EENTH N.4TIOIV.AL R-\DlO SCIEJCE C O U F E R E ~ C I : I\.l;irch 27-29 2001.3I;insour;l [_-niv.. Egvpt

----- .

T ESTIMATION AND . ITS APPLICATION TO ADMISSION CONTROL IN

ATM NETWORKS

G. A. F. M. KHALAF', S. S. K . EL-YAMANY'

ABSTRACT

A new approach to determining the admissibility of a new call request in a buffered ATM network is developed. In this approach, all traffic presented to the network is assumed to be statistically multiplexable sources. This leads to statistical interactions among different trafljc sources. The extent of multiplexing gain (spare capacity) is efficiently estimated in an on-line fashion, and the bandwidth (resource) allocation is traded off in an optimal call admission control policy. The algorithms presented in this paper are tested by simulation, and their effectiveness arc investigated

KEY WORDS: Adaptive estimation, Equivalent bandwidth, Multiplexing gain, Optimal call admission policy.

1. INTRODUCTION

ATh4 (Asynchronous Transfer Mode) networks provide multiservice capability whereby traffic sources with different characteristics share network resources through. the statistical multiplexing of fixed-size 53 byte cells with buffers used to absorb temporary overloads in traffic flow. To provide quality of service (QoS) guarantee to users, such as for cell loss probability (CLP), and delay, input traffic flows need to be controlled. This is a congestion control problem of which an important component is call admission where network resources, such as buffer space and link capacity, are allocated and decision to accept a new call is made. The concept of effective bandwidth (EBW) simplifies resource allocation because it does not include (statistical) interactioii between calls. It associates a bandwidth (BW) value to a gwen connection, independent of other connections in a switch, that is greater than the mean rate but less than the peak rateofthe connection in order to satis@ the required QoS at a given buffer space. Early work on EBW was done in [l]. More recently in [Z] for an unbuffered system and slotted batch models, [3] on a uniform arrival and service (UAS) model, [4] on continuous time Markov modulated fluid sources, [5 ] on discrete-time and Markov fluid sources, and in [6] on Gaussian traffic models. The problem of call admission control, on the other hand, has received considerable attention in the literature: we note in particular thc approaches of [7-IO]. The approach we take in this paper differs, however, in that we extract maximum benefit from statistical multiplexing (SM) over different calls without requiring detailed descriptor of their flows. In the literature, SM has been exploited in the following way: by simple averagmg of trdTic over time in.a large buffer, a call admission control is, then, proposed based on the EBW [3], [SI, [8], 11 I]. One potential drawback of the above approaches lies in their dependence on a relatively long time averaging which might not match the call interarrival times.

' Asscriald Pm&ssor. 1;aculty of Engintxring. Communications Ikprtmcnt. I Iclwan. Fgypl.

Associatcd I'rolcssur. Faculty of Engineering, C w m m " i o n s Ikpartnient. I Ich 'on. I$ypt.

483

2. STATISTICAL MULTlPLEXLNG

The statistical nature of a significant part of the multimedia - trait streams is its burstiness and variability. That is, the peak instantaneous rate is much larger than their long-term average rate. Therefore, allocating BW according to the peak rate results in low resource utilization. A significant saving in BW can be achieved by SM of many streams. BW saving relies on the statistical independence of different sources, and on exploiting the, presumably, noncoincidence of bursts of differen! streams. The buffer stores bursts of cells that arrive faster than they can be transmitted.

2.1 Underlying Theory

When the buffer is large, then under appropriate assumptions on the arrival streams [3], [Z], [12-141, we have

where, ( 1 )

P(W > B) = A e- Ba-I(C) -t O(B) A, is a parameter, assumed to be constant: A=], W, denotes the steady state workload of the buffer, B, is the buffer size in cells, C, link capacity in cells/Sec. a(c), some function of the statistics of the traffic stream and the link capacity,

O(B), is such that lim o(B) = 0 B+oo B

Now, suppose that there are J types of traffic classes differentiated according to their respective statistical characteristic. Consider the following constraint on P(W>B):

6 > 0 1 B+oo B

lim - logP(W>B) 5 -6,

or equivalently,

(3) where the parameter 6 determines the stringency of the buffer overflow pro'bability. It is shown [2], [3], [5] , [12], that this constraint can be satisfied when,

