Tao Dissertation

7/29/2019 Tao Dissertation

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Analysis and Implementation of Multiplexing Techniques in

Connection-Oriented Communication Networks

___________________________________________________________

A Dissertation

Presented to

the faculty of the School of Engineering and Applied Science

University of Virginia

___________________________________________________________

In Partial Fulfillment

of the requirement for the Degree

Doctor of Philosophy

(Electrical Engineering)

____________________________________________________________

by

Tao Li

August 2006


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APPROVAL SHEET

The dissertation is submitted in partial fulfillment of the

requirement for the degree of

Doctor of Philosophy (Electrical Engineering)

________________________________________Tao Li, Author

This dissertation has been read and approved by the examining Committee:

________________________________________Prof. Malathi Veeraraghavan, Advisor

________________________________________Prof. Joanne Bechta Dugan, Chairperson

________________________________________Prof. Stephen G. Wilson

________________________________________

Prof. Mat Brandt-Pearce

________________________________________Prof. Stephen D. Patek

Accepted for the School of Engineering and Applied Science:

________________________________________Dean, School of Engineering and Applied Science

August 2006


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Abstract

Future implementations of wired and wireless communication networks are expected to

support a variety of multimedia applications with diverse traffic characteristics and qual-

ity-of-service (QoS) requirements. To meet these diverse requirements, there are two types

of networking technologies, i.e., connectionless and connection-oriented. While some

applications, such as small data-file transfers, are best served on a connectionless network,

other applications such as large data-file transfers and audio/video applications that have

stringent requirements on data rate, delay, delay jitter, and delay-bound violation probabil-

ity are best served with a connection-oriented network because of its inherent support for

QoS.

This dissertation combines an analytical study of a connection-oriented packet-

switched scheduling mechanism (a data-plane problem) with a hardware implementation

of a signaling protocol for a (connection-oriented) circuit switch (a control-plane prob-

lem). For the analytical study, we model and simulate a polling-based scheduling mecha-

nism for its ability to support real-time applications. This is a connection-oriented packet-

based bandwidth-sharing mechanism because it requires a call admission control phase to

limit the maximum number of communicating endpoints, and it uses packets in the data

plane. The real-time application considered in our study is telephony. We develop models

to determine appropriate values for operational parameters, such as the number of voice

calls that can be simultaneously supported while meeting a predetermined set of quality-

of-service requirements (e.g., delay and loss). Our results can be used to dimension a poll-

ing-based system, such as the polling-based operational mode of an IEEE 802.11 LAN.

The implementation part of this dissertation is focused on demonstrating that signaling


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protocols, needed in all connection-oriented networks for the call admission phase, can, in

spite of their complexity, be implemented in hardware. Advantages of a hardware imple-

mentation are that (a) call setup delay is reduced by at least two-to-three orders of magni-

tude, and (b) call-handling capacity is increased significantly. By reducing call setup

delay, a significant overhead component in connection-oriented networks, resource utili-

zation can be improved. With the high call-handling capacity of a hardware-accelerated

signaling engine, connection-oriented switches can better support end-user applications

with short-duration calls, which typically have high call arrival rates.


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Acknowledgements

First and foremost, I would like to express my sincerest gratitude to my advisor, Profes-

sor Malathi Veeraraghavan, for her inspiring guidance, encouragement, patience, and con-

tinuous support over the past six years. I have benefited tremendously from her unique

blend of energy, enthusiasm, vision, technical insights, and practical sensibility.

I want to thank Professor Joanne Bechta Dugan, Professor Stephen G. Wilson, Profes-

sor Mat Brandt-Pearce, and Professor Stephen D. Patek for serving on the dissertation

committee and providing constructive comments. I would like to particularly thank Dr.

Dimitris Logothetis for his involvement and inspiring direction in part of this research. I

am honored to work together with him.

I want to thank Haobo Wang, Xuan Zheng, Zhifeng Tao, Hojun Lee, Xiangfei Zhu,

Xiuduan Fang, Zhanxiang Huang, Reinette Grobler, Anant P. Mudambi, and Murali Nethi,

with whom I have enjoyed collaboration and numerous discussions. I would like to thank

my fellow graduate students and friends, including Qianling Cao, Qun Xiao, Jun He, Wen-

zhuo Jin, Haijun Fang, Bo Xu, Bin Huang, Xinmin Liu, Shuhao Chen, and Shixiao Zhou.

They have made my life at Charlottesville enjoyable and memorable. I would also like to

take this opportunity to thank Chen Chen, Yaogang Lian, Chengdu Huang, Ying Wang,

Changlong Hu, Hong Xu, Yvan Pointurier, Chad Cole, Yoshihiro Masui, Kirtan Modi,

Shilpa Deshpande, Jinlian Wang, Emily Jinwen Chong, and many others for their friend-

ship and support.

I would like to thank Professor Shivendra S. Panwar for his constructive comments on

my research when I was with Polytechnic University. During my stay at Brooklyn, NY, I

shared many wonderful times with my fellow students and friends. I especially thank Hui


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Lin, Lina Chen, Xiaoan Lu, Zilan Lin, Hua Zhang, Shiwen Mao, Yihan Li, Jiwu Duan,

John Kuo, Hua Tang, Junxu Zhang, Xi Yang, Xueai Bai, Ying Meng, and Yi Qin for their

friendship and support.

I am grateful to Shizhong Xu, Jianhao Hu, Chun He, Xing Li, Xiang Ling, Hongyin Lu,

Hengduan Luo, Jinghua Qian, Xiaoyu Fu, Yingming Lin, Yi Huang, and many others.

Their advice, friendship, and generous support made my stay in Chengdu an enjoyable

and memorable one.

This work is supported by the National Science Foundation under grants ANI-0087487

and ITR-0312376. I also acknowledge the New York State Center for Advanced Technol-

ogy in Telecommunications (CATT) at Polytechnic University, Brooklyn, NY, for provid-

ing funding for my Ph.D. study.

Finally, I dedicate this dissertation to my parents and relatives for their love, support,

and encouragement.


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i

Contents

Chapter 1 Introduction 1

1.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2 Analysis of a Polling System with Application to Wireless LANs 8

2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Polling System Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Analysis of a Single-Queue Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Distribution of Interpoll Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Distribution of Packet Queueing Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Service Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.4 Distribution ofDW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Multiple-Queue Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Validation and Application of Analytical Model to a Wireless LAN . . . . . . . . . 20

2.5.1 Background of IEEE 802.11 and Values Selected for Parameters . . . . . . . . 20

2.5.2 Numerical Results for Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.3 Numerical Results for Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.4 Summary of the Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


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2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 3 Effects of Packetization of Voice Data 33

3.1 An ON-OFF MMF Model with Packetization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Voice Capacity and Delay Bound in Small-N Regime of Operation . . . . . . . . . . 35

3.3 Resource Allocation in Large-N Regime of Operation . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Performance Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 Analytical Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 Analysis of Overflow Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.4 Relationship Between Overflow Probability and Packet Loss Ratio . . . . . . 42

3.3.5 Analysis of Packet Loss Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.1 Delay in Single-call Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.2 Delay in Multi-call Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.3 Distribution ofTsrv in Small-N Regime of Operation. . . . . . . . . . . . . . . . . . 50

3.4.4 Overflow Probability and Packet Loss Ratio in Large-N Regime of Operation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.5 Resource Allocation for Aggregated Flows . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.6 Asymmetry in the Two Directions of Voice Communication. . . . . . . . . . . . 58

3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 4 Implementation of a Signaling Control Card 64

4.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Architecture of Signaling Control Card . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Implementation Details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69


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4.3.1 Gigabit Ethernet Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.2 Hardware Signaling Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.3 PCI Interface Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.4 Configuration Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.5 Power Regulation Module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Chapter 5 Conclusions 96

5.1 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Future Research Directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 100


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iv

List of Figures

Fig. 1.1: (a) An unfolded view of a generic network switch; (b) Output scheduling in an

output-buffered packet switch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Fig. 2.1: Polling system model: An example showing three vacations in one polling cycle.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Fig. 2.2: Timing diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Fig. 2.3: Timing in the small-N regime of operation: the worst-case scenario. . . . . . . 18

Fig. 2.4: Network architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Fig. 2.5: CDF ofDWwith TSas a parameter in the single-call scenario. Twalkand Care set

to 0.23ms and 8.5Kbps, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Fig. 2.6: CCDF ofDWin the small-N regime of operation. , codec rate, and Twalkare set

to 0.5, 64Kbps, and 0.23ms, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Fig. 2.7: CCDF of delay withN'as a parameter in the large-N regime of operation. TS, ,

and codec rate are equal to 30ms, 0.5, and 8.5Kbps, respectively. . . . . . . . . . 26

