REFERENCES

[1) ITU-T Study Group 13. Traffic control and congestion control in B-ISDN. Technical Report 1.371, ITU-T, Geneva, Apr. 29 - May 10, 1996.

[2) R. Ahuja, B. Prabhakar, and N. McKeown. Multicast scheduling for input-queued switches. IEEE JSAC, May 1996.

[3) M.K.M. Ali and M. Youssefi. The performance of an input access scheme in a high-speed packet switch. In Proc. IEEE INFO COM, Miami, pages 454-461, Apr. 1991.

[4) V. Anantharam and T. Konstantopoulos. A methodology for the de­sign of optimal traffic shapers in communication networks. IEEE Trans. Automatic Control, Vol. 44, No. 3:583-586, Mar. 1999.

[5) V. Anantharam and T. Konstantopoulos. Optimality and interchange­ability of leaky buckets. In Proc. 32nd Allerton Conference, Monticello, IL, pages 235-244, Oct. 1994.

[6) T.E. Anderson. S.S. Owicki, J.B. Saxe, and C.P. Thacker. High speed switch scheduling for local area networks. ACM Trans. Compo Sys., pages pp. 319-352, Nov. 1993.

[7) N.G. Aneroussis and A.A. Lazar. Virtual path control for ATM networks with call-level quality of service guarantees. IEEE Trans. Networking, Vol. 6, No.2, April 1998.

[8) S. Asmussen. Applied probability and queues. Wiley, Chichester West Sussex, 1987.

[9) R.Y. Awdeh and H.T. Mouftah. Survey of ATM switch architectures. Computer Networks and ISDN Systems, Vol. 27:pages 1567-1613, 1995.

[10) E. Basturk, A. Birman, G. Delp, R. Guerin, R. Haas, S. Kamat, D. Kand­lur, P. Pan, D. Pendarakis, R. Rajan, D. Saha, and D. Williams. Design and implementation of a QoS capable switch-router. Technical Report RC 20848, IBM Research, Jan. 31 1997.

176 ATM NETWORK PERFORMANCE

[11] V. Benes. General Stochastic Processes and the Theory of Queues. Addison-Wesley, 1963.

[12] J.C.R. Bennett. Implementation of output-buffer switches. Workshop on Packet Scheduling Algorithms (talk), Xerox PARCo CA, Aug. 27, 1996.

[13] J.C.R. Bennett and H. Zhang. Hierarchical packet fair queueing algo­rithms. IEEE/ACM Trans. Networking, Vol. 5, No. 5:pages 675-689, Oct. 1997.

[14] F. Bernabei, R. Ferretti, M. Listanti, and G. Zingrillo. ATM system buffer design under very low cell loss probability constraints. IEEE INFOCOM Proc., pages 8c.3.1-8c.3.10, 1991.

[15] D. Bertsekas and R. Gallager. Data Networks. 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1992.

[16] F. Bonomi and K.W. Fendick. The rate-based flow control framework for the available bit rate ATM service. IEEE Network, Vol. 9, No. 2:pages 25-39, 1995.

[17] D.D. Botvich and N.G. Duffield. Large deviations, the shape of the loss curve, and economies of scale in large multiplexers. Queueing Systems Theory Appl., Vol. 20, 1995.

[18] U. Briem, E. Wallmeier, C. Beck, and F. Matthiesen. Traffic management for an ATM switch with per-VC queueing: concept and implementation. IEEE Comm. Mag., Vol. 36, No.l:pp. 88-93, Jan. 1998.

[19] J.A. Bucklew. Large Deviation Techniques in Decision, Simulation and Estimation. John Wiley and Sons, Inc., New York, NY, 1990.

[20] R. CalIon, P. Doolan, N. Feldman, A. Fredette, G. Swallow, and A. Viswanathan. A framework for multi protocol label switching. Internet Draft, http) /www.ietf.org, July 1999.

[21] T. Chaney, J.A. Fingerhut, M. Flucke, and J. Turner. Design of a gigabit ATM switch. In Proc. IEEE INFO COM, Kobe, April 1997.

[22] C.-S. Chang. Stability, queue length and delay of deterministic and stochastic queueing networks. IEEE Trans. Auto. Control, Vol. 39:pp. 913-931, 1994.

[23] J.S.-C. Chen and R.Guerin. Input queueing of internally nonblocking switch with two priority classes. In Proc. IEEE INFO COM, Miami, pages 529-537, 1989.

References 177

[24] F.M. Chiussi, J.G. Kneuer, and V.P. Kumar. Low-cost scalable switching solutions for broadband networking: the ATLANTA architecture and chipset. IEEE Comm. Mag., Vol. 35. No.12:pp. 44-53, Dec. 1997.

[25] G.L. Choudhury, D.M. Lucantoni. and W. Whitt. On the effectiveness of effective bandwidths for admission control in ATM networks. In The Fundamental Role of Teletraffic in the evolution of Telecommunication Networks, (J. Labetoulle and J. W. Roberts, editors). Elsevier, Amster­dam, pages 411-420, 1994.

[26] ATM Forum Technical Committee. User-Network Interface (UNI) Spec­ification Version 3.1. Prentice-Hall, Upper Saddle River, NJ, 1995.

[27] ATM Forum Technical Committee. Traffic management specification ver­sion 4.0. Technical Report af-tm-0056.000, The ATM Forum, Draft ver­sion 3.0, April 1996.

[28] C. Courcoubetis, F. Kelly, and R. Weber. Measurement-based usage charges in communications networks. preprint, 1998.

[29] C. Courcoubetis, G. Kesidis, A. Ridder, J. Walrand, and R. Weber. Call acceptance and routing using inferences from measured buffer occupancy. IEEE Trans. Comm., Vol. 43, No.2/3/4:1778-1784, Apr. 1995.

[30] C. Courcoubetis and R. Weber. Buffer overflow asymptotics for a switch handling many traffic sources. J. Appl. Prob., 33:886-903, 1996.

[31] R. L. Cruz. A calculus for network delay, Part 1: Network elements in isolation. IEEE Trans. Inform. Theory, 37:114-131, 1991.

[32] R.L. Cruz. Service burstiness and dynamic burstiness measures: a frame­work. J. High-Speed Networks, Vol. 1, No. 2:pages 105-127, 1992.

[33] R.L. Cruz. Quality of service guarantees in virtual circuit switched net­works. IEEE JSAC, Vol. 13, No. 6:pages 1048-1056, Aug. 1995.

