MANY multi-media network services have been introduced

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1 IEEE TRANSACTIONS ON BROADCASTING, VOL. 52, NO. 1, MARCH DOCSIS Performance Analysis Under High Traffic Conditions in the HFC Networks Wei-Tsong Lee, Kun-Chen Chung, Kuo-Chih Chu, and Jen-Yi Pan Abstract Hybrid Fiber Coaxial (HFC) networks provide a wide bandwidth and represent the solution of choice for many residential networks nowadays. The Data Over Cable Service Interface Specification (DOCSIS) protocol is an important standard for HFC networks and is supported by the majority of current vendors. This protocol uses the Truncated Binary Exponential Back-off (TBEB) algorithm to resolve collisions within the network by means of a back-off window. However, the performance of this algorithm tends to deteriorate when the network load is high. Consequently, the present study develops a novel mathematical model for the TBEB algorithm and then uses this model to identify the window size which yields the optimal system throughput and minimum delay time under high traffic conditions. The present numerical results confirm that the performance is improved when the window settings identified from the developed model are applied. Index Terms CATV, DOCSIS protocol, Hybrid Fiber Coaxial (HFC), MAC. I. INTRODUCTION MANY multi-media network services have been introduced in recent years. These services, which include Voice over IP, Video Conferencing, Video on Demand, and Online Gaming permit individuals to communicate with one another more easily, but are bandwidth intensive to such an extent that their transmission requirements cannot be satisfied by traditional telephone networks. Consequently, with its high bandwidth characteristics, the Hybrid Fiber Coaxial (HFC) network has emerged as a highly suitable candidate for residential networks. Fig. 1 presents a schematic illustration of the HFC network architecture. In this arrangement, the Cable Modem Termination System (CMTS) manages all of the Cable Modems (CM) in the network, and the CM s provide the means by which the various stations are connected to the network. The HFC network is constructed using both optical fiber and coaxial cable. Specifically, fiber is utilized between the CMTS and the individual Fiber Nodes, while coaxial cable is used to connect each Fiber Manuscript received February 24, 2005; revised August 25, This work was supported by the National Science Council under Contract NSC E W.-T. Lee is with the Department of Electrical Engineering, Tamkang University, Tamsui, Taipei County, Taiwan 251, R.O.C. ( wtlee@ mail.tku.edu.tw). K.-C. Chung is with the Department of Information Engineering and Computer Science, Feng Chia University, Taichung, Taiwan 40724, R.O.C. ( sean_chuang@ms22.url.com.tw). K.-C. Chu is with the Department of Electronic Engineering, Lunghwa University of Science and Technology, Gueishan Shiang, Taoyuan County, Taiwan 333, R.O.C. ( kcchu@mail.lhu.edu.tw). J.-Y. Pan is with the Department of Communications Engineering and the Center for Telecommunication Research, National Chung Cheng University, Chia-Yi, Taiwan 621, R.O.C. ( jypan@ccu.edu.tw). Digital Object Identifier /TBC Node to its particular set of CM s. Since the transmission distance of the HFC network can extend as far as 80 kilometers, Bi-Directional Amplifiers are used to maintain signal quality. Meanwhile, splitter and tap devices are implemented in the network to distribute the signals to the various CM s. Since the HFC network has a shared network architecture, collisions may occur during the transmission of data from the CM s. Hence, some form of network protocol is necessary to ensure an efficient network performance. The Data-Over-Cable Service Interface Specification (DOCSIS) protocol [1] is currently one of the most influential standards for HFC networks. Under this protocol, transmission collisions are resolved by means of the Truncated Binary Exponential Back-off (TBEB) algorithm, which is similar to that used in the IEEE standard. However, previous studies have reported that the delay time of this algorithm increases rapidly, and that the network throughput decreases severely, when the network load is high [2], [3]. It has been shown that this problem can be resolved to a certain extent by adjusting the length of the back-off window [2], [3]. The TBEB algorithm incorporates the Data_Backoff_Start (DBS) and Data_Backoff_End (DBE) parameters to adjust the initial and maximum values of the back-off window, respectively. In this study, the Markov