An MAP-based Poisson cluster model for Web traffic

Performance Evaluation 49 (2002) 359–370

An MAP-based Poisson cluster model for Web traffic Guy Latouche a , Marie-Ange Remiche b,∗ a

b

Département d’Informatique, Faculté des Sciences, Université Libre de Bruxelles, CP 212, Boulevard du Triomphe, 1050 Bruxelles, Belgium Faculté des Sciences Appliquées, Service Informatique et Réseaux, Université Libre de Bruxelles, CP 165/15, Av. F.D. Roosevelt 50, 1050 Bruxelles, Belgium

Abstract In our Web traffic model, http session requests occur according to a Poisson process and each request induces multiple transmissions of sub-documents, called objects. We use a transient MAP process to model the epochs at which the transmissions of objects associated to a given document are initiated. The lengths of transmission of the objects are modeled as i.i.d. random variables. We analyze the distribution of the total number of objects being in transmission at any given time and determine its auto-correlation function. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Web traffic; Markovian models; MAP processes; Poisson cluster processes

1. Introduction Web traffic modeling has been of crucial interest in the last years in both fixed and mobile networks (see [1] and references therein). One motivation is that new protocols are often validated on simulated networks which should closely resemble real networks. One key parameter of such a simulated network has been clearly identified as the current traffic conditions as met on real network. Our aim is not to provide another statistical analysis of new collected data but to offer a tractable mathematical model integrating well-accepted traffic descriptors. More precisely, we concentrate here on studying a model of http sessions initiated to a unique Web server. Our model is based on two previous analysis carried out by Liu et al. [2]. A common feature in these papers is that one marginal http session is composed of one main object and possibly several embedded objects or in-line objects. Indeed, in some version of http (http 1.1), simultaneous multiple connections can be opened to download in-line objects in parallel. The total traffic generated by one http request at any given time during its duration is made up of the superposition of multiple dependent file transmissions, each corresponding to the downloading of one embedded object. The http session duration is clearly dependent on the size of objects: it finishes when the last downloading process ends. ∗

Corresponding author. Tel.: +32-2-650-29-15; fax: +32-2-650-47-13. E-mail address: [email protected] (M.-A. Remiche). 0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 1 1 9 - 0

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At the Web server, http requests are randomly initialized. We assume that http session requests occur according to a Poisson process. It is indeed accepted that as the rate of new TCP connections increases, the arrival process of connection requests tends to be Poisson (see, for example [2]). There also exist some theoretical justification in support of the assumption: Neuts and Pearce [3] consider a system consisting of a large number of customers, each customer alternating between active and idle periods and intermittently transmitting messages of fixed length during its active periods; the authors show that, under suitable conditions, the system behaves asymptotically, as the number of customers becomes large, as if the customers arrived according to a Poisson process. To each http request is associated a process of object requests: if the http request occurs at time t, K objects will be requested at times t + T1 , t + T2 , . . . , t + TK . We assume that K and T1 , T2 , . . . , TK are determined by a transient MAP. These processes are introduced in [4]; transient MAPs are Markovian processes which terminate after a finite random number of events. The Markovian assumption has the advantage of making the whole process very tractable, and allows for much flexibility in modeling such phenomena as high correlations over long intervals of time [5], even though Markovian models do not exhibit long-range dependence stricto sensu. To each object is associated a mark, namely its downloading duration: if an object is requested at time t, its transmission takes S unit of time where S is a random variable; we write that the object is alive in the interval (t, t + S). We assume that the durations are i.i.d. random variables with common distribution G(·); this implies that the capacity of the system is sufficiently large that a transmission is not slowed down by other downloading processes which might be in progress at the same time. We are interested in the stationary distribution of M(t), that is the total number of objects which are currently downloaded or alive at time t. This is a Poisson cluster process and its characteristics have a simple expression in function of the characteristics of a marginal http session process [6, Section 8.3]. This paper is organized as follows. We briefly recall in Section 2 the definition of transient MAPs and we give a precise definition for the traffic model. We analyze in Section 3 the model for a single http request and in Section 4 the whole traffic model at one Web server. We determine the generating function of its distribution and we show how to numerically evaluate its first two moments. We conclude in Section 5 with some numerical illustrations of these two first moments. 2. Definition of the traffic model Continuous-time transient MAPs are defined in [4] as a generalization of the well-known MAP [7, Chapter 10]. They are two-dimensional processes {(L(t), ϕ(t)) : t ∈ R+ } on the state space {( , i) :

