Traffic conditioner: upper bound for the spacer overflow probability

Performance Evaluation 58 (2004) 15–23 Traffic conditioner: upper bound for the spacer overflow probability Nathalie Omnès a,∗ , Annie Gravey b , Raymond Marie c b a Mitsubishi Electric ITE TCL, 1, allée de Beaulieu, CS 10 806, 35 708 Rennes Cedex 7, France Département informatique, ENST Bretagne, Technopole de Brest Iroise, B.P. 832, 28 285 Brest Cedex, France c Equipe ARMOR, IRISA, Campus universitaire de Beaulieu, 35 042 Rennes Cedex, France Received 3 December 2002; received in revised form 19 November 2003 Available online 20 July 2004 Abstract The current challenge for telecommunication networks is to offer new types of services. To achieve this, somehow traffic has to be conditioned. A spacer is a token bucket traffic conditioner that reshapes flows according to a periodic profile. Packets are delayed until they can be inserted according to this profile. This short paper concentrates on packet discard by overflow in the spacer. An upper bound for the overflow probability is given, valid even when it is fully loaded. It shows the ratio of discarded packets is of the same order than the ratio of out-of-profile packets. © 2004 Elsevier B.V. All rights reserved. Keywords: Traffic conditioning; Leaky bucket; Analytical model; Upper bound; Loss probability 1. Introduction Although Internet Protocol (IP) is currently widely used, it does not provide a straightforward solution to the applications’ needs. Hence it does not permit to offer all the quality of service guarantees on a large scale, and in particular to real-time services such as distributed games. To overcome this problem, more services should appear in the future. The Internet Engineering Task Force (IETF) has standardized in RFC 2475 (see Ref. [7]) an architecture for Differentiated Services (DiffServ). For the sake of scalability, several types of nodes are distinguished. DiffServ ingress and egress nodes are located on the boundary of a DiffServ domain, while DiffServ interior ∗ Corresponding author. Tel.: +33-223-45-58-34; fax: +33-223-45-58-59. E-mail addresses: omnes@tcl.ite.mee.com (N. Omnès), annie.gravey@enst-bretagne.fr (A. Gravey), raymond.marie@irisa.fr (R. Marie). 0166-5316/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.peva.2004.05.001 16 N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 nodes only implement simple functionalities. This is achieved by marking packets in the ingress node with a DiffServ codepoint. Complexity is thus pushed in border routers, which are responsible for offering several types of services. A type of service specifies some significant characteristics of packet transmission, which may include throughput, delay, jitter and loss. To control these characteristics, the network somehow needs to control incoming flows, which is achieved by traffic conditioning functions in ingress and egress nodes. A traffic conditioner measures the temporal properties of a stream of packets to determine whether each packet is in-profile or out-of-profile. Token buckets are commonly used for identifying out-of-profile packets (see Refs. [1, 2 (Subsection 5.3.3), 7 (Subsection 2.3.2)]). They allow out-of-profile packets to be either queued until they are in-profile, or discarded, or marked with a new codepoint, or forwarded unchanged while triggering some accounting procedures. Generally speaking, token buckets allow burstiness in the output stream, up to a given maximum burst size. A leaky bucket can be seen as a particular token bucket that enforces a rigid output pattern, allowing no burstiness in the output stream (see Ref. [2 (Subsection 5.3.3)]). It needs two parameters to identify a constant bit rate source: r, the peak rate, and b, the bucket depth. The peak rate is not sufficient for identifying out-of-profile packets as data go through several layers to reach the physical one, as well as through several network devices. Unless these devices are synchronous, periodicity gets corrupted by them. A spacer is a token bucket whose output is periodic until no more packets are available, thus allowing no burstiness. In Ref. [1 (Corollary 1.2.1, p. 16)], it is shown that spacers and leaky buckets are equivalent in case of constant size packets. To shape the outgoing flow, packets in advance are delayed until they can be inserted according to a periodic profile. Consequently late packets do not suffer from any further delay. It has been proved, in Ref. [1 (Section 1.5.3)], that spacing does not increase delay nor bandwidth requirements. It does not even increase buffer requirements since interior nodes only need to implement very small buffers. Furthermore, limiting the flow to a given rate at the access facilitates management in the core network, which is a key issue (see Refs. [2,4–6]). Let us consider Fig. 1. Each step upwards correspond to a packet arrival, while each step downwards to a departure. Packets 2–6 are in advance, so they are delayed in the spacer. On the contrary, packet 12 Fig. 1. Spacer occupancy level vs. time. N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 17 is late, so it is retransmitted as soon as it enters the spacer, and it induces an idle period. Such a period begins when the spacer is empty and waiting for a new packet to transmit. It ends when a new packet arrives. We see that the output traffic is shaped. The spacer does not need to absorb all bursts, as