Optimal Ranging Algorithms for Medium Access Control in Hybrid Fiber Coax Networks

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PAPER

Optimal Ranging Algorithms for Medium Access Control in Hybrid Fiber Coax Networks

Frank Yeong-Sung LIN^†, Wei-Ming YIN^††, Ying-Dar LIN^†††, and Chih-Hao LIN^††††, Nonmembers

SUMMARY The ranging algorithm allows active stations to measure their distances to the headend for synchronization purpose in Hybrid Fiber Coax (HFC) networks. A practica- ble mechanism to resolve contention among numerous stations is to randomly delay the transmission of their control messages.

Since shorter contention cycle time increases slot throughput, this study develops three mechanisms, fixed random delay, variable random delay, and optimal random delay, to minimize the contention cycle time. Simulation demonstrates that the optimal random delay effectively minimizes the contention cycle time and approaches the theoretical optimum throughput of 0.18 from pure ALOHA. Furthermore, over-estimation reduces the impact on contention cycle time more than under-estimation through sensitivity analysis, and both phenomenon damage slot throughput. Two estimation schemes, maximum likelihood and average likelihood, are thereby presented to estimate the number of active stations for each contention resolution round. Simulation proofs that the proposed estimation schemes are effective even when the estimated number of active stations in initial contention round is inaccurate.

key words: ^ranging,^contentionresolution,throughput,estimation

1. Introduction

Hybrid Fiber Coax (HFC) technology provides coax- ial networks with two-way and broadband transmission capabilities. To make cable modems and headends designed by diﬀerent vendors interoperable, two stan- dards, IEEE 802.14 [1] and Data-Over-Cable Service Interface Speciﬁcations (DOCSIS) [2] specifying Phys- ical and Medium Access Control (MAC) layers have been developed. A two-way HFC network is a point- to-multipoint, tree-and-branch access network in the downstream direction, but a multipoint-to-point, bus access network in the upstream direction. Since a station cannot independently detect upstream collisions, it should not send data at will; otherwise, the perfor-

Manuscript received May 16, 2001.

Manuscript revised January 30, 2002.

†The author is with the Faculty of the Department of Information Management, National Taiwan University, Taipei, Taiwan.

††The author is with the Fastlink Communication Corp., Taipei, Taiwan.

†††The author is with the Faculty of the Department of Computer and Information Science, Nation Chiao Tung University, Hsinchu, Taiwan.

††††The author is with the Department of Information Management, National Taiwan University, Taipei, Taiwan.

mance will be signiﬁcantly impacted by serious collisions. Therefore, all upstream data transmission should be centrally coordinated by the headend. For eﬀec- tive functioning, each station should synchronize with the headend. Owing to the typically large propagation delay in the HFC networks, each station should learn its distance from the headend and compensate for this distance such that all stations and the headend have a consistent system-wide view of time. Therefore, a ranging process is presented to measure the distance between a station and the headend. Collisions occur when multiple active stations simultaneously transmit ranging messages on a slot. Consequently, each collided message must be retransmitted round by round until it is successfully received.

The random delay mechanism is proposed to randomly delay the transmission of the ranging messages and reduces the probability of collisions. To minimize the average time of a ranging process, three algorithms are developed to calculate the optimal random delay.

Since the headend can determine the size of the ranging area, the proposed mechanism is applicable to HFC networks. Furthermore, two estimation schemes for the number of active stations in each round except the initial one are also proposed to eﬀectively adjust the random delay in each round. Many ranging algorithms have been developed for diﬀerent applications [3]–[5]

but were presented to estimate the point-to-point dis- tance. Therefore, those algorithms cannot be directly applied in the HFC environment.

The rest of this paper is organized as follows.

Section 2 gives the statement of the ranging problem and describes the system model. Meanwhile, Sect. 3 proposes a number of algorithms and compares their throughput via simulation and analysis. Section 4 then discusses the implementation issues, including sensitivity analysis and estimation schemes for the number of active stations. Finally, Sect. 5 summarizes this study.

2. Problem Statement

Figure 1 shows how a ranging process performs. If the headend sends a ranging message, M , to invite each station to transmit a message which should arrive at the headend at T1, then the headend needs to send M no later than T1− TX, so that every station can receive

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tion A sends its response immediately, the response will arrive at the headend at T2instead of at T1. The idea is that station A has to wait its Round-Trip Correction time, RT CA, and then send the response, so that the response arrives at the headend at exactly T1. RT CA

is therefore calculated as TX− TA. The headend must help station A to calculate its own round-trip delay, TA, so that station A can adjust its transmission starting time to achieve MAC level synchronization. The gen- eral steps of the ranging process in DOCSIS is listed below and shown in Fig. 2:

1. Obtain global timing reference: After being pow- ered up, the station should listen to the sync mes- sage sent periodically by the headend at an inter- val of tens of milliseconds. Upon receiving sync message, the station should then set its local clock

Fig. 1 Normal ranging process.

