5
3G Wireless Core Network Dimensioning in the presence of Self-Similar Traffics Giovanni Giambene, Claudio Tommasi Dipartimento di Ingegneria dell'Informazione - Universita degli Studi di Siena, Via Roma, 56 - 53100 Siena - Italy Telephone: +39 0577 234603, Fax: +39 0577 233602, Email: [email protected] Abstract-Future AIB-IP 4G wireless networks will be based on the IP protocol and will be characterized by a 3G core network architecture. AIB-IP 4G systems will support multimedia traffics with heterogeneous characteristics. It is anticipated that the integrated traffics at the core network level will exhibit long- range dependent behaviors and self-similar characteristics, as those in the current Internet Backbone. This paper deals with an analytical approach for dimensioning networks with such traffics, modeled as Fractional Brownian Traffic. By imposing a total network cost, we derive the optimum capacity ailocation minimizing an end-to-end delay metric. The results of our dimensioning technique (supported by simulations) highlight significant improvements with respect to the classical capacity alocation problem for Jackson networks. I. INTRODUCTION Future All-IP fourth-generation (4G) wireless networks will be characterized by a 3G core network architecture. Since, multimedia services will be provided to mobile users, the 3G core network will have to support aggregated traffics, exhibiting Long-Range Dependent (LRD) and self-similar characteristics [1], as it occurs in the Internet [2]. A self- similar traffic trace shows structural likeness for a wide range of time aggregations. Moreover, an LRD arrival process has non-summable auto-correlation function, thus causing high delays in the networks. At present, the problem that operators have to face with is the proper dimensioning of core networks on the basis of traffic estimates and traffic models. In particular, it is important that the capacity of the links interconnecting Gateway GPRS Support Node (GGSN) and Serving GPRS Support Node (SGSN) be properly determined in the network. Oversizing capacity leads to good Quality of Service (QoS) levels, but entails high costs. Therefore, a good trade-off solution has to be identified. This paper proposes a dimensioning approach where link capacities are determined to minimize a suitable end-to-end delay metric according to traffic estimates for the different nodes. The problem of allocating capacity to the links of a network is traditionally carried out on the basis of the Jackson theorem [3] under the assumption of Poisson traffics and exponentially distributed service times. This is not the case of IP-based networks [5]. Hence, we adopt the Fractional Brownian Traffic (FBT) model that exhibits self-similar and LRD characteristics [6]. An FBT trace is synthetically described by mean arrival rate, m, variance coefficient, a, and Hurst parameter H E / t (a.,, H,,)) : / ~~Network (%a.~,H.) Fig. 1. The adopted model for the network of queues. q _ rotAod + rfcmuted frf FE f-- ----- z ------- - I. eiernai uamc Exerrily routed traffic Fig. 2. Traffics in a single node of the network. [0.5, 1), that is the triplet (m, a, H). The theoretical study is referred to a general network model that consists of an acyclic network with N nodes (each modeled by a queue) with interconnecting links with a capacity (i.e., packet transmission rate) to be determined. Each node has several input FBT traffics from outside Fig. 1. A packet transmitted by the i-th node is routed to the j- th node with probability qij (routing matrix) or it leaves the network with probability 1- Fij=l qij. If qij = 0, there is no link between the i-th node and the j-th node. In our simplified model [3], all output links from the i-th node have a capacity of Ci pkt/s. The model for the generic i-th node is shown in Fig. 2, where the input external traffic is characterized by parameters (mie), ate), H (e)), the input internal traffic has (mi1), ai ),H(')) and the total (i.e, internal plus external) traffic has the following triplet (mat), at), Hi(t)). 0-7803-9206-X/05/$20.00 ©2005 IEEE 223 (..,a.,H.), (m.,a,.,H.)