[34] B. Davie, J. Lawrence, K. McCloghrie, Y. Rekhter, E. Rosen, G. Swallow, and P. Doolan. MPLS using LDP and ATM VC switching. Internet Draft, http://www.ietf.org, April 1999.

[35] M. de Prycker. Asynchronous Transfer Mode: solution for broadband ISDN. 2nd Ed., Ellis Horwood, New York, 1993.

[36] G. de Veciana. Design Issues in ATM Networks: Traffic Shaping and Congestion Control. PhD thesis, EECS Dept, U. C. Berkeley, 1993.

178 ATM NETWORK PERFORMANCE

[37J G. de Veciana. Leaky buckets and optimal self-tuning rate control. In IEEE Globecom'94 Proc., pages 1207-1211, San Francisco, 1994.

[38J G. de Veciana, C. Courcoubetis, and J. Walrand. Decoupling bandwidths for networks: A decomposition approach to resource management for networks. In Proc. IEEE INFO COM, volume 2, pages 466-473, 1994.

[39J G. de Veciana and G. Kesidis. Bandwidth allocation for multiple qualities of service using generalized processor sharing. IEEE Trans. Info. Th., Vol. 42, No. l:pp. 268-271, Jan. 1996.

[40] G. de Veciana and G. Kesidis. An approach to provisioning for real-time VBR video teleconferencing via end-to-end virtual path connections. In IEEE ICC'96, Dallas, TX, pages 632-636, June 1996.

[41] G. de Veciana, G. Kesidis, and J. Walrand. Resource management in ATM networks using effective bandwidths. IEEE JSAC, Vol. 13, No. 6:pp. 1081-1090, Aug. 1995.

[42J V. Dijk, E. Aanen, and H. van den Berg. Extrapolating ATM-simulation results using extreme value theory. In Proc. ITC-13, Copenhagen, pages pp. 97-104, June 1991.

[43J B.T. Doshi. Deterministic rule based traffic descriptors for broadband ISDN: Worst case traffic behavior and connection acceptance control. In Proc. ITC-14, Antibes, pages 591-600, 1994.

[44] N.G. Duffield. Asymptotic sampling properties of effective bandwidth estimation for admission control. In Proc. IEEE INFO COM '99, New York, Mar. 1999.

[45] N.G. Duffield, J.T. Lewis, N. O'Connell, R. Russell, and F. Toomey. Entropy of ATM traffic streams: A tool for estimating QoS parameters. IEEE JSAC, Vol. 13, No. 6:pp. 981-990, Aug. 1995.

[46] N.G. Duffield and N. O'Connell. Large deviations and overflow probabil­ities for the general single-server queue, with applications. Math. Proc. Camb. Phil. Soc., 1996.

[47] P. Dumortier. Toward a new IP over ATM routing paradigm. IEEE Comm. Mag., Vol. 36, No.1:pp. 82-86, Jan. 1998.

[48] R. Durrett. Probability; Theory and Examples. Wadsworth and Brooks, Pacific Groves, CA, 1991.

References 179

[49] A. Elwalid, D. Heyman, T.V. Lakshman, D. Mitra, and A. Weiss. Fun­damental bounds and approximations for ATM multiplexers with ap­plications to video teleconferencing. IEEE JSAC, Vol. 13, No. 6:pages 1004-1016, Aug. 1995.

[50] A. Elwalid, D. Mitra, and R.H. Wentworth. A new approach for allocating buffers and bandwidth to heterogeneous regulated traffic in an ATM node. IEEE JSAC, Vol. 13, No. 6:pages 1115-1127, Aug. 1995.

[51] A.1. Elwalid and D. Mitra. Effective bandwidth of general Markovian traffic sources and admission control of high speed networks. IEEE/ ACM Trans. Networking, Vol. 1, No. 3:pp. 329-343, June 1993.

[52] N. Feldman, B. Jamoussi, S. Komandur, A. Viswanathan, and T. Worster. MPLS using ATM VP switching. Internet Draft, http://www.ietf.org, Feb. 1999.

[53] D.C. Feldmeier. ATM: A new type ofInternet? IEEE Network Magazine, Vol. 10, No. 4:pages 2-3, July/Aug. 1996.

[54] W.-C. Feng and S. Sechrest. Smoothing and buffering for delivery of prerecorded compressed video. In IS&T/SPIE Multimedia Computing and Networking, San Jose, CA, pages 234-242, Feb. 1995.

[55] N.R. Figueira and J. Pasquale. An upper bound on delay for the Vir­tualClock service discipline. IEEE/ACM Trans. Networking, Vol. 3, No. 4:pages 399-408, Aug. 1995.

[56] V.J. Friesen, J.J. Harms, and J.W. Wong. Resource management with virtual paths in ATM networks. IEEE Network Mag., Vol. 10, No. 5:pp. 10-20, Sept/Oct 1996.

[57] D. Le Gall. MPEG: a video compression standard for multimedia appli­cations. Communications of the ACM, Vol. 34, No. 4:pages 47-58, April 1991.

[58] M.W. Garret and W. Willinger. Analysis, modeling and generation of self-similar VBR video traffic. In Proc. ACM SIGCOMM, pages 269-280, 1994.

[59] L. Georgiadis, R. Guerin, V. Peris, and K.N. Sivarajan. Efficient network QoS provisioning based on per node traffic shaping. IEEE/ A CM Trans. Networking, Vol. 4, No. 4:pages 482-501, Aug. 1996.

[60] R.J. Gibbens and P.J. Hunt. Effective bandwidths for multi-type UAS channel. Queueing Systems, Vol. 9, No. 1:pp. 17-28, 1991.

180 ATM NETWORK PERFORMANCE

[61] P.W. Glynn and W. Whitt. Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. J. Appl. Prob., 31, 1994.

[62] S.J. Golestani. A self-clocked fair queueing scheme for broadband appli­cations. In IEEE INFO COM Pmc., pages 636-646, 1994.

[63] S.J. Golestani. A framing strategy for congestion management. IEEE JSAC, Vol. 9, No. 7:pages 1064-1077, Sept. 1991.

[64] P. Goyal, S. Lam, and H. Vin. Determining end-to-end delay bounds in heterogeneous networks. In Proc. of the 5th International Workshop on Network and Operating System Support for Digital Audio and Video, Durham, NH, Apr. 1995.

