Accurate resource estimation for homogeneous VoIP aggregated traffic

Computer Networks 52 (2008) 2505–2517

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Accurate resource estimation for homogeneous VoIP aggregated traffic q Antonio Estepa, Rafael Estepa * Escuela Superior de Ingenieros, C/Camino de los Descubrimientos s/n, University of Sevilla, Spain

a r t i c l e

i n f o

Available online 26 April 2008

Keywords: Voice over IP (VoIP) QoS CAC On–off model MMPP Fluid model CNG

a b s t r a c t Modern VoIP codecs like G.729, G.723.1 or AMR can generate traffic during voice inactivity periods for Comfort Noise Generation (CNG). This feature alters the classical on–off pattern typically used to model the traffic generated by codecs with a Silence Suppression scheme. Therefore, the traffic generated due to CNG leads to severe inaccuracies in the dimensioning analysis done through traditional models based on multiplexing on–off sources like MMPP or fluid model. This paper addresses the VoIP dimensioning issue. First, we extend the traditional MMPP and fluid analytical models to include those traffic sources which perform the CNG feature. Second, we propose a simple but efficient algorithm which can be applied in dimensioning or admission control to find out the bandwidth reservation required to guarantee delay and loss in a packet-switch multiplexer node for VoIP traffic. Results are validated by simulations and VoIP traces and demonstrate a significant improvement in accuracy with respect to current on–off-based approaches. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Current IP network managers have at their disposal a set of QoS-related mechanisms to achieve VoIP end-toend network performance guarantees within its managed network domain [22]. QoS architectures defined for the IMS [8,31] or large ISP networks [33,30] use different techniques like sharping, classifying, traffic engineering, etc. However, it is important to note that resource reservation still needs to be done in the multiplexing nodes along the media path to ensure minimum service guarantees to realtime traffic. Resource reservation demands accurate estimation of the bandwidth to be granted to VoIP aggregated traffic, which depends on the terminal’s characteristics (e.g., codec, packet size, etc.) and the performance bounds (i.e., loss and delay) admissible for the service at each multiplexer

node [12]. Currently, two main approaches are used for VoIP bandwidth estimation:  CBR approach: it assumes that constant bit-rate (CBR) codecs are used (e.g., G.711) and consequently the aggregated bandwidth is obtained by adding each codec’s peak rate with the corresponding overhead. CBR codecs, although acceptable within LANs1 where network resources are not problematic, are not optimized for WAN links where network resources can be scarce.  VBR approach: it assumes that VoIP codecs can perform Silence Suppression and consequently, act as variable bit-rate (VBR) sources which are often modeled as on– off sources. The Silence Suppression feature allows bandwidth savings of about 50% and takes advantage of statistical multiplexing, which is desirable in WAN links. The modeling of voice traffic with on–off sources started in the early stages of ATM and is backed by two

q This work has been partially supported by grants 1c-006 of Minerva project and 07/130 of Corporacion Tecnologica de Andalucia (CTA). * Corresponding author. Tel.: +34 954487384; fax: +34 954487385. E-mail addresses: [email protected] (A. Estepa), [email protected] (R. Estepa).

1389-1286/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comnet.2008.04.012

1 A remarkable exception is the iLBC codec which is well suited for Internet communications since it performs an outstanding loss recovery mechanism.

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main analytical approaches: fluid model and Markov– Modulated Poisson Process (MMPP) queues [2,14,28]. The MMPP approach is more exact at the cost of demanding higher computational resources when compared with the fluid model, which is typically preferred as a less complex solution. Validation works for current VoIP codecs like G.729 [17,6,9,1] show that, with the proper parameters values (e.g., number of frames per IP packet or the average on and off periods which depends on the codec’s VAD algorithm) and accounting the VoIP overhead (i.e., RTP/UDP/IP headers), the classical on–off multiplexing models [32,3] are still valid for most of current codecs when the silence suppression feature is used. However, today’s VoIP terminals also include new features like Comfort Noise Generation (CNG) to improve the naturalness of conversations. This latter feature is supported by Silence Insert Descriptor (SID) frames which capture the sound of speaker’s background during every voice inactive period. SID frames are generated at the beginning of, and in response to background noise changes, at speech–silence periods [15]. From now on we will use the term Generalized VoIP sources (GVoIP) for VoIP sources which generate SID frames since they constitute the most general case of VoIP traffic pattern. Popular speech codecs like G.729, G.723 or AMR include the CNG feature (i.e., are GVoIP sources) and consequently, send traffic during speech–silence periods. As disclosed in our earlier work [3,10], analytical models based on on–off sources multiplexing are not suited to model the aggregated traffic generated by GVoIP sources since they assume that no traffic is sent during off periods. This results in infra-provisioning WAN links for VoIP usage or equivalently, admitting more VoIP simultaneous connections than those admissible to keep a desired QoS level. In this paper, we define an updated VoIP dimensioning framework which can be used to obtain an accurate estimation of the resources required to guarantee bounds in delay or packet loss for GVoIP sources (i.e., when CNG is used). This implies the fulfillment of two objectives: First, to adapt the current on–off multiplexing models to GVoIP traffic characteristics (i.e., generating traffic during off periods). Second, to find a direct way to estimate the bandwidth requirement in a multiplexer node as a function of the number of multiplexed GVoIP sources and a desired loss and delay performance. The remainder of this paper is as follows. Section 2 overviews current dimensioning approaches based on on–off multiplexing analytical models. Next, in Section 3 we define a new traffic model for our generalized VoIP source and address the extension of previous analytical models to include GVoIP sources. Section 4 proposes an algorithmic dimensioning method based on the analytical models developed in previous section. Sections 5 and 6 provide the experiment and results which validate our proposals and finally, Section 7 concludes the paper.