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Page 1: [IEEE 2005 2nd International Symposium on Wireless Communication Systems - Siena, Italy (05-09 Sept. 2005)] 2005 2nd International Symposium on Wireless Communication Systems - 3G

3G Wireless Core Network Dimensioning in thepresence of Self-Similar Traffics

Giovanni Giambene, Claudio TommasiDipartimento di Ingegneria dell'Informazione - Universita degli Studi di Siena,

Via Roma, 56 - 53100 Siena - ItalyTelephone: +39 0577 234603, Fax: +39 0577 233602, Email: [email protected]

Abstract-Future AIB-IP 4G wireless networks will be basedon the IP protocol and will be characterized by a 3G corenetwork architecture. AIB-IP 4G systems will support multimediatraffics with heterogeneous characteristics. It is anticipated thatthe integrated traffics at the core network level will exhibit long-range dependent behaviors and self-similar characteristics, asthose in the current Internet Backbone. This paper deals withan analytical approach for dimensioning networks with suchtraffics, modeled as Fractional Brownian Traffic. By imposing atotal network cost, we derive the optimum capacity ailocationminimizing an end-to-end delay metric. The results of ourdimensioning technique (supported by simulations) highlightsignificant improvements with respect to the classical capacityalocation problem for Jackson networks.

I. INTRODUCTION

Future All-IP fourth-generation (4G) wireless networks willbe characterized by a 3G core network architecture. Since,multimedia services will be provided to mobile users, the3G core network will have to support aggregated traffics,exhibiting Long-Range Dependent (LRD) and self-similarcharacteristics [1], as it occurs in the Internet [2]. A self-similar traffic trace shows structural likeness for a wide rangeof time aggregations. Moreover, an LRD arrival process hasnon-summable auto-correlation function, thus causing highdelays in the networks.At present, the problem that operators have to face with

is the proper dimensioning of core networks on the basis oftraffic estimates and traffic models. In particular, it is importantthat the capacity of the links interconnecting Gateway GPRSSupport Node (GGSN) and Serving GPRS Support Node(SGSN) be properly determined in the network. Oversizingcapacity leads to good Quality of Service (QoS) levels, butentails high costs. Therefore, a good trade-off solution has tobe identified. This paper proposes a dimensioning approachwhere link capacities are determined to minimize a suitableend-to-end delay metric according to traffic estimates for thedifferent nodes.The problem of allocating capacity to the links of a network

is traditionally carried out on the basis of the Jackson theorem[3] under the assumption of Poisson traffics and exponentiallydistributed service times. This is not the case of IP-basednetworks [5]. Hence, we adopt the Fractional Brownian Traffic(FBT) model that exhibits self-similar and LRD characteristics[6]. An FBT trace is synthetically described by mean arrivalrate, m, variance coefficient, a, and Hurst parameter H E

/ t (a.,, H,,))

:/ ~~Network(%a.~,H.)

Fig. 1. The adopted model for the network of queues.

q

_rotAod

+ rfcmutedfrfFE

f-- ----- z ------- -

I.

eiernai uamcExerrily routed

traffic

Fig. 2. Traffics in a single node of the network.

[0.5, 1), that is the triplet (m, a, H). The theoretical studyis referred to a general network model that consists of anacyclic network with N nodes (each modeled by a queue) withinterconnecting links with a capacity (i.e., packet transmissionrate) to be determined. Each node has several input FBTtraffics from outside Fig. 1.A packet transmitted by the i-th node is routed to the j-

th node with probability qij (routing matrix) or it leaves thenetwork with probability 1- Fij=l qij. If qij = 0, there is nolink between the i-th node and the j-th node. In our simplifiedmodel [3], all output links from the i-th node have a capacityof Ci pkt/s. The model for the generic i-th node is shownin Fig. 2, where the input external traffic is characterizedby parameters (mie), ate),H (e)), the input internal traffic has(mi1), ai ),H(')) and the total (i.e, internal plus external)traffic has the following triplet (mat),at), Hi(t)).

0-7803-9206-X/05/$20.00 ©2005 IEEE 223

(..,a.,H.),

(m.,a,.,H.)

Page 2: [IEEE 2005 2nd International Symposium on Wireless Communication Systems - Siena, Italy (05-09 Sept. 2005)] 2005 2nd International Symposium on Wireless Communication Systems - 3G

In [5], the authors analyze a queuing network where externalself-similar traffics have the same Hurst parameter. In thispaper we deal with external traffics with different Hurstparameter values, we impose a total network cost and wetheoretically determine the link capacities that minimize anestimate of the mean end-to-end transmission delay.

