMMBPn=D=1=K and we compute in particular the rejection probability and the meanwaiting time. For that purpose, a relation between the stationary queue length distri-bution and arrival time distribution is established. This relation adapts the GASTAequality to the arrival process under consideration. We then discuss the relevance ofCv2 for burstiness characterization through an example : the HBP2=D=1=K queue.We show that Cv2 becomes signi�cant only when local overload occurs, i.e., when thearrival rate is momentarily greater than the server rate. The results are then applied totwo basic ATM problems : tra�c characterization and bu�er dimensioning using burstyinputs.

Key words : ATM networks, burstiness, GASTA property, martingales, time averages,

event averages.

1 Introduction

The Asynchronous Time Division (ATD) technique [Cou 91] was introduced at the beginningof the 80's to support the future Broadband Integrated Services Digital Network (B-ISDN)since it provides for exible interfaces. The �rst ATM (Asynchronous Transfer Mode) exper-iment was the PRELUDE project [Dev 88] which proved the feasibility of a network basedupon ATM concepts.

ATM networks handle �xed-length packets, termed cells within CCITT, which are trans-mitted through the network on time slotted media. The PRELUDE experiment has requiredinvestigations in queueing theory and has led in particular to the complete resolution of theGeo=D=1=K queue [Gra 90-a] used to dimension the output bu�ers of the switching element :the COPRIN matrix.

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g g gtra�c. Consequently, is bu�er dimensioning obtained for Poisson or Geometric inputs stillvalid ? Many authors [Sri 86] [Hef 86] [Ros 88] [Dav 90] [Des 88] have addressed this crucialquestion, introducing the notion of burstiness. In the current literature, several parametershave been proposed to quantify the burstiness of a cell arrival process : mean burst length,peak bit rate, variability of the cell arrival process, etc. The idea underlying the burstinessconcept is that the impact of an input process on a queue may depend on many parameters.Some processes such as a Poisson or periodic process are completely characterized by oneparameter. But in general, process characterization calls for a larger set of parameters,which may even be in�nite. The choice of one parameter in preference to another in orderto characterize burstiness is very tricky and in the present study, we rely on the followingqualitative de�nition : a tra�c will be said bursty with respect to a given �nite capacityqueue if it induces a higher loss probability than the Poisson or Geometric process o�eringthe same load. Note that a bursty input tra�c may also induce an increase in mean waitingtime.

Considering di�usion approximations, heavy tra�c approximations [New 82] and Allen-Cunneen approximation ([All 78], p. 221), special attention has been given to the squaredcoe�cient of variation of interarrival (Cv2) for GI=G=1=K queues. The main problem ofburstiness characterization is to relate intrinsic parameters of the input ow to the behaviorof a queue in which the server characteristics are fundamental. In particular, time scaleaspects must be taken into account [Sri 86] [Ros 88]. Especially, what is the in uence of thecoe�cient of variation Cv2 for GI=G=1=K queues when heavy tra�c assumptions are notfull�lled ?

In order to observe the impact of a bursty input tra�c on a queue, a discrete-time model,the n-stage Markov Modulated Bernoulli Process, MMBPn, for short, is developed in Sec-tion 2. The MMBPn tra�c may represent, for example, the superposition of irregular cellstreams at the output bu�er in a switching element based upon an output queueing archi-tecture [Gra 89]. Athough cell interarrival times are in general correlated for the MMBPntra�c, they could be independent under some assumptions (degenerated underlying Markovchain). In such a case, the MMBPn will be called n-stage Hyper-Bernoulli process, HBPn,for short.

For a queue with anMMBPn input process, GASTA (Geometric Arrivals See Time Aver-ages, [Hal 83]) property does not hold. It is however possible to establish a relation betweenstationary probability distribution of the queue length, i.e., at any instant (time averages),and that at cell arrival epochs, by using a martingale formulation. This is performed in Sec-tion 3. That relation adapts GASTA equality to this more general process. Although it canbe viewed as a direct consequence of GASTA, the present proof provides nevertheless a mar-tingale approach to GASTA. It is applied to theMMBPn=D=1=K queue which is numericallysolved in Appendix.