P ( W > B ) < E : = emB6 < I

where a j (6) type j traffic, corresponding to 6. The most general . results is in [ 18-1 91, where the above conclusion is shown to hold for a very larse class of traffic

sources. Assume a source produces a random number, A(t), of cells in t seccinds. Then its EBW can be calculated as,

desiLqates the EBW for n

where A ( 6 ) is its asymptotic log moment generating function,

where E( . ) signifies the expectation. For our purpose, let us assume, for a moment, that 6 < I . Then, one can justify the following approximation,

484

L -1

therefore,

where I 1 h: = lim -E[A(t)] 1

t+mt

D2:= lim l E [ ~ l ( t ) ~ ] ] t+mt

As can be seen from Eq. (7), the

(7)

EBW is in fact, a nondecreasing function of 6, with tlie mean source rate, A, corresponds to a(O), and its peak rate to a(oo), and a(6) lies between the mean and the peak rates. The difference between the peak rate and a(8) is the BW saving due to the statistical muhiplexing. Assume now that we have been able to obtain an estimate for a@), as will be seen later. Thus the QQS constraint (i.e., Eqs. (1-3)) becomes,

P(W>B) =e-Ba-' ( 6 ) < e-B6 (8)

2.2 Adaptive Estimation of the Effective Bandwidth

In this section, we propose an asymptotically consistent estimator for the effective bandwidth (EBW). Assume that time is discretized into intervals of length T seconds such that thc capacity of the queue server is C cells per time interval. Now consider a given cell traffic stream, and assume that AN k cells arrived in time interval K corresponding to {(K-l)T, KT) , The arrival counting process is, therdore, gven by

k

r=l Nk = CANr

substituting Eq. (9) into Eq. (6) gives - -

A(6) = n + m lim n-l*ogEl: .'l..j, and the EBW estimator, a( 6), becomes

&(6) = 6-*log [ L- C18 Ce k = l f-l] (10)

where L is averaging period. Eq. (IO) simplifies the calculation of the EBW, and it also circumvents analytical modeling of the traffic streams. However, one may argue that &(a) will need some time before it converges to the true value of a( 8). In the following we propose a more flexible, adaptive estimation model in order to improve its convergence property, but, of course, at the cost

485

EICrJTKEN7'1-1 3.-lTlOS:\L li.\i)iO SCIESCE C O ~ F E ; I < E N C E h I ; l ~ ~ h 27-29 300 I .31:111S~[Ir:1 Vniv.. Esy,,t

of some delay in determining the estimate as well as more memory in its computation. Define the following innovation process

where m is the mean of AN k assuming that ~ k are independent identically distributed random variables With zero mean. Then, the innovation process, k can be represented as a linear process of order P,

- ARk = ANk - Ill (1 1)

P

r= 1 L?&k = x a r a n - r (12)

and its generating function (Eq. (6) ) is given by

~ ( 6 ) = I I - ~ I O ~ E e k = l (13)

(14)

the EBW estimator is [-I &(6)=6-l A(6).

Thus, in order to use Eq. (14), we still need to find a proper estimate for the mean value, m, as well as the value of the weighting coefficients, ar, in Eq. (1 2). Firstly, the mean, m, can be recursively estimated as follows,

(1.5) where p is the estimator's step size (i.e., its forgetting factor). Secondly, if we assume (by virtue of the central limit theory) that Afik are Gaussian distributed. Then the coefficients, a *, take the

mk = mk-1 + p(ANk-mk-1)

following form ar = h', where h designates the index of dispersion ofthe Gaussian (innovation) process. For the sake of comparison while validating our results, we include the following analytical model which was derived in [ 151.

c

Eq. (1 6) was developed for an ON-OFF, bursty traffic sources with mean rate, m, and peak rate P

2 3 Estimation of the Multiplexing Gain

We now describe an approach for estimating the multiplexing gain fix a set of homogeneous .traffic sources. This approach is developing in [7l, but differs in that it is based on real-time buffer measurements rather than, on (the more complex) analyhcal modeling of traffic streams. Recall now Eci. (1)

1 P(W > B) = A (cl

where we assumed that the parameter A is constant (A=l). A convincing argument is made in [ 161, that the, parameter A, may in fact be small or large, reflecting the extent of the statistical multiplexing (SM), or what is known in the literature as the statistical miultiplexing gain (SMG). . That is A<<1, corresponds to large SMG which means large spare bandwidthhuffer resources. Consequently, the constrains on P(W>B) given by Eqs. (1-3) become

1 P ( W > B ) = ~ e - B a ' (6) < e-B6 and the analytical estimator for the SMG derived in[23] is given by

486

EIGHTEENTH N.4TIONAL RADIO SCIENCE CONFERENCE Rlarch 27-29 2001,Mansoura Univ.. Egypt