2. Dimensioning VoIP networks Our VoIP dimensioning problem has to do with finding the bandwidth which needs to be reserved for the aggregated traffic of N VoIP flows to provide performance guar-

antees (i.e., a maximum delay and packet loss) in a multiplexer node. We will assume that the N voice sources are homogeneous; namely, they have the same peak bit rate (R), and for VBR sources, the same activity rate (qON) and mean sojourn time in the off (or on) state. As stated in the Introduction, one common practice is to reserve the bandwidth equivalent to the sum of conversation’s peak rate (i.e., C = N  R). This approach assumes the use of CBR sources, which can be well suited to high-speed LANs [25], but it leads to an inefficient use of the resources when multiplexing on–off sources. In WAN environments, codecs with Silence Suppression schemes (typically modeled by on–off sources) can be used to achieve bandwidth savings. To obtain dimensioning models for VBR traffic we need to make use of on–off multiplexing analytical models [2,5], which can belong to one of the following families: fluid model or MMPP queues. The solutions of these models offer the overflow probability for a buffer which depletes at a constant rate in a multiplexer node loaded with the traffic from N on–off sources. For the sake of completeness, the basics of fluid and MMPP models and the main approaches used in practise have been summarized in Appendix A. The reader who is not familiar with these models is encouraged to review the appendix at this point since its main concepts and terms will be used in the rest of this paper. However, in the dimensioning problem, we are not directly interested in finding the performance of a multiplexer node given a fixed bandwidth as a function of the buffer size, but in the inverse question: finding the bandwidth reservation and buffer size2 to be set in the multiplexer node. Preliminary applications for VoIP system dimensioning can be found in the validation works done using fluid models [6,26,29] or by Andersson [1] with Baiocchi’s MMPP simplified model [3], which are done over IP networks and provide figures of the packet loss for different output capacities expressed as a percentage of the aggregated traffic peak rate. A direct analytical expression for the capacity has been found in Guerin’s equivalent bandwidth [13], where the author provides an approximation that seems to be valid for a number of cases. His approach is based on the asymptotic behavior of the AMS fluid model (see Appendix A.1) which assumes an infinite-size buffer at the multiplexer. The expression of loss probability G(m) for a buffer of size m is simplified as GðmÞ ¼ ez0 m . By substituting the wellknown expression for z0 (Eq. (A.9)) and solving for C, we find:

C¼

  mNðk þ lÞ RN þ 2lnGðmÞ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 mNðk þ lÞ RN mRN2 k : þ  þ 2lnGðmÞ 2 2lnGðmÞ

ð1Þ

When the effect of statistical multiplexing is of significance, Guerin’s equivalent bandwidth allows the calculation of C by using a bufferless approach which assumes that the instantaneous throughput of the aggregated traffic 2 The buffer size also imposes a maximum for the buffering delay in the node.

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holds a Gaussian distribution centered in the aggregated mean bit rate and, consequently, the packet loss probability is given by the tail area of the Gaussian distribution [13,20]. Guerin’s equivalent bandwidth offers a fair dimensioning approach for a large number of on–off sources at an extremely low computational cost, however it inherits the bandwidth overestimation of the fluid model whenever infinite or zero buffer is considered. 3. New VoIP scenario Generalized VoIP (GVoIP) sources are equipped not only with VAD algorithms but also with Discontinuous Transmission algorithms (DTX) which determine, during silence periods, whether to send SID frames or not to feed the Comfort Noise Generation feature [15]. This section is devoted to the traffic generated by GVoIP sources. First, we describe the traffic pattern followed by one GVoIP source as well as its mean bit-rate as deduced in [11]. Second, we address the more complex task of multiplexing GVoIP sources by introducing changes in the fluid and MMPP models.

H ROFF ¼ RSID OFF þ ROFF :

ð4Þ

As deduced in our earlier work [11] by applying the Elementary Renewal Theorem and considering that SID frames are packetized according to RFC 3551 [27] scheme,3 the analytical expression for ROFF is

ROFF ¼

Nfpp  P1  ðNfpp  1Þ SSID H ; þ T  E½X T  E½X Nfpp

ð5Þ

where X is a random variable that models the inter-arrival time between SID frames (in number of frame periods), P1 is probability of having consecutive SID frames (i.e., P{X = 1}), and SSID is the size of a SID frame. If we define c as the ratio ROFF/R, then R can be expressed as

R ¼ R  ½c þ ð1  cÞpON :

ð6Þ

Therefore, a GVoIP source will be defined by the quadruplet (k, l, R, c). Typical values for c can be calculated from Table 1 and range from 0.1 for the G.729 codec to 0.03 to G.723 codec or 0.05 for the AMR codec, although the actual value depends on the number of frames per packet (Nfpp) configured in VoIP clients.

3.1. Traffic pattern and mean bit-rate of one GVoIP source

3.2. Multiplexing process of GVoIP sources

In VoIP, a sequence of Nfpp consecutive codec frames are sent in a single IP packet every Nfpp  T seconds while the source is in on state, where T is the codec’s frame inter-arrival time in seconds. According to the on–off model, the source bit-rate during on periods (peak bit-rate) is

For dimensioning purposes, we are interested in the study of the multiplexing process of N homogeneous GVoIP sources. From now on, C will denote the multiplexer’s output link capacity (b/s) expressed as a percentage a of the peak rate of the aggregated traffic instead of absolute values:

R¼

Nfpp  SACT þ H ; Nfpp  T

ð2Þ

where H and SACT are the sizes of the packet’s header and of a frame containing speech information (also called active frame or ACT frame), respectively. Taking into account the packets containing SID frames and considering pON as the percentage of time that one source is in active state, the mean bit rate of one source can be expressed as

R ¼ pON  R þ ð1  pON Þ  ROFF ;

ð3Þ

where ROFF is the source bit rate during off periods. The packet generation process of a GVoIP source can be clearly observed in Fig. 1. The computation of ROFF requires the consideration of the contribution of SID frames along with the packet headers that are sent with the SID frames.

ACT frame (SACT) SID frame (S SID )

H T off

H

H Ton

ð7Þ

This notation, already used in [17,6] will let us have a direct view of the statistical multiplexing gain and facilitate the comparison between different codecs. Obviously, the stability condition now imposes the following constraint:

N  R < C;

ð8Þ

where R is that from Eq. (6). With SID frames arise the problem of finding the percentage of the multiplexer’s packet loss that belongs to each type of frames (ACT or SID). Because of the small number and size of SID frames the portion of overall losses (G(m)) due to the ACT packet is almost coincident with the overall packet loss. This is shown in Fig. 2, where we plot the loss distribution found in simulations fed by real traces. Therefore, we adopt the following fluid-based approximation:

GðmÞ 

F ACT L F

ACT

þF

SID

¼

F ACT L F

ACT

ON Þc ð1 þ ð1p Þ p

;

ð9Þ

ON

which can be interpreted as the ratio of the steady-state fluid loss rate (FL) to the steady-state offered fluid rate

X H

C ¼ a  N  R:

H t

Fig. 1. Generalized VoIP source packet generation process during on and off periods (Nfpp = 2).

3 The scheme assumes that one IP packet is sent every time that there is a gap (X > 1) between consecutive SID frames. For consecutive SID frames (X = 1), packets are sent every Nfpp  T seconds.