The numerical routines are applied in Section 5 to the HBP2=D=1=K queue in order toobserve the impact of the squared coe�cient of variation of interarrival time (Cv2) and of theburstiness parameter B (de�ned as peak rate/mean rate) on this queue. This shows that thein uence of Cv2 on the queue behavior depends signi�cantly on local overload, i.e., when thearrival rate is momentarily greater than the server rate. The notion of local overload allows

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2 The MMBPn model

2.1 Presentation

In all this paper we consider a probability space (;F ;P) and an integer n � 1. A n-stageMarkov Modulated Bernoulli Process (MMBPn) de�ned on (;F ;P) is a Bernoulli processwhose paramater is governed by a n-phase Markov chain. Like the Markov Modulated PoissonProcess [Sri 86], the MMBPn is a phase type process. It may be used to model irregular cellarrivals to a queue, for example at the output bu�ers of a switching element when tra�c onthe input links is bursty.

The MMBPn process is a discrete-time stochastic process. An elementary time intervalis called a slot and its duration is taken as time unit in the sequel. In the MMBPn processarrivals occur as follows :

� the process begins just after an arrival,

� there are n phases, namely 0; � � � ; n�1,

� when the process is in phase i, the inter-arrival time I is geometric with parameter1� pi; pi 2]0; 1] : the conditional distribution of I knowing the phase of the interarrivalis given by :

8k � 1; i = 0; � � � ; n � 1; PfI = k= phase = ig = (1 � pi)k�1pi: (1)

Arrivals may take place only at times k�, i.e., just before the epochs k. No more thanone arrival may occur at k� (single arrivals) and inter-arrival time is at least one slotlong.

� just after an arrival, i.e., k+, a new phase is drawn according to a Markov chain. Let!i;j be the probability to jump from phase i into phase j.

Note that inter-arrival times are in general correlated. Let 'r; 'r 2 f 0; � � � ; n�1g, be thephase between the rth and the (r + 1)st arrivals. The initial phase '0 is given. ' = f'rgr2Nis a Markov chain whose transition matrix is given by :

P = ((!i;j))(i;j)2f0;���;n�1g2 withn�1Xj=0

!i;j = 1: (2)

Let � = (�m)m2f0;���;n�1g be the stationary distribution of the process ' ; � veri�es :

� � P = �: (3)

Now, let us observe the phase � = f�kgk�0 of the MMBPn arrival process at time k+.The initial phase �0 = '0 is given. � = (�k)k�0 is a Markov chain whose transition matrixis given by :

� = f�i;jg with

(�i;j = pi!i;j if i 6= j;�i;i = 1� pi(1 � !i;i):

(4)

3

� �� = �: (5)

� is the phase process slot by slot and ' is the phase process between arrivals. We say that' is the underlying Markov chain of the MMBPn process.

Remark : Similar tra�c models have been introduced by Le Boudec [LeB 91], Hirano andWatanabe [Hir 90] and Hashida, Takahashi and Shimogawa [Has 91]. However, in our model,arrivals trigger o� the change of phase in the underlying Markov chain, whereas it is not thecase in those models.

2.2 Parametrizing an HBP2 process

When the underlying Markov Chain of the MMBPn process is degenerated, i.e., each newphase is drawn independently of the current one, the MMBPn is called n-stage Hyper-Bernoulli process (HBPn). Thus, the n-stage Hyper-Bernoulli process is a renewal process.The phase transition matrix P becomes :

P = ((!i;j))(i;j)2f0;���;n�1g2 with !i;j = !j ;n�1Xj=0

!j = 1; (6)

where !i is the probability of drawing phase i.We consider now the HBP2 renewal process. The distribution of interarrival time I is

given by :PfI = kg = !0(1� p0)

k�1p0 + !1(1� p1)k�1p1: (7)

