THE MAC LAYER PACKET SERVICE TIME DISTRIBUTIONS OF DCF IN THE IEEE PROTOCOL

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1 J. Appl. Math. & Computing Vol. 22(2006, No. 1-2, pp THE MAC LAYER PACKET SERVICE TIME DISTRIBUTIONS OF DCF IN THE IEEE PROTOCOL DONG HWAN HAN AND CHUL GEUN PARK Abstract. The IEEE protocol is the most mature technology for WLANs(Wireless Local Area Networks. However, as the number of stations increases, the delay and throughput performance of IEEE MAC(Medium Access Control degrades severely. In this paper, we present the comprehensive performance analysis of IEEE MAC protocol by investigating the MAC layer packet service time when arrival packet sizes have a general probability distribution. We obtain the discrete probability distribution of the MAC layer service time. By using this, we analyze the system throughput and the MAC layer packet service time of IEEE MAC protocol in wireless LAN environment. We take some numerical examples for the system throughput and the mean packet service time for several special distributions of arrival packet sizes. AMS Mathematics Subject Classification : 60K25, 68M20 Key words and phrases : MAC, DCF, IEEE , throughput, mean packet delay 1. Introduction In recent year, the popularity of WLAN(Wireless Local Area Network has generated much interests on improvement and performance analysis of the IEEE protocol. The IEEE protocol defines the MAC(Medium Access Control and the physical layer functions of WLANs. The WLAN MAC protocol employs two medium access methods for packet transmission which are the DCF(Distributed Coordination Function and the PCF(Point Coordination Function[1]. We focus on the DCF scheme in the protocol that is a contention based channel access method and is widely developed in the commercial uses. The mandatory DCF of the MAC protocol provides a CSMA/CA (Carrier Sense Received November 15, Revised February 21, Corresponding author. This work was supported by grant No. R from the Korea Science & Engineering Foundation. c 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 501

2 502 Dong Hwan Han and Chul Geun Park Multiple Access/Collision Avoidance and suits delay insensitive traffic. The optional PCF based on the contention free service is suitable for delay sensitive traffic[2,3]. The DCF defines two access methods for transmitting data packets, namely, the Basic access and the RTS/CTS(Request-To-Send/Clear-To-Send access[4]. In Basic access method, a station senses the channel idle for a specific time interval. If the channel idle, the station transmits data packet. If the channel is sensed busy, to minimize collisions, the station defers its transmission attempts to a later time on the basis of a backoff algorithm[1,4]. On the occasion that the collision probability is high and packet size is larger than a threshold, the RTS/CTS method is used. In this case, short RTS and CTS packets are exchanged to reserve the medium prior to the transmission duration and copes with hidden stations[5]. A large amount of works on the IEEE protocol has been studied for the saturation throughput and delay analysis of CSMA/CA[3-8]. The 2-dimensional MC(Markov Chain model introduced by Bianchi[6] for the analysis of saturation throughput has become a common method to study the performance of the IEEE MAC protocol[4]. Most of the literature start from the proposed in [2]. In the references [3,4,7,8], the delay analysis as well as saturation throughput of the IEEE MAC protocols has been done. Most of the previous studies [3-8] dealt with the throughput and delay analysis of IEEE MAC layer in saturation conditions. The MAC delay analysis of the current works except [5] has been limited to the derivation of mean value while the higher moments and the probability distribution function of the delay are untouched. Fortunately, in [5], they had maid a comprehensive study on throughput and delay of IEEE MAC protocol by obtaining the MAC layer packet service time, when the arrival packets to each station has a uniform probability distribution. The MAC layer packet service time is the time interval between the time instant that a packet starts to contend for transmission and the time instant that the packet either is acknowledged for correct reception by the intended receiver or is dropped. On the other hand, references[9,10] indicate that the trimodal packet size distribution has been demonstrated in subscriber access networks. Hence we can catch an idea that the size of arrival packets have a general probability distribution. In this paper, under this concept and the saturation condition, we deal with the packet delay analysis of IEEE MAC protocol by investigating the MAC layer packet service time distribution of a wireless station. The rest of this paper is organized as follows. In Section 2, we provide an overview of IEEE MAC DCF access methods including the backoff algorithm. Section 3 describes the probability distribution of the MAC layer service time when the size of arrival packets to each station has a general probability distribution. In Section 4, we take some numerical examples on the system