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Dii ¼ R  ði þ ðN  iÞ  cÞ  C;

Table 1 Codec characteristics Codec

Mode

SACT(B)

SSID(B)

T (ms)

E[X]

P1

pon

G.729 G.723.1

– 6.3 5.3 4.75 12.2

10 24 20 12 31

2 4 4 5 5

10 30 30 20 29

7.33 13.05

0 0.27

0.456 0.471

0 0

0.469

AMR

7.47 7.47

Experimental Packet Loss type distribution

100

10—1

Probability

10—2

R  ði þ ðN  iÞ  cÞ > C:

ð13Þ cN bC=R c 1c

Consequently, now there is exactly N  negative eigenvalues. Following the same solution as in AMS, which assumes that the multiplexer node has a infinite-size buffer, the new eigenvalues z are the solution of the following equation:

k ¼ 0; 1; . . . ; N;

ð14Þ

where

AðkÞ ¼ ð1  cÞ2 ðN=2  kÞ2  ½c  ðN=2Þð1 þ cÞ2 ; BðkÞ ¼ 2ðl  aÞðl  kÞðk  N=2Þ2 þ Nð1 þ kÞðc  N=2ð1 þ cÞÞ;

10

CðkÞ ¼ ð1 þ kÞ2 fðN=2  kÞ2  ðN=2Þ2 g:

10—4

The asymptotic behavior results as

—3

j k cN ! N C=R 1 1c Y NR

10—5

10—6

10—7 0

ð12Þ

The boundary equations also change. Now the buffer content will increase during periods in which the number of active sources i is such that:

AðkÞz2 þ BðkÞz þ CðkÞ ¼ 0;

ACT Packet Loss (N=20,α=0.65) Overall Packet Loss (N=20,α=0.65) SID Packet Loss (N=20,α=0.65) ACT Packet Loss (N=40,α=0.65) Overall Packet Loss (N=40,α=0.65) SID Packet Loss (N=40, α=0.65)

i ¼ 0; 1; . . . ; N:

GðmÞ ¼ ez0 m 10

20

30

40

50

60

70

80

90

Fig. 2. Overall packet loss and its components for 20 and 40 sources and a = 0.65.

zi ; zi þ z0

ð15Þ

where zi is the new value of the eigenvalues which can be found from Eq. (14) and the new load of the system is the term in brackets. In particular, the new expression for z0 (larger negative eigenvalue) is

z0 ¼  (F). Both FL and F include (losses or arrivals of) packets with both ACT frames (F ACT and FACT, respectively) and SID L SID SID frames (F L and F , respectively). Therefore, although typical results from traditional analytical models provide the overall packet loss probability G(m), we will use (9) to relate it to the loss probability of those packets which carry voice frames (i.e., GACT  ACT F ACT ). L =F 3.2.1. Fluid model for GVoIP sources Taking the original AMS model [2] as starting point, we extend it to include the GVoIP source model addressed previously. To simplify the description we will only mention the modifications with respect to the approaches overviewed in Appendix A. The basic difference with respect to the original AMS model is that when there are i active sources, the buffer’s fill rate now needs to consider the amount of fluid generated by the N-i sources in the off state. Thus, the diagonal matrix D of depletion rates has now the following entries4

4 Since the fluid model is a balance of incoming fluid versus draining fluid, another option to get the same system dynamic would be adopting the original models but use the following values for R and C:

and adopting the original expressions exposed in Appendx A.

i¼1

100

buffer size

R0 ¼ Rð1  cÞ; C 0 ¼ C  N  ROFF ;

C

ð10Þ ð11Þ

2ð1  cÞðl  kÞðN=2Þ2 þ Nðl þ kÞðc  N=2  ð1 þ cÞÞ ð1  cÞ2  ðN=2Þ2  ðc  N=2  ð1 þ cÞÞ2

:

ð16Þ For the finite buffer case (Tucker’s model) (A.6), the approximation for the packet loss probability is to be replaced by

PN l i¼

GðmÞ ¼

C=RcN 1c

m ½R  ði þ ðN  iÞcÞ  Cui ð17Þ

:

NR

Regarding to the asymptotic behavior of Tucker’s model (Pallares approach [21]), now the buffer occupancy incN creases when more than N  bC=R c sources are active, 1c the new rate of active and inactive sources, and the new value for z0. Therefore,

GðmÞ ¼ ða0  1Þ PN

i¼0

C

pi Rði þ ðN  iÞcÞ

!

1 ;

ð18Þ

where a0 is

1 11 0P Nj k ðRði þ ðN  iÞcÞ  CÞpi cN C C B B i¼ C=R þ1 1c C z0 m C B B j k a0 ¼ 1  B1 þ B Ce C : A A @ @ i¼ C=RcN P 1c ðRði þ ðN  iÞcÞ  CÞpi i¼0 0

ð19Þ Finally, since the loss of ACT frames (and packets) constitute the main impact in the perceived quality, it is worthwhile to consider the loss approximation stated in Eq. (9):

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  ð1  pON Þ  c : GACT ðmÞ  GðmÞ  1 þ pON

ð20Þ

3.2.2. MMPP Model for generalized VoIP sources Following the MMPP approach described in Appendix A, namely the fact that the analysis was based on an associated MMPP/D/1/K queue, which has deterministic service time, and in order to keep the analysis as simple as possible, we are going to assume that all packets (ACT or SID type) have the same service time, viz. all packets have the same length. Accordingly, we set the packet service time, h, to the service time needed to process a ACT packet loaded with Nfpp frames of type ACT, i.e.,

Nfpp  SACT þ H Nfpp  T h¼ ; ¼ C Na

ð21Þ

where we have taken into account (7) and (2). We next derive the model’s packet emission rate during ON and OFF periods, respectively, LON and LOFF. In order to fit the packet emission rate during OFF periods, we should take into account that we are using ACT packets as the packet reference, namely to fit the mean service time, in the model we are building. Accordingly, the real packet (with SID frames) emission rate during OFF periods must be decreased, to address the fact that these packets are of smaller size than the ACT packets, by the factor c, leading to the relation LOFF = c  LON. Thus, the model’s packet emission rate during ON and OFF periods are, respectively,

LON ¼

1 Nfpp  T

and LOFF ¼

c Nfpp  T

:

i þ ðN  iÞ  c : Nfpp  T

ð23Þ

The mean offered load bit rate of the model is

N  R ¼ N  R  ½pON þ ð1  pON Þ  c;

ð24Þ

where we have taken into account (3). In view of (7), this implies that the stability condition for the infinite buffer case, q ¼ N  R=C < 1 is:

q¼

pON þ ð1  pON Þ  c

a

< 1:

