Connection Admission Control for Flow Level QoS in ... - CiteSeerX

Connection Admission Control for Flow Level QoS in Bufferless Models S´andor R´acz, Tam´as Jakabfy, J´anos Farkas and Csaba Antal Traffic Analysis and Network Performance Laboratory Ericsson Research Budapest, Hungary {Sandor.Racz, Tamas.Jakabfy, Janos.Farkas, Csaba.Antal}@ericsson.com Abstract— Admission control algorithms used in access networks for multiplexed voice sources are typically based on aggregated system characteristics, such as aggregate loss probability. Even though the relation of these aggregate performance measures to the performance of specific flows is not trivial, it has gained limited attention so far. We propose an admission control method that provides flow level QoS guarantees. The proposed method is based on Gaussian approximation applied for the bufferless multiplexing model. We show that the flow level packet loss violation probability can be approximated as the quantile of a normal distribution and we give a method to calculate its mean and variance. The obtained admission control formula includes the moments of the activity factor distribution.

I. I NTRODUCTION The most important design objectives of connection admission control algorithms are fast execution time and conservative but accurate operation. The prerequisites of fast algorithms are simplicity, both in terms of traffic model and applied queuing model. Even with simple models, approximations are required to get a feasible algorithm. The cost of simplicity, however, is often reduced accuracy. Admission control mechanisms are crucial in networks carrying delay/jitter sensitive traffic. The most important realtime application for operators is undoubtedly voice. Voice over IP service is provided by both fix and mobile networks. Even though admission control is not needed in the backbone due to the high transmission speed and the applied dimensioning methods, it is still important to control admitted traffic in access networks. In the analysis of aggregated voice sources, a generally accepted traffic model is the periodic ON - OFF source model. Voice coders generate packets periodically when the speaker is talking; and silence suppression algorithms (or voice activity detection) are used to detect silent intervals when only silence indication packets are sent with much less frequency. A simple and efficient queuing model for ON - OFF sources is bufferless multiplexing, which is applicable when the buffers are small so queuing at ON - OFF time scale can be neglected, but they are large enough to avoid packet scale queuing effects [1]. It is intuitively clear that bufferless multiplexing is a reasonable model for aggregated voice sources because strict

delay and jitter requirements usually motivate short buffers and the multiplexing gain due to large buffers is rarely significant. QoS requirements for bufferless multiplexing, which are necessary input to admission control algorithms, can be based on packet loss probability for the total traffic aggregate. In some papers the workload loss ratio is calculated directly [2], but it is even more general to approximate packet loss probability by the system saturation probability [2]–[6]. Previous analytical results on bufferless model are summarized e.g. in [6]. Calculation of aggregate loss probability for bufferless multiplexing allows many simplifications in the traffic description. The distribution of the duration of ON and OFF periods can be arbitrary. Furthermore, it is sufficient to provide the ratio of ON and OFF periods, their separate characterization is not required. A common descriptor to characterize ON OFF sources is the activity factor [7]. Activity factor is defined as the ratio of the total duration of ON periods and the total lifetime of the flow. In case of voice connections, the activity factor means the portion of time when the speaker is busy talking if voice activity detection algorithms operate optimally. The activity factor of a flow depends on several effects and it can vary flow by flow. However, if aggregate loss requirement is applied these variations even out, so a constant activity factor, which typically takes values from 0.4–0.6 [8], [9], provides a sufficient characterization of voice sources. Even though the aggregate loss requirement makes analytic work tractable, it fails to provide flow level QoS assurances. As admission control is designed for aggregate loss requirement, the loss rate of individual flows may violate their requirement even if they are admitted by the admission control. A specific example for such violation is due to the varying activity factor of voice sources [10], [11]. There may be time intervals when the activity factor of many ongoing flows is larger than the average activity factor. Flows in these periods will experience high loss. The goal of this paper is to define a new QoS requirement and a related admission control algorithm that reflects the general need of providing flow level QoS assurances. Due to the new problem formulation, the applied traffic model has to be revised as well. We assume that the ON - OFF operation of voice sources is characterized solely by the activity factor, but we