Chain model is employed to establish a model of the TBEB algorithm. The analytical results obtained from this model are then used to identify suitable settings of the DBS and DBE parameters such that the network throughput and delay time are optimized. The remainder of this paper is organized as follows. Section II provides a detailed description of the DOCSIS protocol and the TBEB algorithm. A two-dimensional model of the TBEB algorithm is then presented in Section III. Using the characteristics of the Markov Chain, equations are developed to predict the delay time and throughput of the network under high traffic conditions. Section IV uses the model generated in Section III to explore the relative influences of the DBS and DBE parameters on the network performance under high traffic conditions. Finally, Section V presents some brief conclusions. II. OVERVIEW OF DOCSIS PROTOCOL A. DOCSIS Operation The DOCSIS protocol, developed by Cable Television Laboratories [4], defines the operations of the Physical and Media Access Control (MAC) layers of the HFC network. The HFC network has a bi-directional network architecture, in which the direction from the CM to the CMTS is referred to as the upstream channel, while the direction from the CMTS to the CM is termed the downstream channel. In North America, the spectrum of the upstream channel ranges from 5 MHz to 42 MHz. An /$ IEEE

2 22 IEEE TRANSACTIONS ON BROADCASTING, VOL. 52, NO. 1, MARCH 2006 Fig. 1. Architecture of HFC network. Fig. 2. Data transmission process of DOCSIS protocol. upstream channel with a spectrum size of 6.4 MHz can provide a bandwidth of Mbps using Synchronous Code Division Multiple Access (S-CDMA) modulation. Meanwhile, the spectrum of the downstream channel ranges from 54 to 860 MHz. A downstream channel with a spectrum size of 6 MHz can provide a bandwidth of 42 Mbps using 256 QAM modulation. The upstream channel is divided into many mini-slots of equal size. The CMTS designates these mini-slots as either request mini-slots (RMS) or data mini-slots (DMS). When a CM wishes to transmit data, it first uses a RMS to transmit a request protocol data unit (PDU) to the CMTS. When the CMTS receives this request PDU, it executes the scheduling algorithm and allocates DMS s to the CM. Following completion of the scheduling process, the CMTS notifies all the CM s of the mini-slot allocation results in the upstream channels by means of a Bandwidth Allocation Map (MAP) message. Essentially, this MAP message informs the CM when it will be given the opportunity to transmit its upstream data PDU. The CM then waits to transmit the data PDU to the CMTS until it receives the DMS assigned to it by the CMTS. The data transmission process of the DOCSIS protocol illustrated in Fig. 2 above can be summarized as follows: 1. At time, the CMTS sends out, which records the mini-slot allocation results for the period to. 2. arrives at the CM at time. If the CM wishes to transmit data, it sends a request PDU to the CMTS via a RMS which arrives at the CM at time. 3. The request PDU from the CM arrives at the CMTS at time. Following the resulting scheduling process, the CMTS generates, which stores the mini-slot allocation results for the period between and. The CMTS then sends through the downstream channels to all the CM s. 4. The CM receives from the CMTS at time. This message notifies the CM of the position of the DMS reserved for its data PDU by the CMTS. The CM then transmits the data PDU in the assigned DMS when it arrives at time. 5. The CM data transmission is completed when the data PDU arrives at the CMTS at time. During the process described above, collisions may occur when the CM sends a request PDU to the CMTS at time.in the DOCSIS protocol, the TBEB algorithm is utilized to minimize the occurrence of such collisions. B. TBEB Algorithm Fig. 3 provides an overview of the operational principles of the TBEB algorithm. The CMTS uses MAP messages to assign the DBS and DBE parameters of all the CM s. The DBS determines the back-off window applied for the initial contention request PDU s, while the DBE specifies the maximum back-off window utilized for the contention requests PDU s. The Back-off Exponent (BE) parameter indicates the number of back-off periods for which a station must wait before the CM transmits the request PDU. Meanwhile, NB (Number of Back-off) refers to the number of times for which the TBEB algorithm is required to back off while attempting the current transmission. Max_backoffs represents the maximum number of back-offs the TBEB algorithm attempts before declaring a failed request PDU contention. Whenever a CM wishes to transmit new data, it sets BE equal to DBS. The CM then randomly selects a number within the range of 0 to. This random value indicates the number of RMS s for