ð25Þ

We now address the simplification to the MMPP with two phases. First we need to change the borderline between the OL and UL states. Q should now be defined as the highest index i of the original MMPP such that the bit arrival rate on phase i is smaller or equal to the multiplexer output bit rate capacity, i.e., Kii[Nfpp  SACT + H] < C. Thus, in view of (23) and (2), we have

Q¼

  C  N  ROFF : R  ROFF

kOL ¼ LON 

N X i¼Q þ1

kUL ¼ LON 

Q X i¼0

½i þ ðN  iÞ  c  PN

bi

i¼Qþ1 bi

bi ½i þ ðN  iÞ  c  PQ

i¼0 bi

:

ð27Þ ð28Þ

Finally, as in the previous section, the rate rUL is computed in such a way that the mean packet arrival rate in the new MMPP equals that of the original MMPP, i.e.,

kOL rUL þ kUL r OL ¼ N  ½LON  pON þ LOFF  ð1  pON Þ: rOL þ r UL

ð29Þ

As, in view of Eq. (22), LOFF = c  LON, the previous equation leads to

r UL ¼ r OL 

N  LON  ½pON þ ð1  pON Þ  c  kUL : kOL  N  LON  ½pON þ ð1  pON Þ  c

ð30Þ

The packet loss rate for a buffer of size K (which in MMPP terminology is named P(K)) given by Eq. (A.7) represents the overall loss rate, including packets containing SID frames. To obtain an approximation for the loss rate of packets containing ACT frames, we apply to Eq. (A.7) the same transformation that was used to obtain Eq. (20), namely



PACT ðKÞ ¼ PðKÞ  1 þ

 ð1  pON Þ  c : pON

ð31Þ

ð22Þ

Moreover, taking into account the emission rate of both ON and OFF sources, and since when there are i ON sources there are conversely N  i OFF sources, the mean arrival rate while the MMPP phase process is in state i is

Kii ¼ i  LON þ ðN  iÞ  LOFF ¼

the OL and UL states, respectively, kOL and kUL, need to take into account the packets with SID frames, which may be done by taking into account (23), leading to

4. Dimensioning the VoIP access node Guerin’s equivalent bandwidth method can be readily updated for GVoIP sources by using expressions deduced in Section 3.2. However, it would still inherit the lack of accuracy of the infinity or zero queue size from the original model, specially for small buffers. In this section, we present a simple but efficient dimensioning method based on inverting the loss expression of any of the models defined in previous section for multiplexing GVoIP sources. 4.1. Problem description Today’s routers allow the creation of several logical queues and the use of scheduling algorithms to guarantee a minimum bandwidth which, in our case, could be the capacity dedicated to VoIP traffic. For the sake of simplicity, we will model a router as a multiplexer with one output queue of size (m) which will determine the maximum delay allowable in the router. The dimensioning problem requires to find the minimum bandwidth reservation (C) that guarantees performance bounds (i.e., a maximum admissible delay, Dmax, and an average packet loss, PLtarget) in our access router to VoIP users.

ð26Þ

After Q is computed by the previous formula, rOL is then computed following the procedure described in Section 2. However, the computation of the packet arrival rates in

4.2. Proposed algorithm We propose to use an algorithmic solution to invert the loss expression, G(m), deduced in Section 3.2. Since

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the relationship between the capacity and the packet loss is a monotonous decreasing function, we will use an iterative algorithm based on the Bisection method [7] to obtain a solution for C, which will be expressed as a percentage (a) of the peak bit rate of the aggregated traffic (i.e., C = aNR). Our algorithm first sets the queue size, m, by imposing a maximum delay on the node, Dmax, and then iterates to find the a which yields an average target packet loss, PLtarget, admitting a relative error less than dmin. The algorithm is as follows: Algorithm 1. Packet loss algorithm Input: PLtarget,dmin,Dmax,N,GVoIP parameters Output: C 1: amin Ü c + (1  c)  pON; amax Ü 1 2: while (abs(d) > dmin) do 3: a Ü (amax + amin)/2 4: C Ü a  NR 5: m Ü C  Dmax 6: G(m) Ü computePacketLoss(m, N, C, GVoIP parameters) 7: d Ü (G(m)  PLtarget)/PLtarget 8: if d > 0 then 9: amin Ü a 10: else 11: amax Ü a 12: end if 13: end while

The maximum number of iterations n required to achieve a certain target packet loss with an error tolerance 1amin of dmin is log2 ðdmin Þ. For example, for a target packet loss PLtarget 3

of 10 with a relative error of 1%, a maximum of 16 iterations would be needed. In each iteration, the computation of the loss probability is done and its complexity depends on the actual method used in this calculus (e.g., fluid or MMPP, asymptotic behavior, finite buffer-size, etc.). We have chosen the Bisection method for simplicity, but our algorithm can be readily adapted to implement any other root-finding method [7] (e.g., Newton’s method, secant method, etc.) to decrease the number of iterations when timing is a constraint. We believe that given our small searching interval and the range of loss of interest, our algorithm offers a fair performance for most practical applications. 5. Validation Our validation process follows a methodology similar to [17] and can be summarized as follows:  We have recorded a set of conversations (a total of 500 min) from an ISDN line. This material (raw format conversations) has been encoded using the SID-capable codecs G.729, G.723.1 and AMR, obtaining a set of text files (named ftype files.) The content of these ftype files is a sequence of identifiers {0, 1, 2} which indicates for every frame encoded (i.e., every T) the type of frame to be generated at every instant of the conversation (i.e., active, sid or no frame, respectively). We used these ftype files for three main purposes:

– To obtain realistic values for the source-related parameters which have been used in the simulations and analytical models (e.g., l, k, pON, E[x], P1, c, etc.). – To measure the actual mean bit rate (R) for each conversation at the codec level and after the packetization process. Note that by ignoring those packets generated with SID frames we can compare the results obtained for GVoIP sources and the traditional on–off sources. – As input for a multiplexing process that will emulate the queue behavior.  Then, we emulate the multiplexing process of N GVoIP sources selected randomly from our ftype pool. In order to do this, we have programmed a discrete-event simulator in Matlab which emulates the behavior of a tokenbucket multiplexer which represents the queue under study. The program follows these basic steps: – Take a random selection of N conversations (ftype files) of 3 min-length each one. – The multiplexing of selected conversations by means of a token bucket system with parameters: C = aN R and m for the token fill-rate and bucket size, respectively. The formation of RTP/UDP/IP packets loaded with Nfpp frames is done previous to the multiplexing process. – The output of the program will be the average packet loss (marking the percentage due to SID packets and ACT packets). These simulations fed with traces are tagged as trace in our results. Finally, in addition to our traces, we believe that it would be interesting to create a GVoIP traffic source which can be used in conventional discrete-event simulators for GVoIP sources multiplexing. Thus, in the opnet [23] simulator, we have modified the default on–off source model defined for VoIP applications so that it generates SID frames during off periods. The inter-arrival time between SID frames has been set to follow an exponential distribution matching our experimental value E[X]. We are interested in finding the validity of using this artificial source for our dimensioning purposes. Fig. 3 shows the results of the multiplexing process (losses in a finite-size buffer multiplexer) when traces and our simulated GVoIP sources are used. Simulations and traces do not exhibit significant differences (especially if we consider that voice does not follow exactly an exponentially distributed on–off pattern [17]). This implies that the exponential distribution is an acceptable approximation for the inter-arrival time between SID frames (Pi). The results from opnet simulations will be tagged as simulation in our results. Each multiplexing process (either in simulations or traces) was repeated 40 times to obtain accurate results.5