A = [I- n(B-l)] eBa-'(6) (18)

n(B-1): = Cn(b) (19)

where, B-1

b=l where, n( * ) denotes the tail distribution of the bdTer workload traffic. In the following, we describe OUT approach to estimating the SMG. Let K denote the set of class-j, j=1,2, . . . K, calls admitted to the (ATM node) buffer. Define the instantaneous load W(t), produced by the K-class calls

now, assume Poisson, ON-OFF traffic sources such that a traffic source-j produces jumps (bursts) of different sizes a j >O with Poisson rate y j , define

the tail distribution of W(t). For the Poisson distributed traffic stream, it follows that Q(Y) = p(W(t)<y)

where C ni ai ~y signifies all jumps of size ai observed in the past interval Ti . Of importance here to note that Eq. (20) is valid for all traffic s t r e a m with steady state probabilities assunkg product form in the holding times Ti . Therefore, our approach is to observe the buffer's workload in a real time manner, and calculate Q(y) according to Eq. (20). Thus, Eqs. (18-19) can easily be calculated, hence, an on-line estimate for the SMG is obtained. An indication of the performance of this approach will be given later in the validation section.

3. AN OPTIMAL CALL ADMISSION POLKCY

In this section, we describe how on-line monitoring of the buffer workload can be used to adjust call admissions at the access port of a given ATM node. Consider the simplified version of the access port shown in Fig. 1. As can be seen, it consists of a single transmission link shared by multiple traffic classes. Different traffic classes are differentiated in terms of their respective statistical characteristics and their QoS requirements. The question is how would the access port decide on the maximum number of calls to be accepted such that the guaranteed QoSs are not degraded. In the following we present our approach to answering this question.

Let us start by writing the QoS constraint (Eq. 17) in the following equivalent form,

+ [*)I 5 1. (21)

Suppose now-that K-independent trafic classes arc sharing a transmission link (SCC Fig. 1 .) with transmission c'yacity C cell/sec., and let 01 j (6 j ) desigate the EBW estimated for a sin&

source belonging to t r f i c class-j and let n j be the maximum number of such sources that can be accepted. In thc following \vc formulate a linear programming (LP), problcm that maximizes the link utilization subject to constraints 011 the QoS requirement by each indi,vidual trafic class.

E I G I3 T E E 3TH 3XT IO 3 .A L I?--! D I 0 S C I E SC E CO 3 F E R E N C E i\l1;1rch 27-29 300 1.3 J:t nso II r:i t -11 iv.. Egypt

. - - On Line Buffer Measurements .- Decisions

lass#2 Traffic

Class # K Traffic Access port in an ATM Node.

K r

Such that

K

+ ?$L]

- - C = C n j a j ( 6 j ) = constant

j=1 Clearly, n j , j=1,2, . . ., K, in the above problem represent the independent (solution) variables. Suppose ‘that a‘ is the optimal “extreme” point for the LP. Then &ch time a new (class-j) call request arrive, the admission system updates the EBW and accept the new call iff the LP finds a new, finite, extreme point. In the following section we validate the findings presented in this paper.

4. SIMULATION RESULTS

The purpose of this section is to obtain qualitative results, and design guidelines. One of the highlights of the results, which has a considerable practical implication, is the ability to determine the SMG for a set of traffic sources sharing a buffered transmission lid.. More importantly, is’ relating the new cal’, acceptancdrejection decisions to the current status of the SMG measurements.

We begin our validation by Fig. 2, showing a simulated sample path of a typical cell arrival process (dashed line) from a set of (20) homogeneous traffic sources along vvith an estimate of the mean t r a c (solid h e ) . The objective in this Figure is to test the adaptability of the recursive estimator (Eq. 15). We see that the estimate is stable and its response to tr&c fluctuations is reasonably slow ( e 3 in Eq. (1 2)). Next, we show the effectiveness of the ElBW estimation model (Eq. (14)) dong the simulation time in Fig. 3,. As can be seen, the EBW estimate is rightly sensitive to v~a t ionS in the cell traffic pattern. In this Figure we did not make use of Eq. (1 2), rather we have assumed that tiat! EBW is being assigned on line in the burst scale. This motivates US to

