MP is time-homogeneous when

( ) ( )

[

Xt xXtn xn

]

P ≤ | = depends only on

(

t−tn

)

and is not function of t and t_n.

When a state space of MP is discrete the MP is called a Markov chain. When the time parameter t it is continuous a Markov chain is called continuous-time Markov chain (CTMC). The CTMC is described by matrix Q which is called the infinitesimal generator of CTMC and is defined as:

[ ]

qij

=

Q , (2)

where q_ij is transition rate (intensity) coefficient and there is

a dependency q q i M Markov chain is called a discrete-time Markov chain (DTMC). DTMC is described by the matrix P which is called one-step transition probabilities matrix and is defined as:

[ ]

pij

P= , (3)

where pij is a probability of transition between a state i and state j, i.e. p_ij =P

(

X_n+1= j|X_n =i

)

.

A Markov-modulated Poisson Process (MMPP) is a doubly stochastic process where the intensity of a Poisson process is defined by the state of a Markov chain. The Markov chain can therefore be said to modulate the Poisson process, hence the name. MMPP is characterized by matrices Q (2) and R, the latter is the matrix of Poisson arrival rates. When the modulat-ing process is in the state X(t_n)=i,i=1,2,K,Mthan events are generated and their interarrival times are described by the exponential distribution

( )

^{t r}i

(

^ri^t

)

^{i M}

i = exp− , =1,2,K,

α . (4)

This distribution is valid during all the time the Markov proc-ess remains in state X(t_n)=i,i=1,2,K,M. The state so-journ times of CTMC are exponentially distributed.

Discrete time MMPP (dMMPP) evolves over time in con-stant time intervals and the number of events in each interval have a Poisson distribution whose parameter is a function of the state of the modulator Markov chain.

Formally, a two-dimensional Markov chain

(

X,J

) (

=

{

X_n,J_n

)

,n=0,1,_K

}

with state space Ν×S is

where R is a matrix of Poisson arrival rates. A graphical in-terpretation of dMMPP was presented on Fig. 2.

i j

Fig. 2 Graphical interpretation of dMMPP

For time-homogeneous CTMC, for a very small interval τ

=

t , there is a linear dependency:

τ

= _ij

ij q

p , (7)

where pij is probability of transition between the states

i

and

j

(3), and qij is transition rate (intensity) coefficient (2) [4].

III. PREVIOUS WORKS

The performance analysis of packet voice traffic usually in-cludes the analysis of an appropriate queuing model. The works related to the analysis can be divided into two groups.

In the first group, authors concentrated on a microscopic view of network traffic and tried to model dependency be-tween subsequent packets (micro scale). Eckberg treated mul-tiplexed voice as the

∑

^Di/D/1 queuing system and derived the exact delay distribution for it in [5]. In [6] it was stated that the multiplexed voice streams may be approximated with quite good results by a Poisson process. In [7] renewal processes were used and voice multiplexer was modelled as G_i/D/1

The second group consist of works in which authors tried to match the behaviour of the VoIP traffic over a relatively long time interval (macro scale) neglecting the dependency be-tween subsequent packets. Usually, statistical properties of a voice source are taken into account. Stern [8] presented a queuing model based on the exponential ON/OFF model and an imbedded CTMC whose states represent the number of currently active speakers. Daigle et. al in [9] investigated three different approximations for aggregated arrival process based on a semi-Markov process model, a CTMC model, and a uni-form arrival and service model. In [10] a multiplexer with infinite buffer was studied with a stochastic fluid flow model but it is shown in [11] that this model only works for a multi-plexer under heavy load. A multimulti-plexer with finite buffer is studied in [12] using the fluid flow model but it does not work well for small buffers. Some authors proposed approximate methods, mainly based on a Markov Modulated Poisson proc-ess (MMPP). A two-state MMPP was used quite succproc-essfully in [13] to estimate the delay in a multiplexer with infinite buffer. In [14] a different method for finding the parameters of the MMPP was developed. Besides authors proposed two other concepts based on renewal processes and fluid models to estimate multiplexer efficiency. In [15] the arrival process is approximated with a two-state MMPP and a method called asymptotic matching is suggested for the calculation of the parameters of the MMPP. However, in all cases above, the number of MMPP states was insufficient to capture a correla-tion of traffic over a longer period.