they correspond to packets that have suffered from a long delay in the network devices they crossed. The larger the burst is, the greater the delay perceived by the receiver. Therefore the spacer lops long bursts by discarding packets when they arrive as the spacer is full. This is represented in Fig. 1, where packets 7 and 10 are lost by overflow. The spacer cannot be reduced to a queuing analysis in a straightforward manner as it consists of a G/D/1 system, which we aim at studying even if the load ρ = 1. This is a particularly delicate point. It cannot either be analyzed as a “(max, +)” system unless we give up the random behavior of the core network, which would take off an essential property. Our aim in this article is to study the impact of the spacer capacity on the overflow probability (see Refs. [3,4]). To do so, we state working assumptions and define the variables that are central to our modeling in Section 2. We then study the overflow probability in Section 3, where our main result is an upper bound presented in Proposition 3. We also prove the upper bound in a deterministic context. We finally conclude in Section 4. Main proofs and intermediate lemmas are presented in the Appendix A. 2. Assumptions and formal description The assumptions made all along this paper are the following: (i) (ii) (iii) (iv) (v) Packets are of constant size. Packets arrive in sequence. The spacer is empty before packet 1 arrives. All packet losses are due to overflow. The spacer load is smaller than or equal to 1. We assume that the packets are of constant size for the sake of simplicity. This is the case for telephony over IP and it also corresponds to the case of an Asynchronous Transfer Mode (ATM) network (see Ref. [2]). Let us consider Fig. 2. TF is a constant defined as the interval between the generation of two consecutive packets at the source level. As packets are of constant size, the parameters r (the peak rate) and b (the bucket depth) of the leaky bucket can be replaced by TF and τF , the delay variation tolerance, also called the jitter (see Refs. [1 (Corollary 1.2.1, p.16), 2 (pp. 466–467)]). If S denotes the packet size in bits, we Fig. 2. Variables. 18 N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 Table 1 Basic notations tatk tk b TF τF TL Zk Wk Nk Packet k’s theoretical arrival time in the spacer Packet k’s arrival time in the spacer Capacity of the spacer Interval between generation of two consecutive packets at source level Delay variation tolerance of the flow entering the spacer Interdeparture time of packets from the spacer Occupancy level of the spacer at tk− Packet k’s waiting time in the network Number of packets that were lost by overflow before packet k’s arrival have TF = S/r. Furthermore, it is assumed that the interval between arrivals of two consecutive packets into the spacer lies between TF − τF and TF + τF . Let tatk be packet k’s theoretical arrival time in the spacer, tk its arrival time, and TL the interdeparture time of packets from the spacer. The sequence of theoretical arrival times is used to generate the output of the spacer. Actually, packet k is taken out of the spacer when tatk is reached. Furthermore, the spacer is in an idle period at time t if and only if the time elapsed since the last packet to enter has left is greater than or equal to TL . Note that tatk is moved forward when a packet arrives late, and backward when a packet is lost. It is computed using the following rule:  for k = 1, t1     tat if packet k − 1 is lost and tk ≤ tatk−1 , k−1 (1) tatk =  tatk−1 + TL if packet k − 1 is not lost and tk ≤ tatk−1 + TL ,    tk otherwise. Furthermore, packet k is conforming to parameters (TF , τF ) (or in-profile) if and only if tatk − tk ≤ τF . (2) As we assume that the spacer load is smaller than or equal to 1, we have TL ≤ TF . If TL = TF then the spacer is fully loaded. We further denote the spacer occupancy level just before packet k’s arrival by Zk , the waiting time packet k is subject to in the network by Wk , and the number of packets that were lost by overflow before packet k’s arrival by Nk . Wk is defined as the difference between the observed and the minimum transmission delay in the network. Due to the periodicity of the initial stream, we have tk = t1 −W1 +(k −1)TF +Wk , ∀k ≥ 1. All these basic notations are depicted in Table 1 and Fig. 2. 3. Study of the overflow probability With the help of the variables defined in Section 2, we are now going to state necessary conditions for overflow, give a formal expression for the sequence of theoretical arrival times, and finally study the probability for a packet to be lost in the spacer. N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 19 3.1. Preliminaries Packet k is lost only if it reaches the spacer in advance. In particular, it implies it reaches the spacer before a particular packet, denoted by P(k), leaves the spacer. Therefore we have (Zk = b) ⇒ (tk < tatP(k) ). (3) Proof. Let CTi denote the ith packet that is not lost by overflow in the spacer. If Zk = b, let J be a random variable taking its value in N such that tatCTJ−1 ≤ tk < tatCTJ . At tk− , k − 1 packets arrived, Nk of which overflowed. Furthermore, if tatCTJ−1 ≤ tk < tatCTJ , then J − 1 packets have left. Therefore there are k − 1 − Nk − J + 1 packets in the spacer. This leads to the following condition on J : Zk = k − Nk − J = b ⇒ J = k − Nk − b. In particular, P(k) = CTk−Nk −b .