Fig. 2.8: N'max versus TS, with Ploss and as parameters. Codec rate, Twalk, and stretch

distribution are respectively set to (a) 8.5Kbps, 0.23ms, and S(t); (b) 8.5Kbps,

0.23ms, and VSmax. The plot for Ploss=0 is obtained from one million samples.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Fig. 2.9: N'max versus TS, with Ploss and as parameters. Codec rate, Twalk, and stretch

distribution are respectively set to (a) 64Kbps, 0.23ms, and VSmax; (b) 64Kbps,


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0.13ms, and VSmax. ROHC is applied in (b). The plot for Ploss=0 is obtained from

one million samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Fig. 2.10: Transmission efficiency as a function ofTS. Codec rate,H, and Twalkserve as

parameters.R is fixed at 11Mbps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Fig. 3.1: Illustration of the packetization of voice traffic. . . . . . . . . . . . . . . . . . . . . . . . 34

Fig. 3.2: Discrete-time ON-OFF Markov model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Fig. 3.3: Timing diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Fig. 3.4: CDF ofDWwith TSas a parameter in a single-call scenario. Packetization period

is set to 10ms. Vacation stretch follows the distribution specified in (2.18). . 46

Fig. 3.5: CDF ofDWwith TSas a parameter in a single-call scenario with clock skew.

Packetization period is set to 10.001ms. Vacation stretch follows the distribution

specified in (2.18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Fig. 3.6: CCDF ofDWin the small-N regime of operation. and Twalkare set to 0.5 and

0.13ms, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Fig. 3.7: CCDF of frame delay withN'as a parameter in the large-N regime of operation.

Vacation stretch is set to VSmax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Fig. 3.8: CCDF ofTsrv forN'=5, 10, 20, and 30, respectively. TSequals 30ms. . . . . . . 51

Fig. 3.9: CCDF ofTsrv forN'=5, 10, 20, and 30, respectively. TSequals 50ms. . . . . . . 52

Fig. 3.10: Overflow probability and packet loss ratio in the large-N regime of operation.

TS= 30ms.Dboundis set to TS+L+2ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Fig. 3.11: Difference between the two service disciplines. . . . . . . . . . . . . . . . . . . . . . . 55

Fig. 3.12: Resource allocation for aggregated voice flows. TSequals 30ms. . . . . . . . . 57

Fig. 3.13: Resource consumption of each voice call. TSequals 30ms. . . . . . . . . . . . . . 57


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Fig. 3.14: Ploss in the two directions assuming the mobile-assisted decision-feedback ap-

proach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Fig. 3.15: Ploss in the two directions assuming the call-based termination with traffic esti-

mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Fig. 4.1: Architecture of a typical switch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Fig. 4.2: System architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Fig. 4.3: Block diagram of the signaling control card. . . . . . . . . . . . . . . . . . . . . . . . . . 68

Fig. 4.4: Block diagram of the Gigabit Ethernet module. . . . . . . . . . . . . . . . . . . . . . . . 70

Fig. 4.5: Block diagram of the hardware signaling accelerator module. . . . . . . . . . . . . 73

Fig. 4.6: State transition diagram of MAC interface unit for the receive path. . . . . . . . 74

Fig. 4.7: State transition diagram of MAC interface unit for the transmit path. . . . . . . 76

Fig. 4.8: Block diagram of the FIFO interface unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Fig. 4.9: State transition diagram for memory segmentx. . . . . . . . . . . . . . . . . . . . . . . . 79

Fig. 4.10: State transition diagram of the FIFO controller. . . . . . . . . . . . . . . . . . . . . . . 81

Fig. 4.11: Block diagram of a switch fabric interface unit. . . . . . . . . . . . . . . . . . . . . . . 81

Fig. 4.12: State transition diagram for PATH message processing. . . . . . . . . . . . . . . . . 83

Fig. 4.13: Block diagram of the PCI interface module. . . . . . . . . . . . . . . . . . . . . . . . . . 87


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List of Tables

Table 2.1: Values for parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Table 2.2: Summary of numerical results at TS= 30ms. . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 4.1: 10-bit PHY interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Table 4.2: MAC interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Table 4.3: MAC register interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Table 4.4: Control output on the receive path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Table 4.5: Interface signals of the RAM controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Table 4.6: Control output of the RAM controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Table 4.7: Interface signals of the FIFO controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Table 4.8: Signals on TCAM and SRAM interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Table 4.9: Control input to the TCAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Table 4.10: Signals to the configuration module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Table 4.11: Memory mapping table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Table 4.12: Configuration procedure for the GbE controller. . . . . . . . . . . . . . . . . . . . . . 91

Table 4.13: Configuration procedure for the TCAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


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List of Acronyms

AP: Access Point

ATM: Asynchronous Transfer Mode

BE: Best Effort

CAC: Call/Connection Admission Control

CCDF: Complementary Cumulative Distribution Function

CDF: Cumulative Distribution Function

CFP: Contention Free Period

codec: coder-decoder

CP: Contention Period

CPU: Central Processing Unit

CRC: Cyclic Redundancy Check

DCF: Distributed Coordination Function

DMA: Direct Memory Access

DOCSIS: Data Over Cable Service Interface Specification

ERT-VT: Extended Real-Time Variable Rate

FFT: Fast Fourier Transform

FIFO: First In First Out

FPGA: Field Programmable Gate Array

FSM: Finite State Machine


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GbE: Gigabit Ethernet

GMPLS: Generalized MPLS

HWSAC: Hardware Signaling Accelerator Core

i.i.d.: independent and identically distributed

IEEE: Institute of Electrical and Electronics Engineers

IP: Internet Protocol

IPv4: Internet Protocol version 4

LAN: Local Area Network

LDP: Label Distribution Protocol

LLC: Logic Link Control

MAC: Media Access Control

MMF: Markov Modulated Fluid

MPLS: Multi-Protocol Label Switching

MUX: Multiplexer

PCF: Point Coordination Function

PCI: Peripheral Component Interconnect

PCM: Pulse Code Modulation

PECL: Positive Emitter Coupled Logic

PNNI: Private Network-Network Interface

PSTN: Public Switched Telephone Network

QoS: Quality of Service

RAM: Random Access Memory

ROHC: RObust Header Compression


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RSVP-TE: Resource ReserVation Protocol - Traffic Engineering

RTP: Real-time Transport Protocol

RT-VR: Real-Time Variable Rate

SONET: Synchronous Optical Network

SRAM: Static Random Access Memory

TCAM: Ternary Content Addressable Memory

TCP: Transmission Control Protocol

TOE: TCP Offload Engine

TTL: Transistor Transistor Logic

UDP: User Datagram Protocol

VC: Virtual Circuit

WDM: Wavelength Division Multiplexing


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1

Chapter 1

Introduction

1.1 Background

Future implementations of wired and wireless communication networks are expected to

support a variety of multimedia applications with diverse traffic characteristics and qual-

ity-of-service (QoS) requirements. For example, mission-critical applications may

demand deterministic (or hard) guarantees on loss and delay. Interactive voice and video

applications require a guarantee on delay but can generally tolerate a small packet loss

rate. Hence, they only require statistical (or soft) QoS guarantees. Data transfers expect

error-free transmission. Some applications such as e-mail do not require explicit QoS

guarantees.

To meet these diverse QoS requirements, there are two types of networking technolo-

gies, i.e., connectionless and connection-oriented. While some applications, such as small

data-file transfers, are best served on a connectionless network, other applications such as

large data-file transfers and voice/video applications that have stringent requirements on

data rate, delay, delay jitter, and delay-bound violation probability are best served with a

connection-oriented network because of its intrinsic support for QoS. There are two types

of connection-oriented networks: circuit-switched networks, such as time-division-multi-


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plexed SONET (Synchronous Optical Network) and WDM (Wavelength Division Multi-

plexing), and packet-switched networks, such as MPLS (MultiProtocol Label Switching)

and ATM (Asynchronous Transfer Mode). Connection-oriented packet-switched networks

are also referred to as virtual-circuit (VC) networks.