5 The results presented for the average packet loss have a sample standard deviation that meets that the confidence interval (ac.i. = 0.05) is always smaller than the 10% of the mean value in the region of interest.

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10

average loss probability

10

10

10

10

0

Traces vs simulations for GVoIP aggregated traffic

Dmax ¼

trace N=30 simulation N=30 trace N=20 simulation N=20 trace N=10 simulation N=10

—1

m¼

—3

—4

10

20

30

40

50

60

ð32Þ

Therefore, the queue size necessary to obtain a given Dmax will be

—2

0

m  ðH þ Nfpp  SACT Þ : aNR

70

Buffer size (m, number of packets)

Fig. 3. Multiplexing process of traces versus simulations with opnet.

6. Results The results shown in this section are aimed to validate the algorithm and models previously presented. We choose the codec G.729 (see Table 1) since it is highly available in VoIP environments and widely referenced in bibliography. The mean on and off sojourn time measured for this codec was 0.3201 and 0.4019 respectively, and the value for c was 0.1073. Nfpp for the results presented is one. Whenever we plot the packet loss it will denote the ACT packet loss from Eq. (9) so that results can be compared with those obtained with the traditional models based on on–off sources multiplexing. Since we have a large range of potential values to use in the system parameters (i.e., buffer size, target packet loss, etc.) we believe that it is adequate to start this section with a preliminary analysis of the range of interest for these parameters. 6.1. Region of interest There is a range of values for end-to-end loss and delay which are generally agreed to be the bounds of admissible QoS for a conversation [16]. For example, an average packet loss larger than 10% makes it extremely difficult to reproduce a normal conversation (even for those codecs equipped with PLC algorithms [24]). Conversely, we consider that losses below .01% (e.g., one packet every 100 s for the G.729 codec with Nfpp = 1) are negligible [16]. Consequently, we will set our output capacity and buffer size to obtain losses within the previous range. The buffer size has a direct implication not only in the losses, but also in the maximum delay possible in the node. Long delays are only feasible with large buffers and small buffers would increase losses for a large number of multiplexing sources. Therefore, we need to analyze reasonable values for the maximum and minimum buffer size:  The maximum delay admissible in a node6 will be given by the time taken to deplete a buffer of size m, which is

  Dmax  a  N : N fpp  T

ð33Þ

We will use this equation as the upper limit of the queue size and hence, limit the maximum delay allowable in the multiplexer node. We take Dmax as a conservative approach of the average delay in the node. In our results we have used a value of 50 ms7 for Dmax.  Minimum buffer size: the buffer size is not only limited in its upper bound by Dmax, but also in its minimum size (mmin) by the own multiplexing process. According to the capacity of the output link, we should be able to transmit up to a  N conversations continuously. However, since there is only one server, if we suppose that the first source content gets the link’s server, then the minimum buffer size (in bits) imposed by the serialization process would be

mmin ¼ a  N  1:

ð34Þ

The quantity above is the minimum buffer size that allows the complete usage of the output link capacity (i.e., m P mmin). For example, for a = 0.65 the mmin values would be 5.5, 12 and 18.5 for N values of 10, 20 and 30, respectively. For a given queue size, the losses in the multiplexer will depend on the output capacity (i.e., C = a  NR). Therefore, we have chosen a values which guarantee that the resulting losses will be approximately in the desired range (i.e., between 0.1 and 1e04). 6.2. Numerical results related to Generalized VoIP sources multiplexing We have worked with two families of analytical models: MMPP and fluid. MMPP based-models are a closer approach to the physical phenomenon since they include the packetization process not included in the fluid-based models. However, this is only relevant for small values of N since as the number of sources grows the effect of the packet arrivals is smoothed. In addition, the computational complexity of the MMPP model grows with both N and buffer size while the fluid model only depends on N. Therefore we will prefer the use of the MMPP model for small values of N (e.g., N 6 30) whereas in other case we will use the fluid-based approaches. To measure the fitness improvement of the proposed adapted models over the original models we will use the Mean Percentage of Absolute Relative Error (MPARE) fitness criteria, which is defined as

Pm MPARE ¼

i¼mmin

jPLtrace PLmodel j i i PLtrace i

m  mmin þ 1

;

ð35Þ

6

Regarding to average delay in the node, our analysis does not offer a direct analytical expression for it. However and expression for average delay can be readily deduced from existing approximations for both the fluid model [32] and MMPP queues [5,34].

7 Although sometimes the complete range of m is not presented since the real measurements drops drastically.

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A. Estepa, R. Estepa / Computer Networks 52 (2008) 2505–2517

Fig. 4. (a) Packet loss probability for proposed MMPP model (GVoIP) vs traces and traditional MMPP (Baiocchi) on–off model for a = 0.65 and N = 10, (b) N = 20 and (c) N = 30. (d) Fluid model approaches for GVoIP vs traces and traditional Tucker on–off model for N = 30 (a = 0.65), (e) N = 40 (a = 0.6) and (f) N = 50 (a = 0.6).