consider its random nature when designing the new connection admission control algorithm. Our proposed algorithm is based on Gaussian approximation applied for bufferless systems [6], extending it to guarantee flow level QoS. The new method considers the variance and higher order statistics of the activity factor. The rest of the paper is structured as follows. The related work is presented in the next section. We describe the considered system and its model in Section III. In Section IV, we introduce the flow level QoS measure and propose a corresponding connection admission control algorithm. Section V shows an example application of the proposed results in a UTRAN system. Finally, in Section VI we conclude the paper. II. R ELATED WORK Methods related to effective bandwidth [6] and network calculus [12] are extensively studied in the literature for determining capacity need of flows in QoS enabled packet networks. The effective bandwidth expression describes the minimum bandwidth that is needed to fulfill the QoS criteria of the flow. The effective bandwidth, which is between the average and peak rate of the flow, is generally used in traditional dimensioning and admission control methods. Effective bandwidth was determined for many traffic types, for example periodic sources, fluid sources, ON - OFF sources [6]. The network calculus also determines the capacity need of flows in a packet network [12]. In the network calculus method, rate envelopes are used for packet arrivals and services in the network. The deterministic network calculus is a worstcase analysis method, which gives upper bounds for packet delay. By introducing stochastic bounds for packet delay, which slightly weakens the QoS requirement, better network utilization can be achieved. Stochastic network calculus was proposed to solve this problem (see e.g. [13]). Li, Burchard and Liebeherr established a link between effective bandwidth and network calculus in [14]. They showed that a general formulation of effective bandwidth can be expressed within the framework of stochastic network calculus, where both arrivals and service are specified in terms of probabilistic bounds. In [15], the authors determined stochastic bounds for the service experienced by a single flow when resources are managed for aggregates of flows. Using statistical network calculus, per-flow bounds are calculated for backlog, delay, and the burstiness of output traffic. The above techniques were developed for the analysis of a wide variety of systems. However, they do not exploit all information that we assume to be available in a voice-only system or in a multi-service system where voice service has differentiated treatment, for example in a DiffServ IP network. In this paper, we utilize traffic and system properties of voice services. Beside the periodic ON - OFF structure of flows, strict delay requirement and short buffer we also consider the random nature of activity factor of flows. Measurement and analysis of ON - OFF parameters of voice sources have been the subject of research, especially in relation to mobile communication systems. In [11], the distribution

of voice activity factor is presented based on measurements performed in a GSM network. Jiang and Schulzrinne measured and analyzed the distribution of ON and OFF durations of voice flows in a VoIP system [16]. Jugl and Boche [17] also took into account the fluctuation of voice activity factor in order to calculate the capacity of the air interface in CDMA based systems. They modeled the activity factor of users as a uniform distribution and showed that the fluctuation of activity factor affects the performance of the air interface. Measurement based analysis was also carried out to find the sources of the fluctuation of the activity factor. The activity factor of a flow depends on the user behavior and also on the applied transport protocol. Furthermore, in case of voice connections it also depends on the voice activity detection algorithm and on the operation of voice coders [16], [18]. Regarding user behavior the sex of the speakers, their language and personality also affect the activity factor [9], [19]. Sun and Ifeachor [20] studied the effect of these differences on the perceived voice quality in IP networks. III. S YSTEM

AND MODEL DESCRIPTION

The subject of the paper is a FIFO queue fed by periodic sources. Firstly, we describe the parameters of the studied system; we then give the characterization of traffic sources. Finally, we introduce the flow level quality of service requirement to be applied at the development of the admission control algorithm and we describe the model of the system. The FIFO server is characterized with capacity C and a buffer of length B. Packets of ON - OFF sources are characterized with a constant packet size b. Periodic ON - OFF sources generate packets with a deterministic inter-arrival time T in state ON, also called as source period; and they do not generate any packets in state OFF. In most practical cases, the duration of ON periods and the duration of OFF periods of ON OFF sources are independent, so we assume this property in the analysis. The random activity factor of flow i is denoted by αi and the common distribution function of activity factors is F (x). The activity factors are assumed to be iid random variables. For example, Westholm and Olin provide the distribution of the activity factor in [11] based on measurements of voice sources in a GSM system ( [α] = 0.58) and the distribution of ON and OFF period lengths. An example system is illustrated in Fig. 1, where call arrivals and departures are marked by vertical lines and packet arrivals are marked by small bars. In this system, a call admission control algorithm limits the maximum number of parallel flows to N = 4. As mentioned above, the actual value of the activity factor can be different for each flow. For example, the activity factor of flow-i is close to 0.6, which means that the source sends packets during 60% of the duration of flow-i. However, the activity factor of flow-j is close to 0.9, meaning that the source almost always sends packets. After the definition of how the system works, the next step is to introduce the flow level quality of service requirement. ON - OFF

flow-i

T

TOff TOn

flow-j

B

Number of parallel flows

2

3

3

4

4

3

2

1

2

2

3

PSfrag replacements δ [Y ≤ x] x

fully loaded system state

Fig. 1.

An example system: the maximum number of parallel flows is 4

We assume that the system that we study has a strict delay requirement D. That is, all packets that have longer delay than D are dropped at the receiver. Under these circumstances a packet is dropped if the buffer is overflown or if it suffers longer delay than D in the buffer. The per flow QoS requirement can now be defined as a per flow packet drop requirement, denoted by ε, considering dropped packets both in the FIFO buffer and in the receiver. The fraction of dropped packets depends on the number of parallel flows. In general, it increases if the number of parallel flows increases, as Fig. 2 shows. In order to get a conservative admission control that operates independently of the call arrival rate, we must assume that the number of flows in the system is always the maximum allowed by the admission control, i.e. the system is always fully loaded. The fraction of dropped packets also depends on the activity factors of ongoing flows even in a fully loaded system. The activity factor can take any value in (0, 1) interval in practifully loaded system state

0.5%

55

0.3% 50

0.2%

45

0.1%

PSfrag replacements

0.0% 10

20

30

40

50

Our system model is as follows. We study a bufferless multiplexer with capacity of serving Z = CT /b packets in a source period, which multiplexes N ON - OFF sources with random activity factor. If more packets arrive in a source period T than the system capacity Z then all packets are considered lost, which is a worst case assumption that is valid when the buffer size B is larger than the delay requirement D. We allow arbitrary activity factor distribution. The parameters used in the model are summarized in Table II.