[65] P. Goyal, H. Vin, and H. Cheng. Start-time Fair Queueing: a scheduling algorithm for integrated services packet switching networks. IEEE/ACM Trans. Networking, Vol. 5, No. 5:pages 690-704, Oct. 1997.

[66] D. Gross and C.M. Harris. Fundamentals of Queueing Theory. Wiley, New York, 1974.

[67] M. Grossglauser, S. Keshav, and D. Tse. RCBR: A simple and efficient service for multiple time-scale traffic. In Pmc. ACM SIGCOMM, Boston, MA, pages 219-230, Aug. 1995.

[68] M. Grossglauser and D.N.C. Tse. A time-scale decomposition approach to measurement-based admission control. In Pmc. IEEE INFO COM '99, New York, Mar. 1999.

[69] E. Guarene, P. Fasano, and V. Vercellone. IP and ATM integration perspectives. IEEE Comm. Mag., Vol. 36, No.1:pp. 74-80, Jan. 1998.

[70] R. Guerin, H. Ahmadi, and M. Naghshineh. Equivalent capacity and its application to bandwidth allocation in high-speed networks. IEEE JSAC, Vol. 9, No. 7:pp. 968-981, 1991.

[71] D. Heyman, A. Tabatabai, and T.V. Lakshman. Statistical analysis and simulation study of video teleconferencing traffic in ATM networks. IEEE Trans. Circuits and Systems for Video Tech., Vol. 2, No. l:pages 49-59, March 1992.

[72] M.G. Hluchyj and M.J. Karol. Queuing in high-performance packet switching. IEEE JSAC, Vol. 6, No. 9:pp. 1587-1597,1988.

[73] J.Y. Hui. Switching and Traffic Theory for Integrated Broadband Net­works. Kluwer Acad. Publ., Boston, 1990.

References 181

[74] J.Y. Hui. Network, transport, and switching integration for broadband communications. IEEE Network, pages 40-51, March 1988.

[75] A. Hung. Bandwidth Scheduling for ATM. PhD thesis, E&CE Dept, University of Waterloo, 1997.

[76] A. Hung and G. Kesidis. End-to-end delay bounds and buffer sizing in ATM networks with applications to video transmission. In Proc. First Workshop on ATM Traffic Management, Paris, pages 157-164, Dec. 1995.

[77] A. Hung and G. Kesidis. Bandwidth scheduling for wide-area ATM net­works using virtual finishing times. IEEE/ACM Trans. Networking, Vol. 4, No. l:pp. 49-54, Feb. 1996.

[78] A. Hung and G. Kesidis. End-to-end delay bounds and buffer sizing in ATM networks. Technical Report 95-08, E&CE Dept, Univ. of Waterloo, June 1995.

[79] A. Hung and G. Kesidis. Performance evaluation of hierarchical round­robin bandwidth scheduling for ATM. In Proc. ITC-15, Washington, DC, pages 1247-1256, June 1997.

[80] T. Ibaraki and T. Kameda. Multi-frame isochronous service for ATM networks: Stop-and-Go revisited. In Proc. 6th International Conf. on Computer Communications and Networks, Las Vegas, Sept. 1997.

[81] R. Jain. Congestion control and traffic management in ATM networks: recent advances and a survey. Computer Networks and ISDN Systems, Jan. 1995.

[82] Y. Joo and N. McKeown. Doubling memory bandwidth for network buffers. In Proc. IEEE INFO COM, San Pransisco, March 1998.

[83] J.Roberts, U. Mocci, and J. Vitramo (Eds.). Broadband Network Tele­traffic: Final Report of Action COST 242. Springer, 1996.

[84] C.R. Kalmanek, H. Kanakia, and S. Keshav. Rate controlled servers for very high-speed networks. In Proc. IEEE Globecom, 1990.

[85] S. Karlin and H.M. Taylor. A Second Course in Stochastic Processes. Academic Press Inc., New York, NY, 1981.

[86] D. Kataria, A. Elwalid, and D. Logothetis. A better cdv accumulation method. Technical Report 97-0293, The ATM Forum, Chicago, April 27 - May 2, 1997.

182 ATM NETWORK PERFORMANCE

[87] F. Kelly. Notes on effective bandwidths. In Stochastic Networks: Theory and Applications (Editors F.P. Kelly, S. Zachary and I.B. Ziedins), Royal Statistical Society Lecture Notes Series, 4. Oxford University Press, pages 141-168, 1996.

[88] F .P. Kelly. Effective bandwidths of multi-class queues. Queueing Systems, Vol. 9, No. l:pp. 5-16, 1991.

[89] G. Kesidis. Estimation of Cell Loss in High Speed Digital Networks. PhD thesis, EECS Dept. University of California at Berkeley, 1992.

[90] G. Kesidis. A traffic regulator for effective bandwidth usage parameter control in ATM networks. Technical Report No. 93-03, E&CE Dept, Univ. of Waterloo, 1993.

[91] G. Kesidis. End-to-end cell delay estimates based on accumulations of per-hop statistics. Nortel Networks technical report, 1999.

[92] G. Kesidis. ATM Network Performance. Kluwer Academic Publishers, Boston, MA, First Edition, 1996.

[93] G. Kesidis. Resource provisioning in ATM networks using virtual path connections. Technical Report No. 95-11, E&CE Dept, Univ. of Waterloo, Oct., 1995.

[94] G. Kesidis. Bandwidth adjustments using on-line packet-level measure­ments. In SPIE Conference on Performance and Control of Network Systems, Boston, Sept. 19-22, 1999.

[95] G. Kesidis and T. Konstantopoulos. Worst-case performance of a buffer with independent shaped arrival processes. to appear in IEEE Commu­nications Letters, 1999.

[96] G. Kesidis and T. Konstantopoulos. Performance bounds for a multiclass queue with deterministically shaped arrival processes. Technical Report No. 98-04, E&CE Dept, U. of Waterloo, http://odysseus.uwaterloo.ca, April 2, 1998.

[97] G. Kesidis and T. Konstantopoulos. Extremal shape-controlled traffic patterns in high-speed networks. Technical Report No. 97-14, E&CE Dept, U. of Waterloo, http://odysseus.uwaterloo.ca, Dec. 1997.

[98] G. Kesidis and T. Konstantopoulos. Shape-controlled traffic patterns that maximize overflow probabilities in high-speed networks. In Proc. IEEE CDC'98, Tampa, Dec. 1998.

