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On Bandwidth Request Mechanism with Piggyback in Fixed IEEE 802.16 Networks Jianhua He, Kun Yang, Ken Guild, and Hsiao-Hwa Chen

Abstract—This paper investigates the random channel access mechanism specified in the IEEE 802.16 standard for the uplink traffic in a Point-to-MultiPoint (PMP) network architecture. An analytical model is proposed to study the impacts of the channel access parameters, bandwidth configuration and piggyback policy on the performance. The impacts of physical burst profile and non-saturated network traffic are also taken into account in the model. Simulations validate the proposed analytical model. It is observed that the bandwidth utilization can be improved if the bandwidth for random channel access can be properly configured according to the channel access parameters, piggyback policy and network traffic. Index Terms—IEEE 802.16, media access control, wireless network, performance evaluation.

I

I. I NTRODUCTION

EEE 802.16 is a standard for broadband wireless access (BWA) in metropolitan area networks [1]. The physical layer and media access control (MAC) layer specifications are defined in 802.16d for fixed BWA, and enhanced with low to moderate mobility support in 802.16e [1]. Flexible bandwidth request and allocation mechanisms have been specified in the standard to support different quality-of-services (QoS) requirements, namely unsolicited grant services (UGS), realtime polling services (rtPS), non-real-time polling services (nrtPS) and best effort services (BE) [1] [2]. A subscription station (SS) can get bandwidth request message for services other than UGS from the base station (BS) by the means of requesting (initiated by SS) or polling (initiated by BS). Requesting can be made by so-called general mechanism that an SS sends bandwidth request header to the BS in collisionfree bandwidth grant or in contention based transmission opportunities (TXOPs) (for nrtPS and BE services only). Alternative requesting mechanisms include contention-based focused Bandwidth Requests (for WirelessMAN-OFDM physical specification only) and contention-based CDMA Bandwidth Requests (for WirelessMAN-OFDMA only) [1]. In this paper we investigate the general requesting mechanism specified in 802.16d for nrtPS and BE services [1]. In the physical layer, WirelessMAN-OFDM specification is considered. The system uses a fixed frame for data transmission. The frame is divided into subframes for downlink Manuscript received November 3, 2007; revised March 9, 2008 and May 12, 2008; accepted May 29, 2008. The associate editor coordinating the review of this letter and approving it for publication was P. Fan. J. He is with the Institute of Advanced Telecommunications, Swansea University, UK (e-mail: [email protected]). K. Yang and K. Guild are with the Department of Computing and Electronic Systems, University of Essex, UK (e-mail: {kunyang, kguild}@essex.ac.uk). H.-H. Chen (corresponding author) is with the Department of Engineering Science, National Cheng Kung University, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/T-WC.2008.071226

and uplink transmissions. Initially an SS sends stand-alone bandwidth request message (abbreviated as SREQ) to indicate required bandwidth (in bytes) in TXOPs, which are allocated by the BS in the uplink subframes. If bandwidth is granted, the SS can send collision-free transmission bursts in the allocated bandwidth. Each burst is associated with a physical burst profile (i.e., modulation and error control coding schemes). A collision-free bandwidth request header (abbreviated as PREQ) can be sent together with data in the allocated bandwidth to request further bandwidth if the SS has more traffic to send. In the 802.16 standard, a contention resolution scheme, called truncated binary exponential backoff (TBEB), has been specified to transmit SREQs to the BS. However, further research is still needed for the requesting mechanism management, such as bandwidth configuration and piggyback policy, to efficiently utilize limited bandwidth. In the literature, Vinel et al. accurately analyzed the TBEB algorithm of the 802.16 general requesting mechanism by the means of Markov chain [3]. Seo et al. studied the queueing performance of the 802.16 contention-based CDMA requesting mechanism with and without piggyback (refer to [4] [5] and the references therein). The attempts of analyzing the random access protocol and finding the optimal access parameters were made in [6] [7], in which however the TBEB scheme was not correctly modeled. In the above works [3] [5] [6] [7], sufficient bandwidth is assumed, meaning that the bandwidth requested by every received SREQ will always be granted by the BS. However, the bandwidth that can be allocated to the TXOPs and bursts is not unlimited in reality. Indeed, the bandwidth should be properly configured jointly with the requesting mechanism to improve bandwidth utilization [8]. It is also observed that the analytical models suggested in [3] [5] can not be directly extended to take into account limited bandwidth. The media access control in DOCSIS of cable TV networks is similar to the 802.16 general requesting mechanism. The contention behavior of DOCSIS was analyzed with TCP traffic in [9]. Unfortunately, the signalling procedure was ignored in the analytical model proposed in [9], e.g., each cable modem (CM) can transmit multiple bandwidth request headers in one frame. The 802.16 random access scheme was analyzed with saturated traffic and limited bandwidth in [8]. In this paper, we analytically model the 802.16 random access scheme with limited bandwidth by taking into account the piggyback policy and non-saturated traffic, for the purpose of improving network performances by properly configuring parameters and bandwidth for the random access mechanism. As to be seen later, the piggyback policy studied in this paper is more general than that used in [5].