A connection-oriented network switch has a data-plane module and a control-plane

module, as shown in Fig. 1.1(a). The data-plane module of a network switch, typically

implemented in hardware, consists of line cards and a switch fabric. The input sections of

line cards demultiplex input signals and process protocol headers. In a circuit switch, the

appropriate output interface for a demultiplexed input signal is identified according to the

position information (e.g., interface number, time slot, and wavelength) of this signal and

a set of table entries of the form {input channel identifier, output channel identifier}, used

to describe the cross-connections. A channel identifier can be a combination of interface,

time slot, and wavelength indexes. In a connection-oriented packet switch, the output

interface of a packet is determined by checking the header information contained in the

packet and a set of table entries of a form similar to the one just mentioned but with differ-

Switch

fabric

Signaling/Routing/Link management

engines

Line card

(a)

Output

scheduler

(b)

Control plane

Data plane

Fig. 1.1: (a) An unfolded view of a generic network switch; (b) Output

scheduling in an output-buffered packet switch.


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3

ent channel identifiers. The switch fabric switches demultiplexed signals or packets

according to table entries {input channel identifier, output channel identifier}, which can

be either statically set up through provisioning, or dynamically established using a signal-

ing protocol such as Private Network-Network Interface (PNNI) [1], Resource ReserVa-

tion Protocol (RSVP) [2], Label Distribution Protocol (LDP) [3], and RSVPTraffic

Engineering (RSVP-TE) protocol [4]. After traversing the switch fabric, signals or packets

are transmitted out through the output section of a line card. Additionally, in an output-

buffered packet switch, packet schedulers are used to select packets for transmission on an

output port based on the QoS commitments made during Call/Connection Admission

Control (CAC). This is shown in Fig. 1.1(b).

The controlplane module is an implementation of signaling protocols, routing proto-

cols, and link management protocols. The chief characteristic of a connection-oriented

network is that resources are reserved prior to data transfer. Resources are reserved in a

circuit/virtual-circuit setup phase. Resources are released in a circuit/virtual-circuit release

phase. Signaling protocols are used to set up and release circuits/virtual-circuits (or con-

nections) dynamically. Setting up a connection typically consists of the following steps:

Determine a route for the connection by consulting routing tables, which are created

by routing protocols.

Determine whether or not available network resources (i.e., bandwidth and buffer

space) are sufficient to meet the declared QoS requirements of a connection request.

If yes, the connection request is accepted and a fraction of resources is reserved

along the path of the connection. Otherwise, the connection request is rejected. This

step is referred to as CAC.


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4

Allocate channel identifiers and construct table entries in the form {input channel

identifier, output channel identifier}.

Program the switch fabric of each switch on the end-to-end path in circuit-switched

networks, or program schedulers in packet-switched networks.

Maintain state information for each connection.

After data transfer completes, the reserved resources are released and the table entries

related to this connection are deleted.

1.2 Motivation

With recent advancements in networking and communication technologies, there have

emerged several communication systems that simultaneously support connection-oriented

mode of operation and connectionless mode of operation. For example, in the IEEE

802.11 wireless Local Area Network (LAN) [5], the Point Coordination Function (PCF)

mode of operation, a scheduling-based access mechanism, coexists with the Distributed

Coordination Function (DCF) mode of operation, a contention-based channel access

mechanism. The PCF mode of operation is a connection-oriented packet-based band-

width-sharing mechanism because it requires a call admission control phase to limit the

maximum number of communicating endpoints, and it uses packets in the data plane. In

the Medium Access Control (MAC) layer of the IEEE 802.16 wireless Metropolitan Area

Network (MAN) [6] and the Data Over Cable Service Interface Specification (DOCSIS)

[7], the Real Time Variable Rate (RT-VR) Service and Extended Real-Time Variable Rate

(ERT-VR) Service, which are used to support real-time applications, coexist with a Best

Effort (BE) Service, which is used to support applications without explicit bandwidth or

delay requirements. The RT-VR, ERT-VR, and BE services are provided through a cen-


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5

tralized scheduler.

The IEEE 802.11, IEEE 802.16, and DOCSIS standards adopt scheduling-based access

schemes to support real-time applications that have stringent QoS requirements. Although

a variety of scheduling and CAC algorithms have been proposed for QoS provisioning in

wireline networks [8][9][10][11], these results can not be directly applied for QoS provi-

sioning in the upstream direction (from mobile stations to a base station) of a shared-

medium system, such as an infrastructured IEEE 802.11 wireless LAN, where queues are

located across wireless stations. This is because that the scheduler of an infrastructured

IEEE 802.11 wireless LAN, which is typically built in the base station, does not have a

central knowledge of the instantaneous status of each queue. Given the limited informa-

tion about data arrivals in a shared-medium environment, the performance of polling-

based channel-sharing schemes has become an important topic of study in literature (see

[12], [13], [14], and [15]).

Motivated by the PCF mode of operation in IEEE 802.11 and other shared-medium

systems, we develop a model for a polling system with vacations, where vacations repre-

sent the time periods in which the resource sharing mechanism used is a non-polling

mode. The real-time application served by the polling mode in our study is telephony.

This dissertation combines an analytical study of a connection-oriented packet-

switched scheduling mechanism (a data-plane problem) with an implementation of a

signaling protocol for a circuit switch (a control-plane problem).

Signaling protocols are traditionally implemented in software due to their complexity

and the requirement for flexibility. While software implementations of signaling protocols

can cope with the complexity and remain flexible, their performance level becomes a con-


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6

cern if high call-handling capacities and small signaling overheads are required. For

instance, a signaling protocol implementation in an off-the-shelf commercial SONET

switch requires around 90ms to process a call setup message [16], which includes running

a call admission control algorithm and configuring the switch fabric. Call setup delay is a

significant overhead component in connection-oriented networks. The longer the delay,

the fewer the applications that can reap the QoS-related benefits of connection-oriented

networks. Furthermore, the longer the delay, the lower the link resource utilization

because during call setup, the bandwidth being allocated for the call is not being used for

user data transport. In addition, software implementations of signaling protocols can

rarely achieve a call-handling capacity beyond the order of 1000-10000 calls per second

[17]. However, the magnitude of call arrival rates at backbone switches can be several

orders higher than the call-handling capacities of current-day switches if future connec-

tion-oriented networks are directly used to support end-user applications such as file trans-

fers and video conferencing.

The implementation part of this dissertation is focussed on demonstrating that signaling

protocols, needed in all connection-oriented networks for the call admission phase, can, in

spite of their complexity, be implemented in hardware. Advantages of a hardware imple-

mentation are that (a) call setup delay can be reduced by at least two-to-three orders of

magnitude, and (b) call-handling capacity can be increased significantly.

1.3 Outline

The remainder of this dissertation is organized as follows. In Chapter 2, we model a

polling system with vacations and apply this model to an IEEE 802.11 LAN. Appropriate

values for operational parameters, codec rates, etc., are determined to obtain the highest


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7

level of performance. Performance metrics include number of calls that can be simulta-

neously supported while meeting a predetermined set of quality-of-service guarantees

(such as delay and loss). We have published the main results of Chapter 2 in a journal

paper [18].

In Chapter 3, we consider the packetization of voice data. We first consider determinis-

tic QoS guarantees and compute the worst-case delay bound for each voice flow. Then we

consider the statistical multiplexing of independent voice flows. We compute analytically

the amount of resources that should be reserved to guarantee a specific overflow probabil-

ity or packet loss ratio for a polling-based scheduling algorithm. Finally, we illustrate

through simulations how different polling-period termination schemes impact the symme-

try in the two directions of voice communication in an IEEE 802.11 wireless LAN.

In Chapter 4, we describe our implementation of a FPGA-based signaling control card,

which supports RSVP-TE with extensions for General MultiProtocol Label Switching

(GMPLS) [19]. Specifically, while a software-based implementation of this signaling pro-

tocol in an off-the-shelf commercial SONET switch takes around 90ms to process a call

setup message [16], our breakthrough hardware implementation demonstrates that the

same set of actions involved in processing a call setup message (with the same signaling

protocol) can be accomplished in 2.4 microseconds. Our signaling control card has a call-

handling capacity of 400,000 calls/second. We have published a part of Chapter 4, the

FPGA implementation of the signaling protocol, in a conference paper [20], and a journal

paper [21]. Finally, in Chapter 5, we summarize this dissertation and discuss directions for

future research.


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8

Chapter 2

Analysis of a Polling System with

Application to Wireless LANs

Polling systems were introduced for various time-sharing computer systems in the

early 70s [22]. The basic concept of a polling system is to have a server poll a set of

queues in a cyclic order. While prior work [22-28] on the performance modeling of polling

systems yielded significant results, they were primarily targeted at computer data applica-

tions and assumed that customers arrive at queues according to a Poisson process.