. 60

0.4%

0

Finally, the parameters of the studied system are summarized in Table I.

65 N=61

number of parallel flows

fraction of delayed or lost packets

0.6%

cal cases [11]. Therefore, if packet loss rate was required to be less than ε for all flows then we would need to calculate with an activity factor of 1 for each flow in the admission control. However, this would inherently exclude any gain from the multiplexing of voice sources. As a reasonable compromise, we assume that the δ probability of the violation of the ε packet drop rate requirement is small.

60

time [sec]

70

80

90

40 100 110

δ [Y ≤ x] x

Fig. 2. Fraction of dropped packets of a flow: N = 61, T = 20ms, D = 8ms, B = 15ms, C = 672kbps, b = 336bit, F (x) = U nif orm(0.1, 0.9)

TABLE I S YSTEM PARAMETERS

C B N D ε δ αi b T

System descriptors server capacity size of the buffer max number of admitted ON - OFF sources packet delay requirement max allowed packet drop rate probability of not fulfilling drop requirement Descriptors of individual sources random activity factor; αi ∼ F (x) packet size packet inter-arrival time (source period)

kbps ms ms

bit ms

TABLE II

where Xp is a Bernoulli random variable with parameter p. The calculation of the distribution of the sum of Bernoulli random variables was extensively studied in the literature. Two well-known approximations of the sum in (3) are the Chernoff bound and the Gaussian model [6]. In this paper, we use the Gaussian approximation, which applies the central limit theorem and approximates the sum of the Bernoulli random variables with normal distribution. This means that

PARAMETERS OF THE MODEL

System descriptors server capacity max number of admitted ON - OFF sources max allowed packet drop rate probability of not fulfilling drop requirement Descriptors of individual sources random activity factor; αi ∼ F (x)

Z = CT /b N ε δ αi

 

N

IV. A DMISSION

CONTROL ALGORITHM FOR FLOW LEVEL

QOS

GUARANTEES

Analysis of models where the traffic descriptor, which is the activity factor in our case, is also random can be performed according to the law of total probability. As a first step a QoS function is evaluated under the condition of fixed activity factors. Using the density function of activity factors we can remove the conditioning and obtain the probability δ of QoS violation. Accordingly, we firstly derive a QoS function that indicates whether the fraction of dropped packets is below ε assuming fixed values of random activity factors α1 = a1 , . . . , αN = aN for the ongoing flows   0, fraction of dropped Q(a1 , . . . , aN ) = packets is below ε;  1, otherwise.

That is, QoS violation is indicated by Q = 1. In the next step, we can take into account the randomness of the activity factor. Denote f (a1 , . . . , aN ) the joint density function of the random activity factors. The probability that the system does not provide flow level QoS requirement for any of the ongoing flows can be calculated as follows:




δ=

Q(a1 , . . . , aN ) f (a1 , . . . , aN )da1 . . . daN .

(1)

A. Evaluating QoS function for fixed activity factors According to the bufferless model, the QoS requirement on the fraction of dropped packets is fulfilled if:



[the number of sources in state

ON

≥ Z] ≤ ε,

(2)

which is the QoS defined on aggregation of flows. The distribution of the number of sources in state ON depends on the values of the activity factors. Denote ai the constant activity factor of source i. The distribution of the number of sources in state ON follows the sum of N independent Bernoulli random variables with parameter ai , i.e.



=

[the number of sources in state

 

N

i=1



Xa i ≤ Z ,

ON





Z−

N i=1

N

ai , σ 2 = i=1

(3)

N

ai (1 − ai )

i=1

N i=1

ai

ai (1 − ai )





= (4)

.

Q(a1 , . . . , aN ) =

 



1



Z−



N i=1

ai

N i=1

ai (1 − ai ) otherwise,

0

≥ Φ−1 (1 − ε) (5)

where the Φ(x) and Φ−1 (x) denote the standard normal distribution function and its inverse. The Φ(x; µ, σ 2 ) denotes the distribution function of a normally distributed random variable with mean µ and variance σ 2 . B. Evaluating QoS function for random activity factors We now provide the second step of the analysis. In this step, we take into account the randomness of the activity factor. The random activity factors of N sources α1 , . . . , αn are iid random variables. The value of the flow level QoS measure δ can be obtained by substituting (5) into (1) and using the joint density function of the activity factors: δ=



Z−



N i=1

N i=1

αi

αi (1 − αi )



≥ Φ−1 (1 − ε) .

(6)

The direct calculation of δ is very complicated and results in very complex formula even for small values of N . This motivates the application of an accurate and fast approximation of δ. Before we derive an admission control algorithm for general distributions of the activity factor, we show an illustrative example for the above equations. Let us consider an example system multiplexing N = 50 connections and having capacity Z = 40. Let ε be 0.1%. The activity factors of the connections are independent and identically distributed Bernoulli variables: Xαi . Let the αi parameters have the following distribution:



[αi = 0.4] =



[αi = 0.6] = 0.5

Applying (3) and (4) for the case when the activity factor of all connections equals to 0.4 we get Packet drop probability = 1 −

≤ Z] =



Finally, we can define the QoS function Q, which indicates whether the fraction of dropped packets is below ε, as follows:

a1 ,... ,aN

In the following two subsections, we propose a method to evaluate the value of δ. In Section IV-A, we present an accurate QoS function based on Gaussian approximation [6]. We then provide a method to evaluate efficiently the integral in (1) in Section IV-B.