The common conclusion of macro scale models is that they lack stochastic properties of a process but they are better for a correlation modelling in comparison to the micro scale mod-els. None of the models took into consideration connection scale, i.e. statistics of connections durations and interarrival times between them.

We proposed the model that took into account both burst and connection scales, neglecting inter-packets dependencies. Also, our modelling methodology was different in comparison to the above-mentioned works. We did not create aggregated model from single sources models but we approximated synthetic traf-fic obtained from trace driven simulation. The advantage of our methodology is simplicity and a good level of accuracy; the disadvantage is a lack of flexibility. When changing traffic pa-rameters, one must generate again the synthetic trace and repeat the fitting procedure to update the model.

IV. TRAFFIC MODELLING A. Traffic generation process

The synthetic trace was generated from real data recorded on both connection and burst level.

The connection level data were captured at the main tele-phone exchange of the Silesian University of Technology in Poland. It contained the record of about eighty thousand con-nections recorded in December 2005 using traditional tele-phone lines. The record included the beginning time of a

con-nection and its duration time with one-second accuracy. Hav-ing excluded the data from holidays and weekends, we ana-lyzed the set generating between 10-14 o’clock, which was a homogenous arrival Poisson process not influenced by time dependencies. In order to get busier multiplexed VoIP traffic, we increased the arrival rate. However, Poisson property of the arrival process was maintained.

Ethernet

Data processing in Matlab

Summation

Fig. 3 The generation process of simulated VoIP traces Than, with the use of Windows Sound Recorder we re-corded one side of several real phone conversations held using popular VoIP software. We connected two computers equipped with OpenH323 library [16] with Ethernet cable.

Previously recorded conversations were played and encoded by G.711 voice coder. Next, they were sent through the net-work to the second computer where we recorded the time-stamps of the voice packets using Ethereal software [17]. We obtained single binary time series, where 0-values corre-sponded to OFF periods and 1-values correcorre-sponded to ON periods. Then, we concatenated the binary time series into one single series.

From these series, for each starting connection, a subset was randomly chosen. Its length equalled the connection dura-tion time. For the all active connecdura-tions we were totalling up the values of the subsets in discrete periods obtaining the time series which represented the traffic intensity, Fig. 3.

We generated several traces, each containing about one hundred thousands elements. The traces were divided into two categories; the model was trained on the traces from the first category (training set) and validated against the traces from the second category (test set). The sets represented the traffic which flows into VoIP gateway, which was capable of servic-ing up to 45 or 90 users, i.e. the VoIP gateway was assumed to

MULTI-SCALE MODELLING OF VOIP TRAFFIC BY MMPP 57

be equipped with 45 or 90 connection lines.

B. Parameters estimation

The inference procedure for model parameter estimation matched both the autocovariance and marginal distribution of the counting process which represented the number of packets in a time unit. The MMPP was constructed as a superposition of L 2-MMPP and one M -MMPP, where L is the number of two-states Markov chains and M is the number of states in Markov chain. The 2-MMPPs were designed to match the autocovariance and the M-MMPP to match the marginal distribution of a traffic trace. Each 2-MMPP modelled a specific time-scale of the data. The procedure started by approximating the autocovariance by a weighted sum of ex-ponential functions that model the autocovariance of the 2-MMPPs. We adjusted the autocovariance tail to capture the long-range dependence characteristics of the traffic, up to the time-scales of interest to the system under study. The proce-dure then fitted the M-MMPP parameters in order to match the marginal distribution, within the constraints imposed by the autocovariance matching. The final MMPP with M2^L states was obtained by superposing the L 2-MMPPs and the

M-MMPP. An important feature of the procedure was that both L and M were not defined a priori, since they were determined as part of the procedure. Detail of the procedure are given in [18]. In the end we obtained two matrices P and

Rrepresenting dMMPP as in equation (6).