Eq. (3) depends on the sequence (tatk )k≥1 , which needs to be expressed in a more tractable way than Eq. (1). To achieve this, we firstly need to introduce DAk , which quantifies the theoretical arrival times shift due to the idle period that occurs just before packet k’s arrival, if any. We secondly introduce δk , which equals 1 when packet k arrives in a idle period and 0 otherwise. Proposition 1. ∀k ≥ 2, if packet k does not overflow, then tatk = t1 + (k − 1)TL + Packet k arrives in an idle period if and only if δk = 1. k j=2 δj DAj − Nk TL . The proof which implies  the introduction of further variables is presented in the Appendix A.1. The proposition shows that kj=2 δj DAj is the total duration of the idle periods that occurred before packet  k’s arrival. Hence kj=2 δj DAj represents the theoretical arrival times shift (delay) due to idle periods arrival times shift (advance) occurring between packet 1 and packet k arrivals, while Nk TL is the theoretical due to packets lost by overflow in the spacer. We can sum up this into Xk = kj=2 δj DAj − Nk TL , which represents the theoretical arrival times shift between packet 1 and packet k arrivals. Let us now give a new necessary condition for packet k to overflow. Proposition 2. (Zk = b) ⇒ (W1 − Wk > (k − 1)(TF − TL ) + bTL − Xk ).  Proof. Using Proposition 1, we have tatP(k) = t1 + (P(k) − 1)TL + P(k) j=2 δj DAj − NP(k) TL . But the number of packets that entered the spacer before packet k is k − 1 − Nk . It also equals P(k) − NP(k) + b − 1 as P(k) is the next packet to leave the spacer, and the spacer is full when packet k arrives. Therefore P(k) − NP(k) = k − Nk − b. Furthermore, δj = 0, ∀j = P(k) + 1, . . . , k, as these packets do not arrive  in an idle period. This leads to tatP(k) = t1 + (k − b − 1)TL + kj=2 δj DAj − Nk TL . Finally, we obtain (Zk = b) ⇒ (tk < tatP(k) ) ⇒ (W1 − Wk > (k − 1)(TF − TL ) + bTL − Xk ).
3.2. Upper bound in a stochastic context Thanks to Proposition 2, we can now give an upper bound for the overflow probability, as shown in the proposition below. This is our essential result, as it proves that the loss ratio is proportional to the out-of-profile packets ratio, even if the spacer is fully loaded. 20 N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 Proposition 3. Let k be a packet of the flow. If (i) P(|Wi − Wj | > τF ) < ε, ∀i, j ∈ N∗ , (ii) (Wk+1 − Wk )k≥1 is a sequence of independent random variables, and (iii) b = max(2, ⌈τF /TL ⌉) where ⌈x⌉ is the smallest integer greater than x, then P(Zk = b) < ε. Proof. Let us define Ai as the index of the first packet of the ith idle period. The key points are to show firstly that (Zk = b) ⇒ (WAi − Wk > bTL ) and secondly that {Ai = j} ∈ σ(W2 − W1 , . . . , Wj − Wj−1 ). Then, on the ith idle period, {Ai = j} and {Wj − Wk > bTL } are independent by assumption (ii). Finally, assumption (iii) implies bTL ≥ τF , and assumption (i) leads to the upper bound. A detailed version of this proof is presented in the Appendix A.2.