≈ Φ Z; µ = =Φ



Xa i ≤ Z ≈

i=1

≈1−Φ



40 − 50 × 0.4 √ 50 × 0.4 × 0.6



 

50

i=1



X0.4 ≤ 40 ≈

≈ 10−4 .



a normal approximation can be applied when the denominator is greater than zero even if the numerator and denominator are correlated. Applying these results we obtain the lemma. 2



Lemma 2: The first and the second moments of Y can be calculated as infinite Taylor:

If 29 connections have an activity i factor of 0.6 then hP P50 29 ≥ 40 ≈ 0.10%, so X + X i=1 0.6 i=30 0.4 flows still fulfill the drop requirement. However, if 30h connections have the  largeri activity factor then P50 P30 ≈ 0.12%, so i=31 X0.4 ≥ 40 i=1 X0.6 + flows violate the drop requirement. Obviously, if even more connections have an activity factor of 0.6 then the drop requirement cannot be fulfilled. For example, if allh connections have i the larger activity factor (0.6) then P50 i=1 X0.6 ≥ 40 ≈ 2.96%. Calculating (5) we get that



Q (α1 , . . . , α50 ) = 1, if #{αi = 0.6} ≥ 30.

That is, the packet loss requirement is not met if more than 29 sources have an activity factor of 0.6 (and the rest have activity factor of 0.4). Now δ can be calculated based on (6) and (1). The probability of each particular realization in case of our example is 1 [α1 = 0.4, . . . , α50 = 0.4] = 2150 . Thus 250 , e.g.



δ

=



α1 ,... ,α50





  

[Y]

Y2

=

A(µ1 ) + C(µ1 )N σ 2 + E(µ1 )N

=

A2 (µ1 ) + N σ 2 ×

1 250

 5030

of δ let us introduce Y =



Z−



N i=1

N i=1

αi

αi (1 − αi )

A(a)

=



(α − µ1 )

3



×

B(a)

=

C(a)

=

E(a)

=

!

Z − Na N a(1 − a) 1 (Z − N a)(1 − 2a) − − 2 [N a(1 − a)]3/2

!

1 N a(1 − a)

Z − Na 3 (Z − N a) (1 − 2a)2 1 + + 3/2 2 [N a(1 − a)] 8 [N a(1 − a)]5/2 1 − 2a 1 2 [N a(1 − a)]3/2 9 (1 − 2a)2 15 (Z − N a)(1 − 2a)3 − − − 5/2 4 (N a(1 − a)) 8 (N a(1 − a))7/2 1 9 (Z − N a)(1 − 2a) − . 3 (N a(1 − a))3/2 2 (N a(1 − a))5/2

. We derive δ in

the following steps: • We show that Y is normally distributed independently of the distribution of activity factors; • We determine the expected value and the variance of Y using the first moments activity factor;  of the Y ≥ Φ−1 (1 − ε) . • and calculate δ = The following two lemmas give the formal description of the above described solution method. PN Z − i=1 αi Lemma 1: The distribution of Y = qP N i=1 αi (1 − αi ) follows a normal distribution with negligible error. Sketch of the Proof: The derivation has three steps: PN PN • The random variables i=1 αi and i=1 αi (1 − αi ) are normally distributed q for large N ; PN • The random variable i=1 αi (1 − αi ) is also normally distributed; • The ratio of two normal random variables approximately follows a normal distribution too. The first step directly follows from the central limit theorem. The proof for the second step is described in Appendix in detail. The ratio of two correlated normal random variables, in our case the numerator and the denominator of Y, is studied in [21]. Hinkley examined the exact distribution and showed that

 

where µi and σ 2 denote the i-th moment and the variance of the activity factor, and

= 4.2%.

Thus, the probability of the violation of the 0.1% drop requirement is 4.2%. We now proceed to the calculation of δ for general distribution of the activity factor. For the sake of easier calculation



2

B (µ1 ) + 2A(µ1 )C(µ1 ) + N



(α − µ1 )3 + . . .

[2A(µ1 )E(µ1 ) + 2B(µ1 )C(µ1 )] + . . .

[α1 = a1 , α2 = a2 , . . . , α50 = a50 ] ×

Q (a1 , a2 , . . . , a50 ) =





Higher coefficients of the Taylor series can also be calculated, due to lack of space we do not present them here. Proof: First, we determine the Taylor expansion of the function f (x1 , . . . , xN ), which is defined as follows:

f (x1 , . . . , xN ) =





Z−

N i=1

N i=1

xi

.

xi (1 − xi )

This function is symmetric in x1 , . . . , xN and takes ∞ if at least one of the xi is 0 or 1. The symmetric property makes the Taylor expansion easy. The structure of the Taylor expansion of f (x1 , . . . , xN ) around x1 = a, . . . , xN = a is the following:

f (x1 , . . . , xN ) = A(a) + B(a) C(a)

E(a)