488

EIGI-IT'EENTH 3.ATIONAL li--\DIO SCIENCE COYFERENCE &Iar-ch 27-29 200 1.3 1;i I I S O I I I - ~ Un iv.. Esypt

examine the sensitivity of the EBW estimation to the traffic intensity as illustrated in Fig. 4. Also, included in Fig. 4, is the results obtained using the analytical model (Eq. (16)) for comparison purpose. These (analytical) results are based on the assumption that the peak and mean rates, P, m, equal, respectively, to the EBW, and mean values. This should not be considered as unreasonable assumption because we already know that, on one hand, the EBW lies between the mean and peak rate, and on the other hand, as commonly known, that the EBW tends to be conservative in its BW estimation. We see that our EBW estimation (solid line) is fairly sensitive to the QoS metric, 6 , over a wide range Recall that our approach to EBW estimation does not depend on a particular modeling of the traffic source as did the analytical model. This should justify the discrepancy between the analytical and simuldion results in Fig. 4. Below, we illustrate the relationship between the EBW and the resultant QoS measured in terms of the probability of buffer overflow. Once again, it should not be unreasonable to consider the buffer overflow probability as expressing the ceil loss probability, CLF'. Fig. 5 , depicts the CLP obtained by simulation for a single traffic source (solid line). At the same time, during simulation, the EBW is estimated and substituted into the left hand side (LHS) of Eq. (8) to obtain the LHS analytx (dash line) results. As can be seen the simulation results is reasonably close to that obtained by the analytical results over a wide range of cell arrival rates. This indicates the efficiency of the EBW estimation for the single source case. The same observation can be seen for the case of (20) homogeneous traffic sources shown in Fig. 6, along with their corresponding (LHS) analytical results. Up to this point, Figs. 5-6, were considered as qualitative results demonstrating the effectiveness of the EBW estimation. We now comparc the same LHS analytic results of Fig. 6, with that obtained by the right hand side (RHS) of Eq. (8). As shown in Fig. 7, this provides us with an indication of the status of the SMG. That is whenever Eq. (8) is satisfied, this indicates the existence of a SMG, i.e., A -= 1. Take for example thecase wherein h =0.82. It sliould not be difficult to conclude that up to a buffer size of (say) 250 cells, there is no SMG, i.e., A > I . However, for larger buffer sizes, a SMG is being achieved. These observations are summarized in Fig. 8, (dashed line) along with our (new) approach to estimating the SMG (solid line) which is based on real time measurements of the tail distribution of the buffer workload through Eqs. (18-20). As can be seen, the two results seem to agree to some cxtent. However, it has been reported in [16], that for large number of sources the EBW, tends not to capture the SMG among different traffic classes. This observation can be seen fiom the bandwidth. decoupling property as stated by Eq. (4). This explains why we decided not to include the rcsults for large number of sources among our results in this paper. In contrast to Figs. 6-8, for the homogeneous tr&c. The following results focus on the hzterogeneous traffic case. Fig. 9, compares the CLP obtained by simulation (solid line) with that obtained by LHS of Eq. (8) (dashed line) for different values of cell arrival rates. We see a good agreement between the two results which, again, confirms the efficiency of the EBW estimation. Next, we examine $e existence of the SMG in Fig. 10, by comparing tlie LHS analyhc results (solid line), with the RHS analpc (dashed line) of Eq. (8). As can be seen, the discrepancy between the two results is relatively larger than its counterpart of Fig. 7. This reflects the effect of ignoring the SMG by simply adding the EBWs of individual traffic sourccs in order to obtain the overall EBW for the set of (20) heterogeneous sources (Eq. (4)). Fig. 1 1, summarizes the behavior of the SMG as seen fiom Fig. 10, along with our new estimation of the SMG (solid line) using the tail distribution of the buffer workload. Again, although the two results seem to agree to some extent. However, we believe that the (for large number of sources) conclusion made in [ 161 remains unchanged. In the remaining of this section, we examine the performance of the call admission policy which we are proposing to be adopted by the access port(s) of an ATM node. Consider a simplified example in which (say) 3-traffic classes share a buffered link with constant capacity as depicted in Fig. 1. Each class has its EBW, SMG estimated, and each class has its QoS requirement unchanged. Fig. 12, shows the maximum number

489

EIGHTEEYTN N.-tTION.AL K \ D 1 0 SCIENCE COSFERENCE March 27-29 2001.~1; insour~ Univ.. Egypt

of admksibie (class-j) tr&c sources versus the SMG as the key controlling factor. Here, the extreme (solution) points are obtained using the revised Simplex linear programming algorithm for different values of QoS, 6. As can be seen, the adrmssion control pollicy adjusts the maximum number of socuces according to the (estimated) value of the SMG over a reasonably wide range of QoS r e q u k “ t s . Take for example the case wherein 6 =1. At SMG=l . the admission policy can admit 100 ( b - j ) sources. However, as the SMG increases (i.e., as!jdgIowervalues),the poky cm admit m e sources, by taking advantage of the spare capacity available, and vlse versa. fn tk fdhwhg sectkm we summarize the findings presented in this paper.