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II. A SSUMPTIONS We consider a fixed PMP network architecture in which a BS serves Nu independent and identical SSs. The duration of an OFDM symbol is tsym in seconds and the frame duration is tf rm in seconds. Each TXOP consists of treq OFDM symbols. Assume that Nr TXOPs are assigned in each uplink subframe, and the bandwidth (in bytes) requested in SREQ and PREQ can be accommodated by one transmission burst. Each burst consists of tdt OFDM symbols and the bandwidth for Nd bursts is allocated for the requesting mechanism in each uplink subframe. The SSs use TBEB algorithm to determine which frame and which TXOP to transmit their SREQs [1]. W0 and Wm denote the parameters of initial and maximal backoff windows associated with the backoff process of the TBEB algorithm, respectively. If a transmitted SREQ is successfully received by the BS and bandwidth is available, the SS sending the SREQ will be granted bandwidth for burst transmission in the subsequent frames [1]. Otherwise, the SS failed to receive bandwidth grant can retransmit SREQs up to a maximal number of m retries. We assume that the SSs know the outcome of SREQ transmissions at the beginning of the subsequent frame. If the available bandwidth in the BS is less than the sum of that requested in the successfully received SREQs, the available bandwidth will be granted to randomly chosen SSs with SREQs successfully received. After an SREQ results in bandwidth successfully granted to an SS, which has more data than that can be sent in one burst, the SS can piggyback a request (PREQ) with the data in a burst to request more bandwidth. If the SS receive bandwidth grant due to the PREQ, another burst can be transmitted without sending SREQ by the backoff process. It has been shown in [5] that piggyback can increase throughput efficiency. Different piggyback policies can be used and will have considerable impacts on network performance. In [5] it is assumed that the BS will grant bandwidth to all received PREQs. One potential problem is that all bandwidth may be occupied by the SSs sending PREQs and unreleased until some SSs have no traffic to send to the BS. In such case, it is unfair for the SSs requesting bandwidth by SREQs because their channel access delay can be unacceptably large. To strike a good balance between bandwidth efficiency, channel access delay and fairness, we can consider a general piggyback policy. Under this policy, if an SS sending an SREQ in the current frame is granted bandwidth for one burst in the next frame, the SS is guaranteed to receive bandwidth for at most L bursts in the consecutive L frames starting from the next frame, provided that the SS sends PREQ to request bandwidth in each of the next L − 1 frames. After bandwidth for L bursts has been granted to the SS with one SREQ and L−1 consecutive PREQs, or if a PREQ is not sent following an SREQ or a PREQ in the previous frame, the SS needs to send SREQs again to request bandwidth. The parameter L can be used to flexibly control the bandwidth efficiency, access delay and fairness. L=1 corresponds to piggyback being disabled, and L approaching ∞ corresponds to the piggyback policy used in [5]. For simplicity, we assume a traffic pattern, which is similar