Increasingly, interest in polling systems is shifting from computer data applications to

multimedia applications, e.g., in IEEE 802.11 wireless LANs [5], Bluetooth [29], and

DOCSIS systems [7]. Particularly in 802.11 MAC, the Point Coordination Function (PCF)

mode of operation, which is a polling mechanism, coexists with the Distributed Coordina-

tion Function (DCF) mode of operation, a contention-based channel access mechanism.

The reason for doing this is that while in the past, local area networks were developed with

only one sharing mechanism, such as Carrier Sense Multiple Access with Collision Detec-

tion or Token Ring, more recently, with the emphasis on multimedia services, local area

networks now support multiple sharing mechanisms simultaneously. We refer to time

intervals in which the medium is used in a non-polling sharing mode as vacations and to


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9

such system as apolling system with vacations. The impact of the coexistence of multiple

sharing mechanisms on the performance of polling mode needs to be understood.

In this chapter, we model and analyze the performance of a polling system for a real-

time application, i.e., telephony, which brings two new dimensions to the modeling prob-

lem addressed in [22-28]. First, telephony data generated within a call has been fitted

quite closely to ON-OFF Markov Modulated Fluid (MMF) models [30, 31], which are not

Poisson. Second, unlike computer data applications, telephony application is delay-sensi-

tive but loss-tolerant. Specifically, it has a stringent end-to-end delay requirement of

150ms for excellent-quality voice and 400ms for acceptable-quality voice, both with echo

cancellers [32]. Meanwhile, it is believed that a packet loss ratio up to 5% is tolerable [33]

for some voice encoding schemes.

Our problem statement is as follows: analyze the delay and voice capacity of the poll-

ing system with vacations. Towards solving this problem we have built an analytical

model for the delay in a single-queue polling system. For a multiple-queue system, we

have identified a small-N regime of operation in which deterministic service is provided,

and a large-N regime of operation in which statistical service is provided. We have com-

puted voice capacity and delay bounds for the small-N regime of operation, and simulated

voice capacity and delay distribution for the large-N regime of operation.

The rest of the chapter is organized as follows. Section 2.1 describes related work. Sec-

tion 2.2 describes our polling system model. Section 2.3 presents the delay analysis of a

single-queue system. Section 2.4 presents the capacity analysis of the small-N regime of

operation. Section 2.5 validates our analytical model with simulation results and also dem-

onstrates how our model can be applied to an IEEE 802.11b wireless LAN. Section 2.6


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10

concludes this chapter.

2.1 Related Work

Broadly speaking, prior work related to this work can be classified into three catego-

ries: papers on general polling systems, papers on QoS provisioning in wired and wireless

networks, and papers on voice support over MAC protocols. In the first category, we have

papers by Takagi [23, 24], which model single-buffer and infinite-buffer systems. The

arrival process is typically Poisson. An MMF model of the type that we assume for tele-

phony traffic is not considered. Other work such as [22, 25, 26, 34] are useful for general

modeling techniques used but they do not consider supporting telephony on a polling sys-

tem.

The literature on QoS provisioning in wired networks is extensive (see [10], [8], [35],

[36], [37], [38], and [39]), and a full review is beyond the scope of this section. These

papers did not specifically address the polling scheme considered in this chapter. Deter-

ministic services have been considered in [40] for wireless LANs and in [12] for wireless

ATM networks. Kim and Krunz studied how to provide statistical QoS guarantees for a

single ON/OFF fluid source or multiplexed ON/OFF fluid sources over a non-shared wire-

less link [41] [42]. They assumed a FIFO scheduler in their study. QoS provisioning in

wireless networks was also investigated in [43], [44], and [45] under different settings.

The scheduling scheme assumed in this chapter was not considered in these papers.

On the topic of how to support voice traffic on MAC protocols there is a very rich liter-

ature [46-48]. Focusing on polling-based MAC systems, there were several proposals to

use polling to support real-time communications in wireless environments (e.g., [49] and

[50]). Furthermore, the industry-standard IEEE 802.11 MAC protocol [5] includes a poll-


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11

ing scheme for real-time communications. This led to a number of papers on voice over

802.11 MAC schemes [51-61]. Most of these papers [51-61] use simulations to determine

how to support telephony on the 802.11 polling mode. Some conclude that it is feasible to

support telephony and provide operating points, such as the number of telephone calls to

admit to the polling list and corresponding delay bounds [e.g., 53, 58, 60]. Others [e.g.,

59] conclude that there are better MAC schemes (when compared to the polling scheme)

for telephony. Equipment vendors such as Cisco and Symbol [62, 63] have proprietary

implementations of access points that support telephony traffic. Even the IEEE 802.11

working group is now specifying an enhanced version of the MAC protocol, labelled

802.11e [64], to support other MAC schemes for quality of service. Nevertheless, our

interest in this question of understanding the behavior of a polling system with vacations

when supporting telephony traffic remains, and given that there are other communication

networks, such as Bluetooth and DOCSIS, proposing polling schemes for real-time traffic,

we decided to study this problem in a general context.

2.2 Polling System Model

The polling system model with vacations is illustrated in Fig. 2.1. In this model, time is

divided into repetitive intervals, each lasting . These intervals are further divided into

alternatingpolling periods and vacation periods. A centralized server schedules transmis-

sion opportunities among all queues during polling periods. After each polling period, the

server goes on vacation for at least a fraction ( ) over interval . If a vacation

period does not finish at the completion instant of a interval, it can stretch into the next

polling period, which is in turn foreshortened by the amount of time that equals the length

of the vacation stretch, a random variable upper-bounded by . The length of a poll-

TS

0 1< < TS

TS

VSmax


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12

ing period would be if not foreshortened by a vacation stretch. The notion of

and are important for applications receiving service in vacation periods. controls

the frequency of occurrence of vacation periods, while governs the partition inside each

.

We assume that voice sources are independent and identical ON-OFF Markov Modu-

lated Fluid (MMF) sources. A constant-rate bit stream is generated while a source is in the

ONstate (see Fig. 2.1). We assume that the server polls all queues in a round-robin fash-

ion. All data accumulated at a queue up to the instant when it is polled is served immedi-

ately after that poll (i.e., the gated-service discipline). With telephony traffic, the limited-k

service discipline is not an option because this could lead to excessive delays. Walk time,

, is the time needed for the server to move from one queue to another (i.e., the over-

head of a polling scheme). A polling cycle is the time taken to poll all queues. With

the assumed service discipline, two situations may happen in a polling period:

b

aON OFF

The queue is filled with acontinuous bit stream when

the source is in the ON state.

source rate

c

1. The ON-OFF model shownabove is assumed to be the sourcefor each of theNqueues.

2. Numbers k, m,p are allarbitrary; in other words,a vacation can occur

at arbitrary points within apolling cycle. It can evenoccur multiple times within onepolling cycle. The only constraint

is that a vacation occurs once inevery TSinterval.

Queue 1

Queue 2

Queue k

Vacation

Vacation

Vacation

Queue k+1 Queue k+2

Queue k+m

Queue k+m+1

Queue k+m+p

QueueN Queue k+m+p+1

TS TSpolling pollingvacation vacation

periodperiodperiodperiod

vacation stretchtime

Fig. 2.1: Polling system model: An example showing three vacations in one polling cycle.

1 ( )TS

TS TS

TS

Twalk

Tc N


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13

The server needs to go on vacation before completing a polling cycle because the

polling period is exhausted. If this happens, the server will continue polling the next

queue when the next polling period starts (i.e., after the vacation), as shown in Fig.

2.1.

The server completes a polling cycle. If this happens, in our system, a vacation

period starts immediately. In other words, a queue is served at most once in each

polling period. The reason for this latter assumption is that sources in our model

generate data at low rates relative to service rates (e.g., if the server is a transmission

link, service rates are likely to be in the order of a few Mbps while voice codec rates

are in the order of tens of Kbps). This implies that the amount of data accumulated

may not be sufficiently large compared to the overhead represented by if a

queue is polled more than once in a polling period.

2.3 Analysis of a Single-Queue Scenario

The mode of operation in this single-queue scenario (as per our system model

described in Section 2.2) is that the server will poll and serve the queue once and then

immediately go on vacation until next polling period. A voice packet, which consists of

the data accumulated in the queue at a polling instant, would experience two delay compo-

nents, i.e., a queueing delay , the waiting time in the queue, and a service delay ,

the time taken to serve the voice packet. Since voice data is created in the form of bit

stream, the data that arrives earlier would have had a longer waiting time at the polling

instant. We define to be the time gap between the arrival instant of the earliest bit in

the voice packet and the polling instant of this voice packet. Fig. 2.2 shows a scenario in

which a voice source enters the ONstate between two consecutive polls.