The mean arrival rate is de�ned by � = limt!1N(t)=t where N(t) is the number of arrivalsprior to t. For a renewal process � is equal to 1=E[I]. This yields for the HBP2 process :

� =p0p1

p1!0 + p0!1: (8)

The HBP2 tra�c is a phase type process and phase 0 can be chosen arbitrarily moreactive than phase 1 (p0 > p1). The peak rate of the process is then de�ned as the largestphase rate, namely p0. In order to quantify the deviation of the peak rate from the mean rate,we introduce the burstiness parameter B = peak rate/mean rate. In the case considered, wehave :

B =p0�: (9)

For the HBP2 process, the squared coe�cient of variation Cv2 can be easily derived fromequation (7) :

Cv2 =V ar[I]

E2[I]= �2

"2!0p20

+2!1p21

�1

��

1

�2

#: (10)

In order to observe the impact of B and Cv2 on the queue characteristics, we parametrizean HBP2 process with the set f�;B;Cv2g, such a ow has only three degrees of freedom. Byusing (8), (9) and (10), we obtain :

p0 = B�; p1 =B�(B � 1)

Q�B; !0 =

Q�B2

Q� 2B + 1; (11)

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3 Relation between probability distribution at cus-

tomer arrival epochs and time averages for theMMBPntra�c

3.1 Main result

We consider a discrete-time system to which customers arrive according to a n-stage MarkovModulated Bernoulli Process - MMBPn - de�ned on (;F ;P). The state of the system maychange only at epochs k+; k 2 N . Let � = (�k)k2N be the stochastic process which takesvalues in a countable measurable space (E; E) and which describes the state of the system atepochs k+; k 2 N .

The objective of this section is to relate the fraction �m(�) of time the system spends insome set � 2 E while the phase of the MMBPn is m to the fraction �m(�) of customerswhich �nd the system in that set. This leads to establish a relation which adapts the GASTAproperty (GASTA : Geometric Arrivals See Times Averages, [Hal 83]) to this more generalprocess. This relation may be established by using GASTA but in the following we adopt amartingale approach.

As for any discrete-time system [Gra 90-b], simultaneity plays a crucial role. Here, eventsoccur according to the following assumptions :

Simultaneity assumptions (S) :

� a customer may arrive at k�,

� a MMBPn's phase change may occur at k+,

� the state of the system may change at k+,

� a customer arriving at k� is counted in the system only at k+.

Thus, the new state of the system is determined at k+. Now, we introduce the notationswhich will be used in the following (� is an element of E)

� �(�) : fraction of customers which �nd the system in �,

� � (�) : fraction of time the process spends in �,

� �m(�) : fraction of customers which �nd the system in � while theMMBPn is in phase m,

� �m(�) : fraction of time the system spends in � while the phase is m,

� Nk : the number of arrivals prior to k+.

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g ( )

8k � 1; Uk(�) = 1f�k�12�g: (12)

For m 2 f0; � � � ; n� 1g, we de�ne Wm = (Wmk )k�1 by :

k � 1;Wmk = 1f�k�1= mg: (13)

Let fAkgk�1 be the sequence of random variables de�ned as follows :

Ak = 1 if a customer arrives at k�; Ak = 0 otherwise.

From the construction of the MMBPn, we have :

k � 1;PfAk = 0=�k�1 = ig = 1� pi; PfAk = 1=�k�1 = ig = pi: (14)

Moreover, Nk =Pkl=1Al.

As in [Wol 82] [Hal 83], we introduce the processes V (�), V m(�), Y (�), Y m(�) , Z(�) andZm(�) de�ned by :

k � 1; Vk(�) =1

k

kXi=1

Ui(�); Yk(�) =kXi=1

Ui(�)Ai; Zk(�) =Yk(�)

Nk

;

V mk (�) =

1

k

kXi=1

Ui(�)Wmi ; Y

mk (�) =

kXi=1

Ui(�)AiWmi ; Z

mk (�) =

Y mk (�)

Nk

: (15)

Vk(�) (resp. V mk (�)) is the proportion of the k �rst slots during which the system is in �

(resp. in � while the phase of the MMBPn is m). With simultaneity assumptions (S),Zk(�) (resp. Zm

k (�)) is the fraction of arriving customers during the k �rst slots who �nd thesystem in � (resp. in � while the phase is m).