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to that used in [10]. We further assume that packets arrive at the MAC layer with probability of ρ per frame. Each packet needs to be transmitted by L bursts. A packet will be dropped, if it arrived at the MAC layer when the transmission of previous packets is not finished. If the bandwidth request attempt for the first burst of a packet fails, the packet will be discarded. The traffic pattern is relatively simple, with ρ=1 representing saturated traffic and ρ < 1 representing nonsaturated traffic, respectively. Extending the analytical model with more complex non-saturated traffic pattern will be our future work. III. A NALYTICAL M ODEL In this section, the steady state performance of the general requesting mechanism will be analyzed. We assume that the probability that an SREQ is transmitted from a tagged SS in a general frame is independent of the outcomes of the previous SREQ transmissions from this SS and the other SSs in the steady state. Consequently, the probability that an SREQ from the tagged SS collides with other SSs’ SREQs is also independent of the outcomes of the SS’s previous SREQ transmissions. Let τ denote the probability that the tagged SS transmits an SREQ in a general frame, and p denote the probability that a transmitted SREQ does not result in bandwidth grant in the steady state. τ and p are assumed to be constants. A. Transmission probability To simplify the calculation of transmission probability τ , an embedded bandwidth request procedure (EBRP) is introduced. An EBRP starts from the tagged SS waiting for a packet and ends at the instance of the whole packet being received by the BS or discarded. According to the previously introduced requesting mechanism, the tagged SS will be in one of the following non-overlapped states at any frame of an EBRP: 1) Wait state: From the beginning of a new EBRP, the SS will wait for packets. Nw denotes the average number of frames in which the SS is in wait state in an EBRP, which is computed by Nw = 1−ρ ρ . 2) Backoff state: After the SS receives a packet, or after an SREQ does not result in bandwidth grant, the SS will start a backoff process, provided that the number of SREQ retries does not exceed m. The SS is said to be in backoff state when the backoff process is active and in the frames it sends SREQs. Let Nb denote the average number of frames in which the SS is in backoff state. 3) Transmit state: After an SREQ from the SS is successfully received in a frame and results in bandwidth grant, the SS will be in the state of transmitting bursts in the next L consecutive frames. Let Ntd denote the average number of frames in which  the SS is in transmit state m in an EBRP. We have Ntd = i=0 (1 − p)pi L. Let preq (i) denote the probability that the number of SREQ transmissions is exactly i in an EBRP, for i ∈ [1, m + 1]. We have preq (i) = (1 − p)pi−1 , for i ∈ [1, m], and preq (m + 1) =

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pm . The average number of SREQ transmissions in an EBRP (denoted by Ntx ) can be computed by

average bandwidth (in unit of data bursts) reserved for PREQs in a frame. In average, Nf rv can be calculated as Nu prv . However, as the average bandwidth reserved for PREQs is m+1  also upper bounded by Nd , we get Nf rv = min(Nu prv , Nd ). Ntx = ipreq (i) Let pr (q) denote the probability that bandwidth for q of the i=1 (1) m    i−1   m 1 − pm+1 Nd bursts in the one frame will be granted to the q SSs which = i 1−p p + m+1 p = send PREQs in the previous frame, with q ∈ [0, min(Nu , Nd )]. 1−p i=1 Define an intermediate variable Nud as min(Nu , Nd ). Without In an EBRP, the average number of frames in which the loss of generality, we can consider the bandwidth of Nud tagged SS is in backoff state depends on the number of tagged bursts, which is the maximum bandwidth that can be b (i) denote the average number reserved for PREQs in one frame. Let φ denote the probability SREQ transmissions. Let N of frames in which the SS is in backoff state in an EBRP, that any one of the Nud tagged bursts is reserved for a PREQ under the condition that the number of SREQ transmissions in a general frame. As the bandwidth reserved for PREQs in a in the EBRP is exactly i, where i ∈ [1, m + 1]. Assume W0 is frame is Nf rv bursts in average, we can calculate φ as Nf rv . Nud dividable by Nr . For the ith SREQ transmission in an EBRP, The probability that q of the Nud tagged bursts are reserved the backoff window size is Wi =min(2i−1 W0 , Wm ), and the for PREQs in the frame is equivalent to pr (q). Then we can   backoff counter Ki will be randomly chosen from [0, Wi -1] get pr (q) = Nud φq (1 − φ)Nud −q , where q ≤ Nud . q with equal probability W1i . The value of Ki means the number Suppose that q SSs send PREQs in one frame. The rest Nu of TXOPs to defer before the SREQ can be transmitted. q SSs can send SREQs to contend only for the left bandwidth b (i) can be computed by Therefore, N of Nd -q bursts in the next frame. The probability that an SS Wj transmits an SREQ in a general frame in which the SS will not Wj −1 Nr i i τ . Let p (l|q) denote the probability  transmit a PREQ is pnrv 1   k + 1   Nr  req b (i) = N = k that l of the Nu -q SSs transmit SREQs in the current frame, W N W j r j j=1 j=1 k=0 k=1 under the condition that q SSs transmit PREQs in the current i  Nr + Wj frame. We have (2) =