References 183

[99] G. Kesidis and T. Konstantopoulos. Extremal traffic and worst-case per­formance for a queue with shaped arrivals. In Workshop on Analysis of Communication Networks: Call Centres, Traffic and Performance, Fields Institute, Toronto, Nov. 9-13, 1998.

[100] G. Kesidis, J. Walrand, and C.-S. Chang. Effective bandwidths for mul­ticlass Markov fluids and other ATM sources. IEEE/ACM Trans. Net­working, Vol. 1, No. 4:pp. 424-428. Aug. 1993.

[101] S. Kumar and D.P. Agrawal. On multicast support for shared-memory­based ATM switch architecture. IEEE Network. Vol. 10, No.1:pp. 34-39, Jan./Feb. 1996.

[102] H.T. Kung and R. Morris. Credit-based flow control for ATM networks. IEEE Network, Vol. 9, No. 2:pages 40-48, March/April 1995.

[103] S.S. Lam, S. Chow, and D.K.Y Yau. An algorithm for lossless smoothing of MPEG video. In Proc. ACM SIGCOMM, pages 281-293, 1994.

[104] C. Lefelhocz, B. Lyles. S. Shenker, and L. Zhang. Congestion control for best-effort service: why we need a new paradigm. IEEE Network, Vol. 10. No. l:pages 10-19. Jan.jFeb. 1996.

[105] F. LoPresti, Z.-L. Zhang, D. Towsley, and J. Kurose. Source time scale and optimal buffer/bandwidth trade-off for regulated traffic in an ATM node. In Proc. IEEE INFO COM, Kobe, Japan, Apr. 1997.

[106] S. Low and P. Varaiya. A new approach to service provisioning in ATM networks. IEEE/ACM Trans. Networking, Vol. 1, No. 5:pp. 547-553, Oct. 1993.

[107] W. Luo and M. EI Zarki. Analysis of error concealment techniques for MPEG-2 video transmission over ATM based networks. In Proc. of SPIE/IEEE Visual Communications and Image Processing, Taipei, Taiwan, May 1995.

[108] N. McKeown. The iSLIP scheduling algorithm for input-queued switches. IEEE/ACM Trans. Networking, Vol. 7, No. 2:188-201, 1999.

[109] N. McKeown, V. Anantharam. and J. Walrand. Achieving 100% through­put in an input-queued switch. In Proc. IEEE INFO COM, pages 296-302, March 1996.

[110] N. McKeown and F. Tobagi. Bridges, routers and switches. preprint, 1997.

184 ATM NETWORK PERFORMANCE

[111] N. McKeown and J. Walrand. SLIP: A fast algorithm for scheduling cells in input queued switches. In LAN MAN Workshop Proc., pages 245-253, 1995.

[112] J.M. McManus and K.W. Ross. Video on demand over ATM: Constant­rate transmission and transport. IEEE JSAC, 14:1087-1098,1996.

[113] J.M. McManus and K.W. Ross. A dynamic programming methodol­ogy for managing prerecorded VBR sources in packet-switched networks. Telecommunications Systems, 9, 1998.

[114] J.M. McManus and K.W. Ross. Video on demand over ATM: Constant­rate transmission and transport. IEEE JSAC, Vol. 14:pages 1087-1098, August 1996.

[115] D. Minoli. Video Dialtone Technology: Digital Video over ADSL, HFC, FTTC, and ATM. McGraw-Hill Inc., New York, 1995.

[116] K. Obraczka. Multicast transport protocols: a survey and taxonomy. IEEE Comm. Mag., Vol. 36, No.1:pp. 94-102, Jan. 1998.

[117] P. Oechslin. Worst case arrivals of leaky bucket constrained sources: The myth of the on-off source. In Proc. IFIP Fifth International Workshop on Quality of Service (IWQoS'97), 1997.

[118] H. Ohsaki, M. Murata, H. Suzuki, C. Ikeda, and H. Miyahara. Rate­based congestion control for ATM networks. ACM SIGCOMM, Computer Communication Review, pages pages 60-72, 1995.

[119] R. Onvural. Signalling in ATM Networks. Artech House, Boston, 1997.

[120] A.K. Parekh and R.G. Gallager. A generalized processor sharing approach to flow control in integrated services networks: The multiple node case. IEEE/ACM Trans. Networking, Vol. 2, No. 2:pages 137-150, Apr. 1994.

[121] A.K. Parekh and R.G. Gallager. A generalized processor sharing approach to flow control in integrated services networks: The single node case. IEEE/ACM Trans. Networking, Vol. 1, No. 3:pages 344-357, June 1993.

[122] S. Rajagopal, M. Reisslein, and K.W. Ross. Packet multiplexers with ad­versarial regulated traffic. In Proc. IEEE INFO COM '9S, San Francisco, March 1998.

[123] K.K. Ramakrishnan and P. Newman. Integration of rate and credit schemes for ATM flow control. IEEE Network, Vol. 9, No. 2:pages 49-56, Marchi April 1995.

References 185

[124] P.V. Rangan, S.S. Kumar, and S. Rajan. Continuity and synchronization in MPEG. IEEE JSAC, Vol. 14, No. l:pages 52-60, Jan. 1996.

[125] E.P. Rathgeb, W. Fischer, C. Hinterberger, E. Wallmeter, and R. Wille­Fier. The MainStreetxpress core services node - a versatile ATM switch architecture for the full services network. IEEE JSAC, Vol. 15, No. 5:pp. 795-805, 1997.

[126] A.R. Reibman and A.W. Berger. Traffic descriptors for VBR video tele­conferencing over ATM networks. IEEE/ACM Trans. Networking, Vol. 3, No. 3:pages 329-339, June 1995.

[127] E. Reich. On the integrodifferential equation of Takacs. i. Annals of Mathematical Statistics, Vol. 29:pages 563-570, 1958.

[128] J. Rexford, A. Greenberg, and F. Bonomi. Hardware-efficient fair queue­ing architectures for high-speed networks. In Proc. IEEE INFO COM, San Jilransisco, pages 638-646, Apr. 1996.

[129] J.W. Roberts. Virtual spacing for flexible traffic control. International Journal of Communication Systems, Vol. 7:pages 307-318, 1994.

[130] J.W. Roberts. What ATM transfer capabilities for the B-ISDN. In Proc. First Workshop on ATM Traffic Management, Paris, pages 183-194, Dec. 1995.