N

i=1

(xi − a)2 + D(a)

N

i=1

(xi − a)3 + F (a)





N

i=1

i6=j

i6=j

(xi − a) +

(xi − a) (xj − a) + (xi − a)2 (xj − a) + . . .

and the value of the coefficients is as follows:

D(a) E(a) F (a)

=

∂ f (x1 , . . . , xN ) ∂x1 ∂x2

=

1 ∂ 3 f (x1 , . . . , xN ) 3! ∂x31

=

1 ∂ 3 f (x1 , . . . , xN ) 2! ∂x21 ∂x2

"" ""

""

""

x1 =a,... ,xN =a

x1 =a,... ,xN =a

""

"" ""

0

.

f 2 (x1 , . . . , xN ) = A2 (a) + 2A(a)B(a)



N

(xi − a) +

i=1 N

2

(xi − a)2 +

# 2A(a)D(a) + B (a)$   i=1 N

2

i=1 i6=j

[2A(a)E(a) + 2B(a)C(a)]



(xi − a) (xj − a) +

N

(xi − a)3 +

i=1

[2A(a)F (a) + 2B(a)C(a) + 4B(a)D(a)] ×

  N

i=1 i6=j

(xi − a)2 (xj − a) + . . .

Now we apply the function f (x1 , . . . , xN ) to the random activity factors. Denote µ1 the expectation value and σ 2 = µ2 − µ21 the variance of the random activity factors αi , i = 1, . . . , N . We calculate [f (α1 , . . . , αN )] using the Taylor expansion of the function f (x1 , . . . , xN ):

%&

Z−



N i=1

N i=1

αi

αi (1 − αi )



=

%

% # (α − µ ) $ 3

2

αi

2

i

2

1

N

1

i

2

1

1

1

3

1

1

1

1

δ [Y ≤1x] x

0.9

1

Exact distribution Fitted distribution Proposed distribution

0.8 0.7

[Y ≤ x]

N i=1

0.8

1 0.9

2

N i=1

0.3 0.4 0.5 0.6 0.7 Activity factor of GSM speech

In Fig. 4, the distribution of α is according to GSM speech measurements [11], while in Fig. 5 it is uniform over the interval [0.48, 0.68]. Fig. 4 and 5 illustrate that the distribution of Y is close to a normal distribution even for small value of N (N = 10). Our proposed distribution is also shown in the figures. The fitted distribution is a normal distribution with expected value and variance got from the exact distribution. It is seen in Fig. 5 that the proposed and the fitted distribution run together very close to the exact distribution. The GSM speech activity factor distribution is spread over much wider interval than the examined uniform distribution, which causes the small difference between our proposed and the exact distribution in Fig. 4. However, the exact and the fitted distributions run together in this figure, which illustrates that Y has normal distribution. Note that Fig. 4 and 5 illustrate the accuracy of the proposed method for N = 10. For larger N the error is even smaller.

+ ...

1

 = % # f (α , . . . , α )$ # α (1 − α )  = A (µ ) + B (µ ) + 2A(µ )C(µ )$ N σ + % # [2A(µ )E(µ ) + 2B(µ )C(µ )] N (α − µ ) $ + . . .

0.2

Fig. 3. Density function of activity factors of GSM speech based on measurements

Similarly, the second moment of f (α1 , . . . , αN ) can be calculated as: Z−

0.1

[f (α1 , . . . , αN )] =

A(µ1 ) + C(µ1 )N σ 2 + E(µ1 )N

%' 

0

PSfrag replacements

x1 =a,... ,xN =a

Evaluating this we get the formulae of the statement. Now we provide the Taylor expansion of the function f 2 (x1 , . . . , xN ) around x1 = a, . . . , xN = a. The first terms of the Taylor expansion are the following:

# B (a) + 2A(a)C(a)$ 

2.04

0.5

x1 =a,... ,xN =a

""

1

0.58

2

1.5

1.10

=

1.04

x1 =a,... ,xN =a

1.33

""

1 ∂ f (x1 , . . . , xN ) 2! ∂x21

0.81

""

2

C(a)

2

0.74

=

0.19

B(a)

f (a, . . . , a) ∂f (x1 , . . . , xN ) ∂x1

0.13

=

Probability density

A(a)

2.5

(

0.6 0.5 0.4 0.3 0.2

PSfrag replacements

0.1

Two example distributions of the activity factor are used to illustrate the numerical properties of the proposed method. The first one is a measured GSM speech activity distribution published in [11] and depicted in Fig. 3. The second one is the uniform distribution family with various parameters.