2N Nu − q  τ l  τ (Nu −q−l) r j=1 (4) preq (l|q) = 1− pnrv pnrv l Counting all the possibilities on the number of SREQ transFor an SREQ transmitted in a general TXOP of a frame, missions in an EBRP, we have a collision will happen if and only if there is at least one m+1  sr (l) SREQ transmitted in the same TXOP from other SSs. N b (i)preq (i) Nb = N denotes the average number of SREQs successfully received i=1 (3) in a frame, under the condition of l SREQs transmitted in the m    m   i−1 b (i) 1 − p p b m + 1 p = N +N frame. pcop (s|l, Nr ) is the probability that exactly s of the i=1 given l SREQs transmitted over Nr TXOPs are successfully Because the three states of the tagged SS in an EBRP do received. pcop (s|l, Nr ) can be computed by the method for a not overlap, the average number of frames Nf rm in an EBRP classic occupancy problem suggested in page 91 of [11] as

can be obtained by Nf rm = Nw +Nb +Ntd . The transmission Nr ! 1 l pcop (s|l, Nr ) = l probability τ can be computed by the average number of Nr s (Nr − s)! SREQs transmitted in an EBRP and the average frames in

min(Nr −s,l−s) (5)   l−s−v v Nr − s . an EBRP, which is give by τ = NNftx × N (−1) − s − v r rm v v=0 B. SREQ collision and dropping probabilities Let pc denote the probability that an SREQ transmission from the tagged SS fails due to collision with other SREQs. pd denotes the probability that a successfully received REQ is dropped by the BS due to insufficient bandwidth. Under the assumption about a clean channel, we have p = 1 − (1 − pc )(1 − pd ). Let Nrv denote the average number of frames in which the tagged SS uses PREQ to request bandwidth in an EBRP, and prv denote the probability that the SS requests bandwidth by PREQ ina general frame of an EBRP, respectively. We m+1 . The have Nrv = i=1 (1 − p)pi−1 (L − 1), and prv = NNfrv rm probability pnrv that the SS does not request bandwidth by PREQ in a general frame can be simply computed by 1 − prv . As SREQs will only contend for the bandwidth not reserved by PREQs, we need compute how much bandwidth will be reserved by PREQs in a general frame. Let Nf rv denote the

In (5), the first factor N1l is the probability of each arrangement r of random transmission of l SREQs over Nr TXOPs; the second and the third factors calculate the number of ways that s of l SREQs are chosen and each of the chosen s SREQs is transmitted over one of the Nr TXOPs without collision; the summation calculates the number of ways that the left l − s SREQs are transmitted over the left Nr − s TXOPs and none of the l − s SREQ  transmissions is successful. From (5), we sr (l) = l spcop (s|l, Nr ). get N s=1 let N tr (q) denote the average number of SREQs transmitted in a frame, and N sr (q) is the average number of SREQs successfully received in a frame, both under the condition that q SSs transmit in the same frame. NPREQs u −q lp (l|q) and compute We can compute N tr (q) as req l=1 Nu −q sr (l)preq (l|q). Ntr and Nsr denote the N N sr (q) as l=1 average number of SREQs transmitted and successfully received in a frame, respectively. We can compute Nsr as

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 ud N sr (q)pr (q), and compute Ntr as N q=0 N tr (q)pr (q). sr The collision probability pc is obtained as N Ntr . Let Nur =min(Nu , Nr ). psr (s|q) stands for the probability that s SREQs are successfully received in a frame under the condition of q PREQs sent in the frame, where Nus−q ∈ [0, min(Nu − q, Nr )]. We have psr (s|q) = l=s preq (l|q)pcop (s|l, Nr ), with preq (l|q) and pcop (s|l, Nr ) obtained by (4) and (5), respectively. In the case of q PREQs sent in a frame, the BS can grant bandwidth in the next frame to at most Nd -q SSs whose SREQs are successfully received in the frame. A received SREQ may be dropped due to insufficient bandwidth, if the sr (q) is larger than number of successfully received SREQs N dr (s|q) denote the number of dropped SREQs Nu -q. Let N due to insufficient bandwidth in the next frame, under the conditions that s SREQs are successfully received and q dr (s|q) = PREQs are sent in the current frame. We have N max(s − (Nu − q), 0). Let Ndr denote the average number of dropped SREQs over all cases of q. Ndr is given by Nud q=0

Ndr =

N u −q,N r) ud min(N   q=1

      dr s|q psr s|q pr q N

(6)

s=Nd +1−q

dr The SREQ dropping probability pd is then computed as N Nsr . With the expressions developed for pc and pd , the values of τ and p are ready to be obtained by solving nonlinear equations with numeric techniques.

