Network Working Group A. Charny Request for Comments: 3247 Cisco Systems, Inc. Category: Informational J.C.R. Bennett Motorola K. Benson Tellabs J.Y. Le Boudec EPFL A. Chiu Celion Networks W. Courtney TRW S. Davari PMC-Sierra V. Firoiu Nortel Networks C. Kalmanek AT&T Research K.K. Ramakrishnan TeraOptic Networks March 2002 Supplemental Information for the New Definition of the EF PHB (Expedited Forwarding Per-Hop Behavior) Status of this Memo This memo provides information for the Internet community. It does not specify an Internet standard of any kind. Distribution of this memo is unlimited. Copyright Notice Copyright (C) The Internet Society (2001). All Rights Reserved.
AbstractThis document was written during the process of clarification of RFC2598 "An Expedited Forwarding PHB" that led to the publication of revised specification of EF "An Expedited Forwarding PHB". Its primary motivation is providing additional explanation to the revised EF definition and its properties. The document also provides additional implementation examples and gives some guidance for computation of the numerical parameters of the new definition for several well known schedulers and router architectures.
1 Introduction ........................................... 2 2 Definition of EF PHB ................................... 3 2.1 The formal definition .................................. 3 2.2 Relation to Packet Scale Rate Guarantee ................ 6 2.3 The need for dual characterization of EF PHB ........... 7 3 Per Packet delay ....................................... 9 3.1 Single hop delay bound ................................. 9 3.2 Multi-hop worst case delay ............................. 10 4 Packet loss ............................................ 10 5 Implementation considerations .......................... 11 5.1 The output buffered model with EF FIFO at the output. .. 12 5.1.1 Strict Non-preemptive Priority Queue ................... 12 5.1.2 WF2Q ................................................... 13 5.1.3 Deficit Round Robin (DRR) .............................. 13 5.1.4 Start-Time Fair Queuing and Self-Clocked Fair Queuing .. 13 5.2 Router with Internal Delay and EF FIFO at the output ... 13 6 Security Considerations ................................ 14 7 References ............................................. 14 Appendix A. Difficulties with the RFC 2598 EF PHB Definition .. 16 Appendix B. Alternative Characterization of Packet Scale Rate Guarantee ......................................... 20 Acknowledgements .............................................. 22 Authors' Addresses ............................................ 22 Full Copyright Statement ...................................... 24 RFC2598  was not sufficiently precise (see Appendix A and ). A more precise definition is given in . This document is intended to aid in the understanding of the properties of the new definition and provide supplemental information not included in the text of  for sake of brevity. This document is outlined as follows. In section 2, we briefly restate the definition for EF PHB of . We then provide some additional discussion of this definition and describe some of its properties. We discuss the issues associated with per-packet delay
and loss in sections 3 and 4. In section 5 we discuss the impact of known scheduling architectures on the critical parameters of the new definition. We also discuss the impact of deviation of real devices from the ideal output-buffered model on the magnitude of the critical parameters in the definition. 6]. Here we restate the formal definition from  verbatim. A node that supports EF on an interface I at some configured rate R MUST satisfy the following equations: d_j <= f_j + E_a for all j>0 (eq_1) where f_j is defined iteratively by f_0 = 0, d_0 = 0 f_j = max(a_j, min(d_j-1, f_j-1)) + l_j/R, for all j > 0 (eq_2) In this definition: - d_j is the time that the last bit of the j-th EF packet to depart actually leaves the node from the interface I. - f_j is the target departure time for the j-th EF packet to depart from I, the "ideal" time at or before which the last bit of that packet should leave the node. - a_j is the time that the last bit of the j-th EF packet destined to the output I actually arrives at the node. - l_j is the size (bits) of the j-th EF packet to depart from I. l_j is measured on the IP datagram (IP header plus payload) and does not include any lower layer (e.g. MAC layer) overhead. - R is the EF configured rate at output I (in bits/second). - E_a is the error term for the treatment of the EF aggregate. Note that E_a represents the worst case deviation between actual departure time of an EF packet and ideal departure time of the same packet, i.e. E_a provides an upper bound on (d_j - f_j) for all j.