∈ N, i ∈ {0, . . . , m}}, where m is finite. The variable L is called the level, ϕ is the phase. The transition rate matrix is   C0 C1 0 0    0 C0 C1 0 . . .      ..   . 0 C0 C1 Q= 0 ,     . .  0 0 0 C0 .    .. .. .. .. . . . .

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so that the possible one-step transitions are from (n, i) to {(n, j ) : 0 ≤ i = j ≤ m}, that is, a phase transition only, and to {(n + 1, j ) : 0 ≤ j ≤ m}: an increase of level, possibly with a phase transition. The blocks C0 and C1 have the following structure:     0 0 0 0 C0 = and C1 = d0 D0 d1 D1 with D1 ≥ 0, d0 , d1 ≥ 0, the off-diagonal entries of D0 non-negative, and the diagonal entries strictly negative so that D0 1 + D1 1 + d0 + d1 = 0.

(1)

Briefly stated, in addition to the blocks D0 and D1 which characterize traditional MAPs, we introduce the rates d0 and d1 which lead to states of the form (n, 0) from which no transition is possible. Such a transition is termed a catastrophe and marks the end of the evolution of the process. Transitions from (n, ·) to (n + 1, ·) are called events. We denote by αi the probability that the phase at time 0 is equal to i and we define the vector α = (α1 · · · αm ). Thus, a transient MAP is characterized by the 4-tuple (α, D0 , D1 , d0 ), α0 is implicitly defined by α0 = 1 − α1 and the vector d1 by (1). Starting with L(0) = 0, we define T0 = 0 and Tk = inf{t ≥ 0 : L(t) ≥ k} if there exists such a t for k ≥ 1; this is the collection of epochs at which events occur. We also define the total number of events K = sup{k : Tk < ∞}. If D, defined as D = D0 + D1 is non-singular, then the process enters one of the absorbing states in a finite time and K is finite a.s.; this we assume throughout. It is shown in [4] that K has a phase-type distribution (see [8, Chapter 2]). Thus, transient MAPs form a versatile class of processes with respect to the distribution of the number of events as well as the pattern of their occurrence in time. We now define a Poisson cluster process which will be our model of http traffic at one Web server. We first consider a Poisson process {Wn : n ∈ N} with rate λ; this is called the request process and Wn is the arrival time of the nth request. We next associate to each request its own copy of a transient MAP (α, D0 , D1 , 0) which starts at the time when the request happens: if T1(n) , T2(n) , . . . , TK(n) are the epochs of events for the nth copy of the transient MAP. We say that objects of the nth request are demanded at time Wn +T1(n) , Wn +T2(n) , . . . . To each object is associated an interval of time during which the object is said to be alive: the kth object of the nth request is alive during the interval of time (Wn + Tk(n) , Wn + Tk(n) + Sk(n) ). The copies of the transient MAP are independent and the interval lengths Sk(n) are i.i.d. with distribution G(·). This is not a restrictive assumption as one may extend our analysis to a model where the interval length Sk(n) depends on the arrival time Tk(n) . In our model, the requests correspond to http requests, the objects to embedded object requests within a same http session. The time between the request arrival and the first object correspond to the time for downloading the main object of our Web request as proposed in [9], this implies that α0 = 0. Moreover, we assume that d0 = 0, i.e. the objects request arrival process stops evolving with the arrival of the final demand. We also assume that there is no bandwidth limitation, so that the time needed to complete the transmission of an object is independent of the number of other objects simultaneously transferred. Denote by M(t) the total number of downloaded objects in the system at time t, that is, the total number of objects for which Wn + Tk(n) ≤ t < Wn + Tk(n) + Sk(n) .