C. Throughput and channel access delay Let Nsuc denote the average number of SREQs that are successfully received and result in bandwidth grant  in a general frame. Nsuc can be computed as  Nud min(Nu −q,Nr ) min(s, Nd − q)pr (q)psr (s|q) q=1 s=1 Define channel bandwidth efficiency η as the ratio of the average number of OFDM symbols in the bursts used to transmit data to the total number of OFDM symbols allocated to TXOP and transmission bursts for the requesting mechanism in a frame. η can be computed by η=

(Nf rv + Nsuc )tdt Nr treq + Nd tdt

(7)

Similarly, define throughput of a single SS (denoted by θ) as the average number of bits transmitted from an SS to the BS in one second. The throughput of a single SS depends largely on the physical burst profile. Let Bsym denote the number of uncoded bits that can be transmitted with an OFDM symbol for an SS. Then the throughput θ of an SS can be computed by ηBsym (Nr treq + Nd tdt ) Bsym (Nf rv + Nsuc )tdt = Nu tf rm Nu tf rm (8) Define channel access delay D as the average time between the beginning of a frame in which the first backoff process is initiated to the end of the frame in which the last backoff process ends in an EBRP. We have D = Nb tf rm . θ=

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IV. N UMERICAL R ESULTS A discrete event driven simulator has been implemented in OPNET for the 802.16 bandwidth request mechanism to verify the proposed analytical model. The analytical model is general. But only presented are the network performances with some selected configurations of channel access parameters, piggyback parameter and traffic arrival rate. In the physical layer, we use configurations similar to those presented in [12]. Channel bandwidth is set to 20 MHz. The frame duration is set to 10 ms. Number of subcarriers in an OFDM symbol is 256. The total number of OFDM symbols in a frame is 844 [12]. For the mandatory OFDM based random channel access mechanism without subchannelization, each TXOP takes two OFDM symbols (treq =2). We assume that each transmission burst takes 16 OFDM symbols (tdt =16). The burst profile of 16QAM modulation and 1/2 coding rate is used. Therefore the number of uncoded bits Bsym that can be transmitted in an OFDM symbol is 384 [12] [1]. The backoff parameters are configured with W0 =Nr , Wmax =28 W0 . Setting W0 to Nr is observed to be a good choice. The minimal value of m is 16 as specified in the 802.16 standard [1]. The number of bursts that can be used for random access traffic in a frame is configurable and can be set according to the real-time and best-effort traffic in the network. In this paper, the number Nd is set to 24. The number of TXOPs Nr in a frame has two configurations, or Nr =1.5Nd and Nr =3Nd . We have observed that the network performances with Nr > 3Nd will generally be worse than those with Nr =3Nd . There are also two configurations of L, L=1 and L=3. L=1 means bandwidth request will not be piggybacked. And L=3 represents a reasonable piggybacked policy by considering both bandwidth efficiency and channel access delay. With the above configurations, we investigate how the network performances change with the number of SSs and traffic arrival rate. Bandwidth efficiency, channel access delay and throughput of single SS versus the number of SSs are presented in Figs. 1, 2, and 3, respectively. The packet arrival rate ρ is set to one. The results with the configurations on Nr and L are presented. Throughput of a single SS versus packet arrival rate ρ with Nu =40 is presented in Fig. 4. Several interesting points can be observed from these analytical and simulation results. 1) For all investigated configurations, the proposed analytical model shows high accuracy. The analytical results (solid lines) match closely to the simulation results (dashed lines), with less than 1.5% difference. 2) Piggybacked request can be used to effectively improve the bandwidth efficiency, but at the cost of possible increased channel access delay. For example, bandwidth efficiency with Nr =3Nd and Nu =80 is 0.65 for L=1 and 0.72 for L=3. But channel access delay increases from 23 to 63 ms accordingly. 3) Channel access delay and throughput are better with Nr =3Nd than those with Nr =1.5Nd in most of the investigated scenarios. However, in the case of L=3, bandwidth efficiency obtained with Nr =1.5Nd is higher by up to 15% than that obtained with Nr =3Nd . The above results demonstrate that the 802.16 random channel