Twalk

DQ DS

DQ


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14

We are interested in total delay , which is defined as the sum of and . The

size of a voice packet, which impacts its service time , is affected by the queueing

delay. Thus, and are dependent random variables. The queueing delay also

depends on the interpoll time between two consecutive polls, which is in turn exclu-

sively determined by and the two vacation stretches. With this observation, we start

with analyzing the distribution of in Section 2.3.1. Then we derive the conditional dis-

tribution of and in Section 2.3.2 and 2.3.3, respectively. Finally we combine all

together and obtain the distribution of in Section 2.3.4.

2.3.1 Distribution of Interpoll Time

Consider two consecutive polls, th poll and th poll. As shown in Fig. 2.2, the

interpoll time equals , a constant, plus the difference between two consecutive vacation

stretches. We assume that vacation stretches are i.i.d. random variables with known Prob-

ability Density Function (PDF) , . Then the PDF of , denoted by

, can be computed by convolution

. (2.1)

2.3.2 Distribution of Packet Queueing Delay

The source state at the th polling instant can be either OFF, denoted by , or ON,

kth pollTI= t

OFF ONSource

DQ

(k+1)th poll

TS time

state

Eventsstretch

stretch

Fig. 2.2: Timing diagram.

DW DQ DS

DS

DQ DS

TI

TS

TI

DQ DS

DW

k k 1+( )

TS

s x( ) x 0 VSmax,[ ] TI

i t( )

i t( ) s t TS +( )s ( ) d0

VSma x

=

k A


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15

denoted by . Consider a packet created at the th polling instant. Let denote

that the packet is nonempty. Then for and , we obtain the following con-

ditional probability

, (2.2)

where is the transition rate of the voice source out of the OFFstate as shown in Fig. 2.1.

If the source is in the ONstate at the th polling instant, the resulting queueing delay is

. Thus we have

, (2.3)

where is the unit step function defined by

. (2.4)

Removing conditioning on the source state at the th polling instant, we obtain

, (2.5)

where

, (2.6)

A k1

+( ) B

TI t= 0 q t


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16

and . Parameters and are the transition rates shown in Fig.

2.1. Unconditioning on , we obtain as follows

, (2.7)

where is given by (2.1).

2.3.3 Service Time Distribution

Service time depends upon the amount of voice data accumulated in the time period

that equals . The reason for considering the queueing time instead of the interpoll time

is that no data is generated before the voice source transitions into the ONstate (see Fig.

2.2). Let be the total amount of time spent in the ONstate during . By using a uni-

formization technique [65], we obtain the conditional Cumulative Distribution Function

(CDF) of as

. (2.8)

Applying our polling model to a shared communication link with rate , where service

means transmission on the link, we define service time as , where

denotes the source rate. This leads to the following relationship between the conditional

CDFs of and

. (2.9)

2.3.4 Distribution ofDW

Let denote the joint PDF of and . Given , the CDF

P A B{ } 1 P A B{ }= a b

TI P DQ q B{ }

P DQ

q B{ } P DQ

q B T,I

t={ } i t( )dtTS VSma x

TS VSma x+

=

i t( )

DQ

Z DQ

Z

FZ DQz q( ) P= Z z DQ q={ } e

a b+( )q a b+( )q( )n

n!---------------------------

nk 1

ba b+------------

k1

aa b+------------

n k1

+ ni

zq---

i1

zq---

n i

i k=

n

k 1=

n

n 1=

=

R

DS DS Z c R= c

DS Z

FDS DQs q( ) FZ DQ

R s

c---------- q

=

DS DQ,s q,( ) DS DQ DW DQ DS+=


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17

of can be obtained by

, (2.10)

where is given by (2.9). The PDF of the queueing delay, , can be

found by taking the derivative of (2.7).

2.4 Multiple-Queue Scenario

In this section, we focus on the voice capacity of a multiple-queue scenario. Given that

source rates and vacation stretches are upper bounded, we recognize that there is a param-

eter such that if , the number of queues, is no more than , all queues are guaran-

teed to be served in each polling period even in the worst-case scenario. Although all

queues enjoy guaranteed service when , the problem is that, in most cases, a size-

able fraction of the time allocated to polling periods would be left unused because of the

OFFstate of the source model. A strategy that can improve utilization is to increase

beyond to fill in blanks in polling periods that would otherwise be left to vacation peri-

ods at the expense of the absolute service guarantee. This leads to statistical multiplexing.

If , as illustrated in Fig. 2.1, large interpoll times would occur if a polling period is

exhausted before a polling cycle completes. Large interpoll times will bring about large

queueing delays, which impose a negative impact on human perceived voice quality.

Therefore, should be controlled such that the frequency of occurrence of such large

delays is kept to within a tolerable limit.

We refer to the regime of operation in which as the small-N regime of opera-

tion, and to the regime in which as the large-N regime of operation. The choice of

DW

FDWw( ) P DW w{ }= fDS DQ,

s q,( ) sd qd

w q

fDS DQ s q( ) fDQ q( ) sd qd

w q

FDS DQ w q q( ) fDQ q( ) qd0

w

=

= =

FDS DQw q q( ) DQ

q( )

Np N Np

N Np

N

Np

N Np>

N

N Np

N Np>


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18

the regime of operation is up to implementation. Next we compute , the voice capacity

of the small- regime of operation.

We consider two consecutive intervals, the th interval and the th interval, as

shown in Fig. 2.3. In the worst-case situation, all polls in the th interval result in an

empty response while all polls in the th interval result in a maximum-sized packet.

This can happen if every voice source transitions into the ONstate right after a poll in the

th interval. The th interval is shown to have a stretch, which is of maximum length

in the worst-case. Since the total time spent on all queues should not exceed the

remaining time of a polling period, the inequality

(2.11)

holds, where denotes the maximum service time for the th queue.

Denote as the worst-case interpoll time of the th queue. If the voice source

stays in the ONstate during the whole interpoll time, the service time reaches its maxi-

mum value

, for , (2.12)

where and are the source rate and service rate regarding the th queue, respectively.

Np

N

k k1

+( )

1 2 i-1 i... 1 2 i-1

Timeinterpoll time for the ith queue

kth interval (k+1)th interval

stretchwalk times

... ivacation

service timesTS

Fig. 2.3: Timing in the small-N regime of operation: the worst-case scenario.

k

k1

+( )

k k

VSmax

DSmax i, Twalk+( )i

1=

N

1 ( )TS VSmax

DSmax i, i

TIm ax i, i

DSmax i, ciTIm ax i, Ri= i 1

ci Ri i


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19

As shown in Fig. 2.3, we have

, for . (2.13)

It immediately follows that

, (2.14)

for , given . Using (2.12) and (2.14), (2.11) can be rewritten

as

. (2.15)

Interestingly, we see that the value of is not impacted by the

polling order. If source rates and service rates are homogeneous, i.e., and ,

(2.15) can be simplified as

. (2.16)

The parameter equals the greatest for which (2.15) or (2.16) holds. We can thus

compute iteratively. The running time of each stage of iteration is only . For call

admission control, the inequality (2.15) or (2.16) needs to be tested only once at a given

. The number of multiplications is in the heterogeneous scenario, or in the

homogeneous scenario.

Denote as the worst-case delay experienced by a packet from the th queue.

TIm ax i, TIm ax i 1,= DSmax i 1,+ i 2

TIm ax i, 1 ci 1 Ri 1+( )TIm ax i 1, TS VSmax+( ) 1 cj Rj+( )j 1=

i 1

= =

i 2 TIm ax 1, TS VSmax+=

DSmax i, Twalk+( )i

1=

N

ci TIm ax i, Rii

1=

N

N Twalk+

TIm ax N 1+, TIm ax 1, N Twalk+

TS VSmax+( ) 1cj

Rj-----+

1

j 1=

N

N Twalk+

1 ( )TS VSmax

=

=

=

DSmax i, Twalk+( )i1=

N

ci c= Ri R=

TS VSmax+( ) 1 c R+( )N

1

[ ] N Twalk 1 ( )TS VSmax+

Np N

Np O 1( )

N O N( ) O 1( )

DWmax i, i


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20

We obtain as the sum of the worst-case queueing delay, which equals ,

and the maximum service time, i.e.,

, for . (2.17)

2.5 Validation and Application of Analytical Model to a Wireless LAN

In this section, we describe how we validate our analytical model with simulations, and

obtain numerical results for a polling system based on the IEEE 802.11 wireless LAN

standard [5]. We assume a network architecture shown in Fig. 2.4, which is typical in cur-

rent-day implementations. In Section 2.5.1, we briefly review the IEEE 802.11 Media

Access Control protocol and list parameters for our simulation study and analytical model

validation. Then we present numerical results for delay and voice capacity in Section 2.5.2

and 2.5.3, respectively.