We have obviously : Zk(�) =Pn�1m=0 Z

mk (�). In order to deal with systems of pratical

interest, we make the following ergodicity assumptions :

Assumptions (E) :

� the MMBPn is ergodic, that is : 8m 2 f0; � � � ; n � 1g; limk!11k

Pki=1W

mk = �m > 0

a.s.,

� 8m 2 f0; � � � ; n� 1g; limk!1 V mk (�) = �m(�) > 0 a.s. and limk!1 Zm

k (�) = �m(�) > 0a.s.

Under assumptions (E), we have : � (�) � limk!1 Vk(�) =Pn�1m=0 �

m(�) a.s. and �(�) �limk!1 Zk(�) =

Pn�1m=0 �m(�) a.s.

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Theorem 1 With the above notations and under assumptions (E) and (S), we have :

8m 2 f0; � � � ; n� 1g;8� 2 E; �m(�) =pm�

m(�)

�: (16)

It follows that :

8� 2 E; �(�) =n�1Xm=0

�m(�) =1

�

n�1Xm=0

pm�m(�); (17)

where � =Pn�1m=0 pm�m and � veri�es (5).

Remark : If n = 1, the MMBPn reduces to a Bernoulli process and Theorem 1 is thenGASTA. Equations (16) and (17) can be viewed as direct consequences of Papangelou's for-mula revisited in [Bre 91] which holds for the case considered in the present paper underassumptions (E) and (S). However, we present hereafter a direct proof of Theorem 1 byusing a martingale formulation and which gives in particular a sample-path proof of GASTA.Moreover, equation (16) adapts the so-called Conditional PASTA [Koe 90] to the discrete-timecase (PASTA : Poisson Arrivals See Times Averages).

Proof of Theorem 1 : We introduce the increasing family I of sub-�-�elds (�ltration,[Bre 91]) as follows :

I0 = �(�0;�0) ; k � 1;Ik = �(�`;�`; ` � k) _ I0;

I = (Ik)k2N : (18)

Note that �(X1; � � � ;Xn) denotes the �-�eld generated by the random variables X1; � � � ;Xn.Let ~N = ( ~Nk)k�1 be the process de�ned by :

k � 1; ~Nk =kXi=1

n�1Xm=0

pmWmi : (19)

N � ~N is an I-martingale (i.e., 8k � 1; E[Nk � ~Nk=Ik�1] = Nk�1 � ~Nk�1), ~N is called thecompensator of N relatively to the �ltration I [Bre 91]. Moreover, it is easy to check that(N � ~N )2 � ~N is also an I-martingale.

From assumptions (E), ~N1 = 1 a.s., it follows from the Strong Law of Large Numbersfor martingales (SLLN - [Nev 72], proposition VII.2.4, p.150) that : limk!1

Nk~Nk

= 1 a.s. From

assumptions (E), we have then :

� � limk!1

Nk

k= lim

k!1

~Nk

k=

n�1Xm=0

pm�m a.s. (20)

Usually, � is called the mean arrival rate of the MMBPn. Let m 2 f0; � � � ; n � 1g;� 2E and ~Y m(�) be the process de�ned by :

8k � 1; ~Y mk (�) = kpmV

mk (�):

7

1 p ( ), k!1 Ymk(�)

and therefore, by using (20) :

�m(�) � limk!1

Zmk (�) = lim

k!1

~Y mk (�)

k

k~Nk

=pm�

m(�)

�a.s.