[131] J.W. Roberts. Traffic modeling and engineering for B-ISDN (tutorial). In Proc. ITC-15, Washington, DC, June 1997.

[132] E.C. Rosen, A. Viswanathan. and R. CalIon. Multiprotocollabel switch­ing architecture. Internet Draft, http://www.ietf.org, Aug. 1999.

[133] S. M. Ross. Introduction to probability and statistics for engineers and scientists. Wiley, New York. NY, 1987.

[134] J. Salehi, Z.-L. Zhang, J. Kurose, and D. Towsley. Supporting stored video: Reducing variability and end-to-end resource requirements via op­timal smoothing. In ACM SIGMETRICS, May 1996.

[135] M. Schwartz. Broadband Integrated Networks. Prentice-Hall, Englewood Cliffs, NJ, 1996.

[136] W. Stallings. IPv6: The new Internet protocol. IEEE Communications Magazine, Vol. 10, No. 4:pages 2-3, July/Aug. 1996.

[137] R. Steinmetz. Human perception of jitter and media synchronization. IEEE JSAC, Vol. 14, No. l:pages 61-72, Jan. 1996.

186 ATM NETWORK PERFORMANCE

[138) D. Stiliadis and A. Varma. A general methodology for designing efficient traffic scheduling and shaping algorithms. In Proc. IEEE INFO COM, Kobe, pages 326-335, Apr. 1997.

[139] S. Suri, G. Varghese, and G. Chandranmenon. Leap Forward Virtual Clock: a new fair queueing scheme with guaranteed delays and through­put fairness. In Pmc. IEEE INFO COM, Kobe, pages 558-566, Apr. 1997.

[140) L. Tassiulas. Linear complexity algorithms for maximum throughput in radio networks and input queued switches. In Pmc. IEEE INFO COM, San Fransisco, March 1998.

[141] B.L. Tierney, W.E. Johnston, J.R. Lee, and G. Hoo. Performance analysis in high-speed wide area IP-over-ATM networks: top-to-bottom end-to­end monitoring. IEEE Network Magazine, Vol. 10, No. 3:pages 26-39, May / June 1996.

[142] F.A. Tobagi. Fast packet switch architectures for broadband integrated services digital networks. Proc. of the IEEE, Vol. 78, No.1:pp. 133-167, Jan. 1990.

[143) D. Tse, R.G. Gallager, and J.N. Tsitsiklis. Statistical multiplexing of multiple time-scaled Markov streams. IEEE JSAC, Vol. 13, No. 6:pages 1028-1038, Aug. 1995.

[144] J. Walrand and P. Varaiya. High-Performance Communication Networks. Morgan Kaufman Publishers Inc., San Fransisco, CA, 1996.

[145] P.P. White. RSVP and integrated services in the Internet: A tutorial. IEEE Communications Magazine, Vol. 35, No. 5:pages 100-106, May 1997.

[146] G.G. Xie and S.S. Lam. Delay guarantee of Virtual Clock server. IEEE/ACM Trans. Networking, Vol. 3, No. 6:pages 683-689, Dec. 1995.

[147] H. Zhang. Service disciplines for guaranteed performance service in packet-switching networks. Pmc. of the IEEE, Vol. 83, No. 10, Oct. 1995.

[148] H. Zhang and S. Keshav. Comparison of rate-based service disciplines. In Pmc. ACM SIGCOMM, pages 113-121, Sept. 1991.

[149] L. Zhang. Virtual Clock: A new traffic control algorithm for packet­switched networks. ACM. Trans. Compo Sys., Vol. 9, No. 2:pp. 101-124, May 1991.

Refe1'ences 187

[150) Z.-L. Zhang, J. Kurose, J.D. Salehi, and D. Towsley. Smoothing, statisti­cal multiplexing and call admission control for stored video. IEEE JSAC special issue on Real- Time Video Services in Multimedia Networks, 1997.

A GLOSSARY OF ACRONYMS AND

SHORT FORMS

AAL: ATM Adaptation Layer

ABR: Available Bit Rate

ABT: ATM (Asynchronous Transfer Mode) Block Transfer

ACR: Allowed Cell Rate

ATM: Asynchronous Transfer Mode

B-frame: short for Bidirectional Frame (MPEG Video)

B-ISDN: Broadband Integrated-Services Digital Network

CAC: Connection Admission Control

CBR: Constant Bit Rate

CDV: Cell Delay Variation

CI: Congestion Indication

CLP: Cell Loss Priority

CTD: Cell Transfer Delay

CS: Convergence Sublayer

190

Demux: short for Demultiplex

diffserv or DS: Differentiated Services (Internet)

EBCI: Explicit Backward Congestion Indication

EFCI: Explicit Forward Congestion Indication

ER: Explicit Rate

FEC: Forwarding Equivalence Class

FIFO: First-In-First-Out (queue or buffer)

FRM: Fast Resources Management

GCRA: Generic Cell Rate Algorithm

GFR: Guaranteed Frame Rate

GPS: Generalized Processor Sharing

GR: Guaranteed-Rate

HRR: Hierarchical Round Robin

I-frame: short for Intracoded Frame (MPEG Video)

IBT: Intrinsic Burst Tolerance

ICR: Initial Cell Rate

IETF: Internet Engineering Task Force

IN: IN-or-out of profile bit

intserv: Integrated Services (Internet)

IP: Internet Protocol

IPP: Input Port Processor (of a switch)

ApPENDIX A

Glossary of Acronyms and Short Forms

ITU: International Telecommunications Union

IVC: Idling Virtual Clock

JPEG: Joint Photographic Experts Group

LAN: Local-Area Network

LSR: Label Switching Router

MBS: Maximum Burst Size

MCR: Minimum Cell Rate

MFS: Maximum Frame Size

MPEG: Motion Pictures Experts Group

MPLS: Multiprotocol Label Switching

Mux: short for Multiplex

OPP: Output Port Processor (of a switch)

P-frame: short for Predicted Frame (MPEG Video)

PCR: Peak Cell Rate

per-VC: short for per-virtual-circuit or per-virtual-channel

PGPS: Packetized Generalized Processor Sharing

PHB: Per-hop Behavior

PSN: Processor Sharing Node

PTI: Payload Type Identifier

QoS: Quality-of-Service

RCBR: Renegotiated Constant Bit Rate

191

192

RED: Random Early Detection

RM: Resource Management (cells)