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Let Γ (t; z) be the probability generating function defined as Γ (t; z) = E[zM(t) ] with Γ (z) its stationary version, that is, Γ (z) = lim E[zM(t) ]. t→∞

By Daley and Vere-Jones [6, Proposition 8.3.1], we have that t  Γ (t; z) = exp λ (E[zNu (x) ] − 1) dx , 0

and



Γ (z) = exp λ

∞

 (γ (x; z) − 1) dx ,

(2)

0

where Nu (x) is the number of objects alive at time x for a single http session which has been initialized at time u and γ (x; z) = E[zN0 (x) ]. Moments give less detailed information but are easier to manage. We thus define the stationary mean µ = lim E[M(t)], t→∞

and auto-covariance function ρ(h) = lim E[(M(t) − µ)(M(t + h) − µ)]. t→∞

We have



∞

µ=λ

m(x) dx,

(3)

0

where m(t) = E[N(t)] and N(t) = N0 (t),

∞ ρ(h) = λ r(x; h) dx, 0

where r(t; h) = E[N(t)N(t + h)]. In order to proceed with the analysis, the distribution of the number of objects of an isolated http session needs to be characterized. This is the purpose of the next section. 3. Characteristics of a page request We restrict our attention in this section to a single transient MAP: a session starts at time 0, and evolves until the last of its children is born. We are interested in the number of its children who are alive at time t and at time t + h, for a fixed value of t and h. Each object is thus marked with a “1” or a “0” according to whether it is alive or not at time t. This is a time-dependent mark since the probability of a “1” depends on the time at which the object is requested.

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Let us denote by N a (x, t) the number of children who are born before time x ≤ t and are still alive at time t and by ψ(x, t; z) its generating function. Clearly, N(t) = N a (t, t) and γ (t; z) = ψ(t, t; z). We have ψ(x, t; z) = αB(x, t; z)1 + αb(x, t; z),

(4)

where Bij (x, t; z) = E[zN

a

(x,t)

bi (x, t; z) = E[zN

I{ϕ(x) = j }|ϕ(0) = i],

a

(x,t)

I{ϕ(x) = 0}|ϕ(0) = i]

for 1 ≤ i, j ≤ m and I{·} is the indicator function. We now prove the following lemma. Lemma 1. The matrix B(x, t; z) is a solution of the equation

x B(x, t; z) = exp(D0 x) + ζ (t − u; z)B(u, t; z)D1 exp(D0 (x − u)) du,

(5)

0

where ζ (x; z) = G(x) + z(1 − G(x)). The vector b(x, t; z) is given by

x b(x, t; z) = ζ (t − u; z)B(u, t; z)d1 du.

(6)

0

Proof. The proof is obtained by conditioning on the last object to arrive before time x. Two cases are possible. Either there is no child born in (0, x) or there is at least one. The generating function can thus be decomposed into Bij (x, t; z)= E[I{ϕ(t) = j }I{M(x) = 0}|ϕ(0) = i]+E[zN

a

(x,t)

I{ϕ(x) = j }I{M(x) > 0}|ϕ(0) = i],

where M(x) is the number of children born in (0, x). The first term is equal to [exp(D0 x)]ij . If there is at least one object, that is if [M(x) > 0], then there is some time at which a birth occurs which is the last before x. We condition on that time u, on the phase k before and on the phase k  after that birth, and find that

x a a E[zN (x,t) I{ϕ(x) = j }I{M(x) > 0}|ϕ(0)=i]= E[zY (u,t) ]E[zN (u,t) I{ϕ(u)=k}|ϕ(0)=i] 0

1≤k,k  ≤m

× (D1 )kk E[I{M(x) − M(u) = 0}I{ϕ(x) = j }|ϕ(u) = k  ] du, where Y (u, t) = 1 if the object requested at time u is still downloaded at time t, 0 otherwise. This proves (5) since E[zY (u,t) ] = Pr[S ≤ t − u] + z Pr[S > t − u], where S is a random variable with distribution G(·). The proof of (6) is similar: the only difference is that, since ϕ(x) = 0, we know that the catastrophe occurs upon the birth of the last child. 䊐 Having characterized the generating function, we may determine the mean number of objects alive at time t.