3 LEE et al.: DOCSIS PERFORMANCE ANALYSIS UNDER HIGH TRAFFIC CONDITIONS IN THE HFC NETWORKS 23 Fig. 3. Operating principles of TBEB algorithm. which the CM must defer before transmitting. As an example, if the back-off exponent parameter is set to, the CM can select a back-off window between 0 and 15. For the sake of illustration, this study assumes that it selects a value of 11. Accordingly, the CM must defer for a total of 11 contention transmission opportunities. If the transmitted request PDU collides, the CM executes the following steps: the value of NB is increased by 1. If BE is still less than DBE, then BE is also increased by 1. Finally, the previous random selection and deferral steps are repeated. The complete collision resolving algorithm is repeated until either the request PDU is transferred successfully or the retransmission times exceed the Max_backoffs parameter and are therefore discarded. The performance of the TBEB algorithm deteriorates under high traffic conditions [2]. To improve the network performance, the DOCSIS protocol allows the CMTS to adjust the DBS and DBE parameter values of all the CM s in accordance with the current network conditions by means of a MAP message. The following section of this paper presents a method to set the values of the DBS and DBE parameters under high network traffic conditions such that the network throughput and data PDU delay time performance criteria are optimized. A. Model of TBEB Algorithm As stated above, the performance of the TBEB algorithm deteriorates under high network traffic conditions. In analyzing this problem, the current study considers the network to be in a saturation condition. In other words, an assumption is made that there are n stations in the system and that each station has a data PDU immediately available for transmission. In constructing the TBEB algorithm model, the following assumptions are made. First, an integer and discrete time scale is employed. Hence, and are two continuous mini-slot times and the length of each mini-slot is a constant,. Second, a stochastic process,, is specified to represent the size of the back-off counter of a station at mini-slot time. The back-off counter is defined as the number of RMS s for which the station must wait prior to commencing transmission. This stochastic process is non-markovian since possible values of the back-off counter of each station depends on its particular transmission history. The second stochastic process,, is defined using Markov Chain rules to represent the back-off stage of the station at mini-slot time, where R represents the retry back-off stage. In Fig. 3, the back-off stage is equal to BE and is increased by a value of 1 when an unsuccessful transmission occurs. From and, it is possible to establish the two-dimensional Markov Chain model for the TBEB algorithm shown in Fig. 4. Fig. 4 represents a two-dimensional process in the form of a discrete-time Markov Chain. In this figure, the back-off period unit is expressed by and is of the same size as the mini-slot time. The model adopts the notation, where is the back-off stage and. Meanwhile, represents the conditional collision probability, which is an independent event probability with a constant value. Furthermore, expresses the collision probability of a request PDU during transmission in the upstream channel. The non-null single step transition probability of the model is now discussed. The following notations are adopted: When 1. The back-off counter decreases unconditionally at the beginning of each mini-slot time. 2. The station enters the state if it verifies a successful transmission and the back-off counter is randomly chosen from. III. MODEL AND ANALYSIS This section commences by constructing a model of the TBEB algorithm and then discusses the behavior of a single station within the proposed Markov Chain model. Following a series of mathematical derivations, the steady state probability,, of a single station transmitting a request PDU is obtained. Finally, this steady state probability parameter is utilized to predict the data PDU delay time and the throughput of the complete network. 3. The station chooses a back-off counter for the next stage after an unsuccessful transmission at stage and the back-off counter is randomly chosen from.