5. SuRaMARY AND CONCLUSHQNS

In this paper, we developed number of simple and robust approaches to solve the problez of resource allocation and dynamic admission control in high-speed ATM networks. The first approach relies on monitoring the traffic offered to the buffered output link and deriving an adaptive estimate for the equivalent bandwidth based on the index of dispersion of‘a Gaussin linear process. Next, we proposed an approach to estimating the statistical multiplexing gain for a set of multiplexed tr&c sources based on measuring the tail distribution of tlne buffer workload (in the present. time, we are investigating a way of using this approach to inforce a real time traffic policing. strategy). Both of the equivalent bandwidth and the multiplexing gain estimates are, then, considered as the main ingredient of a new dynamic call admission corttrol policy proposed to be adopted by the access port(s) of a given ATM node. A linear programming problem is formulated that maximizes the link utilization, constraint to both the QoS and the estimated equivalent bandwidth/multiplexing gain. The result is the maximum number of admissible traffic sources such that the guaranteed QoSs are not degraded. Based on the simulation and analytical results, we claim that our approaches have proven to be effective in providing reasonably accurate estimation of the key controlling factors of the network. And that the, linear progrmrning based, call admission’ policy is tremendously important for practical, real time control of the ATPd environment.

1.

2.

. 3.

4.

5.

6.

7.

REFERENCES

Hui, J. Y., “Resource Allocation for Broadband Networks”, IEEE Joumal on Selected Areas in Communications, Vol. 6, No. 9, December 1988, pp. 1598-1 608. Kelly, F. P., ‘Effective Bandwidth at Multi-class Queues”, Queuing Systems, Vol. 9,

Gibbers, R J. and Hunt, P. J., “Effective Bandwidths for lululti-type UAS Channel”, Queuing Systems, Vol. 9, 1991, 17-28. Elwalid, A. I. And Metra, D., ‘Effective Bandwidth of General Markovian Traffic Sources and Admission Control of High Speed Networks”, IEEE/ACM Trans. On Netwwking, Vol. 1, No. 3, June 1993, pp. 329-343. Kesidis, G., Walrand, J., and Chang, C. S . , “Effective Bandwidths for Multiclass Fluids and other ATM SOU~C~S”, IEEE/ACM Trans. On Networking, Vol. 1, No. 4, August 1993, pp. 424-428. Courcoubetis, C., Fouskas, G. and Weber, R, “On the pedormanceofanEffective Bandwidths Formula”, Proceedings of ITC-14, 1994, pp. 201-212. Courcoubetis et al., “Admission Control and Routing in ATM Networks Using Inferences From Measured Buffer Occupancy”, IEEX Trans. Commun., Vol. 43, No. 2/3/4, Febr.March/April 1995, pp. 1778-1784.

1991, pp. 5-16.

490

GI-ITEENTH 3VXTIO3.AL li.-\DIO SCIENCE COSFERENCE hlnrcli 27-29 2001.,“rI;1nsoura Univ.. Egypt

8 Guerin, R, Ahmadi, H, and Naghshineh, M., “Equivalent Capacity and its Application to Bandwidth Allocation in High-speed Nekorks”, XEE Select. Areas Commun., Vol. 9,

Rasmussen, C., Sorensen, H., Kvols, K. S., and Jacobsen, S . B., “Source Independent. Acceptance Procedures in ATM Networks”, IEEE J. Select. Areas Commun., Vol. 9,

Saito, H., and Shiomoto, K., “Dynamic Call AdmissionControlin ATMNetworks”, IEEE Select. Areas Commun., Vol. 9, No.7, 1991, pp. 982-989. De Vaci na, G., Kesidis, G., and Walrand, J., ‘‘Resource Management in Wide Area ATM Networks Using EEective Bandwidths”, Submitted to EEE J. on Select Areas commun.

12. Veciana, G., “Design Issues in ATM Networks: Traffic Shaping and Congestion Control’’, Ph D thesis, Dept. of EECS, Univ. of Califomia, Berkeley, 1993.