3.2 Application to the MMBPn=D=1=K queue

In this sub-section we consider a single server queue to which customers arrive according toa MMBPn process. Each customer requires a constant service time equal to D slots. Thequeue capacity is �nite and denoted by K ; the queue discipline is FIFO and the queue loadis � = �D where � is given by (8). As in Section 3 , assumption (S), we adopt the followingsimultaneity principle : arrivals see departures. More precisely, a customer arriving at k� iscounted in the queue only at k+ if space is available in the queue ; a customer �nding Kcustomers in the queue at k� is rejected even if a service end occurs at k+. Departures canonly occur at instants k+.

Since events can only occur at instants k; k � 0, we observe the system at each instantk. Thus, let Qk denote the queue length (including the customer being processed) and �kbe the phase of the MMBPn process at time k+. We introduce a supplementary variable inorder to obtain a Markov chain as follows. Let dk be the position of the server in the currentservice. More precisely, dk = d; d 2 f0; � � � ;D� 1g, if the service of the cell being processed isalready d slots long at k+. If dk = D� 1, the cell ends its service at (k+1)+ and is no longercounted in the queue at that time. If dk = 0 and Qk > 0, a cell begins its service at k+. IfQk = 0 then dk = 0, the server is idle.

(Qk;�k; dk)k2N is a Markov chain taking values in E = f(`;m; d) ; ` = 0; � � � ;K ;m =0; � � � ; n�1 ; d = 0; � � � ;D�1g and whose transition matrix is denoted byM . The initial state isgiven by : (Q0;�0; d0) = (0; '0; 0). The analysis of this Markov chain and the characteristicsof the MMBPn=D=1=K queue are given in Appendix. Let E be the �-�eld generated bythe sets f(`;m; d)g for (`;m; d) 2 E, T = (� (`;m; d)) be the stationary distribution and� = (�(`;m; d)) be the distribution of the state of the system seen by an arriving customer.As a direct consequence of Theorem 1, � and T are related as follows :

` = 0; � � � ;K ;m = 0; � � � ; n� 1 ; d = 0; � � � ;D � 1 ; �(`;m; d) =pm� (`;m; d)

�: (21)

The rejection probability Ploss in the MMBPn=D=1=K queue is given in Appendix by equa-tion (24) and the mean waiting time W by equation (25).

4 Results

In this section, we qualitatively study the impacts of the burstiness parameter B and thesquared coe�cient of variation Cv2 (de�ned by (9) and (10)) through a simple example,namely the HBP2=D=1=K queue. The HBP2 model may represent cell arrivals at an outputqueue in a switch. Note however it has never be claimed that this model corresponds to

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queue and then, to capture \physical phenomena".We now introduce the parameters which are used to explain the behavior of the

HBP2=D=1=K queue. The distribution of the number of consecutive, identical phases isgeometric with parameter !i. The mean duration ESi of a sojourn in phase i and the meannumber ni of cells generated during a sojourn of the process in phase i are given by :

i = 0; 1; ni =1

1 � !i; ESi =

1

pi(1� !i): (22)

The system HBP2=D=1=K is parametrized by using the set (�;B;Cv2;D). The parame-ters pi and !i; i = 0; 1, are given by equations (11) and � = �D. The queue in which the inputprocess is a Bernoulli (or Geometric) process with intensity p, service times are deterministicand the queue capacity is K, is denoted by Geo(p)=D=1=K. The loss probability in such a

queue will be denoted by PGeo(p)loss and is given in [Gra 90-a].

Impact of the squared coe�cient of variation Cv2.

Figure 1 shows Ploss with respect to the server load for B = 5, a queue capacity K = 32cells, a service time D = 8 slots and several values of Cv2. The loss probability for theGeometric tra�c o�ering the same load is plotted as a dotted line and the loss probability inthe Geo(p0)=D=1=K queue is plotted as a dashed line.

It is clear from Figure 1 that the loss probability in the HBP2=D=1=K queue is greaterthan the loss probability for the Geometric process o�ering the same load and therefore, thetra�c under consideration is bursty from the de�nition given in Introduction. Moreover, thelarger the Cv2, the larger Ploss for a given o�ered load. Consequently, the larger the Cv2, themore bursty the HBP2 tra�c.