RSVP: Resource Reservation Protocol

SAR: Segmentation and Reassembly

SCFQ: Self-Clocked Fair Queueing

SCR: Sustainable Cell Rate

TCP: Transmission Control Protocol

TDM: Time-Division Multiplexing

TFT: Target Finishing (departure) Time

UBR: Unspecified Bit Rate

UNI: User-Network Interface

UPC: Usage-Parameter Control (policing)

VBR: Variable Bit Rate

VCC: Virtual Channel Connection (virtual circuit)

VCI: Virtual Channel Identifier

VFT: Virtual Finishing Time

VN: Virtual Network

VPC: Virtual Path Connection

VPI: Virtual Path Identifier

VT: Video Teleconferencing

WAN: Wide-Area Network

ApPENDIX A

Glossary of Acronyms and Short Forms 193

WRR: Weighted Round Robin

B SOLUTIONS AND REFERENCES

FOR SELECTED EXERCISES

Chapter 2

5. Proof of Theorem 2.1.3: The following proof is by T. Kameda. A brute-force proof is given in Appendix B of [92] (the first edition of this book).

We are required to prove that

r max Z(rn)l O<m<n

where Z(rn) = L;~~(Aj - pl. Define

Zmax = max Z(rn) = Z(rna) O<m<n

where rno is the largest index satisfying the equality.

We can easily show the desired result if rno = n. If rna :S n - 1,

n'

2..:: (Aj-p) > 0 Vn'E{rna,rna+1, ... ,n-l}. (B.l) j=mo

If rno 2: 1,

mo-l

2..:: (Aj - p) < 0 V rn E {O, 1, ... , rna - I}. (B.2) j=m

196 ApPENDIX B

By Equation (B.2),

mo-1

L Aj -p(mo-m-2) < 0 j=m

mo-1

==> L Aj < lp(mo - m - 2)J . (B.3) j=m

Note that 1::7;;;;1 Aj is the total number of arrivals over [m, mol. Also, lp(mo - m - 2)J is the number of departures over [m, mol provided [m, mol is part of a busy period. Thus, Equation (B.3) implies that X mo = 0 if mo ~ 1; In case mo = 0 recall that Xo = 0 by assumption. Equation (B.1) implies that [mo + 1, n] is the initial part of a busy period having

2::;':-';'0+1 Aj arrivals and lp(n - mo)J departures. Thus,

n-1

Xn L Aj - lp(n - mo)J = r Z(mo)l j=mo+1

as desired.

6. If the initial contents are Xo = 0"0, then the maximum backlog is 0"0 + 0".

12. First note that m m-1

-Xn + LAn n=O

(i.e., the state of the queue equals the cumulative arrivals minus the cu­mulative departures) and then apply Theorem 2.1.1.

16. It is required to show that the departures of the cell spacer are (0", p) constrained. Let D,,(s, t) and Dp(s, t) respectively be the cumulative de­partures from the cell spacer and leaky bucket over the interval of time (s, t). First note that the cumulative arrivals to the cell spacer over (s, t) is Dp(s, t). Let X" be the queue occupancy of the cell spacer. Conse­quently,

D,,(s, t) -X,,(s) + X,,(t) + Dp(s, t) 0"

< 0 + X,,(t) + "2 + p(t - s)

by the fact that Dp is (I' p) constrained and X" ~ O. Since X" ::; Xp ::; I' 0" 0"

D,,(s,t) < "2+"2+ p(t-s)

as desired.

Solutions and References for Selected Exercises 197

Chapter 3

2. Alternate GR parameter for HRR [79): To obtain the desired result, use the following lemma instead of Lemma 3.2.1.

Lemma B.0.1 For all t ~ sand all 1 ~ l ~ L,

IKds, t) - (t - s)p,1 < (1 - p,)k,

Proof: Since there are exactly k, reserved level-l slots every k, pi 1 units of time,

K,(s, t) - (t - s)p, = K,(s, s + r) - rp, (BA)

where r = (t - s) mod(k,pi 1 ) (note, by assumption (3.7), k,pi 1 E Z+).

Therefore,

K,(s, t) - (t - s)p, ~ 0 - (k,pi 1 - k,)p, = - (1 - p,)k"

i.e., a lower bound when r = k,pi 1 - k, and K,(s, s + r) = O. Also,

K,(s, t) - (t - s)p, < k, - k,p, = (1 - p,)k"

i.e., an upper bound when r = k, = K,(s, s + r). o 9. Induction will be used to show that the leaky bucket guaranteed-rate pa­

rameter is J.L = -1.

For the first cell, Fl = al + p-l. Since there is a token initially in the queue, the first cell will arrive to the leaky bucket to find a token, consume the token and immediately depart, i.e., d1 = al. So, as desired,

d1 = Fl _ p-l.

Assume that dk ~ Fk - p-l up to k = n. To complete the proof, we must show that dn+1 ~ Fn+1 - p-1. First note that

Fn+1 = max{Fn,an+d+p-1 ~ Fn+p-1. (B.5)

Since the nth cell departed at time dn consuming a token and the token arrival rate is p, the next token will be generated at at timetn +1 ~ dn +p-1. Thus, the departure time of the (n + 1) th cell

dn +1 < tn+l ~ dn + p-1

< Fn by the inductive assumption

< Fn+1 - p-1 by (B.5)

o

198 ApPENDIX B

10. SCFQ versus HOL-SCFQ: Assume the same arrival processes to both schedulers and the same queue bandwidth allotments. For clarity, we assume that there is no best-effort queue. Let S:' and :FE' respectively be the start time and VFT of ci under HOL-SCFQ. Let 9r be the VFT of ci under SCFQ. We use a proof by induction to show that :FE' = 9r for all n, i. Let Vs be the virtual time function of SCFQ and let VH be the virtual time function of HOL-SCFQ.

Let a be the first time that a cell (or cells) arrive to the PSN s. Since the cell (or cells) arrive to empty queues at time a (i.e., a = h), the virtual time function v for both SCFQ and HOL-SCFQ is zero at this time. Consequently, the VFT of the cell (or cells) arriving at time a are the same under SCFQ and HOL-SCFQ.

Now consider cell ci arriving at time ai. We make the inductive assump­tion that the VFTs of all cells arriving before ai are the same under SCFQ and HOL-SCFQ.