0

δ 0.5

1

1.5

2

2.5

3

x Fig. 4. Distribution of Y: activity factors have the GSM speech distribution shown in Fig. 3, N = 10, Z = 8, ε = 0.001

The proposed admission control has fast execution time because the obtained formula includes only basic operations. Table III also shows the parameters of the usual aggregated QoS models in addition to the proposed flow level model. The admission criteria for fulfilling the aggregate QoS requirement using Gaussian approximation is expressed in [6] (see (3.13)). It has the following form with our notations:

1 0.9

Exact distribution Fitted distribution Proposed distribution

0.8

[Y ≤ x]

0.7

(

0.6 0.5 0.4

ε ≥ 1−Φ

0.3 0.2

PSfrag replacements

0.1 0

δ 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

x Fig. 5. Distribution of Y: α ∼ U (0.48, 0.68), N = 10, Z = 8, ε = 0.001

C. The proposed connection admission control algorithm Applying the results of the previous sections, the new admission control algorithm can be defined. Firstly, the input parameters of the admission control algorithm should be determined. As Table III shows, the activity factor should be characterized by its mean, variance and the third moment of the distribution, i.e. µ1 , σ 2 and µ3 . To determine these parameters measurements are required, or the distribution published in [11] can be used. The remaining input parameters are the drop requirement ε, the allowed violation probability δ of this requirement and the system capacity Z, which all are fixed system parameters. The only variable input of the admission control is the number of actual connections N . Once the input parameters for the flow level admission control are determined, we are ready to define the realtime part of the algorithm. The admission control algorithm should calculate the probability of larger drop rate than the requirement at the actual number of flows. If it is larger than its allowed maximum then the new connection is blocked, otherwise it is admitted. The actual probability of drop rate violation, δN , can be calculated according to (6) and using the results proposed in Lemma 1 as follows: δN = 1− Φ Φ−1 (1 − ε) ; µ =

)

%

[Y] , σ 2 =

% #Y $ −% 2

*

[Y]2 .

(7)

The expected value and variance of Y should be determined based on Lemma 2. If δN < δ then the QoS decision is ”Accept”, otherwise it is ”Reject”.

!

Z − N µ1 N µ1 (1 − µ1 )

QoS measure

Model parameters QoS input Activity desc. Output

MODELS

Aggregation level model The probability that a packet suffers higher delay than a predefined target delay value should be less than ε

ε µ1

Flow level model The probability that activity factors of connections are ”selected” such that the aggregated QoS requirement is violated should be less than δ N, Z ε, δ µ1 , σ 2 , µ 3 Accept or Reject

(8)

.

This expression reinforces our previous statement that activity factor is considered only with its mean value in the aggregate QoS model. It is interesting to see the δaggr probability of violating the packet loss requirement ε if the aggregate admission control algorithm is used by the same ε setting. The probability δaggr can be obtained from (7) by substituting ε from (8) assuming equality: δaggr = 1− Φ 0; µ =

)

%

[Y] − A (µ1 ) , σ 2 =

% #Y $ −% 2

*

[Y]2 ,

where A(µ1 ) is defined in Lemma 2. If [Y] equals to the first term of its Taylor series A (µ1 ) then δaggr = 50%, which means that flows violate the drop requirement with 50% probability. Now we show for two sample activity factor distributions how the capacity requirement changes if we change the number of multiplexed sources and the value of δ. Table IV and Table V show the capacity requirements if the system has to carry N multiplexed voice flows. In Table IV uniform activity factor distribution over (0.2–0.8) is used. Table V belong to the case when the activity factor distribution is based on measured GSM speech. The packet drop requirement ε = 1% and the δ probability of violating packet drop requirement runs from 0.001% to 50%. We can see that the results of the two tables are similar. We found that the small difference between the two tables can be attributed to the different mean values of the activity factor distributions (0.5 in Table IV; 0.58 in Table V). For example, if N = 40 and δ = 50% then the difference is 3 units. The capacity requirement significantly changes if small δ values (< 1%) are used instead of the δ = 50% setting. For TABLE IV C APACITY REQUIREMENT Z

FOR

N

SOURCES WITH UNIFORM ACTIVITY

FACTOR DISTRIBUTION OVER

TABLE III C OMPARISON OF THE



0.2-0.8 AND ε = 1%

Uniformly distributed activity factor over 0.2-0.8 δ 50% 1% 0.1% 0.01% 0.001% N=20 15 17 17 18 18 N=40 27 30 30 31 31 N=60 39 42 42 43 43 N=80 50 53 54 55 55 N=100 61 65 66 67 67 N=120 73 76 77 78 79 N=140 83 87 89 89 90 N=160 94 99 100 101 102 N=180 105 110 111 112 113 N=200 116 121 122 123 124

TABLE V C APACITY REQUIREMENT Z

FOR

N

UE

Node B

SOURCES WITH ACTIVITY FACTORS

DISTRIBUTION BASED ON MEASURED

GSM speech activity factor δ 50% 1% 0.1% N=20 17 18 19 N=40 30 33 33 N=60 44 46 47 N=80 56 59 60 N=100 69 73 73 N=120 82 86 87 N=140 94 98 100 N=160 107 111 112 N=180 119 124 125 N=200 132 137 138

RNC

UTRAN

GSM SPEECH AND ε = 1%

Core Network MGW

distribution 0.01% 19 33 47 61 74 87 100 113 126 139

MGW

Iub

PSfrag replacements

0.001% 19 34 48 61 75 88 101 114 127 140

example, if N = 200 and δ changes from 50% to 0.001% then the difference is 8 units at both activity factor distributions. Fig. 6 shows the plot of δ against the variance of activity factor at three capacity values (Z = 40; 41; 42). The activity factors in different cases have the same expected value (0.5) and they have uniform distribution in different ranges. The figure shows that higher variance of the activity factor causes larger δ value, i.e. more capacity is needed to achieve the same QoS. For example, if the variance grows from 0.003 to 0.03 then δ increases from 0.8% to 9.3% at Z = 40.