13. Courcoube, C., and Weber, R., “Buffer Overflow Asymptotic for a Switch Handling Many Traffic Sources”, Preprint, 1993.

14. , Weiss, A., “Technique for Analyzing Large Traffic Systems”, Adv. Appl. Prob., 1985,

1 . Kelly, F. P., “On Tariffs and Admission Control for Multiservice Networks”, Oper. Resea. Letters, 15, 1994, pp. 1-9.

16. Choudhury, G. L., Lucantoni, D. M., and Mitt, W., “Squcezing the Most out of A m ’ , IEEE Trans. Commun., Vol. 44, No. 2, Feb. 1996, pp. 203-2 17.

NO..7, pp. 968-98 1, 199 1. 9.

NO. 3,1991, pp. 351-358. 1 .

11.

- pp. 506-532.

0 50 100 150 200 250 300 Simulation Time

Fig. 2. Recursive estimation ofthe mean traffic, in, during simulation model.

491

E I G 1-1 T E E 3 T 11 3 .-! TI 0 3 .-! L R--! D IO SCIENCE CO 5 F 1: R E S C E .Ilarc!i 27-29 2001.3Iansour:t Univ.. Egypt

10 4 i

- 0 50 100 150 200 250 3 00 Simulation Time

F.g. 3. Estimation of the equivalent bandwidth (EBW).

0.0 0.2 0 -4 0.6 0.8 1 .o Tr&ic intensity

F.g. 4. Mean equivalent bandwidth estimqtjQp model.

492

E?GI-ITEE.\ITII 3'.4TIO3.4L I?.-!DIO S C I E S C E CONFERENCE e' Itinrcli 27-27 2001.31:insour:~ Cniv.. Egypt

1 .OE-1

1 .OE-2

1 .QE-3

1 .OE-4

1 .OE-5

1 .OE-6

1 .OE-7

0 20 40 60 80 100 120 140 160 180 200 220 Buffer size in cells.

F.g. 5. Cell loss probability (single source).

1 .OE-2

1 .OE-3

1 .OE4

1 .OE-5

1 .OE-6

1 .OE-7

1 .OE-8

0 50 100 150 200 250 300 350 400 450 500 550

F.g. 6. Cell loss probability ( homogeneous traffic sources).

' Buffer size in cells

493

1 .OE-2

1 .OE-3

1 BE-4

1 .OE-5

1 .OE-6

1 .OE-7

1 .OE-8 0 50 100 150 200 250 300 350 400 450 500 550

Buffer size in cells

F.g. 7. Cell loss probability : indicating the multiplexing gain.

2.25.

2.00

1.75

1 S O

1.25

1.00

0.75

0.50

0.25

0 50 100 150 200 250 300 350 400 450 500 550 Buffer size in cells

F.g. 8. Estimation of tlie multiplexing gain.

494

EIGHTEENTH NXTlOlUXL ILAD10 SCIENCE CONFERENCE e RInrcli 27-29 2001,iVIansot1r:1 Univ.. Egypt

1 .OE-2

1 .OE-3

1 .OE-4

1 .OE-5

1 .OE-6

1 .OE-7 0 40 80 120 160 200 240 280 320 360 400 440 480 520

Bufzer sizc in cells

F.g. 9. Ccll loss probability ( heterogeneous traffic sources).

1.OE-1

1 .OE-2

1 .OE-3

1 .OE-4

1 .OE-5

1 .OE-6

1 .OE-7

. \ Analytic LHS

Analytic RHS k0.64 To 0.84 , --. - - -. 1 --&!i-- k0.4 To 0.6 1

----_----_.___________

i 0 40 80 120 160 200 240 280 320 360 400 440 480 520

Buffer sizc in cells

F.g. 10. Ccll loss probability ( heterogeneous traffic sources): indicating tlic multiplexing gain.

495

EIGHTEENTH N-ATIONAL RADIO SCIESCE COSFERENCE March 27-29 2001,hIansuura C'niv.. Egypt

3 .OO 2.75 2-50 2.25 2.00 1.75 1 S O 1.25 1 .oo 0.75 0.50 0.25

h=0.22 TO 0.34

- . .\ I I I I l ~ l ~ l ~ l , l , l ' l

0 50 100 150 200 250 300 350 400 450 500 550 Buffer size in cells

100

10

F.g. 1 1. Estimation of the multiplexing gain.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.4 1.8 2.0 2.2 Multiplexing gain

F.g. 12. Admission control policy.

496