II. SELF-SIMILAR TRAFFIC GENERATION

An FBT process represents the number of packets, At,arriving at an hypothetical queue from instant 0 to t E R+ s[6]:

At = mt +v/Z ()tu/

(1)

where tu, is a normalizing constant assumed equal to 1 s,m > 0 is the mean packet arrival rate (pkts/s), a > 0 isthe variance coefficient (pktsxs), Z(t/t,) is the FractionalBrownian Motion (FBM) with Hurst parameter H.

We can easily verify that Z(t) is strictly self-similar (ss)with Hurst parameter H, but it is not LRD, because it is notstationary [7]. Whereas, Z(t+h)-Z(t) is stationary, V h E R+.Let Xn denote the number of arrived packets due to the FBTprocess in a time interval (nA, (n + 1)A] (i.e., incrementalprocess):

X M=m? +VaAHXn (2)where n E Io, mA is the mean number of packets in A, ais the variance coefficient, X$ = Z((n + 1)A) - Z(nA) isthe Fractional Gaussian Noise (FGN) with Hurst parameter H.

The FGN X,n is stationary, ergodic and Gaussian with X0Z(1), E(Xn) = 0 and E(X1 ) 1 [7]. We can easily verifythat X' is both exactly second order - self-similar (es-ss) andLRD. Unfortunately, Xn defined in (2) is a real-valued processthat may have negative values. However, the number of packetsarrived in A needs to be integer and non-negative to performsimulations. Hence, we generate Xn with parameters (m, a,H) and we 'adjust' such Xn values as follows:

Xn = max(0, [Xn- 0.51) (3)

where [.] denotes the ceiling function.

Xn is characterized by a different triplet (m?, a, H) withrespect to (m, a, H) of Xn. Several algorithms are available togenerate Xn traces; we adopt the Paxson algorithm [8], sinceit allows to deal with long sequences with significantly lowcomputational loads.

III. SOLUTION OF A NETWORK OF QUEUESIn this Section we study a GID/I queue (G = general arrival

process; D = deterministic transmission time; '1' = one server)with infinite rooms, input FBT traffic and packet service rateC. Moreover, we characterize the traffics at the different nodesof a network of queues (Fig. 1).

A. FBT/D/J QueueWe refer to a transmission queue of the G/DI/ type with

input FBT traffic. The complementary distribution of thenumber Q of packets in the queue can be approximated as[9]:

Pr.(Q > x) exp (-12X2(1-H)) (4)

1 (C m) K(H) = HH(l - H)1Hwhere 6 = HKH

On the basis of distribution (4), we derive an estimate ofthe mean number of packets in the queue:

[pkts] (5)

where r(.) is the Gamma function.

Correspondingly, an estimate of the mean queuing delay, d,can be derived from the Little theorem as [4]: d =E(Q)/m.

In order to validate the approximation in (5), Fig. 3 com-pares measures and estimates of E(Q) for an FBTID/l queuewhere the input traffic has (m. a, H) = (41.78, 9.42, 0.77), forincreasing C values. These results highlight that (5) yields agood approximation for intermediate traffic values from 0.4to 0.7 Erlang. Note that link traffic loads close to 0.7 Erlang,could be an acceptable design criterion for network planning.We have also verified that the estimate of E(Q) given by (5)is quite good in the presence of almost Poissonian FBT inputtraffics (i.e., H 1 0.5 and a 1).

D 01 02 03 04 05 06 07 08 09

p [loa

Fig. 3. Simulation (dotted line) and analysis (continuous line) for E(Q)against the load p for an FBTIDI/ queue.

B. Merging of FBT Input TrafficsWe consider a packet arrival process to the G/DIl queue

that is the sum of two independent FBTs:

At = m1t + (a1mM(t), Bt = m2t + (/aiimi2N(t) (6)

where M(t) and N(t) are FBMs with Hurst parameters H1and H2, respectively.

224

-o- Ssmsoulan .DArnlysis

e

XI,,

A

11

'A'~ ~ ~ s

/;/l

2 1/(2-2II) 2H-31-1E(Q) 1-- T2 r

2H 2

8

6

&2

-2

4~

Page 3: [IEEE 2005 2nd International Symposium on Wireless Communication Systems - Siena, Italy (05-09 Sept. 2005)] 2005 2nd International Symposium on Wireless Communication Systems - 3G

Applying definition (2) to both processes in (6), we obtain thenumber of packets generated by both processes in the n-th slotof duration A:

=1+ aimiAHIXn Y mA+ ra2M2AH2YnXn = mIlA+ t H Xn, Yn = Tn2A+ H2n

(7)where Xn and Yn are FGN with Hurst parameters H1 andH2.