- d_0 and f_0 do not refer to a real packet departure but are used purely for the purposes of the recursion. The time origin should be chosen such that no EF packets are in the system at time 0. - for the definitions of a_j and d_j, the "last bit" of the packet includes the layer 2 trailer if present, because a packet cannot generally be considered available for forwarding until such a trailer has been received. An EF-compliant node MUST be able to be characterized by the range of possible R values that it can support on each of its interfaces while conforming to these equations, and the value of E_a that can be met on each interface. R may be line rate or less. E_a MAY be specified as a worst-case value for all possible R values or MAY be expressed as a function of R. Note also that, since a node may have multiple inputs and complex internal scheduling, the j-th EF packet to arrive at the node destined for a certain interface may not be the j-th EF packet to depart from that interface. It is in this sense that eq_1 and eq_2 are unaware of packet identity. In addition, a node that supports EF on an interface I at some configured rate R MUST satisfy the following equations: D_j <= F_j + E_p for all j>0 (eq_3) where F_j is defined iteratively by F_0 = 0, D_0 = 0 F_j = max(A_j, min(D_j-1, F_j-1)) + L_j/R, for all j > 0 (eq_4) In this definition: - D_j is the actual departure time of the individual EF packet that arrived at the node destined for interface I at time A_j, i.e., given a packet which was the j-th EF packet destined for I to arrive at the node via any input, D_j is the time at which the last bit of that individual packet actually leaves the node from the interface I. - F_j is the target departure time for the individual EF packet that arrived at the node destined for interface I at time A_j.
- A_j is the time that the last bit of the j-th EF packet destined to the output I to arrive actually arrives at the node. - L_j is the size (bits) of the j-th EF packet to arrive at the node that is destined to output I. L_j is measured on the IP datagram (IP header plus payload) and does not include any lower layer (e.g. MAC layer) overhead. - R is the EF configured rate at output I (in bits/second). - E_p is the error term for the treatment of individual EF packets. Note that E_p represents the worst case deviation between the actual departure time of an EF packet and the ideal departure time of the same packet, i.e. E_p provides an upper bound on (D_j - F_j) for all j. - D_0 and F_0 do not refer to a real packet departure but are used purely for the purposes of the recursion. The time origin should be chosen such that no EF packets are in the system at time 0. - for the definitions of A_j and D_j, the "last bit" of the packet includes the layer 2 trailer if present, because a packet cannot generally be considered available for forwarding until such a trailer has been received. It is the fact that D_j and F_j refer to departure times for the j-th packet to arrive that makes eq_3 and eq_4 aware of packet identity. This is the critical distinction between the last two equations and the first two. An EF-compliant node SHOULD be able to be characterized by the range of possible R values that it can support on each of its interfaces while conforming to these equations, and the value of E_p that can be met on each interface. E_p MAY be specified as a worst-case value for all possible R values or MAY be expressed as a function of R. An E_p value of "undefined" MAY be specified. Finally, there is an additional recommendation in  that an EF compliant node SHOULD NOT reorder packets within a micorflow. The definitions described in this section are referred to as aggregate and packet-identity-aware packet scale rate guarantee ,. An alternative mathematical characterization of packet scale rate guarantee is given in Appendix B.
section 2.1 is stronger than the well-known rate-latency curve  in the sense that if a scheduler satisfies the EF definition it also satisfies the rate-latency curve. As a result, all the properties known for the rate-latency curve also apply to the modified EF definition. However, we argue below that the definition of section 2.1 is more suitable to reflect the intent of EF PHB than the rate- latency curve. It is shown in  that the rate-latency curve is equivalent to the following definition: Definition of Rate Latency Curve (RLC): D(j) <= F'(j) + E (eq_5) where F'(0)=0, F'(j)=max(a(j), F'(j-1))+ L(j)/R for all j>0 (eq_6) It can be easily verified that the EF definition of section 2.1 is stronger than RLC by noticing that for all j, F'(j) >= F(j). It is easy to see that F'(j) in the definition of RLC corresponds to the time the j-th departure should have occurred should the EF aggregate be constantly served exactly at its configured rate R. Following the common convention, we refer to F'(j) as the "fluid finish time" of the j-th packet to depart. The intuitive meaning of the rate-latency curve of RLC is that any packet is served at most time E later than this packet would finish service in the fluid model. For RLC (and hence for the stronger EF definition) it holds that in any interval (0,t) the EF aggregate gets close to the desired service rate R (as long as there is enough traffic to sustain this rate). The discrepancy between the ideal and the actual service in this interval depends on the latency term E, which in turn depends on the
scheduling implementation. The smaller E, the smaller the difference between the configured rate and the actual rate achieved by the scheduler. While RLC guarantees the desired rate to the EF aggregate in all intervals (0,t) to within a specified error, it may nevertheless result in large gaps in service. For example, suppose that (a large number) N of identical EF packets of length L arrived from different interfaces to the EF queue in the absence of any non-EF traffic. Then any work-conserving scheduler will serve all N packets at link speed. When the last packet is sent at time NL/C, where C is the capacity of output link, F'(N) will be equal to NL/R. That is, the scheduler is running ahead of ideal, since NL/C < NL/R for R < C. Suppose now that at time NL/C a large number of non-EF packets arrive, followed by a single EF packet. Then the scheduler can legitimately delay starting to send the EF packet until time F'(N+1)=(N+1)L/R + E - L/C. This means that the EF aggregate will have no service at all in the interval (NL/C, (N+1)L/R + E - L/C). This interval can be quite large if R is substantially smaller than C. In essence, the EF aggregate can be "punished" by a gap in service for receiving faster service than its configured rate at the beginning. The new EF definition alleviates this problem by introducing the term min(D(j-1), F(j-1)) in the recursion. Essentially, this means that the fluid finishing time is "reset" if that packet is sent before its "ideal" departure time. As a consequence of that, for the case where the EF aggregate is served in the FIFO order, suppose a packet arrives at time t to a server satisfying the EF definition. The packet will be transmitted no later than time t + Q(t)/R + E, where Q(t) is the EF queue size at time t (including the packet under discussion).