where mmin and m are the maximum and minimum queue size of the region interest, and PLtrace and PLmodel denote the i i trace simulated loss ratio and the model predicted steadystate loss probability for the value i of the buffer size inside the range [m, mmin]. Fig. 4 plots the average packet loss predicted by the multiplexing models proposed in Section 3.2 for GVoIP sources and the average packet loss of simulations fed with traces8 (conversations recorded with SID-capable codecs). Sub-plots (a), (b) and (c) show the predicted loss rate of the proposed MMPP-based analytical model for N values of 10, 20 and 30, versus the values found for the traces recorded. The MPARE criteria yields values of 35.71(%), 36.42(%) and 39.12(%), respectively, within the buffer region of interest. The same sub-plots also include the original on– off-based MMPP analytical model which offers MPARE values of 61.45(%), 69.24(%) and 87.15(%), respectively. Sub-plots (d), (e) and (f) in Fig. 4 use the fluid-based analytical models for GVoIP sources for N values of 30, 40 and 50 sources, respectively. We plot all fluid approaches (finite – Tucker and Pallares – and infinite buffer – AMS) but we only use the more accurate finite buffer approaches for MPARE calculations. Results present a MPARE of 37.32(%), 41.29(%) and 53.87(%) for the extended Tucker GVoIP model in sub-plots (d), (e) and (f), respectively whereas the values obtained for the original on–off Tucker model yields a MPARE of 88.21(%), 121.57(%) and 212.43(%), respectively, and Pallares GVoIP extension yields MPARE values of 49.86(%), 38.79(%) and 93.20(%), respectively. Tucker’s finite buffer model offers the most accurate results. However, Pallares approach offers also 8 The 95% confidence interval for the trace simulation is one order of magnitude less than the average value so we do not need to plot this confidence interval.

accurate results at a lower computational cost. In addition, sub-plots (c) and (d) let us compare the goodness of MMPP and finite buffer fluid models for GVoIP for N = 30. The MPARE criteria shows that both perform a fair approach to the traces although the fluid model (especially Pallares) has lower computational cost. We can conclude that the multiplexing models proposed in Section 3.2 are a good approximation for those cases where SID frames are being transmitted, especially if we take into account that real conversations do not follow strictly the assumption of exponentially distributed on–off periods [17]. 6.3. Numerical results related to dimensioning Generalized VoIP sources The results presented in this subsection let us answer the dimensioning problem raised in this paper: find the resources (in terms of bandwidth C) necessary to guarantee performance bounds to GVoIP aggregated traffic. The capacity C(b/s) = aNR let us have a direct view of the bandwidth reservation required for VoIP users whereas a (which can be viewed as the statistical multiplexing gain) let us compare different codecs and have a direct view of the bandwidth savings that represents our dimensioning algorithm versus the common approach of reserving the peak bit-rate (i.e., a = 1). Therefore, since both values (i.e., C and a) are useful, we will use both indistinctly in the results presented. In addition, we will use a Dmax value of 50 ms as input in the dimensioning algorithm presented in Section 4, which is used for the results offered in next figures. Fig. 5 answers the dimensioning problem addressed in this paper. The plot sets the bandwidth required for N GVoIP sources to have a certain loss probability. We have

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A. Estepa, R. Estepa / Computer Networks 52 (2008) 2505–2517 (a) Dimensioning example (Fluid GVoIP and traces)

0

trace (N=10) trace (N=20) trace (N=30) trace (N=40) trace (N=50) Tucker GVoIP (N=10) Tucker GVoIP (N=20) Tucker GVoIP (N=30) Tucker GVoIP (N=40) tucker GVoIP (N=50)

—2

10

trace (N=10) trace (N=20) trace (N=30) trace (N=40) trace (N=50) GVoIP (N=10) GVoIP (N=20) GVoIP (N=30) GVoIP (N=40) GVoIP (N=50)

—1

Average Packet Loss (PL)

—1

10

Average Packet Loss

(b) Packet Loss vs capacity(α)

0

10

10

—3

10

—4

10

—2

10

—3

10

—4

10

10

0.2

0.4

0.6

0.8 C (Mbps)

1

1.2

1.4

0.55

0.6

0.65 α

0.7

0.75

Fig. 5. Dimensioning example expressed in C (a) and a (b) for the G.729 codec, Dmax = 50 and different number of multiplexed sources N.

(a) Number of simultaneous sources

1.2

(b) Capacity required for PL=1e03 and Dmax=50ms

1.1

45

1 40 0.9 35

C (Mpps)

N Number of simultaneous sources

50

30 25 Tucker GVoIP (PL=1e02) Tucker GVoIP (PL=1e03) Tucker GVoIP (PL=1e04) Simulation PL=1e02 Simulation PL=1e03 Simulation PL=1e04

20 15 10 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.8 0.7 0.6

Trace Tucker GVoIP AMS GVoIP Pallares GVoIP Tucker onoff MMPP GVoIP

0.5 0.4 0.3 1.2

C (Mbps)

0.2 10

15

20

25

30

35

40

45

50

N (number of simultaneous sources)

Fig. 6. (a) Number of sources admitted as a function of C, (b) variation of C with N for different models for PLtarget = 1e03 and dmin = 0.00001.

used Tucker’s model and traces for GVoIP in the algorithm and N values of 10, 20, 30, 40 and 50. The dmin used was the 1% of the target loss rate input of the algorithm. Subplot (a) shows results expressed in C (output of the algorithm) whereas subplot (b) represent the same results expressed in a (complementary of the statistical gain, a = C/(NR)). Thus, Fig. 5(a) can be used to set the queue size (related to Dmax) and bandwidth reservation at a multiplexer node (i.e., access router) for GVoIP traffic. The algorithm and models previously presented for GVoIP can also be applied in admission control (CAC) or, equivalently, dimensioning the number of users (N) accessing simultaneously to the VoIP service in the access router. This is shown in Fig. 6a where we plot the maximum number of simultaneous GVoIP sources that can be fit in a multiplexer node as a function of C for different packet loss probabilities. The analytical model used has been Tucker’s model. In this plot we have used the simulated GVoIP sources for validation instead of traces. Effectively, finite buffer Tucker and Pallares fluid models exhibit a good matching of experimental behavior independently of N value as shown in subplot (b) for a loss probability of 103 and dmin = 1e05.

As can be observed, our GVoIP dimensioning model offers accurate results when generalized VoIP sources are being used. However, it is important to notice that the proposed models are still valid for any VoIP codec: by setting c = 0 the model is valid for VAD codecs which do not send SID frames. By setting q = 1 the model is valid for CBR-type codecs like G.711 or iLBC.