From Figure 1, we can distinguish two domains in the impact of Cv2 on the queue behavior.The mean number of cells generated during a sojourn in state 0 is :

n0 =1

1� !0=

B2

2 (Cv2 + �+ 1):� 2B + 1

(B � 1)2

Therefore, for given values of B and �, as Cv2 increases, n0 increases too, so that phase 0can be viewed as an active phase for the arrival process : cells are generated at rate p0 > �,giving rise to cell clusters called bursts in the literature. Correlatively, the mean number ofcells generated during a sojourn in state 1 is decreasing to 1, so that phase 1 can be viewedas an idle phase for the arrival process. Since n0 ! 1 as Cv2 ! 1, this implies that thelimiting behavior of the HBP2=D=1=K queue when Cv2 !1 is that of the Geo(p0)=D=1=Kqueue, as shown in Figure 1.

Let �? be the critical load de�ned by �? = 1=B.

� when � > �?, i.e., p0 > 1=D, the queue is overloaded during bursts. We say that it islocally overloaded. Since the limiting behavior of theHBP2=D=1=K queue as Cv2 !1

is that of the Geo(p0)=D=1=K queue which is overloaded, Ploss ! PGeo(p0)loss � 1,

9

( )= = =

PGeo(p0)loss � 1.

Figure 1: Loss probability vs. server load for di�erent values of Cv2. Logarithmic scale.Burstiness B = 5 ; queue capacity K = 32 cells ; service duration D = 8 slots.

Figure 3 shows the mean waiting timeW with respect to the server load � for given valuesof B, K and D, and several values of Cv2. The same phenomenon as for Ploss, i.e., theexistence of a threshold value in the impact of Cv2, can be observed for W . A more burstytra�c which induces by our de�nition a higher loss probability than the Geometric processo�ering the same load is also responsible for a higher mean waiting time.

The notion of local overload allows time scale aspects to be taken into account : theimpact of a bursty tra�c depends signi�cantly on the server characteristics. Rossiter in[Ros 88] provides another approach to the same problem. Moreover, the mean number ofcells generated in overload situation (i.e., n0 for � > �?) seems to be a relevant parameter forthe model considered in this paper.

The impact of Cv2 on the queue behavior depends signi�cantly on the domains f� < �?gand f� > �?g. This appears more clearly in Figure 2 where Ploss is depicted in linear scale.It follows that Cv2 cannot by itself characterize the impact of a bursty tra�c on a queue. Inthe case considered, B must also be taken into account.

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Figure 2: Loss probability vs. server load for di�erent values of Cv2. Linear scale. BurstinessB = 5 ; queue capacity K = 32 cells ; service duration D = 8 slots.

Figure 3: Mean waiting time vs. server load for di�erent values of Cv2. Queue capacityK = 32 cells, burstiness B = 5, service duration D = 8 slots.

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Figures 4 and 5 show the impact of the burstiness parameter B respectively on the lossprobability Ploss and on the mean waiting timeW in a HBP2=D=1=K queue for several valuesof the server load � ; the queue capacity K = 32 cells, the service time duration is D = 8slots and Cv2 = 20. Considering that p0 � 1, B = p0=� must satisfy B � D=�.

Figure 4: Loss probability vs. burstiness parameter B for di�erent values of the server load.Queue capacity K = 32 cells, service duration D = 8 slots, Cv2 = 20.

We can distinguish a threshold in the impact of B : B? = 1=�. Since p0 = B:�=D, thisthreshold can be explained by using the same arguments as for the impact of Cv2 :

� if B < B? then p0 < 1=D, the queue is not overloaded during bursts : there is no localoverload ; Ploss and W are increasing but remain very low,

� if B � B?, the queue is overloaded when the HBP2 process is in phase 0 (local over-load). In the neighbourhood of B?, the loss probability Ploss and the mean waitingtime W increase rapidly for � large enough because the queue becomes locally over-loaded. For su�ciently large values of B, the mean numbers de�ned above verify :n0 � (Cv2+�+1)=2 and n1 � (Cv2+�+1)=(Cv2+��1). Moreover, n0 � n1 for highvalues of Cv2. Consequently, most of cells are generated in phase 0 at a rate greaterthan the server rate.