Fig. 2 The DOCSIS ranging process.

step (b).

2. Identify the ranging area: The headend also peri- odically broadcasts a ranging invitation message to invite all unranged stations to join the network. The starting point of the ranging area is de- scribed by explicitly identifying the starting min- islot number in the bandwidth allocation message, MAP. The headend must make the ranging area suﬃciently large to accommodate the possible long propagation delay from the stations.

3. Transmit the ranging message: The ranging process consists of initial ranging and station ranging.

After determining the ranging area, the station at- tempts to obtain a temporary service identiﬁer to facilitate other initialization operations in initial ranging, and the station uses the station maintenance area to perform periodical ranging. Simi- larly, the headend may calculate the timing oﬀset for station A as T5− T4. The adjustment parame- ters are sent back to station A through the ranging response message.

4. Make adjustments according to the feedback message: The station is roughly ranged after adjusting its parameters including timing offset, power level, frequency offset, and center frequency, according to the values offered in the feedback message, as in steps (d) and (e). The ranging process is re- peated until the headend considers that the station requires no more adjustment.

This study seeks to minimize the average time of a ranging process and thus enhances slot throughput, which is deﬁned as the ratio of the number of involved stations over the average time of a ranging process. The mecha-

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Fig. 3 A contention cycle with blocked arrivals.

nism employed for resolving contention among numerous stations is to randomly delay the transmission of their control messages. The system involves a ﬁnite- population stations and a blocked-access scheme where stations that are activated in the current contention resolution cycle are prohibited from participating in this contention cycle.

Figure 3 illustrates the aforementioned contention resolution process. A contention cycle consists of several contention rounds that completely resolve a group of unranged stations. A contention round corresponds to the maintenance area in Fig. 2. Notably, the maintenance area exists in upstream and the time intervals between consecutive maintenance areas are allocated to ranged stations for data transmission. Moreover, this study focus on contention slots throughput instead of overall upstream bandwidth throughput. Therefore, those time intervals for data transmission are disre- garded in average cycle time evaluation. Assume that when the system is in the contention resolution cycle i- 1, stations a, b, c, d, e, f and g become active. All these active stations are serviced in the next cycle, cycle i. In round 1 of cycle i, each active station must wait for a random delay before it begins transmitting its ranging message, which is assumed to have a length of 1 unit.

The range of random delay in round 1 is between 0 and the duration of round 1. In this illustration, the re- sult of round 1 is that stations d, f , and g successfully transmit their ranging messages, but the transmissions of the other stations fail due to collisions. Stations a, b, c and e will thus continue the ranging process, con- tention resolution process, in the next round of cycle i until all of the active stations have successfully transmitted their ranging messages. In this illustration, the 7 active stations require 4 rounds in cycle i to complete the ranging process.

Notably, the range of the random delay employed in each contention round may subtly inﬂuence the cycle time. A long random delay results in a long round time, but increases the probability of successful message transmission as well as the probability of a contention cycle containing a small number of rounds. On the other hand, a short random delay reduces round time but increases the number of rounds in a contention cycle owing to frequent collisions. Consequently, to minimize the cycle time, the range of random delay in each round

should be selected carefully.

3. Random Delay Algorithms

To resolve contention efficiently, three algorithms are proposed to calculate the random delay in each contention round based on the number of contending stations to minimize the average cycle time. These algorithms are for headend to allocate size of maintenance area effectively; therefore, no modification on DOCSIS protocol is needed.

3.1 Fixed Random Delay: FRD

This algorithm assumes that the number of active sta- tions is given and a fixed random delay for each con- tention round is employed throughout the contention cycle. The fixed random delay is calculated from a state transition diagram depicted in Fig. 4. This state transition diagram defines states as the number of active stations in a round. The contention is assumed to initially involve n active stations. Meanwhile, the transition from State i to State i− j means that j sta- tions successfully transmit their packets among i active stations; therefore, the remaining i− j stations must contend in the following rounds. The probability of this transition is denoted by p(i, j, d) given the random delay is d and is calculated by exhaustive simulation. A resolution scenario is defined as consecutive states from initial state to state 0 in a contention cycle. For each d, the average cycle time is obtained from

ACTdn = d∗ p(n, n, d)

+2d∗ [p(n, n, d)p(n, 0, d) +

n−2

i=1

p(n, i, d)p(n− i, n − i, d)]

...