7. Conclusions The SID frames sent by those VoIP codecs which perform the CNG feature generate additional traffic which is not captured by traditional on–off sources. Therefore, dimensioning methods based on multiplexing on–off sources, namely, fluid and MMPP are not accurate for these kind of codecs and ultimately, underestimate the bandwidth requirement of the aggregated traffic. We have defined a Generalized VoIP source (GVoIP) whose traffic pattern includes not only the silence suppression feature but also the SID frames generation process. Then, we have studied the multiplexing process of GVoIP sources by extending current on–off multiplexing models:

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fluid and MMPP. The new analytical models have been used in a dimensioning algorithm which provides accurate bandwidth requirement estimation for link dimensioning in order to achieve a certain loss rate and maximum delay. Our findings can also be applied in dimensioning the maximum number of simultaneous VoIP users admissible at the access node, in a WAN communication scenario, to achieve performance guarantees (i.e., CAC). The results are validated by simulations and real traces and demonstrate the accuracy of our proposal. Additionally, we have shown that the inter-arrival time between SID frames can be approximated by an exponential distribution. We are currently working on modeling the multiplexing process of heterogeneous GVoIP sources and also on modeling the aggregated traffic pattern. Appendix A. Review of fluid and MMPP models for multiplexing on–off sources This appendix summarizes the current state-of-the-art in on–off sources multiplexing. First, we overview the basics of the two main analytical models for multiplexing on–off sources used in VoIP: the fluid model and MMPP queues. Then, we offer a brief summary of the main approaches used in practice which reduce the complexity of the original models. The voice source activity can be modeled by an elemental on–off source which alternates between active and inactive periods. In the active period, data are emitted at a constant rate R, while in the idle state, no data is generated. The sojourn times in the active and idle states are assumed to be exponentially9 distributed with means 1/l and 1/k, respectively. The activity factor of a source can be defined as the percentage of time the source is active, pON = k/ (k + l). Therefore, an individual on–off source is defined by the triplet: k, l and R. The multiplexing process of N independent homogeneous on–off sources have been studied since the mid eighties by a number of authors [2,14,28] to model voice dimensioning in ATM networks. The studies are supported by two basic analytical models: Fluid Model and MMPP. In both cases, the dynamics of the number of voice sources in active state are that of a continuous-time birth–death process with birth (death) rates proportional to the number of OFF (ON) voice sources. The reader is encouraged to review the cited bibliography for a deeper understanding of the model’s insight. To facilitate this issue, we have kept the original terminology of each model in the referenced papers whenever possible. A.1. Fluid model This model was first proposed by Anick Mitra and Shondy (AMS) [2] and approximates packet-speech multiplexing of N independent homogeneous on–off sources by

9 Other distributions like Weibull are demonstrated to be possible and even more accurate for one voice source. However for aggregated traffic, the exponential assumption has been demonstrated to be valid and simpler for mathematical tractability.

assuming that information10 from each source in the active state arrives at constant rate R to a buffer. If there are i sources active, the buffer content changes at a rate of Ri-C, being C a constant service rate (output link capacity). For system stability, we assume that the average fluid generated by all the sources is less than the output capacity: N  R  pON < C. Let Fi(x) denote the steady-state probability that i sources are on and the buffer content does not exceed x, and let m be the buffer capacity – which is infinity in the AMS model. Therefore, the steady-state probabilities are governed by the set of ordinary differential equations:

D

dFðxÞ ¼ M  FðxÞ; dx

0 < x < m;

ðA:1Þ

where F(x) = (Fi(x))i2S, D = diag{C, R  C, . . . , NR  C} and M = (Mij)i,j2S is the tridiagonal matrix such that

8 j ¼ i  1 P 0; > < ðN  i þ 1Þk M ij ¼ ½ðN  iÞk þ il j ¼ i; > : ði þ 1Þl j ¼ i þ 1 6 N:

ðA:2Þ

The solution of (A.1) is of the form:

FðxÞ ¼

N X

ak /k ezk x ;

0 < x < m;

ðA:3Þ

k¼0

where zk is an eigenvalue of D1M, /k is its corresponding eigenvector, and the ak’s are coefficients that must be found by defining and solving boundary equations. Closed-form expressions for ak, /k and zk are presented in [2] for the infinite queue case, where the number of negative (stable) eigenvalues and boundary equations11 is

N  CR . Thus, the remaining eigenvalues can be ignored in the infinite queue capacity case, and the steady-state packet loss probability for a buffer with size x can be approximated by NbCRc1 0

GðxÞ ¼ 1  1  FðxÞ ¼ 

X

ak ð10 /i Þezk x ;

ðA:4Þ

k¼0

where 1 is a (column) vector of ones and 0 denotes transpose. In the finite queue capacity case, Tucker [32] defines the probability ui of i sources being active and the queue being held at the upper limit m as

ui ðmÞ ¼ pi  F i ðm Þ ¼



N i



ki lNi ðk þ lÞN

 F i ðm Þ;

ðA:5Þ

where pi represents the steady-state probability of i sources being active. Thus, this case leads us to solving N + 1 boundary equations, to obtain the ak’s in (A.3), and requires the computation of all eigenvalues of D1M. In addition to increasing when i > bC/Rc, the queue length decreases when the number of active sources is smaller than

10 In the fluid model, an active source generates fluid at a uniform rate R where accumulations in fractions of packets are allowed. 11 The queue size increases when the number of active sources i is greater than the normalized output link capacity, (i.e., i > C/R). In this case, the



queue length cannot be zero, so that Fi(0) = 0, i > CR , and N  CR boundary equations can be established.

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the normalized output link capacity. Consequently, ui = 0

and Fi(m) = pi, for i 6 CR . Since packet losses occur only when the queue is held at its capacity, the information loss probability may be approximated by the following fraction of the total information:

PN GðmÞ ¼

i¼dCRe ðiR

 CÞui

N  R  pON

ðA:6Þ

:

The fluid model approach cannot capture the short-term queue increments that occur when two or more packets arrive almost simultaneously, since it does not include the concept of packetization, but this fact is likely to be relevant only when the queue size is small. The AMS solution, based on an infinite queue, is simpler to apply than Tucker’s model. By contrast, the AMS model leads to less accurate approximations, being an upper bound of the packet loss probability obtained with Tucker’s model, whose solution is however computationally more demanding. A.2. MMPP model An active voice source produces a stream of fixed size packets with a fixed inter-arrival time L1 [1]. Thus, an approximation to describe the multiplexing of on–off voice sources would be the Markov Modulated Deterministic source model, with phase equal to the number of active voice sources and the number of packet arrivals being that of the number of active sources emitting deterministically one packet every L1 units of time.12 The MMPP model, first introduced for on–off sources multiplexing in [14,19,3], approximates this behavior by merely replacing deterministic arrivals by Poisson arrivals with rate proportional to the number of active voice sources, namely rate i  L when i voice sources are active. Assuming that all packets are of the same size, the associated queuing model is therefore the MMPP/D/ 1/K system, where K is the packet queue size capacity. The corresponding steady-state packet loss probability can be expressed as [3]