arrivals. The aggregate part of the definition measures the accuracy of the output rate provided to the EF aggregate as a whole. The smaller E_a, the more accurate is the assurance that the reservoir is drained at least at the configured rate. Note that in this reservoir analogy packets of EF aggregate may be arbitrarily reordered. However, the definition of EF PHB given in  explicitly requires that no packet reordering occur within a microflow. This requirement restricts the scheduling implementations, or, in the reservoir analogy, the order of pulling packets out of the reservoir to make sure that packets within a microflow are not reordered, but it still allows reordering at the aggregate level. Note that reordering within the aggregate, as long as there is no flow-level reordering, does not necessarily reflect a "bad" service. Consider for example a scheduler that arbitrates among 10 different EF "flows" with diverse rates. A scheduler that is aware of the rate requirements may choose to send a packet of the faster flow before a packet of the slower flow to maintain lower jitter at the flow level. In particular, an ideal "flow"-aware WFQ scheduler will cause reordering within the aggregate, while maintaining packet ordering and small jitter at the flow level. It is intuitively clear that for such a scheduler, as well as for a simpler FIFO scheduler, the "accuracy" of the service rate is crucial for minimizing "flow"-level jitter. The packet-identity-aware definition quantifies this accuracy of the service rate. However, the small value of E_a does not give any assurances about the absolute value of per-packet delay. In fact, if the input rate exceeds the configured rate, the aggregate behavior definition may result in arbitrarily large delay of a subset of packets. This is the primary motivation for the packet-identity-aware definition. The primary goal of the packet-aware characterization of the EF implementation is that, unlike the aggregate behavior characterization, it provides a way to find a per-packet delay bound as a function of input traffic parameters. While the aggregate behavior definition characterizes the accuracy of the service rate of the entire EF aggregate, the packet-identity- aware part of the definition characterizes the deviation of the device from an ideal server that serves the EF aggregate in FIFO order at least at the configured rate. The value of E_p in the packet-identity-aware definition is therefore affected by two factors: the accuracy of the aggregate rate service
and the degree of packet reordering within the EF aggregate (under the constraint that packets within the same microflow are not reordered). Therefore, a sub-aggregate aware device that provides an ideal service rate to the aggregate, and also provides an ideal rate service for each of the sub-aggregates, may nevertheless have a very large value of E_p (in this case E_p must be at least equal to the ratio of the maximum packet size divided by the smallest rate of any sub aggregate). As a result, a large value of E_p does not necessarily mean that the service provided to EF aggregate is bad - rather it may be an indication that the service is good, but non- FIFO. On the other hand, a large value of E_p may also mean that the aggregate service is very inaccurate (bursty), and hence in this case the large value of E_p reflects a poor quality of implementation. As a result, a large number of E_p does not necessarily provide any guidance on the quality of the EF implementation. However, a small value of E_p does indicate a high quality FIFO implementation. Since E_p and E_a relate to different aspects of the EF implementation, they should be considered together to determine the quality of the implementation. appendix B). Because the delay bound depends on the configured rate R and the input burstiness B, it is desirable for both of these parameters to be visible to a user of the device. A PDB desiring a particular delay bound may need to limit the range of configured rates and allowed burstiness that it can support in order to deliver such bound. Equation (eq_7) provides a means for determining an acceptable operating region for the device with a given E_p. It may also be useful to limit the total offered load to a given output to some rate R_1 < R (e.g. to obtain end-to-end delay bounds ). It
is important to realize that, while R_1 may also be a configurable parameter of the device, the delay bound in (eq_7) does not depend on it. It may be possible to get better bounds explicitly using the bound on the input rate, but the bound (eq_7) does not take advantage of this information. 5]. This result implies that the ability to limit the worst case burstiness and the resulting end-to-end delay across several hops may require not only limiting EF utilization on all links, but also constraining the global network topology. Such topology constraints would need to be specified in the definition of any PDB built on top of EF PHB, if such PDB requires a strict worst case delay bound.