+Kd∗

j

(probability of a resolution scenario j with cycle time Kd)

where K is determined if the sum of probabilities of resolution scenarios with cycle time (K + 1)d is less than 10⁻⁵. Therefore, a d with minimal average cycle

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Fig. 4 State transition diagram for developing FRD.

Fig. 5 State transition diagram for VRD.

time is chosen as random delay in each contention round given n initial active stations, dn.

3.2 Variable Random Delay: VRD

VRD stands for variable random delay for each round thus minimizing the average cycle time. In each contention round, the random delay corresponds to that in initial state in FRD. For example, if there are k ac- tive stations in a contention round, the random delay in that round is set to dk. The length of each round can thereby be dynamically controlled. Figure 5 illustrates the contention resolution process adopting the VRD mechanism. There are three active stations in- volved in the ﬁrst round of cycle i; thus, the round time is d3. One active station successfully transmits its control message in this round, therefore, the round time for the second round is set to d2. Since no active station successfully transmits its control message, the round time of the third round must be equal to d2. All active

stations after round 3 have been successfully resolved.

If FRD is used, the cycle time of cycle i is d3+ d3+ d3. However, by adopting VRD, the cycle time of cycle i is reduced to d3+ d2+ d2. Since VRD considers random delay round by round, it intuitively outperforms FRD, in which random delays of rounds other than the ﬁrst round cannot be determined dynamically based on the contention results.

3.3 Optimal Random Delay: ORD

ORD involves a globally optimal random delay for dif- ferent numbers of active stations in each round thus minimizing the average cycle time. The bottom-up approach is adopted herein to derive optimal variable ran- dom delay for each round. di,OP T is denoted as the globally optimal value of d for each p(i, j, d) in Fig. 4 to minimize the average cycle time. Consequently, all di,OP T’s are calculated as follows:

1. d0,OP T = d1,OP T = 0.

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2. d2,OP T denotes the optimal value of d that min- imizes the average cycle time modeled by Fig. 6.

For each d, the average cycle time is obtained from

ACTd2 =

K i=1

i∗ d ∗ p(2, 0, d)ⁱ⁻¹p(2, 2, d) ,

where K is determined if the sum of probabilities of resolution scenarios with cycle time (K + 1)d is less than 10⁻⁵.

3. d3,OP T denotes the optimal value of d that min- imizes the average cycle time modeled by Fig. 7.

For each d, the average cycle time is obtained from

ACTd3=

K i=1

p(3, 0, d)ⁱ⁻¹[i∗ d ∗ p(3, 3, d) + ACTd_{2,OP T}∗ p(3, 1, d)],

where K is determined if the sum of probabilities of resolution scenarios with cycle time (K + 1)d is less than 10⁻⁵.

4. d4,OP T, and so on, can be calculated by the itera- tive procedure as above.

3.4 Numerical Observation

Three random delay algorithms are conducted via simulation. Without losing generality, the time resolution is 10⁻⁴ unit to approximate continuous, i.e., unslotted, behavior of transmitting ranging messages. Moreover, this study only simulates the ranging contention resolution process instead of overall upstream transmission behavior. Hence, no network parameters are consid- ered. Although after receiving several sync messages, one station might align to slot boundary at the station

Fig. 6 State transition diagram for ORD to calculate d2,OP T.

Fig. 7 State transition diagram for ORD to calculate d3,OP T.

side. However, it is very likely for a station’s message to collide with other station’s messages at the head- end side without learning its RT C. Therefore, con- sidering the propagation delay, it makes sense to treat the upstream ranging process as an unslotted system throughout the ranging process. Its behavior can hence be modeled as a pure ALOHA system.

Figures 8 and 9 display the average cycle time and slot throughput, respectively, for the proposed random delay algorithms. Since FRD lacks the ﬂexibility to adjust the random delay round by round, average cycle time is the longest among all. Additionally, by sys- tematically calculating the value of random delay for a given number of active stations ORD produces a shorter average cycle time than VRD. Regarding slot through-

Fig. 8 Average cycle time for FRD, VRD, and ORD algorithms.

Fig. 9 Slot throughput for FRD, VRD, and ORD algorithms.