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Analysis: L=1; Nr/Nd=1.5 Simulation: L=1; Nr/Nd=1.5 Analysis: L=3; Nr/Nd=1.5 Simulation: L=3; Nr/Nd=1.5 Analysis: L=1; Nr/Nd=3 Simulation: L=1; Nr/Nd=3 Analysis: L=3; Nr/Nd=3 Simulation: L=3; Nr/Nd=3

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access mechanism is robust for a wide range of system configurations. For example, even when there are a large number of competing SSs, the system can maintain a stable bandwidth efficiency and the total network throughput. But the channel access delay will increase significantly with the increased number of SSs. The bandwidth efficiency with piggybacked request can be saturated more quickly with the increased number of SSs than that without piggybacked request. V. C ONCLUSION In this paper an accurate analytical model was proposed for the random channel access mechanism specified in the IEEE 802.16 standard. We investigated the impacts of bandwidth configuration, channel access parameters and piggyback policy on the network performances. The impacts of physical burst profile and non-saturated traffic have also been taken into account. It is observed that the random channel access mechanism is robust with regard to the number of contending stations. And there is no single set of configurations that is always the best for all the network scenarios. The bandwidth

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for the random access traffic should be properly configured with the channel access parameters and piggyback policy to adapt to the changes in the number of SSs and network traffic. ACKNOWLEDGEMENTS This work was supported partly by the U.K. Department of Trade and Industry (DTI) and the Engineering and Physical Sciences Research Council (EPSRC) under the HIPNET project (EP/E002382/1), by the European Union through the Welsh Assembly Government, and the grants (NSC96-2221-E006-345 and NSC96-2221-E-006-346) from National Science Council of Taiwan. R EFERENCES [1] “IEEE Standard for local and metropolitan area networks part 16: air interface for fixed broadband wireless access systems,” 2004. [2] C. Cicconetti, A. Erta, L. Lenzini, and E. Mingozzi, “Performance evaluation of the IEEE 802.16 MAC for QoS support,” in IEEE Trans. Mobile Computing, vol. 6, no. 1, pp. 26-38, Jan. 2007.

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[3] A. Vinel, Y. Zhang, M. Lott, and A. Tiurlikov, “IEEE international symposum on personal, indoor and mobile radio communications,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Commun., 2005. [4] J. Seo, H. Lee and C. Cho, “Queueing behavior of IEEE 802.16 random access protocol for sporadic data transmissions,” in Proc. International Conference on Computer Commun. and Networks, pp. 351-357, 2006. [5] H. Lee and J. Seo, “Queueing performance of IEEE 802.16 random access protocol with bulk transmissions,” in Proc. IEEE International Conference on Commun., pp. 5963-5968, 2007. [6] J. Yan and G. S. Kuo, “Cross-layer design of optimal contention period for IEEE 802.16 BWA systems,” in Proc. IEEE International Conference on Commun., pp. 1087-1812, 2006. [7] A. Doha, H. Hassanein, and G. Takahara, “Performance evaluation of reservation medium access control in IEEE 802.16 networks,” in Proc. IEEE International Conference on Comp. Syst. and Applicat., pp. 369374, 2006.

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[8] J. He, K. Guild, K. Yang, and H. H Chen, “Modeling contention based bandwidth request scheme for IEEE 802.16 networks,” IEEE Commun. Lett., vol. 11, no. 8, pp. 689-700, Aug. 2007. [9] K. Chang and W. Liao, “The contention behavior of DOCSIS in CATV networks,” in IEEE Trans. Broadcast., vol. 53, no. 3, pp. 660-669, Sept. 2007. [10] S. Tasaka and Y. Ishibashi, “A reservation protocol for satellite packet communication–a performance analysis and stability considerations,” in IEEE Trans. Commun., vol. 32, no. 8, pp. 920-927, 1984. [11] W. Feller, An Introduction to Probability Theory and its Applications, vol. I. John Wiley, 1957. [12] H. Shetiya and V. Sharma, “Algorithms for routing and centralized scheduling to provide QoS in IEEE 802.16 mesh networks,” in Proc. ACM Workshop on Wireless Multimedia Networking and Performance Modeling, 2005.

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