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Theorem 2. The expected number of objects alive t units of time after the start of the transient MAP characterized by (α, D0 , D1 ) and d0 = 0 is given by

t (1 − G(t − u))φ(u) du, (7) m(t) = 0

where φ(t) = α exp(Dt)(d1 + D1 1), and D = D0 + D1 . Proof. Define ma (x, t) = ∂z ψ(x, t; z)|z=1 . This is the expected number of objects requested before time x and still alive at time t. We have that m(t) = ma (t, t). Further, define H (x, t) = ∂z B(x, t; z)|z=1 . One readily finds, using the fact that B(x, t; 1) = exp(Dx), that

x H (x, t) = {Gc (t − u) exp(Du) + H (u, t)}D1 exp(D0 (x − u)) du, (8) 0 c

where G (x) = 1 − G(x), so that

x a m (x, t) = αH (x, t)1 + {Gc (t − u)α exp(Du) + αH (u, t)}d1 du.

(9)

0

Consider that t is a fixed quantity and take the derivative of ma (x, t) with respect to x. One obtains that ∂x H (x, t) = {Gc (t − x) exp(Dx) + H (x, t)}D1 + H (x, t)D0 ,

(10)

so that ∂x ma (x, t) = α∂x H (x, t)1 + {Gc (t − x)α exp(Dx) + αH (x, t)}d1 = Gc (t − x)α exp(Dx)(d1 + D1 1) + αH (x, t)(D0 1 + D1 1 + d1 ) = Gc (t − x)α exp(Dx)(d1 + D1 1) by (1). Since ma (0, t) = 0, we have that

t

t a a ∂x m (x, t) dx = m (t, t) = Gc (t − x)α exp(Dx)(d1 + D1 1) dx, 0

(11)

0

which proves (7). We observe for future reference that

x H (x, t) = Gc (t − u) exp(Du)D1 exp(D(x − u)) du.

䊐

(12)

0

The proof is by direct verification that the expression on the right-hand side furnishes a solution of (10). We might compute the second moment of N(t) by evaluating the second derivative of γ (t; z) with respect to z; instead, we determine the function r(t; h) = E[N(t)N(t + h)], which yields the second moment for h = 0.

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Theorem 3. The covariance of the number of objects alive at time t and at time t + h, for the transient MAP characterized by (α, D0 , D1 ) and d0 = 0 is given by

t r(t; h) = Gc (t − v)α exp(Dv)D1 exp(D(t − v))γ (h) dv 0

t + αK(u, t, h){d1 + D1 exp(D0 (t − u))1} du, (13) 0

where



γ (h) =

h

Gc (h − v) exp(Dv)(d1 + D1 1) dv,

(14)

0

K(u, t, h) = Gc (t − u)H (u, t + h) + Gc (t + h − u)(H (u, t) + exp(Du)),

(15)

and m(t) is given by (7). Proof. We decompose N(t + h) as the sum of N(t, t + h) and N a (t, t + h) where N(t, t + h) is the number of objects requested after t and alive at t + h and N a (t, t + h) is as previously defined, the number of children born before t and surviving at time t + h. In order to determine E[N(t)N(t, t + h)], we condition on the phase at time t and use the fact that the process before t and the process after t are conditionally independent, given the phase at time t. Thus, E[N(t)N(t, t + h)] = E[N(t)I{ϕ(t) = i}N(t, t + h)] 1≤i≤m

=



E[N(t)I{ϕ(t) = i}]E[N(t, t + h)|ϕ(t) = i]

1≤i≤m

=



(αH (t, t))i E[N(h)|ϕ(0) = i]

1≤i≤m

using the definition of H (x, t) and that of N(t, t + h). Defining γi (h) = E[N(h)|ϕ(0) = i], we find that γ (h) is given by (14) following the same argument as led us to (11), the only difference is that we do not pre-multiply by α since we condition on the phase at time 0. Using (12), we conclude that

t E[N(t)N(t, t + h)] = Gc (t − v)α exp(Dv)D1 exp(D(t − v))γ (h) dv, (16) 0

the first term in the right-hand side of (13). Determining E[N(t)N a (t, t + h)] is more involved because the correlations are stronger: the same children born before time t contribute to both N(t) and N a (t, t + h). We proceed in the same fashion as we did in the proof of Theorem 2 and analyze the function r a (x, t, h) = E[N a (x, t)N a (x, t + h)], which only takes into consideration the objects requested before time x. Note that E[N(t)N a (t, t + h)] = r a (t, t, h).