C. Model evaluation

We evaluated our model against trace-driven simulation. In Table 1 we presented comparison of mean and variance be-tween the model, test and training traffic sets for traces pro-duced by up to 48 and 96 users. Values after “±” symbol cor-respond to 95% confidence interval for simulated measure-ments.

size Model Simulation

Mean 515 532±3

At Fig. 4 we presented the comparison of packet arrivals density function between the simulated traffic and model. At Fig. 5 we presented similar comparison for an autocorrelation function.

V. VOIPGATEWAY PERFORMANCE

In this section we analyse queuing behaviour of VoIP

gate-way buffer using the fluids models theory. In these models, fluid flows into a fluid reservoir according to a stochastic process. In our case, fluid buffer was either filled or de-pleted, or both, at rates which are determined by a state of a background Markov process.

0 200 400 600 800 1000 1200

0

Number of packets per 1 second

Pdf of a number of VoIP packets, number of sources = 48 Model Simulation

0 500 1000 1500 2000

0

Number of packets per 1 second

Pdf of a number of VoIP packets, number of sources = 96 Model Simulation

Fig. 4 Probability density function of a number of VoIP packets in a time unit Let C

( )

t denote the amount of fluid at time tin this reser-voir. Furthermore, let X

( )

t be a continuous time Markov process. X

( )

t is said to evolve "in the background". The con-tent of the reservoir C

( )

t is regulated in such a way that the net input rate into the reservoir (i.e. the rate of change of its content) is r~_i =r_i−C_l at times when X

( )

t is in state i∈N.

A graphical interpretation of Markov fluid model was pre-sented on Fig. 6.

The stability condition is given,

∑

~<0

∈N

bution as t→∞. Hence, the stationary joint distribution of

( )

^t

X and ^C

( )

^t exists and is given by

( )

^{y =P}

[

^X=i,C≤y

]

^i∈N,y≥0

F_i (9)

0 20 40 60 80 100 120

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time [1 second]

Autocorrelation, number of sources = 48 Model

Training data Simulation

95% confidence interval

0 20 40 60 80 100 120

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time [1 second]

Autocorrelation, number of sources = 96 Model Training data Simulation

95% confidence interval

Fig. 5 Comparison of autocorrelation functions obtained from the model and trace-driven simulations

( )

t i X r_i, =

( )

t X

( )

t C

Fig. 6 Graphical interpretation of Markov fluid model

It can be shown that the vector F

( )

y =

[

F1

( ) ( )

y,F2 y,K,Fn

( )

y

]

^T

satisfies the differential equation

( )

y QF

( )

y F

R ′ = ^T , (10)

where prime denotes differentiation and superscript

T

de-notes transpose. R is a diagonal matrix R=diag

(

~r₁,Kr~N

)

,

Q is the generator of the Markov process X

( )

t of size n×n. By assuming that R is non-singular, i.e. ~r_i ≠0 for i∈N, the solution of (10) is given by

( )

y R QF

( )

y

F′ = ^{−1 T} (11)

In case the eigenvalues are simple, it follows that

∑

=

= ^N

i

i y

ie v

a ⁱ

1

F(y) ξ (12)

where the

( )

ξ_i,v_i are the eigenvalue-eigenvector pairs of the matrix R⁻¹Q^T and ci are constants that can be determined by boundary conditions. Further details of the above method can be found in [19]

0 1000 2000 3000 4000 5000 6000 7000 0.7

0.75 0.8 0.85 0.9 0.95 1

Number of packets, C = 1.3 mean = 805 packets / second Cdf of buffer occupancy, number of sources = 48

Model Simulation

95% confidence interval

0 500 1000 1500 2000 2500 3000

0.7 0.75 0.8 0.85 0.9 0.95 1

Number of packets, C = 1.2 mean = 1469 packets / second Cdf of buffer occupancy, number of sources = 96

Model Simulation

95% confidence interval

Fig. 7 Cumulative distribution of packets number in a VoIP gateway buffer Through a few simple computations we transformed the ma-trix P (3) of dMMPP into matrix Q (2) representing CMTC, which was the argument of (10). Solving the mentioned equa-tion we obtained the soluequa-tions as in (12), which were com-pared with results obtained during simulation. The comparison was presented at Fig. 7 for the traffic generated up to 48 and

MULTI-SCALE MODELLING OF VOIP TRAFFIC BY MMPP 59

96 sources. The output line capacity was set at 130% and 120% of mean traffic intensity respectively.