2.5.1 Background of IEEE 802.11 and Values Selected for Parameters

The 802.11 MAC protocol supports two modes of operation: a random-access mode,

called DCF, and a polling mode, called PCF. The time period during which the LAN oper-

ates in the DCF mode is known as Contention Period (CP) while the time period in which

DWmax i, TIm ax i,

DWmax i,

TIm ax i,

= DSmax i,

+ TS

VSmax

+( ) 1 cj

Rj

+( )j 1=

i

= i 1

Wireless

Internet

Access

station

Point (AP)

Wireless

station

Wireless

station

AccessPoint (AP)

PSTN

Ethernet

Voicegateways

Fig. 2.4: Network architecture.


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21

the LAN operates in the PCF mode is known as Contention-Free Period (CFP). Details of

the polling mode of operation such as the number of stations to be admitted or the polling

order, are not specified in the standard, but left up to implementations. A superframe of

length consists of a CFP and a CP. If a station begins to transmit a frame just before the

end of a CP, it may result in a stretched CP. Compared to our model described in Fig. 2.1,

the CFP and CP correspond to the polling period and vacation period, respectively.

The values selected for the parameters of this LAN are shown in Table 2.1. We choose

in the (20ms, 50ms) range to balance delay and efficiency. Two values are used for

to show the impact of the size of vacation periods. For vacation stretch, we assume the fol-

lowing distribution

, (2.18)

where parameter and are listed in Table 2.1. Other than this simple model, measure-

ment-based distributions can be developed to characterize vacation stretches in practice.

We assume that voice data for a wireless station is sent in a Poll+Data frame to that

station. The parameters of telephony traffic model (see the Markov chain included in Fig.

2.1) are set to the values in the May and Zebo model [31]. Two values of codec rate , the

Truespeech codec rate of 8.5Kbps [66] and the Pulse Code Modulation (PCM) codec rate

of 64Kbps [67], are used for sensitivity analysis. Source rate equals the sum of voice

codec rate and the overheads of all layers above the 802.11 MAC. Specifically, the

overheads come from the Real-time Transfer Protocol (RTP) [68], the User Datagram Pro-

tocol (UDP), the Internet Protocol (IPv4), and the Logical Link Control (LLC) protocol.

The data rate of the overheads is calculated as , where the value of is 43 bytes if

TS

TS

S x( )

0 x 0,

=

p1

p2

C

c

C

H TS H


38/126

22

assuming one set of RTP/UDP/IP/LLC headers in each interval. This value is large

since it is comparable to the typical payload size of a voice packet. To improve efficiency,

the RObust Header Compression (ROHC) [69] protocol can be optionally applied to

reduce down to 4 bytes. We assume homogeneous source rate and link rate for

simplicity although (2.15) allows for heterogeneous rates. We assume a link rate of

11Mbps and error-free transmission.

The value of equals the time spent on an empty 802.11 frame. If we assume the

IEEE 802.11b physical layer with the standard long preamble, the value of is

approximately 0.23ms. If the optional short preamble is adopted at the physical layer, the

value of can be reduced down to 0.13ms. Table 2.1 also lists the transmission times

ofBeacon and Contention-Free-End (CF-End) frames, which are overheads added to each

polling period according to the IEEE 802.11 MAC protocol.

In this application of our model, the total number of queues is twice the number of calls

since telephony is bidirectional. We use , which equals , to denote the voice

Table 2.1: Values for parameters.

Parameter Symbol Value

Superframe length 20ms to 50ms

Minimum fraction allocated to

vacation period

0.5 or 0.7

Maximum vacation stretch 2.8ms

Parameters for CP stretch , 0.6, 0.4

Average duration of the ONstate 352ms

Average duration of the OFFstate 650ms

Voice codec rate 8.5Kbps or 64Kbps

Overhead of RTP/UDP/IP/LLC 43 bytes or 4 bytes

Source rate

Service rate 11Mbps

Walk time 0.23ms or 0.13msTime spent on a Beacon frame 0.512ms

Time spent on a CF-end frame 0.352ms

TS

H c R

Twalk

Twalk

Twalk

TS

VSma x

p1

p2

1

a

1 b

C

H

c H TS C+

R

TwalkTBe ac on

TC F e nd

Np' Np 2


39/126

23

capacity in units of calls. We implemented a custom event-driven simulation of our poll-

ing system with vacations. Each simulation run is long enough for the system to reach

steady state.

2.5.2 Numerical Results for Delay

The delay result for the single-call scenario is presented in subsection A. The simula-

tion results for the delays in the multiple-call scenario are shown in subsections B and C.

A. Single-call scenario

Fig. 2.5 shows the CDF of delay obtained for four values of . We observe that

is more likely to have a value around than other values. For example, when

equals 50ms, the probability of being less than 47ms is only 0.12. We explain this as

follows. First, since all four values are much smaller than the mean ONstate holding

time (352ms), it is quite likely that the voice station stays in the ONstate for the entire

duration of . Second, data created after a poll will have to wait for the next poll since

the queue is polled only once every (see Section 2.2). Therefore, is highly corre-

DW TS

DW TS

Fig. 2.5: CDF ofDW with TS as a parameter in the single-call scenario. Twalk

and Care set to 0.23ms and 8.5Kbps, respectively.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

x (ms)

Probability{DWx

}

Simulation: TS = 20ms




Analysis: TS = 20ms

Analysis: TS = 30ms

Analysis: TS = 40ms

Analysis: TS = 50ms

TS

DW

TS

TS

TS DW


40/126

24

lated to interpoll time.

Another observation is that queueing delay dominates in most cases because of the

relatively low source rate and high service rate. With an interpoll time of 52.8ms (50ms

for and 2.8ms for ), at a codec rate of 8.5Kbps, the length of the maximum-

sized voice payload is only 56 bytes. The transmission time of an 802.11 frame with 56-

byte payload is about 0.3ms if assuming 43 bytes for and 0.23ms for .

B. Multiple-call scenario: the small-N regime of operation

Fig. 2.6 shows the empirical Complementary CDF (CCDF) of obtained for three

values: 20ms, 35ms, and 49ms. The corresponding voice capacities in units of calls are

7, 13, and 17, respectively, according to (2.16). Fig. 2.6 illustrates that calls further down

in the polling list is not impacted much by the variability in the service times of calls in the

front of the polling list. For a total of 17 calls, the plots corresponding to the first call and

the 17th call almost coincide with each other in Fig. 2.6 although their maximum delays in

the worst-case scenario can be quite different (see (2.17)). Here is a qualitative explana-

DW

TS VSmax

H Twalk

DW

TS

Fig. 2.6: CCDF ofDWin the small-N regime of operation. , codec rate,

and Twalkare set to 0.5, 64Kbps, and 0.23ms, respectively.

0 5 10 15 20 25 30 35 40 45 50 5510

-3

10-2

10-1

100

x (ms)

Probability{DW>

x}

TS=20ms, the 1st

call

TS=20ms, the 7th

call

TS=35ms, the 1st

call

TS=35ms, the 13th

call

TS=49ms, the 1st callTS=49ms, the 17

thcall


41/126

25

tion for this behavior.

We have observed from subsection A that delay is highly correlated to interpoll time.

The difference between the interpoll time of the th and that of the first queue can be

expressed as

, (2.19)

where denotes the service time of the th queue in the th interval. First, it is likely

that most voice sources do not change state during two consecutive interpoll times since

the average ONand OFFtimes of voice sources are much larger than the values selected

for . This means that and roughly cancel out for these queues. Second, for

the voice sources that do change state, the result of could be either posi-

tive or negative. Given that voice sources are assumed to be independent of each other, the

random variable is expected to be Normal-like with zero mean when

is large. Finally, the maximum value of , which is

(see Fig. 2.3), could be just a small fraction of because of the

large values. The significance of this observation is two-fold: first, the delay distri-

bution derived for the single-queue scenario in (2.10) can be used as a fair approximation

for the delay distribution in the small-N regime of operation; second, the value of pursuing

an accurate analytical answer of the delay distribution in the small-N regime of operation

is low, at least for the parameter values listed in Table 2.1.