topology which are outside the scope of inherently local definition of a PHB. Thus, it must be possible to establish whether a device conforms to the EF definition even when some packets are lost. This can be done by performing an "off-line" test of conformance to equations (eq_1)- (eq_4). After observing a sequence of packets entering and leaving the node, the packets which did not leave are assumed lost and are notionally removed from the input stream. The remaining packets now constitute the arrival stream and the packets which left the node constitute the departure stream. Conformance to the equations can thus be verified by considering only those packets that successfully passed through the node. Note that specification of which packets are lost in the case when loss does occur is beyond the scope of the definition of EF PHB. However, those packets that were not lost must conform to the equations definition of EF PHB in section 2.1. 7]) where the packet sits after being selected by the output scheduler but before it is transmitted. In cases such as these, the extra delay a packet may experience can be accounted for by absorbing it into the latency terms E_a and E_p. Implementing EF on a router with a multi-stage switch fabric requires special attention. A packet may experience additional delays due to the fact that it must compete with other traffic for forwarding resources at multiple contention points in the switch core. The delay an EF packet may experience before it even reaches the output- link scheduler should be included in the latency term. Input- buffered and input/output-buffered routers based on crossbar design may also require modification of their latency terms. The factors such as the speedup factor and the choice of crossbar arbitration algorithms may affect the latency terms substantially.
Delay in the switch core comes from two sources, both of which must be considered. The first part of this delay is the fixed delay a packet experiences regardless of the other traffic. This component of the delay includes the time it takes for things such as packet segmentation and reassembly in cell based cores, enqueueing and dequeuing at each stage, and transmission between stages. The second part of the switch core delay is variable and depends on the type and amount of other traffic traversing the core. This delay comes about if the stages in the core mix traffic flowing between different input/output port pairs. Thus, EF packets must compete against other traffic for forwarding resources in the core. Some of this competing traffic may even be traffic from other, non-EF aggregates. This introduces extra delay, that can also be absorbed by the latency term in the definition. To capture these considerations, in this section we will consider two simplified implementation examples. The first is an ideal output buffered node where packets entering the device from an input interface are immediately delivered to the output scheduler. In this model the properties of the output scheduler fully define the values of the parameters E_a and E_p. We will consider the case where the output scheduler implements aggregate class-based queuing, so that all EF packets share a single queue. We will discuss the values of E_a and E_p for a variety of class-based schedulers widely considered. The second example will consider a router modeled as a black box with a known bound on the variable delay a packet can experience from the time it arrives to an input to the time it is delivered to its destination output. The output scheduler in isolation is assumed to be an aggregate scheduler where all EF packets share a single FIFO queue, with a known value of E_a(S)=E_p(S)=E(S). This model provides a reasonable abstraction to a large class of router implementations.
1]. A class-based WF2Q scheduler, in which all EF traffic shares a single queue with the weight corresponding to the configured rate of the EF aggregate satisfies the EF definition with the latency term E = MTU/C+MTU/R. 12], E can be shown to grow linearly with N*(r_max/r_min)*MTU, where r_min and r_max denote the smallest and the largest rate among the rate assignments of all queues in the scheduler, and N is the number of queues in the scheduler. 9] and Self-Clocked Fair Queuing (SCFQ)  E can be shown to grow linearly with the number of queues in the scheduler. 13]). Recall from the discussion of section 3 that bounding input burstiness B may not be easy in a general topology. In the absence of the knowledge of a bound on B one can bound E_p as E_p = E(S) + D*C_inp/R (see ). Note also that the bounds in this section are derived using only the bound on the variable portion of the interval delay and the error bound of the output scheduler. If more details about the architecture of a device are available, it may be possible to compute better bounds.