4 24 IEEE TRANSACTIONS ON BROADCASTING, VOL. 52, NO. 1, MARCH 2006 Fig. 4. Two-dimensional Markov Chain model for TBEB algorithm.. When 1. The back-off counter decreases unconditionally at the initialization of each mini-slot time. 3. The station chooses a back-off counter for the next stage after an unsuccessful transmission at stage. When, the back-off stage is not increased, but remains equal to. Hence, the back-off counter is randomly chosen from. 2. The station enters the state if it verifies a successful transmission and the back-off counter is randomly chosen from. 4. The station reaches the final stage of the back-off procedure. If it fails, the back-off counter is reset.

5 LEE et al.: DOCSIS PERFORMANCE ANALYSIS UNDER HIGH TRAFFIC CONDITIONS IN THE HFC NETWORKS 25 Conversely, if it is successful, the back-off counter is restarted. where: From observation, it is found that the model is irreducible (i.e. each state can be reached from any other state), aperiodic (i.e. a state cannot be reached periodically from itself) and recurrent non-null. Hence, a stationary distribution of the current model exists. From the seven single-step transition probabilities defined above, the stationary distribution,, can be expressed as: where: Owing to chain regularities, a closed-form solution for this Markov chain can be readily obtained using the following procedure. First, can be rewritten as shown in (3) at the bottom of page. Using (2), (3) can be rewritten as: (4) From (2) and (4), it follows that the values of can be expressed as functions of the value of and the conditional collision probability,. Finally, is determined by imposing the probability conservation law, i.e. (1) (2) (5) From the above, is as shown in the second equation at bottom of page. Given the values of,,, and, the steady state probability of the current model can be computed from (1) to (5). Let be the steady state probability of a station sending a transmission during any mini-slot time. In the network, a station only transmits when its back-off counter is equal to zero (i.e. the station transmits at any of ). Therefore, see (6) at bottom of the page. A collision occurs when two or more stations transmit during the same mini-slot time. Hence, the collision probability,, of a transmitted request PDU is given by: Eqs. (6) and (7) represent a nonlinear system with two unknown parameters, i.e. and. This system can be solved using the Newton-Raphson method [5]. Substituting (7) into (6) yields one equation with a single unknown parameter,. Solving this equation for, yields the probability and enables the subsequent derivation of the stationary distribution by substituting and into (4). (See (8) at bottom of the next page.) B. Throughput Analysis In this paper, it is assumed that each transmission, successful or otherwise, is a renewal process [6]. Consequently, it is sufficient to calculate the throughput of the DOCSIS protocol during a single renewal interval between two consecutive transmissions. However, before doing so, it is necessary to define the complete set of parameters used in the throughput analysis. Let be the probability of system transmission during any minislot time. In other words, denotes the probability of at least one station transmitting during a randomly chosen mini-slot time. It is noted that this value differs from, which indicates the probability of any particular station transmitting during a (7) (3) (6)

6 26 IEEE TRANSACTIONS ON BROADCASTING, VOL. 52, NO. 1, MARCH 2006 Fig. 5. Upstream transmission model of DOCSIS protocol. randomly chosen mini-slot time. Since stations contend on the channel and each transmits with probability, it can be shown that: Moreover, the probability is defined as successful transmission probability, which represents the case exactly one station transmits on the channel, conditioned by the probability, i.e. The system throughput can be expressed as: (9) (10) (11) In (11), represents the average data PDU size and denotes the duration of a mini-slot. The probability of a successful transmission occurring during a mini-slot time is given by. The mean value of the data PDU s successfully transmitted during a mini-slot time is given by, where denotes the time expended in conducting a successful transmission. If no transmissions are attempted on the channel (i.e. no request PDU s exist to be transmitted), the probability of a successful transmission becomes.itis noted that the system requires one mini-slot time,, to verify that this situation exists. Finally, if a collision occurs, the probability of a successful transmission is given by.as before, the system requires one mini-slot time to verify this condition. The various situations discussed above are illustrated in