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Secondly, we condition again on the last object requested in (0, x), using the fact that if an object is requested at time u < x and no other object request arrives in (u, x), then N a (x, t)N a (x, t + h) = (N a (u, t) + I{S > t − u})(N a (u, t + h) + I{S > t + h − u}), where S is the length of time during which the object requested at u is alive. Finally, we consider separately the case where the phase at time x is different from 0 (the transient MAP process has not terminated yet) and the case where the catastrophe occurs before time x. Defining H˜ ij (x, t) = E[N a (x, t)N a (x, t + h)I{ϕ(x) = j }|ϕ(0) = i] for j ≥ 1, we obtain that

x {H˜ (u, t) + K(u, t, h)}D1 exp(D0 (x − u)) du, H˜ (x, t) = 0

where K(u, t, h) is given by (15) and so find that

x {H˜ (u, t) + K(u, t, h)}{d1 + D1 exp(D0 (x − u))1} du. r a (x, t, h) = 0

Taking derivatives with respect to x, we find after some simple manipulations that

t αK(u, t, h){d1 + D1 exp(D0 (t − u))1} du, E[N(t)N a (t, t + h)] = 0

䊐

the second term in (13), which completes the proof.

4. Numerical illustration The stationary distribution of the number of simultaneously downloaded objects is completely characterized by its generating function Γ (z) given in (2). In this section, we observe some transient measurements of the system, namely the mean number µ(t) = E[M(t)] of objects downloaded at time t as well as the covariance function ρ(t, h) = E[(M(t) − µ(t))(M(t + h) − µ(t + h))]. Parameters of Web requests traffic are summarized in the following table: Parameter

Mean

S.D.

Distribution

Number of in-line objects In-line inter-arrival time Object transfer time

6.43 1 0.77

11.13 2.5 1.53

Discrete phase-type (σ, S) Continuous phase-time (τ, T ) log-normal

These parameter values are taken from Choi and Limb [9]. The MAP process is then chosen to be a Ph-renewal process since the correlation between successive intervals are not given in this set of data. The two phase-type distributions are chosen to fit the usual gamma-distribution assumption for those two

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367

parameters. We choose 

σ = 0.15 0

0

0.85

 0 ,

0.85

  0   S= 0   0 

0.15

0

0

0.9

0.1

0

0

0.9

0.1

0

0

0.2

0

0

0

0 

τ = 0.07 0

0

0.93

 0 ,

    T = 4.8    

0



 0    0 ,  0.8   0.15

−0.075

0.075

0

0

0

−0.075

0.075

0

0

0

−0.075

0.075

0

0

0

−1

0

0

0

0

Fig. 1. The expected number of downloaded objects E[M(t)] against t.

0



 0    0 .  1   −1

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Using the Kronecker product ⊗, we obtain the following matrices that characterize the transient MAP used to model one marginal http session: D0 = I ⊗ T ,

D1 = S ⊗ t · T ,

d1 = (1 − S1) ⊗ t,

α = σ ⊗ τ,

where t = −T 1 and 1 is a vector full of 1 of appropriate size. Clearly, one needs to compute

t

t µ(t) = λ m(x) dx, ρ(t, h) = λ r(x; h) dx. 0

0

This is equivalent to solving the following differential equations systems: ∂x µ(x) = λm(x),

∂x ρ(x; h) = λr(x; h)

for x ∈ [0, t), µ(0) = 0 and ρ(0, h) = 0 for h ∈ R+ . Again the two functions m(x) and r(x, h) are obtained by solving differential equations systems, since

x

x m(x) = ma (x, x) = ∂y ma (y, x) dx, r(x; h) = r(x, x; h) = ∂y r(y, x; h) dy 0

0

Fig. 2. The auto-covariance function ρ(t, h) against h, where t = 10.