6]. It adds no new functions to it. As a result, it adds no security issues to those described in that specification.  J.C.R. Bennett and H. Zhang, "WF2Q: Worst-case Fair Weighted Fair Queuing", INFOCOM'96, March 1996.  J.-Y. Le Boudec, P. Thiran, "Network Calculus", Springer Verlag Lecture Notes in Computer Science volume 2050, June 2001 (available online at http://lcawww.epfl.ch).  Bradner, S., "Key Words for Use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997.  J.C.R. Bennett, K. Benson, A. Charny, W. Courtney, J.Y. Le Boudec, "Delay Jitter Bounds and Packet Scale Rate Guarantee for Expedited Forwarding", Proc. Infocom 2001, April 2001.  A. Charny, J.-Y. Le Boudec "Delay Bounds in a Network with Aggregate Scheduling". Proc. of QoFIS'2000, September 25-26, 2000, Berlin, Germany.  Davie, B., Charny, A., Baker, F., Bennett, J.C.R., Benson, K., Boudec, J., Chiu, A., Courtney, W., Davari, S., Firoiu, V., Kalmanek, C., Ramakrishnan, K.K. and D. Stiliadis, "An Expedited Forwarding PHB (Per-Hop Behavior)", RFC 3246, March 2002.  T. Ferrari and P. F. Chimento, "A Measurement-Based Analysis of Expedited Forwarding PHB Mechanisms," Eighth International Workshop on Quality of Service, Pittsburgh, PA, June 2000.  S.J. Golestani. "A Self-clocked Fair Queuing Scheme for Broad- band Applications". In Proceedings of IEEE INFOCOM'94, pages 636-646, Toronto, CA, April 1994.  P. Goyal, H.M. Vin, and H. Chen. "Start-time Fair Queuing: A Scheduling Algorithm for Integrated Services". In Proceedings of the ACM-SIGCOMM 96, pages 157-168, Palo Alto, CA, August 1996.  Jacobson, V., Nichols, K. and K. Poduri, "An Expedited Forwarding PHB", RFC 2598, June 1999.
 Jacobson, V., Nichols, K. and K. Poduri, "The 'Virtual Wire' Behavior Aggregate", Work in Progress.  M. Shreedhar and G. Varghese. "Efficient Fair Queuing Using Deficit Round Robin". In Proceedings of SIGCOMM'95, pages 231-243, Boston, MA, September 1995.  Le Boudec, J.-Y., Charny, A. "Packet Scale Rate Guarantee for non-FIFO Nodes", Infocom 2002, New York, June 2002.
10] states: "The EF PHB is defined as a forwarding treatment for a particular diffserv aggregate where the departure rate of the aggregate's packets from any diffserv node must equal or exceed a configurable rate. The EF traffic SHOULD receive this rate independent of the intensity of any other traffic attempting to transit the node. It [the EF PHB departure rate] SHOULD average at least the configured rate when measured over any time interval equal to or longer than the time it takes to send an output link MTU sized packet at the configured rate." A literal interpretation of the definition would consider the behaviors given in the next two subsections as non-compliant. The definition also unnecessarily constrains the maximum configurable rate of an EF aggregate. RFC 2598. Note that this forwarding pattern can occur in any work-conserving scheduler in an ideal output-buffered architecture where EF packets arrive in a perfectly clocked manner according to the above pattern and are forwarded according to exactly the same pattern in the absence of any non-EF traffic. Trivial as this example may be, it reveals the lack of mathematical precision in the formal definition. The fact that no work-conserving scheduler can formally comply with the definition is unfortunate, and appears to warrant some changes to the definition that would correct this problem.
The underlying reason for the problem described here is quite simple - one can only expect that the EF aggregate is served at configured rate in some interval where there is enough backlog of EF packets to sustain that rate. In the example above the packets come in exactly at the rate at which they are served, and so there is no persistent backlog. Certainly, if the input rate is even smaller than the configured rate of the EF aggregate, there will be no backlog as well, and a similar formal difficulty will occur. A seemingly simple solution to this difficulty might be to require that the EF aggregate is served at its configured rate only when the queue is backlogged. However, as we show in the remainder of this section, this solution does not suffice. RFC 2598 definition of EF PHB. As in the previous example, the underlying reason for this problem is that the scheduler cannot forward EF traffic faster than it arrives. However, it can be easily seen that the existence of internal delay causes one packet to be inside the router at all times. An external observer will rightfully conclude that the number of EF packets that arrived to the router is always at least one greater than the number of EF packets that left the router, and therefore the EF aggregate is constantly backlogged. However, while the EF aggregate is continuously backlogged, the observed output rate is nevertheless strictly less that the configured rate.