As detailed in [3], the computation of the vector pK(0) requires the solution of a linear system with N + 1 equations. Moreover, the coefficients of that linear system need to be obtained by performing matrix operations whose computation time grows with the queue capacity K. A.3. Approximations of the two basic models A number of simplified models of the fluid and MMPP approaches have been proposed by different authors with the objective to reduce the computational complexity of the original models. To the best of our knowledge, the main proposals applicable to our WAN communication scenario are the following: A.3.1. Fluid-based approaches Since G(x) (from Eq. (A.4)) is a linear combination of exponential terms, the steady-state packet loss probability is dominated for large values of x by the exponential term ezk x with the largest exponent coefficient i.e., z0 the largest negative eigenvalue of D1M. In this case, the asymptotic behavior of the AMS model is given by

8 9  N NbCRc1 pON NR < Y zi = z0 x e : GðxÞ ¼ : i¼1 zi þ z0 ; C

Note that the term in brackets in Eq. (A.8) is the load in the system q. Closed form expressions for eigenvalues are given in [2], which in the z0 case is

z0 ¼ 

l þ kð1  NR=CÞ R  ðC=NÞ

ðA:9Þ

:

Pallares [21] proposed a salient approximation for the asymptotic behavior of Tucker’s fluid model, extending its approximation to the heterogeneous sources case. The author first rewrites Eq. (A.6) as

GðmÞ ¼ ða0  1Þ  PN

!

C

pi

i¼0 iR

PðKÞ ¼ 1 

1

q½1 þ pK ð0ÞðK  MÞ1 h1 10 

;

ðA:7Þ

where M is the infinitesimal generator matrix of the number of active sources, defined in (A.2), and:  q is the mean offered load;  h is the mean packet service time;  K is a diagonal matrix whose (i,i)th element is i  L, the mean packet arrival rate when i voice sources are active; and  pK(0) is a row vector whose j-th element, j = 0, 1, . . . , N, is the steady-state probability that j sources are active and no packets remain in queue at a packet service completion epoch.

12 We do not consider the fact that active sources can start asynchronously and therefore the packets sent by active sources may arrive at different times.

ðA:8Þ

1 :

ðA:10Þ

Then, he proposes a closed form expression which approximates a0 as

0

0Pþ1N 1 11 C ðiR  CÞpi b c R z m Ae 0 A : a0 ¼ @1 þ @ P C b Rc 0 ðiR  CÞpi

ðA:11Þ

From all the fluid-based approaches, we point out the accuracy of those which allow the usage of a finite-size buffer (e.g., Tucker and Pallares solutions) versus the original AMS model, which overestimates the loss probability since it uses a infinite-size buffer. A.3.2. MMPP-based approaches Baiocchi [4] proposed the substitution of the phase MMPP with N + 1 states (0, 1, . . . , N) by an MMPP with only the following two states:  OverLoad (OL) state that substitutes the original set of states SO = {Q + 1, Q + 2, . . . , N}, where Q denotes the maximum number of on on sources that can be

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accommodated in the multiplexer output, corresponding to the region of the phase space where the queue tends to build up.  UnderLoad (UL) state that substitutes the original set of states SU = {0, 1, . . . , Q}, corresponding to the region of the phase space where the queue tends to decrease.

[5]

[6]

[7]

We note that the value of Q can be readily calculated as Q = bC/Rc. The new MMPP process possesses four parameters: the transition rates out of the OL and UL states, respectively, rOL and rUL, and the packet arrival rates in the OL and UL states, respectively, kOL and kUL. The computation of rOL is based on the decay-rate of the (phase-type) distribution of the sojourn time of the original MMPP in the set of overload states SO per visit to the set. kUL are made equal to the mean packet emission rate of the original MMPP in the sets SO and SU, respectively, i.e.,

kOL ¼ L 

N X i¼Qþ1

kUL ¼ L 

Q X i¼0

i  PN

j¼0 bj

;

[10]

[11]

[12]

ðA:12Þ [14]

:

ðA:13Þ

Finally, the rate rUL is computed in such a way that the mean packet arrival rate in the new MMPP, (kOLrUL + kULrOL)/(rOL + rUL), equals that of the original MMPP, N  L  pON, giving

r UL ¼ r OL 

[9]

[13]

bi

j¼Qþ1 bj

bi i  PQ

[8]

N  L  pON  kUL : kOL  N  L  pON

ðA:14Þ

As described in [4], once the four parameters of the new MMPP process are computed, the associated MMPP/D/1/K system may be solved numerically in a simple way since it involves elementary calculus of eigenvalues and matrix inversions of 2  2 matrices. Other MMPP-based proposals [18,34,19] carry out different matches but, in essence they do not add any significant contribution to Baiocchi’s approach for a VoIP scenario. As in the fluid model case, Andersson’s [1] work extends the validity of earlier ATM validations to VoIP. Results demonstrate the accuracy of the MMPP model, being more precise than the fluid model for on–off sources multiplexing mostly for small buffers (e.g., when multiplexing a small number of sources). For large buffers (or equivalently, when multiplexing a large number of on–off sources) the fluid model offers good accuracy at less computational cost than MMPP-based solutions. References [1] B. Ahlgren, A. Andersson, O. Hagsand, I. Marsh, Dimensioning links for IP telephony, Sweden Institute of Computer Science, Technical Report T2000:09, 2000. [2] D. Anick, D. Mitra, M. Shondi, Stochastic theory of a data-handling system with multiple sources, Bell System Technical Journal 61 (1982) 1871–1894. [3] A. Baiocchi, N. Melazzi, A. Roveri, Queueing performance and control in ATM, in: Proceedings of the 13th International Teletraffic Congress, Copenhagen, 1991, pp. 13–18. [4] A. Baiocchi, N. Melazzi, A. Roveri, R. Winkler, Loss performance analysis of an ATM multiplexer loaded with high-speed ON–OFF

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Antonio Estepa is associate professor in the Department of Automatics Robotics, and Telematics Engineering at the University of Sevilla. He received his Ph.D. in Telecommunication Engineering from the University of Sevilla in 2004. He has also been a visitor in the Department of Electrical Engineering and Computer Science at the University of Minnesota in 2004. His research interests are in the areas of multimedia and quality of service.

2517 Rafael Estepa is associate professor in the Department of Automatics Robotics, and Telematics Engineering at the University of Sevilla. From 1998 to 2000 he worked as systems engineer in Alcatel Spain. He received his Ph.D. in Telecommunication Engineering from the University of Sevilla in 2002. He has also been a visitor in the Department of Applied Mathematics at the IST at Lisbon in 2006. His research interests are in the areas of multimedia and quality of service.