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Figure 5: Mean waiting time vs. burstiness parameter B for di�erent values of the serverload. Queue capacity K = 32 cells, service duration D = 8 slots, Cv2 = 20.

B and Cv2 in mutual exclusion.

Figure 6 displays the maximum value of Cv2 which meets a given loss probability Ploss fora given value of the burstiness parameter B ; the queue capacity is K = 64 cells, the o�eredload � = 0:8 and the service time D = 8 slots.

It is clear that B and Cv2 are in mutual exclusion to achieve a given Ploss, i.e., if B islarge, Cv2 must be drastically limited and conversely, if Cv2 is large, B must be kept verylow. Therefore, from a practical point of view, if the queue is dimensioned to absorb a bursty ow, for example HBP2, the characteristics of the actual ow must �t exactly those of themodel because the queue is very sensitive to the time structure of the input process. Thismay lead to unreliable queue dimensioning.

Queue dimensioning with a bursty input.

Figure 7 shows the queue capacity expressed in cells required to achieve a given lossprobability Ploss = 10�9 with respect to the squared coe�cient of variation Cv2 ; burstinessparameter B = 3 and service time D = 8 slots. We recall that the critical load �? is equal to1=B.

� if � = 0:25; � < �?, the queue is not overloaded during bursts. The queue ca-pacity required increases as Cv2 increases because bursts become more and morelonger, but tends to a limiting value which can be determined by dimensioning the

13

Figure 6: Admissible value of the squared coe�cient of variation vs. burstiness to achieve agiven loss probability. O�ered load � = 0:8, queue capacity K = 64 cells, service time D = 8slots, loss probability Ploss = 10�9.

14

( )= = =

� if � > 0:5; � > �?, the queue is overloaded during bursts. The queue capacity requiredincreases rapidly as Cv2 increases. No realistic queue capacity can be found as Cv2 !1since the Geo(p0)=D=1=K is overloaded.

Figure 7: Queue capacity vs. squared coe�cient of variation to achieve a given loss probability.Loss probability Ploss = 10�9, burstiness parameter B = 3, service time D = 8 slots.

Figure 8 displays the queue capacity required to achieve a given loss probability Ploss =10�9 with respect to the burstiness parameter B, the o�ered load is � = 0:25, squared coe�-cient of variation Cv2 = 20 and service time D = 8 slots. We recall that the critical value ofB is B? = 1=� beyond which local overload occurs. It is clear from Figure 8 that the queuecapacity required increases rapidly when B is in the neighbourhood of B? or B is greater thanB?.

Figures 7 and 8 show clearly that small variations in the input model lead to large devi-ations for the queue capacity required to achieve a very low loss probability, say, 10�9, whenlocal overload occurs. This must be taken into account when queue dimensioning is performedwith bursty inputs.

15

Figure 8: Queue capacity vs. burstiness parameter to achieve a given loss probability. Lossprobability Ploss = 10�9, service time D = 8 slots, o�ered load � = 0:25, squared coe�cientof variation Cv2 = 20.

16

In this paper we have developed a bursty tra�c model, the MMBPn arrival process and itspeculiar case, the HBP2 process. As GASTA property does not hold for the model consid-ered, we have established a new relation between stationary queue length distribution andarriving customer's distribution, whose proof is based upon a martingale approach. TheMMBPn=D=1=K queue has been studied (loss probability and mean waiting time) in Ap-pendix. Then, the HBP2=D=1=K queue has been used to observe the impact of an inputprocess on a queue : the notion of local overload is introduced, which allows to distinguishdi�erent domains in the impact of tra�c characteristics.