This example indicates that the simple addition of the condition that EF aggregate must receive its configured rate only when the EF aggregate is backlogged does not suffice in this case. Yet, the problem described here is of fundamental importance in practice. Most routers have a certain amount of internal delay. A vendor declaring EF compliance is not expected to simultaneously declare the details of the internals of the router. Therefore, the existence of internal delay may cause a perfectly reasonable EF implementation to display seemingly non-conformant behavior, which is clearly undesirable. RFC-2598-compliant configurable rate for an EF aggregate cannot exceed C/2 . This is because an MTU-sized EF packet may arrive to an empty queue at time t just as an MTU-sized non-EF packet begins service. The maximum number of EF bits that could be forwarded during the interval [t, t + 2*MTU/C] is MTU. But if configured rate R > C/2, then this interval would be of length greater than MTU/R, and more than MTU EF bits would have to be served during this interval for the router to be compliant. Thus, R must be no greater than C/2. It can be shown that for schedulers other than PQ, such as various implementations of WFQ, the maximum compliant configured rate may be much smaller than 50%. For example, for SCFQ  the maximum configured rate cannot exceed C/N, where N is the number of queues in the scheduler. For WRR, mentioned as compliant in section 2.2 of RFC 2598, this limitation is even more severe. This is because in these schedulers a packet arriving to an empty EF queue may be forced to wait until one packet from each other queue (in the case of SCFQ) or until several packets from each other queue (in the case of WRR) are served before it will finally be forwarded. While it is frequently assumed that the configured rate of EF traffic will be substantially smaller than the link bandwidth, the requirement that this rate should never exceed 50% of the link bandwidth appears unnecessarily limiting. For example, in a fully connected mesh network, where any flow traverses a single link on its way from source to its destination there seems no compelling reason to limit the amount of EF traffic to 50% (or an even smaller percentage for some schedulers) of the link bandwidth. Another, perhaps even more striking example is the fact that even a TDM circuit with dedicated slots cannot be configured to forward EF packets at more than 50% of the link speed without violating RFC 2598
(unless the entire link is configured for EF). If the configured rate of EF traffic is greater than 50% (but less than the link speed), there will always exist an interval longer than MTU/R in which less than the configured rate is achieved. For example, suppose the configured rate of the EF aggregate is 2C/3. Then the forwarding pattern of the TDM circuit might be E E x E E x E E x ... |---| where only one packet is served in the marked interval of length 2T = 2MTU/C. But at least 4/3 MTU would have to be served during this interval by a router in compliance with the definition in RFC 2598. The fact that even a TDM line cannot be booked over 50% by EF traffic indicates that the restriction is artificial and unnecessary.
In any interval of time (t1, t2) in which EF traffic is continuously backlogged, at least R(t2 - t1 - E) bits of EF traffic must be served, where R is the configured rate for the EF aggregate and E is an implementation-specific latency term. The "continuously backlogged" condition eliminates the insufficient- packets-to-forward difficulty while the addition of the latency term of size MTU/C resolves the perfectly-clocked forwarding example (section A.1), and also removes the limitation on EF configured rate. However, neither fix (nor the two of them together) resolves the example of section A.2. To see this, recall that in the example of section A.2 the EF aggregate is continuously backlogged, but the service rate of the EF aggregate is consistently smaller than the configured rate, and therefore no finite latency term will suffice to bring the example into conformance. 13]. These proofs use an algebraic characterization of the aggregate definition given by (eq_1), (eq_2), and packet identity aware definition given by (eq_3), (eq_4). Since this characterization is of interest on its own, we present it in this section.
Bill Courtney TRW Bldg. 201/3702 One Space Park Redondo Beach, CA 90278 EMail: firstname.lastname@example.org Shahram Davari PMC-Sierra Inc 411 Legget Drive Ottawa, ON K2K 3C9, Canada EMail: email@example.com Victor Firoiu Nortel Networks 600 Tech Park Billerica, MA 01821 EMail: firstname.lastname@example.org Charles Kalmanek AT&T Labs-Research 180 Park Avenue, Room A113, Florham Park NJ EMail: email@example.com K.K. Ramakrishnan TeraOptic Networks, Inc. 686 W. Maude Ave Sunnyvale, CA 94086 EMail: firstname.lastname@example.org
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