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pure ALOHA. That is, the proposed ORD algorithm can achieve near optimal slot throughput in resolving the contention of multiple active stations.

4. Implementation Issues

Notably, in previous simulation, the number of unranged active stations in each contention round is assumed to be known. However, only the number of collision clusters and the number of active stations that successfully transmitted their messages are known in real world. The number of initial active stations remains unknown because each collision clusters may consist of any number of messages transmitted by active stations.

To eﬀectively adopt the above random delay mechanisms, sensitivity analysis is ﬁrst conducted and then schemes for estimating the number of active stations in each round are presented and investigated.

4.1 Sensitivity Analysis

To determine the influence of the error in estimating the number of active stations on average cycle time, sensitivity analysis is conducted for FRD and ORD algorithms and Figs. 10 and 11 display the results. In these figures, the notation “err = k” denotes the num- ber of active stations with k units of estimation error, namely k = estimate(i)−i where i and estimate(i) are the numbers of actual and estimated active stations, respectively. From these figures, it is observed that greater estimation error increases average cycle time and thus damages the throughput. In addition, negative estimation error increases average cycle time more than positive estimation error does. Therefore, if the number of active stations cannot be estimated accu- rately, the best strategy is to deliberately overestimate the number of active stations. Two estimation schemes

Fig. 10 Average cycle time of FRD with estimation errors.

4.2 Maximum Likelihood Scheme

A maximum likelihood estimation scheme is developed herein based on the probability model. According to historical information regarding contention patterns, the most possible number of active stations in the next round is calculated. Let p(i, j, k, d) denote the prob- ability of that j out of i active stations successfully transmit their control messages and k collision clusters are observed, given that the random delay ranges be- tween 0 and d. These p(i, j, k, d)’s are calculated by exhaustive simulation. Because the precise number of active stations i in the network can never be known, the number can be only estimated by observing the num- ber of successfully transmitted active stations j and the number of collision clusters k. The historical informa- tion is the contention pattern of the previous contention round, and the window size is the number of previous contention rounds that need to be observed for estimat- ing i of the next round. This scheme, with a window size of 2, works as follows:

1. In the ﬁrst round of each cycle, if j = j0, k = k0

and d = d0 are observed, take i0,e= argi max{p(i, j0, k0, d0)}

as the estimate of the number of active stations in the round. Then, max{i0,e− j0, 2k0} can be taken as the estimate of active stations in the second round.

2. In the second round, if j = j1, k = k1 and d = d1

are observed, take

i1,e = argi max{p(i + j0, j0, k0, d0)

∗ p(i, j1, k1, d1)}

as the estimate of the number of active stations in the round. Then, max{i1,e− j1, 2k1} can be taken as the estimate of active stations in the third round.

Fig. 11 Average cycle time of ORD with estimation errors.

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3. Follow above procedures round by round until all active stations in the cycle are resolved.

4.3 Average Likelihood Scheme

Another estimation scheme is based on an average likelihood approach to estimate the number of active stations in the next round. Based upon the expectation approach, the expected value of i is used to estimate the number of active stations in the current round. The basic derivation of the average likelihood algorithm is the same as that of the maximum likelihood algorithm.

This scheme is formally stated, with a window size of 2, as follows:

1. In the ﬁrst round of each cycle, if j = j0, k = k0

and d = d0are observed, take i0,e=

i

(i∗ p(i, j0, k0, d0))

as the estimate of the number of active stations in the round. Then, max{i0,e − j0, 2k0} can be taken as the estimate of active stations in the second round.

2. In the second round, if j = j1, k = k1 and d = d1

are observed, take i1,e=

i

(i∗ p(i+j0, j0, k0, d0)∗ p(i, j1, k1, d1))

as the estimate of the number of active stations in the round. Then, max{i1,e − j1, 2k1} can be taken as the estimate of active stations in the third round.

3. Follow above procedures round by round until all active stations in the cycle are resolved.

4.4 Eﬀect Analysis for Estimation Schemes

For analytical and comparative purpose, the performance of the globally optimal random delay algorithm is combined with the maximum likelihood and average likelihood schemes with a window size of 2. Mean- while, “initial(i) = x” denotes that the estimation num- ber of active stations for the ﬁrst round in contention cycle is equal to x given that the real number of ac- tive stations is i. From Fig. 12, the average likelihood scheme produces a slightly shorter cycle time than that of the maximum likelihood scheme. In Fig. 13, when n ≤ 10, overestimation produces the same results as previously. However, underestimation does not signif- icantly increase the average cycle time in either estimation scheme. This phenomenon occurs because the estimation schemes are based on historical information, not mere guesses, and thus they can rapidly accommodate themselves to the real situation. Consequently,

Fig. 12 Average cycle time for ORD with initial(i) = i.