C. Multiple-call scenario: the large-N regime of operation

We simulate the large-N regime of operation by choosing , the number of voice calls

admitted to the polling list, to be larger than , which is 19 for a of 30ms and a of

TI j,k 1+( )

TI 1,k 1+( )

DS i,k 1+( )

DS i,k( )

( )i 1=

j1

=

DS i,k( )

i k

TS DS i,k

1+( )DS i,

k( )

DS i,k

1+( )DS i,

k( )( )

TI j,k

1+( )TI 1,

k1+( )

( ) j

TI j,k

1+( )TI

1,k

1+( )

1 ( ) TS NpTwalk[ ] TS

Twalk

N'

N'p TS


42/126

26

0.5 according to (2.16). For , we see staircase-like regions in the CCDF plots

shown in Fig. 2.7 where two drops occur around 30ms and 50ms, respectively. These

staircases suggest that a packet may experience a delay spike around 50ms, which is sig-

nificantly larger than the normal delay values around or below 30ms. This behavior is

caused by the strong correlation between queueing delay and interpoll time, which has

been discussed in subsection A. Interpoll times normally appear in the neighborhood of

. However, a larger interpoll time may occur if a queue is missed in a polling period. As

increases, such large interpoll times appear more often, leading to an increase in the

frequency of occurring of the delay spike.

2.5.3 Numerical Results for Capacity

As shown in Fig. 2.7, the cost of operating in the large-N regime of operation is the

occurring of delay spike. Although delay spikes have negative impact on perceived voice

quality, telephony applications could typically tolerate such delay degradation if delay

spikes do not appear very often [33]. We set a delay threshold , the maximum

delay in the small-N regime (see (2.17)), and declare a packet that has a delay greater than

N'1 9>

TS

N'

Fig. 2.7: CCDF of delay with N'as a parameter in the large-N regime of operation.

TS, , and codec rate are equal to 30ms, 0.5, and 8.5Kbps, respectively.

0 10 20 30 40 50 6010

-4

10-3

10-2

10-1

100

x (ms)

Probability{DW>

x}

19 calls20 calls

22 calls

23 calls

24 calls

21 calls

DWmax 2 Np',


43/126

27

as a loss. Loss ratio, , is defined as . Voice capac-

ity in the large-N regime of operation, , is defined as the maximum number of calls

that allows to be kept to within a tolerable value such as 1% or 3%. is equal to

when the tolerable value is 0.

We use simulations to obtain under various , , , and stretch dis-

tribution assumptions. In Fig. 2.8, values are compared against values com-

puted from (2.16). We observe that increases with for all values because

DWmax 2 Np',Ploss P DW DWmax 2 Np',

>{ }

N'max

Ploss N'max

N'p

N'max TS Ploss Twalk

20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

TS (ms)

Maximumnumbe

rofcalls,

N'm

ax

Simulation: Ploss 0.01, = 0.7

Analysis/Simulation: Ploss = 0, = 0.7





20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

TS (ms)

Maximumnumberofcalls,N'm

ax




Analysis/Simulation:Ploss = 0, = 0.5

Simulation:Ploss 0.01, = 0.5


Fig. 2.8:N'max versus TS, with Ploss and as parameters. Codec rate, Twalk, and stretch distri-

bution are respectively set to (a) 8.5Kbps, 0.23ms, and S(t); (b) 8.5Kbps, 0.23ms, and

VSmax. The plot for Ploss=0 is obtained from one million samples.

(b)

(a)

N'max N'p

N'max TS Ploss


44/126

28

more data can accumulate in a queue at a larger . Let us take the computation of as

an example. Since is much smaller than 1 for the parameter values listed in Table

2.1, the term in (2.16) can be approximated to . Thus (2.16) can

be simplified as

. (2.20)

The first term of the denominator, i.e., , is much smaller than

. Therefore we see that and appear to have a linear-like relationship in Fig.

2.8(a). This suggests a trade-off between voice capacity and delay.

There are two sources leading to the capacity gain shown in Fig. 2.8(a) when is

relaxed from 0 to 0.01, and further to 0.03. First, vacation stretches can be exploited to

carry more voice traffic if loss is allowed. Second, statistical multiplexing gain can be

obtained by exploiting the OFFstate of voice sources. To evaluate the gain from the first

source, we need statistical information about vacation stretches, which, however, varies

with load pattern in vacation periods. Thus we focus on the second source. To isolate the

statistical multiplexing gain, we replot against in Fig. 2.8(b) for the same set of

parameter values used in Fig. 2.8(a) except that vacation stretches are fixed at . We

observe that the multiplexing gain is small because the polling overhead dominates

over the service time of voice payload. The gain is only 3 calls (from 19 to 22) when

and are set to 30ms and 0.5, respectively.

Fig. 2.9 shows the results for a codec rate of 64Kbps. Although the voice capacities are

smaller compared to the counterpart shown in Fig. 2.8(b), the multiplexing gain is more

noticeable in Fig. 2.9(a). For example, when is relaxed from 0 to 0.03,

TS N'p

c R

1

c R+( )N

1

N c R+( )

N1 ( )TS VSmax

TS VSmax+( ) c R Twalk+------------------------------------------------------------------

TS VSmax+( ) c R

Twalk N'p TS

Ploss

N'max TS

VSmax

Twalk

TS

Ploss N'max


45/126

29

increases from 11 to 18, which is roughly a 64% gain in ratio, as opposed to 3, or 16%, in

Fig. 2.8(b).

Fig. 2.9(b) shows that the multiplexing gain is significant when is set to 0.13ms.

We see that increases from 17 to 30, which is a 76% gain, at a of 30ms and a

of 0.5, when is relaxed from 0 to 0.03.

2.5.4 Summary of the Numerical Results

Table 2.2 lists a summary of the numerical results obtained at a of 30ms. We see

Fig. 2.9:N'max versus TS, with Ploss and as parameters. Codec rate, Twalk, and stretch

distribution are respectively set to (a) 64Kbps, 0.23ms, and VSmax; (b) 64Kbps,

0.13ms, and VSmax. ROHC is applied in (b). The plot for Ploss=0 is obtained

from one million samples.

20 25 30 35 40 45 500

5

10

15

20

25

30

TS (ms)

Maximumnumberofcalls,

N'm

ax







20 25 30 35 40 45 505

10

15

20

25

30

35

40

45

TS (ms)

Maximumnumberofcalls,

N'm

ax







(a)

(b)

Twalk

N'max TS

Ploss

TS


46/126

30

that by accepting a certain probability of loss, we can run the system in the large-N regime

of operation, accommodating more calls while still allowing a large ratio of link band-

width to be used for other sharing modes. We observe that by reducing the polling over-

head and header overheads, we can use higher-quality voice (64Kbps instead of 8.5Kbps)

and yet accommodate a greater number of voice calls (28 vs. 22, or 30 vs. 22) if we are

willing to accept some loss (1%, or 3%). The maximum delay is shown for all

cases in 30-40ms range. Given the end-to-end delay requirement of 150ms, a delay in this

range is considered to be acceptable since the remaining links are likely to be higher speed

wired links.

Voice capacity can be increased in a number of ways: 1) accept some packet loss, 2)

increase at the cost of larger delay, 3) decrease at the cost of a smaller vacation

period, 4) decrease codec rate at the cost of lower voice quality, 5) adopt ROHC at the cost

of signaling overhead, and 6) decrease at the cost of robustness. We conclude that 1

and 6 are the most effective methods; 2, 3, 4 and 5 could also be considered if necessary.

We define transmission efficiency as the ratio of average service time of voice pay-

load to the average total time spent on a frame. If we ignore large interpoll times, which

occur infrequently, the average interpoll time of a queue is roughly . Then the average

Table 2.2: Summary of numerical results at TS= 30ms.

Codec

rate

ROHC Vacation

stretch ( )

( )

( )

0.23ms 8.5Kbps no randoma

a. Vacation stretch length is a random variable with the distribution function defined in

(2.18).