V. A PPLICATION Our results can be applied for example in Universal Mobile Telephony System (UMTS), of which architecture is illustrated in Fig. 7. A UMTS network consists of three main components: Core Network (CN), UMTS Terrestrial Radio Access Network (UTRAN) and User Equipment (UE). Core network provides switching, routing, management functions and transit for user traffic. A Node B of UTRAN terminates the air interface and forwards the traffic to the Radio Network Controller (RNC). Voice connections then enter the core network via a Media

PSTN/ISDN/ PLMN

δ [Y ≤ x] x

Admission Control

Fig. 7.

UMTS architecture

Gateway (MGW). They are terminated for example in a PSTN after leaving CN as shown in the figure. UTRAN is an access network for which our results can be applied directly, for example for the Iub interface which is the interface between the RNC and the Node B. The radio protocols that are used to convey the data flows result in deterministic inter-arrival times with random offsets between packets transferred through the Iub interface. The delay requirement of packets is strict and typically below the packet inter-arrival time which is 20 ms in the case of voice connections. A voice packet has approximately 5−8 ms delay requirement in UTRAN [22]. This strict delay requirement has to be cut back by the transmission delay and the remaining part can be the queuing delay. The queuing delay is small enough that only a buffer with length of some ms worth to be used. The strict QoS requirement allows us the application of bufferless models at ON - OFF time scale as explained for example in [23], [24]. In this section, we investigate the effect of the application of the proposed method in UTRAN. For this purpose the activity factor distribution of [11] is applied. This distribution is displayed in Fig. 3. Fig. 8 shows the capacity need of a flow versus the number of flows for different values of δ. The loss requirement is ε = 0.001. In the aggregated QoS case the activity factor is constant (0.58), which is the mean of the GSM speech distribution. This case is shown by dashed line in the figure,

1 1 -2

10

δ = 1% δ = 50% Aggregated QoS

-4

δ

10

-6

Z=40 Z=41 Z=42

10

-8

10

PSfrag replacements

Capacity need of a flow (Z/N)

0.95 0.9 0.85 0.8 0.75

PSfrag replacements

0.7 -10

10

0

0.005

0.01 0.015 0.02 Variance of the activity factor

0.025

[Y ≤0.03x] x

Fig. 6. δ probability vs. the variance of activity factor. Activity factor is uniformly distributed and N = 60, ε = 0.1%

0.65

Fig. 8.

0

50

100 150 200 Number of flows (N)

250

δ [Y ≤300x] x

Effect of difference in QoS requirements: ε = 0.001

where (2) gives the QoS requirement. The two other lines show the cases when the flow level QoS is considered with predefined flow level requirements, i.e. δ = 50% or δ = 1%. The jumps in these curves are due to the fact that the result is calculated as the ratio of two integer numbers (Z/N ). It can be seen in the figure that the ignorance of flow level QoS practically gives the same results as setting δ = 50% in the flow level model. That is, if aggregate QoS measures are considered only then practically half of the flows do not meet the requirement. Setting δ = 1% means that 99% of the flows have to fulfill the QoS requirement. The uppermost curve shows the results for this setting. It can be concluded from the figure, that the capacity requirement of a flow increases slightly, but the majority of the flows meet the requirements. These results show that the application of flow level QoS is needed in call admission control in order to ensure that most of the flows fulfill the QoS requirements. Our proposed method can be easily implemented in existing admission control algorithms in UTRAN. VI. C ONCLUSION The aim of our work was to develop an admission control algorithm that relies on a flow level QoS definition. We modeled the aggregation of ON - OFF voice sources with the bufferless multiplexing model. The applied flow level QoS requirement allows the violation of the predetermined packet loss requirement with a small probability. Packet loss violation occurs at the rare cases when ongoing flows have large activity factors. We proposed an admission control method that guarantees per flow QoS. The proposed method is based on Gaussian approximation applied for the bufferless multiplexing model. We have shown that the packet loss violation probability can be approximated as the quantile of a normal distribution. The parameters of the normal distribution are obtained with the Taylor expansion of a random variable, which is the function of the random activity factors αi . The obtained admission control formula includes the moments of the activity factor distribution. We analyzed the relation of the flow level admission control to the conventional aggregate level admission control method for the example of UMTS Terrestrial Radio Access Networks. ACKNOWLEDGMENT The authors would like to thank Lars Westberg, Istv´an Szab´o and Bal´azs Ger˝o for their helpful suggestions and constructive comments. R EFERENCES [1] F. L. Presti, Z.-L. Zhang, J.F. Kurose, D.F. Towsley, ”Source time scale and optimal buffer/bandwidth tradeoff for heterogeneous regulated traffic in a network node,” IEEE/ACM Transactions on Networking, Vol. 7, No. 4, pp. 490-501, August 1999. [2] J. B´ır´o, F. N´emeth, Z. Heszberger, M. Martinecz, ”Bandwidth requirement estimators for QoS guaranteed packet networks,” International Network Optimization Conference, Evry/Paris, 2003. [3] N. Benameur, S. Ben Fredj, S. Oueslati and J. Roberts, ”Quality of service and flow level admission control in the Internet,” Networks, Vol 40, pp. 57-71, 2002.