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with r(y, x; h) = Gc (x−y)α exp(Dy)D1 exp(D(x−y))γ (h)+αK(y, x, h)(d1 +D1 exp(D0 (x−y)))1. A plot of the transient mean number of downloaded objects against the time is given in Fig. 1. We assume that the system is empty at time 0. Each curve corresponds to a given intensity λ for the Poisson process describing the http session request arrivals. As is shown in this plot, for each λ, the transient mean converges to a particular value as explained in the following theorem. Theorem 4. The mean of the total number M of objects, in steady state, is given by µ = λE[S]E[K],

(17)

where E[S] is the expected transfer time and E[K] the expected total number of objects requested within a single http session. Proof. This can be shown analytically from (3) and (7); it also results from Little theorem (see [10]). 䊐

We illustrate the auto-covariance function in Fig. 2 against h where t = 10. One should explore the behavior of the auto-covariance function according to different traffic parameters but this is beyond the scope of this paper. 5. Conclusion This paper presents a first analysis of a particular Poisson cluster process family on the real line. This is a family where the clusters are obtained using transient MAP. This model is used to explain arrivals of in-line objects requests corresponding to different http sessions, where http session starting times form a Poisson process. To each object is associated the time needed to download the object. This work is concerned with computing some of the total traffic parameters: we derive both analytical solutions and corresponding algorithms. Other marks could be associated to the objects according to the modeling problem one is interesting in. It is a simple matter to generalize our equations to non-homogeneous Poisson cluster process. Using a same approach as the one developed here, other measures could be derived according to performance evaluation purposes. Another direction for further research is to include the elasticity of TCP connections into this traffic model. References [1] P. Tran-Gia, D. Staehle, K. Leibnitz, Source traffic modeling of wireless applications, Int. J. Electron. Commun. 55 (2001) 1–10. [2] Z. Liu, N. Niclausse, C. Jalpa-Villanueva, Traffic model and performance evaluation of Web servers, Perform. Eval. 46 (2001) 77–100. [3] M.F. Neuts, C.E.M. Pearce, The superposition of independent discrete Markovian packet streams, J. Appl. Probab. 28 (1991) 84–95.

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[4] G. Latouche, M.-A. Remiche, P. Taylor, Transient Markov arrival processes, Ann. Appl. Probab, in press. [5] A. Anderson, B. Nielsen, A Markovian approach for modeling packet traffic with long-range dependence, IEEE J. Select. Areas Commun. 16 (5) (1998) 196–204. [6] D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer, New York, 1988. [7] M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and their Applications, Marcel Dekker, New York, 1989. [8] G. Latouche, V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA–SIAM Series in Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. [9] H.-K. Choi, J. Limb, A behavioral model of Web traffic, in: Proceedings of the ICNP’99, Toronto, Canada, 1999. [10] R.D. Nelson, Probability, Stochastic Processes, and Queueing Theory. The Mathematics of Computer Performance Modeling, Springer, New York, 1995. Guy Latouche received his Ph.D. degree in mathematics from the Université Libre de Bruxelles in 1976. He has held various positions at the Université Libre de Bruxelles where he is Professor in the Department of Informatics. At present he teaches classes on stochastic processes and their applications. His research interests include various aspects of computational probability: matrix methods in Markov models, traffic models for telecommunication systems, and nearly completely decomposable systems.

Marie-Ange Remiche received her Ph.D. degree in mathematics in 1999 from the Université Libre de Bruxelles (ULB), Belgium. Currently she is a Chargée de Cours in networking at ULB, in the School of Engineering. Her current research interests are traffic modeling and analytical performance evaluation of fixed and wireless networks using matrix-analytical methods or stochastic geometry. During 2000–2002 she was an EC Marie Curie fellow at RWTH-Aachen working on UMTS modeling.