From this study it is clear that the behavior of a queue depends signi�cantly on the timestructure of the input process and not only on the o�ered load. The squared coe�cient ofvariation of interarrival Cv2 and the burstiness parameter B have both a signi�cant in uenceon the queue behavior when local overload occurs. The notion of local overload allows us tocapture time scale aspects related to the notion of burstiness. The model under considerationshows the existence of a threshold �? for the load �, related to the burstiness parameter andbeyond which local overload occurs. The notion of overload states is used in [Nor 91] to study uid queue models. In [Nor 91] and in the present paper, the conclusion is clear : the impactof a bursty input tra�c on a queue depends on whether local overload occurs or not.

For the purpose of tra�c characterization, it seems to be more suitable to consider thetime structure of the input process relatively to the server characteristics (local overload) :the more cells generated in overload situation, the worse the queue performance in termsof loss probability and mean waiting time. In the model considered, Cv2 and B must besimultaneously considered in order to describe the impact of the input process on the queue.One parameter cannot by itself completely describe the queue behavior.

Curves show that small variations in the input process characteristics induce large devi-ations for the loss probability and the mean waiting time in a queue. Also, they show thatfor practical purposes, an usage of any multiplexer with � > �? must be avoided in order toful�l cell loss rate requirements. This implies a rather ine�cient link utilization. Moreover,from an engineering point of view, bu�er dimensioning using bursty tra�c models is stronglyquestionable : the tra�c characteristics of the actual stream must �t exactly those of themodel used for bu�er dimensioning.

For a better network utilization and a more reliable bu�er dimensioning, the results of thepresent paper show that the burstiness parameter B must be drastically limited. This canbe achieved if network input stages reshape and smooth the entering tra�c, for instance byinsuring a minimal spacing between consecutive cells as described in [Boy 90].

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Resolution of the MMBPN=D=1=k queue

With the notations of Section 3.2, the stationary distribution T satis�es the equilibriumequation :

T �M = T: (23)

This leads to the following Chapman-Kolmogorov equations given (in general) by :

� ` = 0.pm� (0;m; 0) = (1� pm)� (1;m;D � 1):

� ` = 1.

d = 0; � (1;m; 0) = (1� pm)� (2;m;D � 1) +Xm0

pm0!m0;m� (1;m0;D � 1)

+Xm0

pm0!m0;m� (0;m0; 0);

d � 1; � (1;m; d) = (1� pm)� (1;m; d� 1):

� 2 � ` � K � 2.

d = 0; � (`;m; 0) = (1� pm)� (`+ 1;m;D � 1) +Xm0

pm0!m0;m� (`;m0;D � 1);

d � 1; � (`;m; d) = (1� pm)� (`;m; d� 1) +Xm0

pm0!m0;m� (`� 1;m0; d� 1):

� ` = K � 1

d = 0; � (K � 1;m; 0) = (1� pm)� (K;m;D � 1) +Xm0

pm0!m0;m� (K � 1;m0;D � 1)

+Xm0

pm0!m0;m� (K;m0;D � 1);

d � 1; � (K � 1;m; d) = (1� pm)� (K � 1;m; d� 1)

+Xm0

pm0!m0;m� (K � 2;m0; d� 1):

� ` = K.

d = 0; � (K;m; 0) = 0;

d � 1; � (K;m; d) = (1 � pm)� (K;m; d� 1) +Xm0

pm0!m0;m� (K � 1;m0; d� 1)

+Xm0

pm0!m0;m� (K;m0; d� 1):

These equations are solved by usual numerical methods in order to compute the stationarydistribution T .

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From simultaneity assumptions, the rejection probability Ploss is the probability that an ar-riving customer �nds K customers in the queue and therefore, by using equation (21), wehave :

Ploss = �K =1

�

n�1Xm=0

D�1Xd=0

pm� (K;m; d): (24)

The mean queue length EN is given by :

EN =KX`=0

n�1Xm=0

D�1Xd=0

`� (`;m; d): (25)

From Little's formula, the mean waiting time W is related to Ploss and EN as follows :

W =EN

�(1 � Ploss)�D: (26)

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