Assumption (ii) means that the sequence of waiting times (Wk )k≥1 has independent increments. Intuitively, this is valid firstly in a large network which is in a stationary state. Secondly, it is valid if the network load varies with time and the period of the flow is large compared to the network load variation scale. Thirdly, it is valid if the period of the flow is small compared to the network load variation scale, as the network is then almost seen as stationary by this flow. Therefore assumption (ii) applies to a large range of networks, including high-speed packet networks such as the Internet. 3.3. Upper bound in a deterministic context The theory of Network Calculus has already solved this problem in a deterministic context. In particular, in Ref. [1 (Fig. 1.10, p. 30)], the buffer bound is given. We will now show that our modeling leads to the same result. Proposition 4. If b = max(2, ⌈τF /TL ⌉) and the flow is conforming with parameters (TF , τF ), then no packet is lost by overflow in the spacer. Proof. Assume we are in the first busy period, and there exists an index k such that packet k is the first packet to overflow. We have k ≥ b + 2. By Eq. (3), we have Zk = b ⇒ tk < tatP(k) . As packet k is the first to overflow, P(k) = k − b. Furthermore, during the first busy period, tatk−b = tat1 + (k − b − 1)TL . Hence Zk = b ⇒ W1 − Wk > bTL ≥ τF . But applying Eqs. (1) and (2), t1 + (k − 1)TF − tk ≤ τF , therefore W1 − Wk ≤ τF . Finally, k does not exist. Applying the same reasoning, it is straightforward to prove by induction that no packet overflows during the jth busy period.
4. Conclusion Traffic conditioning at the ingress of a DiffServ domain can be achieved by a spacer, which is a particular token bucket. We have seen in this article that it is possible to model it with simple mathematical tools. Applying our model, we have presented in Proposition 3 an upper bound for the overflow probability. It proves that the probability of losing packets is of the same order than the probability that the network jitter exceeds the jitter tolerance. Although this result is intuitively valid, it had never, to our knowledge, N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 21 been proved. Our proof takes into account the random nature of the core network, and is valid even if the spacer is fully loaded. This is important, as introducing either a random context or a load equal to 1 may have unexpected consequences. From a practical point of view, this result can be used to offer statistical guarantees to stringent flows. As the spacer size increases linearly with the jitter, it seems crucial to reduce this jitter. This can be achieved by implementing traffic control functions, and in particular appropriate schedulers. Future work shall include application to variable size packets. Appendix A A.1. Proof of Proposition 1 By definition of DAk and δk we have DAk = W2 − W1 + TF − TL + N2 TL for k = 2, DAk = k−1 Wk − W1 + (k − 1)(TF − TL ) − j=2 δj DAj + Nk TL for k ≥ 3, and δk = 1DAk >0 , for k ≥ 2. Let Ek be the last packet that entered the spacer before packet k. Packet k arrives in an idle period if and only if packet k’s arrival time, tk , is greater than tatEk + TL . Any packet reaching the spacer in an idle state is immediately retransmitted. Therefore we have, ∀k ≥ 1, tatk = tk if tk > tatEk + TL , and tatEk + TL otherwise. We now prove Proposition 1 by induction. We also prove that, if Ek < k − 1, then δEk +1 = δEk +2 = · · · = δk−1 = 0. For k = 2, we have tat2 = tat1 + TL if t2 ≤ tat1 + TL and Z2 < b, and t2 otherwise. As Z2 ≤ 1 < b, N2 = 0, tat1 = t1 and t2 = t1 − W1 + TF + W2 , we have tat1 + TL ≥ t2 ⇔ W2 ≤ W1 + TL − TF ⇔ δ2 = 0. If t2 ≤ tat1 + TL , we have tat2 = tat1 + TL = t1 + TL + δ2 DA2 − N2 TL . Furthermore, if t2 > tat1 + TL , tat2 = t2 = t1 − W1 + TF + W2 = t1 + DA2 + TL . Hence in both cases we have tat2 = t1 + TL + δ2 DA2 − N2 TL . If the result is true up to k − 1, as Zk < b, we have tatk = tatEk + TL if tk ≤ tatEk + TL , and tk otherwise.  