Fig. 5. In the DOCSIS protocol, the data transmissions from the various CM s of the network all take place in the upstream channels and hence the present analysis focuses exclusively on these particular channels. Fig. 5 presents the upstream transmission model of the DOCSIS protocol. This model can be discussed from two different perspectives. From the CM perspective, when a request PDU is transmitted successfully, the CM sends a data PDU when the assigned data mini-slot arrives. Moreover, in addition if the transmission fails, the CM simply attempts to retransmit the request PDU. Considering the model from the upstream channel perspective, observing Fig. 5 we find if (1) a collision is encountered, (2) the upstream channel is empty, or (3) the upstream channel simply receives a request PDU, the upstream channel occupies just one mini-slot time. However, if the upstream channel receives a payload, the length of the channel occupancy clearly depends on the length of the data PDU. Since the DOCSIS protocol defines that the maximum retry count of the back-off stage is less than or equal to 16, then R is defined as. The stationary distribution of the current model can then be computed from (6). From (9) and (10) with known parameters, the throughput of the upstream transmission of the DOCSIS protocol can then be obtained from (11). (8)

7 LEE et al.: DOCSIS PERFORMANCE ANALYSIS UNDER HIGH TRAFFIC CONDITIONS IN THE HFC NETWORKS 27 C. Data PDU Delay Time Analysis In general, the delay time of a data PDU is defined as the elapsed time between its generation and its successful reception. Since collisions may occur during the transmission process, the data PDU delay time can be considered to comprise two components, namely a failure component and a successful transmission component. The mean data PDU delay time is given by: times Failure one time Success (12) where is the random variable representing the data PDU delay time and is the mean value of. Eq. (12) is discussed below in terms of its two basic components. Failure Component:: The total delay time of a data PDU may reflect the result of several collisions prior to its successful reception by the CMTS. The delay time is given by the term in (12), where indicates the expected value of number of collisions experienced by a request PDU before successful reception by CMTS. Meanwhile, is the average delay time of the back-off counter specified by a station before accessing the channel under busy conditions. Finally, is the time duration of the collision and is expressed as round trip delay time CMTS schedule time. From the behavior of a transmission (i.e. it collides continually before successful receipt) and the definition of mean value, it is known that the random variable conforms to a geometric distribution with a parameter. The mean value of is given by: Hence (13) The average data PDU delay time attributable to the back-off counter depends on the value of the counter and the duration for which it is frozen when the DMS period begins. When the counter of a station is at state, a time interval of mini-slots is required for the counter to reach state. This interval is denoted by the random variable, whose mean value is given by equation (14) at bottom of page. The time for which the counter of a station remains frozen is denoted by. When the counter freezes, it remains inactive for the duration of one DMS period. The value of this duration depends on the length of the data PDU. In order to calculate the time for which the counter remains inactive, it is necessary to establish, i.e. the average number of times a station must wait for other stations transmission opportunity (i.e. the DMS period) before its counter reaches 0. is based on, i.e. the average back-off delay time of each station, and on, i.e. the length of the RMS period. Therefore, it can be shown that. Further: (15) According to the DOCSIS protocol, is a constant parameter. Additionally,, i.e. the time of a DMS period, is defined as: mean data PDU size mini-slot size mini-slot time From (14) and (15), it can be shown that: (16) Successful Component:: This section considers the second component of the data PDU delay time, i.e. the successful transmission component. This component is given by the second part of (13) (i.e. ), where is the elapsed time for a successful transmission and is expressed by: round trip delay time CMTS schedule time mean data PDU size mini-slot size mini-slot time Substituting (13) and (16) into (12) permits the mean data PDU delay time to be calculated. Note that all the time intervals in the relations above are measured in the same units. IV. NUMERICAL RESULTS AND DISCUSSIONS A. Parameter Definitions This section uses the equations developed above to predict the network throughput and the data PDU delay time under saturation load conditions. For convenience, the discussions which follow adopt the expression (DBS, DBE) to represent the values of the DBS and DBE parameters in the TBEB algorithm. For example, (0, 10) indicates that and. A total of 55 different combinations are possible in the TBEB algorithm. In the current analysis, the throughputs generated from different combinations of (DBS, DBE) under saturation load (14)