[4] G. Mao, D. Habibi, ”Loss performance analysis for heterogeneous OnOff sources with application to connection admission control,” IEEE/ACM Transactions on Networking, vol. 10, no. 1, February 2002. [5] A. M. Makowski, ”Bounding On-Off sources - variability ordering and majorization to the rescue,” Technical research report, 2001. [6] F. P. Kelly, ”Notes on effective bandwidths,” Stochastic Networks: Theory and Applications, Oxford Univ. Press, 1996. [7] 3GPP, ”Methodology for measuring impact on voice activity factor,” TR 26.9783 V4.0.0, March 2001. [8] P. Tran-Gia, K. Leibnitz, ”Teletraffic models and planning in wireless IP networks,” IEEE Wireless Communications and Networking Conference, New Orleans, 1999. [9] M. Arvedson, M. Edlund, O. Eriksson, A. Nordin, A. Furusk¨ar, ”Packet or circuit switched voice radio bearers – A capacity evaluation for GERAN,” MPRG Wireless Symposium, Blacksburg, June 2001. [10] S. Ni, ”Network capacity and quality of service management in F/TDMA cellular systems,” Ph.D. Dissertation, Helsinki University, February 2001. [11] T. Westholm, B. Olin, ”A model for GSM speech,” 2000 Symposium on Performance Evaluation of Computer and Telecommunication Systems, pp. 458-62, 2001. [12] J.Y. Le Boudec, ”Application of network calculus to guaranteed service networks,” IEEE Transactions on Information Theory, Vol. 44, pp. 10871096, 1998. [13] A. Burchard, J. Liebeherr, S.D. Patek, ”A calculus for end-to-end statistical service guarantees,” Technical Report, University of Virginia, CS-2001-19, May 2002. [14] C. Li, A. Burchard, J. Liebeherr, ”Network calculus with effective bandwidth,” Technical Report, University of Virginia, CS-2003-20, November 2003. [15] J. Liebeherr, S.D. Patek, A. Burchard, ”Statistical Per-Flow Service Bounds in a Network with Aggregate Provisioning,” IEEE Infocom, San Francisco, April 2003. [16] W. Jiang and H. Schulzrinne, ”Analysis of On-Off patterns in VoIP and their effect on voice traffic aggregation,” ICCCN, 2000. [17] E. Jugl, H. Boche, ”Effect of the fluctuation of the voice activity factor on CDMA system capacity,” International Workshop on Mobile Communications IWMC’99, Chania, Crete, pp. 310-320, June 1999. [18] F. Beritelli, S. Casale, G. Ruggeri, S. Serrano, ”Performance evaluation and comparison of G.729/AMR/Fuzzy voice activity detectors,” IEEE Signal Processing Letters, Vol. 9/3, pp. 85 -88, March 2002. [19] F. Beritelli, S. Casale, A. Cavallaro, ”A robust voice activity dDetector for wireless communications using soft computing,” IEEE Journal on Selected Areas in Communications, Special Issue on Signal Processing for Wireless Communications, vol. 16, N. 9, December 1998. [20] L. Sun, E.C. Ifeachor, ”Perceived speech quality prediction for voice over IP-based networks,” IEEE International Conference on Communications, New York, pp. 2573-2577., April 2002. [21] D. V. Hinkley, ”On the ratio of two correlated normal random variables,” Biometrika, 56, pp. 635-639, 1969. [22] 3GPP, ”Delay budget within the access stratum,” TR 25.853 V4.0.0, March 2001. [23] S.-Q. Li, ”Study of information loss in packet voice systems,” IEEE Transactions on Communications, vol. 37. no. 11, November 1989. [24] S. Malomsoky, S. R´acz, S. N´adas, ”Connection admission control in UMTS radio access networks,” Computer Communications, Vol. 26, pp. 1907-1917, 2003.

A PPENDIX Assume an XN normally distributed random variable with 2 expectation µN√and variance σN . We are interested in the distribution of XN if µN → ∞. The Taylor expansion of the square root function at µN is

)

*

√ √ 1 (x − µN )2 1 x − µN x = µN + + + O (x − µN )3 . √ √ 2 µN 8 µN µ N

Substituting x with the random variable XN in the series we get

)

√ √ 1 (XN − µN )2 1 XN − µ N + + O (XN − µN )3 XN = µ N + √ √ 2 µN 8 µN µ N 2 √ σN σN ∼ µN + √ Φ(x) + χ2 , √ 2 µN 8µN µN 2

where the χ22 denotes the chi-square distribution with two degrees of freedom. PN In our case XN has to be substituted with i=1 αi (1−αi ). 2 The coefficients of χ2 in (9) tends to zero if N → ∞: 2 N (µ2 − 2µ3 + µ4 − µ21 + 2µ1 µ2 − µ22 ) σN → 0, = √ µN µ N N 3/2 (µ1 − µ2 )3/2

where µj denotes the j-th moment of the activity factor. Finally thisPmeans that for large N the square root transN formation of i=1 αi (1 − αi ) tends to a normal distribution √ σN with expectation µN and variance 2√ µN .

*