k But ZEk < b and 1 ≤ Ek ≤ k − 1, therefore tatEk = t1 + (Ek − 1)TL + E i=2 δi DAi − NEk TL . Firstly, if tk ≤ tatEk + TL , we have tk ≤ tatEk + TL ⇔ t1 − W1 + (k − 1)TF + Wk ≤ t1 + (Ek − 1)TL + Ek Ek j=2 δj DAj − NEk TL + TL ⇔ Wk ≤ W1 − (k − 1)TF + j=2 δj DAj + (Ek − NEk )TL . Let us show that, if Ek < k − 1, then δEk +1 = δEk +2 = · · · = δk−1 = 0. As packets Ek + 1, Ek +  k 2, . . . , k − 1 overflow, tEk +1 ≤ tatEk ⇒ t1 − W1 + Ek TF + WEk +1 ≤ t1 + (Ek − 1)TL + E j=2 δj DAj − Ek NEk TL ⇒ WEk +1 − W1 + Ek (TF − TL ) − j=2 δj DAj − NEk +1 TL ≤ −TL , as NEk +1 = NEk . Therefore tEk +1 ≤ tatEk ⇒ δEk +1 = 0. If δEk +1 = δEk +2 = · · · = δEk +i−1 = 0 for Ek + i ≤ k − 1, then tEk +i ≤  k tatEk ⇒ t1 − W1 + (Ek + i − 1)TF + WEk +i ≤ t1 + (Ek − 1)TL + E j=2 δj DAj − NEk TL ⇒ δEk +i = 0, as NEk +i = NEk + i − 1. Hence δEk +1 = δEk +2 = · · · = δk−1 = 0.  k Therefore tk ≤ tatEk + TL ⇔ Wk ≤ W1 − (k − 1)TF + E j=2 δj DAj + (Ek − NEk )TL ⇔ Wk ≤ k−1 W1 − (k − 1)(TL − TF ) + j=2 δj DAj − Nk TL ⇔ δk = 0. This means an idle period occurs before tk− if  k k−1 and only if δk = 1. So we have tatk = t1 + (Ek − 1)TL + E δi DAi − NEk TL + TL = t1 + i=2 δi DAi + i=2 k k (Ek − NEk )TL = t1 + i=2 δi DAi + (k − 1 − Nk )TL = t1 + (k − 1)TL + i=2 δi DAi − Nk TL . Secondly, if tk > tatEk + TL , then δk = 1, tatk = tk = t1 − W1 + (k − 1)TF + Wk and tatk = tk =  t1 − W1 + (k − 1)TF + Wk = t1 + (k − 1)TL + ki=2 δi DAi − Nk TL . 22 N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 A.2. Proof of Proposition 3 Lemma 5. Ai = inf{k > Ai−1 : Wk −WAi−1 +(k −Ai−1 )(TF −TL )+(Nk −NAi−1 )TL > 0}. Furthermore, i j=1 DAAj = WAi − W1 + (Ai − 1)(TF − TL ) + NAi TL . Proof. The result is straightforward fori = 2. If the result is correct for i − 1, then for k > Ai−1 , DAk = Wk − W1 + (k − 1)(TF − TL ) − i−1 j=1 DAAj + Nk TL = Wk − W1 + (k − 1)(TF − TL ) − WAi−1 + W1 − (Ai−1 − 1)(TF − TL ) − NAi−1 TL + Nk TL , therefore packet k arrives in an idle period if and only  if (DAk > 0) ⇔ (Wk − WAi−1 + (k − Ai−1 )(TF − TL ) + (Nk − NAi−1 )TL > 0). Finally, ij=1 DAAj = i−1 WAi − WAi−1 + (Ai − Ai−1 )(TF − TL ) + (NAi − NAi−1 )TL + j=1 DAAj = WAi − W1 + (Ai − 1)(TF − TL ) + NAi TL .
Lemma 6. ∀k ∈ N∗ , Nk+1 ∈ σ(Wk − Wk−1 , Wk−1 − Wk−2 , . . . , W2 − W1 ). Proof. We prove this result by induction. We know that N2 = N3 = · · · = Nb+2 = 0, so this is true for k ≤ b+1. We know Nb+3 ∈ {0, 1}, and Nb+3 = 1 ⇔ tb+2 < tatCT(b+2−b−1) +TL ⇔ t1 −W1 +(b+1)TF +Wb+2 < t1 + TL ⇔ W1 − Wb+2 > (b + 1)TF − TL . Hence the result is true at index b + 2. Assume this is true for all i ≤ k. We have Nk+1 = Nk if packet k does not overflow, and Nk+1 = Nk + 1 otherwise. Hence Nk+1 − Nk = 1 ⇔ tk < tatCT(k−Nk −b−1) + TL ⇔ W1 − Wk + (k − Nk − 2 − k−Nk −b−1+NCT(k−Nk −b−1) b)TL + j=2 δj DAj > (k − 1)TF . But CT(k−Nk −b−1) < k, so by applying the induction assumption we have NCT(k−Nk −b−1) ∈ σ(Wk − Wk−1 , . . . , W2 − W1 ). Furthermore, DA2 ∈ σ(W2 − W1 ). It is straightforward that DAj ∈ σ(Wj − Wj−1 , . . . , W2 − W1 ), and so for δj = 1DAj >0 . Finally, Nk+1 − Nk ∈ σ(Wk − Wk−1 , . . . , W2 − W1 ).