8 28 IEEE TRANSACTIONS ON BROADCASTING, VOL. 52, NO. 1, MARCH 2006 Fig. 6. Network throughputs for different combinations of (DBS, DBE). TABLE I PARAMETERS USED TO CALCULATE NUMERICAL RESULTS conditions are obtained from (7)to (12). The system data PDU delay time under saturation load conditions for these combinations of (DBE, DEB) are then calculated from (6) and (13)to (16). The network parameters used to calculate the current numerical results are listed in Table I. B. Numerical Results and Discussions 1) Throughput: Fig. 6 indicates the network throughputs obtained from different combinations of (DBS, DBE) under saturation load conditions. It is noted that since some of the throughputs are relatively small, e.g. those obtained for (0, 1) and (0, 2), they are not visible in this figure. From Fig. 6, it is clear that the optimal throughput is generated by (7, 9) while the worst throughput is obtained from (0, 1). Additionally, it is noted that when (DBS, DBE) is small, the resulting network throughput is also small. The reason for this is that when the back-off counters are very small, a large number of collisions will result and hence the number of successful data PDU transmissions will be limited. Finally, Fig. 6 reveals that when the value of (DBS, DBE) is increased, the throughput increases accordingly until a certain value is reached, at which point the throughput decreases as (DBS, DBE) is increased further. For example, it can be seen that the throughput obtained from (8, 10) is worse than that generated from (7, 9). The principal reason for this is that when (DBS, DBE) is too high, the station spends an excessive time in backing off rather than in attempting data PDU transmissions and hence the actual network throughput is reduced. 2) Data PDU Delay Time: Fig. 7 shows the data PDU delay times under saturation load conditions for the 55 different combinations of (DBS, DBE). It is clear that the optimal data PDU delay time is generated when (DBS, DBE) is set to (2, 3), whereas the worst case is obtained with (0, 1). Additionally, the data PDU delay times associated with (0, 1), (0, 2), and (1, 2) all exceed mini-slots. The principal reason for this is that the maximum back-of counters of these sets are very small and therefore the number of collisions and subsequent retransmissions increases. These additional retransmissions obviously prolong the overall delay time of the data PDU. Furthermore, those sets with larger DBE values collide with less frequency since they have larger back-off counters. Meanwhile, these stations will wait for more RMS s as well as more DMS s due to more successful transmission requests, and causes the data PDU delay time is increased. According to Fig. 7, the value of the DBE parameter should be set between 2 and 5 if the data PDU delay time bounds to mini slots. 3) Rank of Throughput and Data PDU Delay Time: The previous results have indicated that most (DBS, DBE) combination provide either an acceptable throughput or an acceptable data PDU delay time, but do not generally provide satisfactory results for both performance criteria. To compare the performances of the complete set of (DBS, DBE) parameter combinations, this study ranked the 55 (DBS, DBE) sets according to their relative performances in terms of throughput and data PDU delay time. In the ranking process, it was assumed that the lower the ranking number, the better the performance. The ranking numbers of the throughput and data PDU delay time were then summed for each particular (DBS, DBE) combination. The corresponding results are presented in Fig. 8, in which the x-axis denotes the various (DBS, DBE) combinations, while the y-axis presents the combined ranking of the throughput and data PDU delay time. The results reveal that the (7, 8) combina-