To complete our inductive argument, we will now prove that :FE' = 9r. First note that, by the inductive assumption, the number of cells in the nth queues of both PSNs is the same at time ai. Also, the same cell c in both PSNs was the last cell to be served before time ai. For the cell c: let S be its start time, :F = 9 be its VFT, and p be its queue's bandwidth allotment.

Take the first case where ci arrives to an empty queue (in both PSNs). Consequently, for HOL-SCFQ, hi = ai and, therefore, vH(hi) = S. Also, for SCFQ, vs(ai) =:F. The departure time of cell Ci-l is less than or equal to that of c (ci arrived to an empty queue and c was the last cell to depart the PSN before the arrival of ci). Thus, 9:'-1 ::; 9 and :FE'_1 ::; :F. Therefore, for SCFQ,

9r max{vs(af),9r_d+p;;1 max{9, 9r- d + p;; 1

9 + p;;l

and, for HOL-SCFQ,

:FE' Sr + p;; 1

max{vH(hf),:Fl'_d + p;;l

max{:F, :Ft_l} + p;;l

max{9, 9r.-l} + p;;l (inductive assumption)

9r

Solutions and References for Selected Exercises 199

as desired.

Now take the second case where c7 arrives to a nonempty queue (in both PSNs). By definition of h7 and VH,

:Fr-1 g:'-1 (inductive assumption).

Consequently,

:Fr Sf + p;; 1 (defini tion of :Fr)

max{vH{hi), :F:'-1} + p;;t (definition of Sf) maxWf_1' :Fr-tl + p;;1 (previous equation)

gf-1 + p;;1 (inductive assumption).

Under SCFQ, since cell c7 arrives to a non empty queue, the departure time of C7_1 is greater than that of c. So, vs{a7) = :F ~ g:'-1. Consequently,

gf max{ vs{ai), gf-d + p;;1 (definition of gr) gn + -1

i-1 Pn Thus, :Fr = gr as desired.

11. See [65].

Chapter 4

o

4a. For (cut-through) WRR, let p be the bandwidth allotment of a queue so that p = kj f for some integer k. If ai ~ di - b the ith cell will either depart at time di -1 + 1 (if the next slot is also assigned to the queue under consideration) or at time di - 1 + f - k + 1 (i.e., it will have to wait until the next frame). If ai > di - 1 (i.e., the ith cell arrives to a nonempty queue), di ~ ai + f - k (i.e., in the worst case, the cell arrives just after its queue's block of assigned slots and has to wait until the next frame). So, the fairness parameter for WRR is

a f - k + 1 = {I - p)f + 1.

6. (0-, p) bound of departures from an idling mode queue under idling HRR: By Lemma 3.2.1,

L

~)KI(S,t) - {t - s)pt} < ~ 1=1

200 ApPENDIX B

K(s, t) - (t - s)p ;s; e.

where K(s, t) is the total number of reserved slots over (s, t] for the queue under consideration. Let D( s, t) = I::l 1 {di E (s, t]) be the number of departures from this HRR queue over (s, t]. Since each departure must be in an HRR reserved slot, D(s, t) ;S; K(s, t). Therefore,

D(s, t) - (t - s)p < e

=> D(s,t) < (t-s)p+e

as desired.

7a. That Bi = l (I' + 1 - P + /-Li P J is sufficient to prevent cell loss is an immedi­ate consequence of the buffer sizing Theorem 4.2.2. By Theorem 4.2.1, the departure process of Bl (which is the arrival process to the leaky bucket) is ((I' + 1 - P + /-LP, p) constrained. The backlog in the cell queue of the leaky bucket at time n is less than or equal to the maximum number of cell arrivals over [0, n] minus the minimum number of tokens generated over [0, n]. Thus, a sufficient cell buffer size for the leaky bucket is

as desired.

Chapter 6

BL2 max(pn + (I' + 1 - P + /-LIP) - (pn) n~O

(I' + 1- p+ /-LIP

1. WRR with shuffled slot assignments: Consider Figure B.1 depicting two consecutive frames. The slot assign­ments of a single queue with bandwidth allotment P are shaded in; note that the positions of these slots have been shuffled from one extreme end of the frame to the other. Suppose a cell arrives to an empty queue at time k = pi indicated in Figure B.1 (just after its k = pi slot assign­ments). This cell, with a TFT of :F = k + p- 1 and a departure time of d = 21 - k, represents a "worst-case" scenario, Le., d - :F is maximal for this cell. Thus, for WRR with shuffled slot assignments,

as desired.

/-L d-:F = (21-k)-(k+p- 1 )

2(1 - p)1 - p-l = p-l(26 - 1)

Solutions and References for Selected Exercises 201

f f ~E~------~------~>~~E~------~------~~~

k k E E

t t t time time time o k 2f - k

Figure B.1 WRR Frames with ShufHed Slot Assignments

2. HRR with shuffled queue slot assignment positions: First note that (3.12) and (3.13) do not hold because, in general,

if the queue slot assignment positions can be shuffled from frame to frame (where r is the smallest integer ~ s satisfying NI(r, t)mod/l = 0). For an upper bound (i.e., instead of (3.13)), consider the case where the queue under consideration has kl level-l assigned slots at the very beginning and at the very end of interval (s, t]; also assume that these two blocks of kl slots respectively terminate and initiate a level-l frame. Thus,

<

This leads to the desired upper bound.

For the lower bound (i.e., instead of (3.12)), consider the case where the first Ii - kl and last II - kl slots of (s, t] belong to level-l and are not assigned to the queue under consideration; also assume that these two blocks of II - kl slots respectively terminate and initiate a level-l frame. Thus,

>

This leads to the desired lower bound.

202 ApPENDIX B

Note: if the positions of the (n,) slots assigned to subsequent levels can also change from frame to frame, a similar argument gives tl = 2~1 which implies t = 2~.

3. See [72].

5. Consider an IPP. In any given cell time, the memory block that does not transmit into the switch fabric is the one that a cell arriving to the IPP is written to. That is, the memory block that does not experience a read operation is the one that is eligible to experience a write operation. So, a FIFO queue of the IPP is implemented as a singly-linked list that might "weave" between the two memory blocks. See, for example, [82].

Chapter 8

6. Recall that encoding a B-frame requires an additional delay of one frame­time because of its dependency on the (temporally) next P-frame. Also, the use of B-frames will not reduce the peak cell rate (PCR) of the con­nection: the PCR will be determined by the larger I-frames. Finally, using B-frames instead of a P-frames will produce a smaller average cell rate but give poorer image quality. In summary, using B-frames results in poorer image quality and additional delay without any reduction in required trans­mission bandwidth (PCR).