For the merged traffic Zn = Xn + Yn we use the followingtheorem [5].

Theorem 1: Xn and Yn are two independent es-ss (as-ssl)processes with Hurst parameters H1 and H2, respectively.We consider the process Zn = Xn + Yn-. If H1 = H2, thenZn is es-ss (as-ss) with Hurst parameter H = H1 = H2.Whereas if H1 5$ H2, Zn is as-ss (as-ss) with Hurst parameterH = max{HI, H2}.

If H1 :& H2, from the previous theorem, we know that Zn isnot es-ss, but as-ss with Hurst parameter H = max {H1, H2}.However, we propose to approximate Z,, (for network dimen-sioning purposes so as to use (5) for evaluating the mean delay)with an FBT process Zn with the same mean and variance ofZnm

Zn = 'm/\A Zn (8)

through the queue (this is a worst-case assumption, sincehigher traffic variance entails higher queuing delays):

ao -~- a(t) (10)

If the load is small, the input process passes trough the queuingsystem practically unchanged with the same (m, a, H). If theload is very high, the output process tends to be deterministic(i.e., a sequence of packets uniformly spaced). In this casethe Hurst parameter tends to 1. We assume that for typicalload values of network planning, the H value does not changethrough the queue:

Ho ~ H, . (11)

Each output packet from the queue of the i-th node canbe assigned to one of the N + 1 output links according to arandom splitting process (statistical routing used here as amodel of datagram routing in IP networks), with probabilitiesqij. Therefore, the output process X,, is divided amongN + 1 sub-processes Xn1), ..X(,N+l), according to thepacket-by-packet splitting operation, that is characterized bythe following theorem [5]:

Theorem 2: If Xn is es-ss with Hurst parameter H, thenn1 . ..., Xn 1) are as-ss with Hurst parameter H.

where m = m1 + m2, a = (am1l + a2M2)/(ml + M2) andH = max {Hi,H2}.

The adoption of the max {Hl, H2} value for the mergedprocess is correct only if long packet traffic traces are con-sidered. The extension of the above considerations to the caseof n merged traffics is straightforward. In sub-Section D, wecompare simulation results of the queue occupancy distributionfor an as-ss process and analytical results of the corresponding(es-ss) FBT process showing that theory yields a conservativeestimate suitable for network planning purposes.

Also for the traffic resulting from splitting we make the FBTapproximation in order to apply formula (5). Simulation resultsin sub-Section D support such assumption.We can characterize X(J by means of the triplet

(imj, aj, Hj) derived from the triplet (mi, a0, HO) of X,:

mj = qijmO

aj =qij 2aomo + qijmO(1 - qij)

(12)

(13)

(14)Hj = Ho.

C. Output Process

Let us assume that an FBT traffic with characteristics(m(t), a(t), Ht)) is at the input of the i-th node modeled as anFBT/DI/ queue. Let (ino, ao, H0) denote the parameters ofthe output process to be split (at most) among N + 1 links. Ifwe consider a work-conserving FBTID/l queue with infinitecapacity, the following equality holds:

mO = m(t) (9)

Moreover, the variance of the output traffic cannot exceedthe variance of the input traffic. In fact, when the FBTID/lqueue is congested, input traffic fluctuations are filtered out bythe passage through the queue. Whereas, in typical operatingconditions the traffic variance does not significantly change

'Asymptotically second order - self-similar

D. Resolution Algorithm for Network Traffic

On the basis of the decomposition ofnetwork ofqueues withself-similar traffics shown in [5], our network can be studied asN independent FBT/DI/ queuing systems. Let m = { mik },a = { aik }, H = { Hik }, i = I ... N, k = I ... t, denote thematrices for the mean arrival rate, the variance coefficient andthe Hurst parameter of t external FBTs for all the N nodesof the network (Fig. 1). We indicate with m(e) = { m e) },a(e) { a(e) } and H(e) - { I} the vectors characterizingthe merged external traffics for each node. We denote withm(= { rn }, a(')={ a(') } and H(i) = { H(i) } the vectorscharacterizing the merged internal traffics at each node. Weindicate with m(t) { m }, a(t) = { a(t) }, H(t) { Hi(t)the vectors characterizing the total (i.e., internal plus external)traffics for all the nodes. On the basis of these notations wecan characterize network traffics as follows.