0.5 19 (0) 24 (0.01) 26 (0.01) 35.1ms

0.23ms 8.5Kbps no 0.5 19 (0) 22 (0.01) 22 (0.01) 35.1ms

0.23ms 64Kbps no random 0.5 11 (0) 21 (0.01) 22 (0.01) 38.1ms

0.23ms 64Kbps no 0.5 11 (0) 17 (0.01) 18 (0.01) 38.1ms

0.13ms 64Kbps applied random 0.5 17 (0) 32 (0.01) 34 (0.01) 39.6ms

0.13ms 64Kbps applied 0.5 17 (0) 28 (0.01) 30 (0.01) 39.6ms

DWmax 2 Np',

Twalk Np'Ploss

N'ma xPloss

N'maxPloss

DWmax2

Np',

S x( )

VSma x

VSma x

VSma x

TS

Twalk

E

TS


47/126

31

payload size of a frame is , where and are the codec rate

and the probability of a source being in the ONstate, respectively. We compute as fol-

lows

, (2.21)

where is the overheads of all protocol layers above the IEEE 802.11 MAC. Fig. 2.10

shows that transmission efficiency is low (less than 0.05) when and

. If larger codec rate and smaller are assumed, transmission effi-

ciency can be improved significantly (up to 0.44) for the range of under consideration.

The numerical values shown in Fig. 2.10 is mainly determined by the relatively high over-

head of the IEEE 802.11b when supporting low-data-rate applications such as telephony.

2.6 Chapter Summary

In this chapter, we modeled a polling system with vacations. We started by deriving an

analytical solution for delay distribution in a single-queue scenario, which has later been

TS C b a b+( ) C b a b+( )

E

ETS C b a b+( ) R

TS C H+( ) b a b+( ) R Twalk+-------------------------------------------------------------------------------------=

H

Fig. 2.10: Transmission efficiency as a function ofTS. Codec rate,H,

and Twalkserve as parameters.R is fixed at 11Mbps.

20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

TS (ms)

TransmissionEfficiency

C = 64 KbpsH = 4 bytesTwalk = 0.13 ms

C = 64 KbpsH = 44 bytesTwalk = 0.23 ms

C = 8.5 KbpsH = 4 bytesTwalk = 0.13 ms

C = 8.5 KbpsH=44 bytesTwalk = 0.23 ms

C 8 . 5 Kbps=

Twalk 0 . 2 3 ms= Twalk

TS


48/126

32

found to be a fair approximation for delay distribution in a multiple-queue scenario, i.e.

the small-N regime of operation. For voice capacity, we established a procedure to calcu-

late the number of telephone calls that can be supported with a guarantee of being polled

every polling period. To admit more calls and yet keep the delay guarantee, we allow for

some packet loss. We demonstrate that for the IEEE 802.11b wireless LAN and a codec

rate of 8.5Kbps, we are not able to obtain much statistical multiplexing gain by exploiting

the ON-OFF characteristics of telephony traffic because of large overheads. However, by

decreasing these overheads by half (which is feasible with an optional short preamble and

a header compression technique), we demonstrate that the system can indeed exploit the

silences in telephony traffic and accommodate a greater number of voice calls even with a

higher-rate codec (64Kbps). The numbers identified appear to be acceptable because the

range of an 802.11b access point is small (on the order of 100m).


49/126

33

Chapter 3

Effects of Packetization of Voice Data

We assumed the ON-OFF Markov Modulated Fluid (MMF) model for voice sources in

one analysis in Chapter 2. Although this model is considered suitable for characterizing

digitized human speech, it does not take into account the packetization of voice traffic that

exists in many voice communication systems [70][71]. In these systems, the output of a

voice encoder is a stream of data blocks filled with compressed voice data. Then several

blocks (or one block) are encapsulated into a packet for transmission over a packet-

switched network [72]. Packetization delay typically ranges from 10ms to 30ms. There-

fore, the incoming voice data seen by the MAC layer of an end point is a stream of voice

packets instead of the constant-rate bit stream assumed in the ON-OFF MMF model. In

order to better predict the performance of such voice communication systems, we study

this packetization effect in this chapter.

3.1 An ON-OFF MMF Model with Packetization

Fig. 3.1 illustrates how voice packets are created when assuming an ON-OFF MMF

model with packetization. This model assumes that the ON and OFF states still have expo-

nentially distributed holding times. Time is divided into repetitive periods of fixed length,

which is shown to be 10ms in Fig. 3.1. A voice packet is generated at the end of a period


50/126

34

only if the source is detected active during some portion of this period. As an example, see

Fig. 3.1, which shows five voice packets created at ms, respec-

tively. No voice packet is generated at ms or ms because the voice source

was silent in the entire previous period.

This packetization of voice traffic is important to us since it not only introduces a pack-

etization delay, but also impacts the number of voice packets waiting in a queue. The latter

affects both the voice capacity and the queueing delay. For example, in Fig. 3.1, a server

polls a queue at time ms and ms, consecutively. The first queueing delay of

20ms shown in Fig. 3.1 is caused by the poll arriving 20ms after the transition from the

OFF to the ON state. The second queueing delay is 35ms because the last three packets,

which contain all voice data generated in the time period between time ms and the

end of the ON period, are served immediately after the second polling instant. This delay

would have been 30ms instead of 35ms had there been no packetization. We refer to the

first and the second queueing delays as a queueing delay of the first type, and a queueing

delay of the second type, respectively.

Other delay components introduced by voice compression include look-ahead delay

and processing delay. These delays are not considered in this model because they are of

fixed length and hence can be easily dealt with by adding a fixed component to the end-to-

OFF ONVoice activity OFF

Packet creationinstant

10 20 30 40 50 60 70t (ms)

Polling instant

first queueing

delay = 20ms second queueingdelay =35ms

Fig. 3.1: Illustration of the packetization of voice traffic.

t2 0 3 0 4 0 5 0 6 0 , , , ,=

t1 0

= t7 0

=

t3 5

= t6 5

=

t3 0

=


51/126

35

end delay.

3.2 Voice Capacity and Delay Bound in Small-N Regime of Operation

We assume the same polling system model as the one described in Section 2.2. Follow-

ing the approach used in Section 2.4, we divide the multiple-queue scenario into a small-N

regime of operation and a large-N regime of operation. In order to compute , the largest

number of queues that can be admitted in the small-N regime of operation, we construct a

worst-case scenario similar to the one described in Section 2.4, but for the ON-OFF MMF

model with packetization. Let and represent packetization delay and the size of a

voice packet, respectively. Assume all queues have the same service rate . For the first

queue, the maximum length of an interpoll time is . Thus the greatest possi-

ble number of voice packets accumulated in the queue at a polling instant is the smallest

integer that is greater than . This could happen if a voice packet is created

immediately after the beginning instant of an interpoll period of in length.

For example, for an interpoll time of 25ms and an of 10ms, there can be up to three

voice packets waiting in the queue at the end of this interpoll period. Let denote

the service time for the th queue in the worst-case scenario. can be computed by

. (3.1)

Similarly, a largest interpoll period for the th queue occurs if all polls for the previous

queues in the previous polling interval result in zero payload but all polls in the

following polling period find a maximum number of voice packets. Thus is given

by

, for . (3.2)

Nl

L S

R

TS VSmax+( )

TS VSmax+( ) L

TS VSmax+( )

L

DSmax i,

i DSmax 1,

DSmax 1, TS VSmax+( ) L S R=

i

i1

( )

DSmax i,

DSmax i, TS VSmax DSmax k,k 1=i 1

+ +( ) L S R= i 2 3 4 , , ,=


52/126

36

Since the total time spent on serving all queues should not exceed the length of a poll-

ing period, equals the greatest integer for which the following inequality holds

. (3.3)

The number of multiplications or additions involved in computing is .

We can compute the worst-case queueing delay, or the delay bound, for the first queue

as

. (3.4)

Similarly, the worst-case queueing delay for the th queue is

, for . (3.5)

Equations (3.4) and (3.5) apply to both frame queueing delay and packet queueing

delay. Moreover, it is worth noting that (3.1)-(3.5) can be easily adapted to a polling sys-

tem with heterogeneous , , , and .

3.3 Resource Allocation in Large-N Regime of Operation

So far we have investigated a voice capacity problem, i.e., finding the largest number

of admissible calls if a fraction of each superframe is allocated to carry voice traf-

fic. We can alternatively formulate a resource allocation problem, where the resource

being shared is the time in each superframe. Given queues, we need to compute the

minimum time needed to transmit voice packets accumulated in all queues. For the

small-N regime of operation, this minimum time is the sum of the worst-case service times

plus walk times, i.e.,

. (3.6)

Nl N

DSmax i, Twalk+( )i 1=N

1 ( )TS VSmax

Nl O N( )

Dbound1, T= S VSmax L+ +

i

Dbound i, T= S VSmax DSmax k,

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