9 LEE et al.: DOCSIS PERFORMANCE ANALYSIS UNDER HIGH TRAFFIC CONDITIONS IN THE HFC NETWORKS 29 Fig. 7. Data PDU delay time. Fig. 8. Ranking the throughput and data PDU delay time. tion generates the lowest combined ranking for the throughput and data PDU delay time. The figure provides an understanding of the throughput and data PDU delay time trends for all the (DBS, DBE) combinations. Using this information, a network administrator has the ability to specify appropriate (DBS, DBE) values according to the users requirements. For example, users tend to demand shorter data PDU delay times and generally ignore throughput under saturation load conditions. V. CONCLUSIONS AND FUTURE WORK Due to its high bandwidth capacity, the HFC network has evolved into the preferred solution for residential networks nowadays. The DOCSIS protocol is the most commonly adopted protocol for HFC networks. Since this network has a tree-like architecture, when more than one CM transfers data to the CMTS simultaneously, collisions inevitably take place. Although the DOCSIS protocol utilizes the TBEB algorithm to resolve these collisions, previous research has indicated that the performance of this algorithm deteriorates significantly under high traffic conditions. To improve the performance of the algorithm, the DOCSIS protocol provides a control message, which permits the DBE and DBS parameters in the TBEB algorithm to be changed at will at the CM side. This study has discussed the settings of the (DBS, DBE) parameters which generate the optimal system throughput and data PDU delay time. This paper has constructed a two-dimensional Markov model of the TBEB algorithm. A number of equations have been derived to predict the throughput and data PDU delay time generated by different (DBS, DBE) settings. It has been shown that (7, 9) generates the optimal throughput, while (2, 3) yields the optimal data PDU delay time. The current results have demonstrated that the performance of the network is greatly influenced by the specification of the (DBS, DBE) parameters. Finally, this study has ranked the various combinations of (DBS, DBE) according to their relative throughputs and data PDU delay times. The ranking numbers ofthe throughputanddata PDU delaytime have beensummedfor each (DBS, DBE) combination. The resulting combined ranking values provide network administrators with the ability to choose the most appropriate (DBS, DBE) parameter values for different network requirements. In the future, we will measure the actual performance of equipment and compare to these theoretical results. It would be interesting and still an open issue.

10 30 IEEE TRANSACTIONS ON BROADCASTING, VOL. 52, NO. 1, MARCH 2006 ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments. REFERENCES [1] Cable Television Laboratories, Inc., Data-over-cable service interface specifications, in Radio Frequency Interface Specification, Apr [2] Y. D. Lin, C. Y. Huang, and W. M. Yin, Allocation and scheduling algorithm for IEEE and MCNS in hybrid fiber coaxial networks., IEEE Trans. on Broadcasting, vol. 44, no. 4, pp , Dec [3] Y. D. Lin, W. M. Yin, and C. Y. Huang. (2000, Third Quarter) An investigation into HFC MAC protocols: mechanisms, implementation, and research issues. IEEE Communication Surveys [Online] Available: [4] Cable Television Laboratories, Inc.. [Online] Available: [5] H. Khalil, Nonlinear Systems, 3rd ed: Prentice Hall, [6] L. Kleinrock, Queuing System Vol I: Theory. New York: John Wiley & Sons, Kun-Chen Chung received the B.S.and M.S. degrees in Information Engineering and Computer Science from Feng-Chia University, Taichung, Taiwan, R.O.C., in 2002 and 2004, respectively. His research interests include performance evaluation of broadband and wireless networks. Kuo-Chih Chu received the B.S. degree in Information Engineering and Computer Science from Feng-Chia University, Taichung, Taiwan, R.O.C., in 1996 and received M.S. and Ph.D. degrees in Electrical Engineering from National Cheng Kung University, Taiwan, R.O.C., in 1998 and 2005, respectively. He is currently an assistant professor with the Department of Electronic Engineering, Lunghwa University of Science and Technology, Taoyuan, Taiwan R.O.C.. His research interests include protocol design and analysis of HFC networks. Wei-Tsong Lee received the B.E.E.E., the M.S. and the Ph.D. degrees in Electrical Engineering all from National Cheng Kung University, Taiwan, R.O.C., in 1984, 1986, and 1995, respectively. He is currently an Associate Professor in the Department of Electrical Engineering of Tamkang University, Taiwan, R.O.C.. His current interests include high speed network, cable modems, embedded system and stochastic ordering. Dr. Lee is a member of IEEE and IEICE. Jen-Yi Pan received the B.S. and Ph.D. degrees in computer science from National Tsing-Hua University, Hsinchu, Taiwan, R.O.C., in 1995 and 2002, respectively. He is currently an assistant professor with the Department of Communications Engineering, National Chung Cheng University, Chaiyi, Taiwan, R.O.C. His research interests include performance evaluation of medium access control, voice over WiFi, and Internet telephony. Dr. Pan is a member of ACM, IEEE, and IEICE.