VI. CONCLUSION

In this paper, we examined the suitability of MMPP for modelling of multiplexed VoIP traffic, which flows into a VoIP gateway. We stated that MMPP might approximate the second order statistics of the traffic with good level of accu-racy, although it made errors in variance estimation. We ap-plied the model to evaluate a VoIP gateway performance by computing cumulative distribution of packets number in the VoIP gateway buffer. The results were in good agreement with the simulation.

REFERENCES

[1] T. D. Dang, B. Sonkoly, and S. Molnár, "Fractal Analysis and Modelling of VoIP Traffic," presented at NETWORKS 2004, Vienna, Austria, 2004.

[2] A. Biernacki, "Analysis of VoIP Traffic Produced by Coders with VAD,"

presented at 4th Polish-German Teletraffic Symposium, Wroclaw, Po-land, 2006.

[3] M. Grossglauser and J. C. Bolot, "On the relevance of long-range de-pendence in network traffic," Networking, IEEE/ACM Transactions on, vol. 7, pp. 629-640, 1999.

[4] T. Czachórski, Modele kolejkowe w ocenie efektywności pracy sieci i systemów komputerowych (in Polish only). Gliwice: Wydawnictwo Politechniki Śląskiej, 1999.

[5] A. Eckberg, Jr., "The Single Server Queue with Periodic Arrival Process and Deterministic Service Times," Communications, IEEE Transactions on [legacy, pre - 1988], vol. 27, pp. 556-562, 1979.

[6] K. Byung, "Characterization of Arrival Statistics of Multiplexed Voice Packets," Selected Areas in Communications, IEEE Journal on, vol. 1, pp. 1133-1139, 1983.

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[9] J. Daigle and J. Langford, "Models for Analysis of Packet Voice Com-munications Systems," Selected Areas in ComCom-munications, IEEE Journal on, vol. 4, pp. 847-855, 1986.

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61, pp. 1871-1894, 1982.

[11] S. Zheng, "Capacity Study of Statistical Multiplexing for IP Telephony,"

Department of Mathematics, Linkoping University, Sweden. LiTH-MAT-EX-98-12 1998.

[12] R. C. F. Tucker, "Accurate method for analysis of a packet-speech multi-plexer with limited delay," Communications, IEEE Transactions on, vol.

36, pp. 479-483, 1988.

[13] H. Heffes and D. Lucantoni, "A Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statistical Multiplexer Performance," Selected Areas in Communications, IEEE Journal on, vol.

4, pp. 856-868, 1986.

[14] R. Nagarajan, J. F. Kurose, and D. Towsley, "Approximation techniques for computing packet loss in finite-buffered voice multiplexers," Selected Areas in Communications, IEEE Journal on, vol. 9, pp. 368-377, 1991.

[15] A. Baiocchi, N. B. Melazzi, M. Listanti, A. Roveri, and R. Winkler,

"Loss performance analysis of an ATM multiplexer loaded with high-speed on-off sources," Selected Areas in Communications, IEEE Journal on, vol. 9, pp. 388-393, 1991.

[16] Vox-Gratia, "OpenH323," 2005.

[17] E. S. Inc., "Ethereal, A Network Protocol Analyzer," 2005.

[18] A. D. Nogueira, P. S. Ferreira, R. Valadas, and A. Pacheco, "Modeling Self-Similar Traffic through Markov Modulated Poisson Processes over Multiple Time Scales," presented at IEEE International Conf. on High-Speed Networks and Multimedia Communications - HSNMC, Estoril , Portugal, 2003.

[19] W. Scheinhardt, "Markov-modulated and feedback fluid queues," vol.

Ph.D. thesis: University of Twente, the Netherlands, 1998.

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