(0", p) constraint, 23 ABR, 3, 41, 99, 142

ACR,100 flow control, 99 ICR,100 MCR,100 PCR,100 virtual source/destination, 105

ABT,4 arrival process, 19 arrival times process, 22 ATM

ATM Adaptation Layer, 6 connection-oriented, 6 service classes, 2

back-pressure, 135 bandwidth granularity, 47, 65, 126 bandwidth scheduling, 43

clumping, 73, 76 bandwidth-delay product, 104, 107 beat-down problem, 110 best-effort, 41,61-62 bottleneck bandwidth, 109 buffer sizing, 75, 78, 88-89

flow control, 107 burstiness curve, 38 burstiness, 24 busy period, 21 CBR,3

piecewise-CBR, 141 CDVT,3 cell delay variation, 80 cell spacer, 39, 93, 99 cell, 5 Chernoff bound, 28, 38

CLP bit, 9 clumping, 47, 73

INDEX

theorem, 74 congested state, 104 congestion indicator (CI) bit, 101 connection admission control, 9 control flow direction, 101 cu~through, 36,41, 66 data flow direction, 101 delay bound, 71, 89, 92 delay jitter, 65, 78, 80 departure times process, 22 diffserv, 14 EBCI, 101, 109 EFCI bit, 101 EFCI, 101, 109 effective bandwidths, 27, 29, 65

augmenting, 166 curve, 30 mean rate, 30 on-off sources, 166 peak rate, 35 theorem, 28

effective guaranteed-rate property, 89, 92

empirical distribution, 170 equivalent bandwidths, 27 excess bandwidth, 56, 108 explicit rate, 103 fair share, 108 fairness, 56, 58, 60, 63, 94, 108, 128

beat-down problem, 110 dynamic, 109 Static Equal Share, 108 Static Proportional Share, 109

204

flow control, 99, 142 buffer sizing, 107 control loop segmentation, 105 credit-based, 111 explicit cell rate

off rule, 104 explicit rate, 103 hop-by-hop, 105 rate-based, 99

fluid model, 31, 39, 55 frame

Weighted Round-Robin, 45 frames (video)

MPEG, 144, 162 segmentation and reassembly,

139, 162 GCRA, 9, 37 Generalized Processor Sharing, 55,

65 GFR, 4,10 goodput, 110 guaranteed service, 65 guaranteed-rate property, 43

effective, 89, 92 parameter, 43 reference FIFO queue, 44, 66 scheduler, 44 single FIFO queue, 43

head-of-line blocking, 119, 121 Head-of-Line Self-Clocked Fair

Queueing, 67 Hierarchical Round-Robin (HRR)

Admission Control, 125 Hierarchical Round-Robin, 47, 64

bandwidth granularity, 65 hop, 8 IBT, 3, 10 idle bandwidth, 62, 65, 107 Idling Virtual Clock, 61, 78, 82, 96

guaranteed-rate property, 61 indicator function, 22 Internet, 11

intserv, 13 IP,l1 Kullback-Leibler, 174

INDEX

large buffer approximation, 166 leaky bucket, 24, 99

token rate, 103 Leap-Forward Virtual Clock, 97 Lindley, 19 MBS, 3,10 MPEG, 144, 159

traffic model, 164 MPLS,13 multicast, 11, 133, 135 overbooking, 42, 107 packetization delay, 5 PCR,3 PGPS, 56, 59, 65

guaranteed-rate property, 56 piecewise-CBR, 141

schedule of bandwidth allotments, 141

windows, 141 pipelining, 66 playback buffer, 139 playout buffer. 139 policing, 9, 151

peak-rate, 39, 93 VBR,173

processor sharing node, 41 cut-through, 66 pipelined, 66

q,21 rate-proportional fair, 56, 58, 60 RED,12 Resource management (RM) cells,

4 resource management (RM) cells,

100 round-robin scheduling, 45 RSVP, 13 SCR, 3,10

Index

segmentation and reassembly, 6, 139, 162

Self-Clocked Fair Queueing, 58, 64 guaranteed-rate property, 59, 67

service curve, 65 set-top box, 139 Shaped Virtual Clock, 97 shaping schedulers, 94-95, 97 Slepian-Duguid algorithm, 124 stable queue, 27 statistical multiplexing, 167 Stop-and-Go Scheduling, 83 switch

bidirectional, 101 fabric, 115 input-buffered, 117

arbitration, 123 head-of-line blocking, 119 per-output-port queueing, 121 per-VC queueing, 121 throughput, 119 virtual-output queueing, 121

input/output-buffered, 131 memory block, 114 multicast, 133 output-buffered, 116

memory bandwidth, 117 per-VC queueing, 115 scalability, 129 single-stage, 113

Target Finishing Times, 43 TCP, 11 throughput, 100 tightly (0', p) constrained, 72 time-division multiplexing, 46 token buffer, 24 traffic contract, 2 traffic measurements, 169 traffic shaping, 10 UBR, 4, 41 unit of memory, 8, 140 unit of time, 7

unreserved bandwidth, 107 usage parameter control, 9 VBR, 3,159

policing, 173 VCC, 5 video

decoder, 139 JPEG,145 MPEG, 144-145, 159, 164 prerecorded, 159

205

deadline function, 143, 145 initial playback delay, 143 source transmission function,

143 relative deadlines, 143 synchronization with voice, 139 teleconferencing, 159

end-to-end VPC, 160 trace, 139

virtual channel identifier, 6, 115 virtual circuit, 5, 8 Virtual Clock, 57, 64

guaranteed-rate property, 57 Virtual Finishing Times, 54 virtual network, 10 Virtual Path Connection, 10

arbitrary, 90 buffer sizing, 89 delay bound, 92

effective guaranteed-rate property, 89, 92

end-to-end, 70, buffer sizing, 78, delay bound, 71

single node, 88 buffer sizing, 88 delay bound, 89

virtual path identifier, 6, 10, 115 virtual time function, 55, 67 voice, 5, 162

synchronization with video, 139 Weighted Fair Queueing, 56

206

Weighted Round-Robin, 45, 64, 95-96, 121

bandwidth granularity, 47 frame, 45 guaranteed-rate property, 46

worst-case traffic, 31 zero buffer approximation, 35, 165,

172 augmenting, 166