225

Page 4: [IEEE 2005 2nd International Symposium on Wireless Communication Systems - Siena, Italy (05-09 Sept. 2005)] 2005 2nd International Symposium on Wireless Communication Systems - 3G

External traffics:

mie) Zmjk i=1,..,Nk=1

(e) = tk=1 for m(e) om(e)

miH(e) max{Hik}.

k

Internal traffics:N

m(i) = Em7t)qjj i=l,..,IVj=1N

_ (t) (t) 2 + qjM(t) qj,)

i (i)

miHi = max{H(t)}

jESeo i

where Si is the set of nodes linked to the i-th node.

Ir-~total trafiss120

8E -

6040[~0 2 44~1 atot2J traffic$s 34

(15)

,0

6

2(16)

(17)

(18)

(19)

(20)

Total traffics:

m m= (e) +m(i) (21)

a(t) = a(e)m(e) +a m')(a

i i

(t)(2mi

H(t = max IH (e), Ht() (23)In Fig. 4 we show a 4-node network where we apply the

above network traffic resolution algorithm to characterize thetotal traffics at each node. The results in Fig. 5 for the networkin Fig. 4 highlight a good agreement between the valuesof (m(t), a(t),Ht))measured from simulations and thoseestimated according to previous formulas.

(24.96.15.58.0.92) r 250

(29.11.2.0.84) 1005 200

~~~~~~4(34.9,0.9L0.49) \ 4

4"5

(20.9.17.4,0.86) 3

Fig. 4. The 4-node network.

In order to illustrate the effectiveness of our assumptions inthe case of merged traffics, Figs. 6, 7, 8 show the occupancydistribution for queues 1, 2 and 3 of the network in Fig. 4.These figures show that the theory yields a worst-case analysisthat allows a safe network dimensioning.

IV. OPTIMAL CONSTRAINED CAPACITY ALLOCATIONAn estimate of the mean end-to-end packet transmission

delay T (performance metric) can be determined by applyingthe Little theorem both to the network as a whole and to eachsingle queue. We have:

N

T .v *n (mz di) (24)Zmm(e)

0.1

0.1

u- , _. .. I_,41 2 H-t~otal tratffcs 3 4I

1 2 (;ueu 3

Fig. 5. Comparison between simulation and theoretical results for(m(t), a(t), H(t)) of the total traffic at different nodes.

IvI

, -4o ^------ .____ _ __ _ __ _

4 smulat-S . Theory I"K1-

x[pkts]

Fig. 6. Queue occupancy distribution for queue #1.

where di is an estimate of the mean packet delay at node ifrom (5).

In our simplified study, we consider that all the links from thei-th node have the same capacity, Ci, and that the capacity costdoes not depend on distance. Hence, the cost corresponding tothe i-th node is wiCi, if such node has wi output links. Thetotal network cost results as:

N

w = E wicii=l

(25)

Thus, the network sizing problem is allocating the Cicapacity to minimize T under the constrain W = Wma,.Such method can be generalized to the case where the delaymetric is represented by a percentile of the end-to-end packettransmission delay. By means of the Lagrange Multipliermethod, we achieve the following result:

(~~~~~1Htl~~~~IH(t) a(ici = m(t) +( ;; )

where

2H()-3

ai = r H 2a, mif(f()/2H(t 2

and A is the solution of the following equation:

Wmax (Wim't)+ (iI,() N (e)

(26)

(27)

226

1-1--.".

1.41-20

Page 5: [IEEE 2005 2nd International Symposium on Wireless Communication Systems - Siena, Italy (05-09 Sept. 2005)] 2005 2nd International Symposium on Wireless Communication Systems - 3G

(4.01.0.98,0.5)

(16.9,25.01,0.92) c

(24 73, 20 49,0.97) CZV

(41.63,1.15A05) =

Fig. 7. Queue occupancy distribution for queue #2.

x [pkts]

Fig. 8. Queue occupancy distribution for queue #3.

When all the traffics are Poissonian (i.e., a(t) = 1 and H(t)0.5, for i = 1, ... , N), (26)-(27) yield the classical capacityallocation formula for Jackson networks [4]. If extemal trafficshave different Hurst parameters, (27) cannot be solved in a

closed form and a recursive method is adopted, where thestarting point is the capacity allocation for the correspondingJackson network.