Proof of Proposition 3. Let k be a packet of the ith busy period. We have by Lemma 5: XAi = WAi − W1 +(Ai −1)(TF −TL ). Furthermore, knowing packet k belongs to the ith busy period, Xk = WAi −W1 + (Ai − 1)(TF − TL ) + (NAi − Nk )TL . Hence Zk = b ⇒ Wk < W1 − (k − 1)TF + (k − b − 1)TL + Xk ⇔ WAi − Wk > bTL + (k − Ai )(TF − TL ) + (Nk − NAi )TL > bTL ≥ τF . Using Eq. (2), we conclude that P(Zk = b) ≤ P(Wk < W1 − (k − 1)TF + (k − b − 1)TL + Xk ) ≤ P(WAi − Wk > τF ) =  j≤k−1 P(Wj − Wk > τF |Ai = j)P(Ai = j). By Lemmas 5 and 6, {Ai = j} ∈ σ(W2 − W1 , . . . , Wj − Wj−1 ), and {Wj − Wk } ∈ σ(Wk − Wk−1 , . . . , Wj+1 − Wj ). By (ii) σ(W2 − W1 , . . . , Wj − Wj−1 ) and σ(Wk − Wk−1 , . . . , Wj+1 − Wj ) are  independent. Therefore P(Wj − Wk > τF |Ai = j) = P(Wj − Wk > τF ). Hence P(Zk = b) ≤ P (W − W > τ ) P (A = j) < ε
j k F i j≤k−1 j≤k−1 P(Ai = j) ≤ ε. References [1] J.-Y. Le Boudec, P. Thiran, Network Calculus, A Theory of Deterministic Queuing Systems for the Internet, Lecture Notes in Computer Science, vol. 2050, Springer, Berlin, 2001. [2] A.S. Tanenbaum, Computer Networks, 3rd ed., Prentice-Hall International, Englewood Cliffs, NJ, 1996. [3] N. Omnès, Thèse no. 2416, Analyse d’outils de contrôle de la qualité de service dans les réseaux de paquets haut débit, Ph.D., Université de Rennes 1, France, October 2001. [4] P. Boyer, F. Guillemin, M. Servel, J.P. Coudreuse, Spacing cells protects and enhances the utilization of ATM network links, IEEE Network 6 (5) (1992) 38–49. N. Omnès et al. / Performance Evaluation 58 (2004) 15–23 23 [5] F. Guillemin, P. Boyer, J. Boyer, O. Dugeon, C. Mangin, Regulation of TCP over ATM via cell spacing, in: Proceedings of ITC’16, Edinburgh, June 1999. [6] G. Karlsson, F. Orava, The DIY approach to QoS, in: Proceedings of IWQoS’99, IEEE Catalog Number 98EX354, 1999. [7] S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, An Architecture for Differentiated Services, RFC 2475, IETF, December 1998. Nathalie Omnès received in 1996 a Master of Science in probability from the University of Rennes, France. She then joined the CNET, France Telecom research center, and received in 2001 a PhD degree in applied mathematics. She is now working as a research engineer in Mitsubishi Electric Telecommunication Laboratory, ITE TCL, located in France. She focuses on wireless LANs and their interconnection with access networks, more specifically on traffic handling mechanisms and their mathematical modeling. Annie Gravey received the French “Agrégation de Mathématiques” in 1978 and a PhD degree in automatics and signal theory from Paris-Sud University in 1981. She is currently Professor and Head of the Computer Engineering Department at the ENST-Bretagne, in Brest (France). Her current research interests include the design and evaluation of traffic engineering methods for broadband and wireless networks, and of procedures related to QoS specification and management. Raymond Marie received the Doctorat d’Ingénieur and the Doctorat d’Etat es-Sciences Mathématiques from the University of Rennes, France, in 1973 and 1978, respectively. From 1977 to 1999, he was a Research Manager of an INRIA group in modeling. He spent the 1981–1982 academic year as a Visiting Associate Professor at North Carolina State University, Raleigh, NC, USA. Since 1983, he is a Professor at the Computer Science Department of the University of Rennes. His active research interests include performance evaluation of computer systems and quality of service of telecommunication networks. He also works on dependability for highly reliable complex systems.