Fig. 13 Average cycle time for ORD with initial(i) = 10.

the novel estimation schemes presented herein can help ranging algorithms to enhance HFC network performance.

5. Conclusion

A ranging algorithm calculates the distance between the headend and the station for medium access control in HFC networks. The ranging processes for the DOCSIS standard is detailed herein. To enhance slot throughput, minimizing the ranging cycle time is essen- tial. Therefore, a practical and eﬃcient mean of resolving collisions is to randomly delay the transmission of ranging messages.

Three algorithms, FRD, VRD, and ORD, are developed to determine the optimal random delay for each contention round so as to minimize the average cycle time. The fundamental idea to the algorithms is to model the contention resolution process in a ﬁnite state machine in which the state transition probabilities are exhaustively calculated by simulation. The ORD is demonstrated via simulation to eﬀectively minimize the contention cycle time and approach theoretically optimal throughput from pure ALOHA. For purposes of practicality in implementation, the sensitivity of the estimation error is assessed. Negative estimation error results in lower throughput than positive estimation

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derestimate them. Additionally, based on the historical information about the pattern of the contention results, the maximum likelihood and average likelihood schemes are developed to estimate the number of active stations. The simulation indicates that the proposed estimation schemes are eﬀective even when the estimate of the number of initially active stations is inaccurate. In summary, the proposed random delay mechanisms assisted with proposed estimation schemes can eﬀectively help ranging algorithms to enhance HFC network performance.

References

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[3] M.Y. Jin, “Optimal range and Doppler centroid estimation for a ScanSAR system,” IEEE Trans. GeoScience & Remote Sensing, vol.34, no.2, pp.479–488, March 1996.

[4] D.T. Batarseh, N. Timothy, and G.M. McFadyen, “An ul- trasonic ranging system for the blind, biomedical engineering conference,” Proc. 1997 Sixteenth Southern, pp.411–413, April 1997.

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Frank Yeong-Sung Lin received his B.S. degree in electrical engineering from the Electrical Engineering Department, National Taiwan University in 1983, and his Ph.D. degree in electrical engineering from the Electrical Engineering Depart- ment, University of Southern California in 1991. After graduating from the USC, he joined Telcordia Technologies (formerly Bell Communications Research, abbrevi- ated as Bellcore) in New Jersey, U.S.A., where he was responsible for developing network planning and ca- pacity management algorithms. In 1994, Prof. Lin joined the faculty of the Electronic Engineering Department, National Taiwan University of Science and Technology. Since 1996, he has been with the faculty of the Information Management Department, National Taiwan University. His research interests include network optimization, network planning, performance evaluation, high-speed networks, wireless communications systems and dis- tributed algorithms.

tively. Currently, he works as a senin- ior engineer for Fastlink Communication Corp. targetting on providing mobile handset design service. His research interests include protocol design and analysis of residential networks, and bandwidth scheduling over QoS guaranteed networks.

Ying-Dar Lin was born in Hsi- Lo, South Taiwan, in 1965. He received the Bachelor’s degree in Computer Sci- ence and Information Engineering from National Taiwan University in 1988, and the M.S. and Ph.D. degrees in Computer Science from the University of California, Los Angeles in 1990 and 1993, respectively. At UCLA Computer Science De- partment, he worked as a Research Assis- tant from 1989 to 1993 and worked as a Teaching Assistant from 1991 to 1992. In the summers of 1987 and 1991, he was a technical staﬀ member in IBM Taiwan and Bell Communications Research, respectively. He joined the faculty of the Department of Computer and Information Science at National Chiao Tung University in August 1993 and is Professor since 1999. His research interests include design, analysis, and implementation of network protocols and algorithms, wire-speed switching and routing, quality of services, and intranet services.

He has authored two books. Dr. Lin is a member of ACM and IEEE.

Chih-Hao Lin is currently a can- didate Ph.D. student in the Informa- tion Management Department of National Taiwan University in Taiwan, R.O.C. He received his B.S. degree in computer science from the Department of Informa- tion Engineering, Tam Kang University in 1994 and the M.S. degree in information engineering from the Institute of Informa- tion Engineering, National Cheng Kung University in 1996. His research interests include network optimization, network planning, performance evaluation, network monitoring, network servicing, network ex- pansion, and wireless communication networks.