(9.69,32.86,0.965)

Fig. 9. Network example.

0.

.S~

-0.

-1

550 60) 650 700 750 800 850 900 950

Total Cost [pkts/s]

Fig. 10. Mean and 95-th percentile of the end-to-end packet transmissiondelay.

V. RESULTS

In order to support our capacity allocation approach, we

show in this Section a network example with given routingmatrix q, extemal traffic matrices m, a, H and the totalnetwork cost Wmax. The results obtained with our approachwill be compared with those produced by minimization ofthe mean delay for the corresponding Jackson network (i.e.,approximating the input processes as Poisson -with the same

mean arrival rates-) under the same total cost constraint(classical method) [4].

Let us consider the network in Fig. 9. Then, Fig. 10shows the network performance (in terms of mean and 95-th percentile of the end-to-end packet transmission delay) forcapacities assigned according to both the classical method andour approach considering the total cost value in abscissa. Wecan note that our dimensioning technique allows lower delayvalues than the classical one in the presence of FBT traf-fics. The classical method allocates the following capacities:(Cl, C2, C3, C4, C5) = (78,78,103,109,176) pkts/s under theconstraint Wmax = 700 pkts/s. Hence, this method assignsthe same capacity to nodes 1, 2 that have about the same

mean arrival rates. In our method, the capacity allocationis optimized for (C1, C2,C3, C4, C5) = (114,44,106,102,176)pkts/s. Actually, self-similar traffic enters nodes 1 and 3;whereas, Poissonian traffic enters nodes 2 and 4 (the Poissontraffic coming from node 2 is much heavier that the local ss

input traffic). Hence, this approach assigns higher capacity tonodes 1 and 3 that can be more congested and reduces thecapacity for nodes 2 and 4.

VI. CONCLUSIONSThis paper has dealt with the dimensioning of IP-based

core networks for 3G cellular systems. Our general analyticalapproach is based on the FBT traffic model and optimizes thelink capacity to minimize an end-to-end delay metric.

REFERENCES

[1] Y. Koucheryavy, A. Krendzel, S. Lopatin, J. Harju, "PerformanceEstimation of UMTS Release 5 IM-Subsystem Elements", Proc. of the4th IEEE Conference on Mobile and Wireless Communication Networks,Stockholm, Sweden, September 9 - 11, 2002.

[2] W. Willinger, M. S. Taqqu, R. Sherman, D. V. Wilson, "Self-similaritythrough high-variability: statistical analysis of Ethernet LAN traffic at thesource level", IEEE/ACM Trans. Networking, Vol. 5, No. 1, pp. 71-86,1997.

[3] G. Giambene, Queuing Theory and Telecommunications Networks andApplications, Springer, New York, 2005.

[4] J. F. Hayes, Modeling and Analysis of Computer CommunicationsNetworks, Plenum Press, New York, 1984.

[5] T. K. Chan, V. 0. K. Li, "Decomposition of networks of queues withself-similar traffic", IEEE Globecom, Nov., 1998.

[6] I. Norros, "A storage model with self-similar input", Queueing Systems,Vol. 16, pp. 387-396, 1994.

[7] P. Abry, P. Flandrin, M. S. Taqqu, D. Veitch, "Wavelets for the analysis,estimation and synthesis of scaling data", Self Similar Network TrafficAnalysis and Performance Evaluation K. Park and W. Willinger Eds.,Wiley, 2000.

[8] V. Paxson, "Fast, approximate synthesis of Fractional Gaussian Noisefor generating self-similar network traffic", Computer CommunicationReview, Vol. 27, pp. 5-18, 1997.

[9] R. Addie, P. Mannersalo, I. Norros, "Performance formulae for queueswith Gaussian input", Proc. ITC-16, Vol. 3b, pp. 1169-1178, 1999.

227

s ,¢. Ol~~~~eandelay fJackson net.,i95-th percent. delay!acks.. net.)

.4g. Mean delay lFBT traf-fic", v6 95-th percent. delay 'FBT traffic',

13.. ....I

.,, . 'K.