3 The MAC layer packet service time distributions of DCF 503 Sender Receiver Slots DATA SIFS ACK DIFS Busy med. Slots Other Busy medium DIFS Frozen backoff time Data Figure 1. Basic access mechanism throughput and the mean packet delay for several special distributions of arrival packet sizes. We finally draw conclusions in Section System model The DCF based on the CSMA/CA protocol provides two access methods for transmitting data packets. The essential method used in DCF is called Basic access method, which is depicted in Figure 1. In the IEEE , three different IFS(Inter-frame Space time intervals have been specified to provide various priority levels for access to the medium, namely, SIFS(Short IFS, PIFS(PCF IFS and DIFS(DCF IFS. The SIFS is the smallest one followed by PIFS and DIFS. After a SIFS, only ACK(Acknowledgements, CTS and data packet may be sent. In order to minimize collisions, after an idle DIFS, a station is allowed to transmit only at the beginning of a slot time, which is equal to the time needed to by any station to detect the transmission of a packet from any other station and is denoted by δ[4,6]. The other way of transmitting data packets is called RTS/CTS method[2,3], which is illustrated in Figure 2. Two types of carrier sensing functions can be managed. The physical carrier sensing is done by detecting any channel activity on the physical layer by other stations. The virtual carrier sensing is provided by the NAV(Network Allocation Vector, which is a timer that indicates the amount of time the medium will be reserved. All stations that hear the data or RTS update their NAV field based on the value of duration field in the received packet which includes the SIFS and the ACK packet transmission following the data packet, before sensing the medium again. When a packet arrives at the head of the transmission buffer, it will first monitor the channel activity. If the channel is scanned busy, the MAC wait until the medium become idle, then defers for an extra time interval DIFS. If the channel stays idle during the DIFS deference, the MAC then starts the backoff process. The DCF uses a slotted binary exponential backoff technique. To begin the backoff process, each station maintains a contention window size

4 504 Dong Hwan Han and Chul Geun Park Sender RTS Receiver SIFS CTS SIFS DATA SIFS ACK DIFS Time Slots Other NAV(RTS DIFS NAV(CTS Access to medium deferred Backoff Figure 2. RTS/CTS access mechanism CW, which takes CW min as an initial value and doubles its value before it reaches a maximum upper limit CW max. The backoff counter is measured in terms of slot time. The backoff counter is uniformly chosen in the range of [0,CW, where CW is current contention window. If the channel becomes busy during a backoff process, the backoff is frozen. When the channel becomes idle again and the backoff counter reaches zero, the station attempts to retransmit the packet. If the maximum transmission failure limit is reached, the retransmission shall stop, CW max shall be reset to CW min and the packet shall be discarded. 3. The distribution of the MAC layer service time 3.1 The MAC layer packet service time The MAC layer packet service time depends on the number of active stations, the probability distribution of packet sizes and the number of retransmission attempts(backoff stages based on the backoff mechanism of CSMA/CA. The collision probability p c is defined by the probability that there is at least one of other stations which will transmit at the same backoff time slot. We assume that this probability does not change and is independent during the transmission regardless of backoff stages. We also assume that packet sizes are generally distributed. Then we know that the MAC layer service time is a non-negative random variable, which will be denoted by T B. When the MAC layer transmits a packet, it experiences three basic processes: the decrement process of the backoff counter, the two frozen backoff counter processes that take time slots of T S and T C, respectively. Hereafter we use a slot time δ as a unit service time. Thus T S is a random variable representing amount of time slots that channel is sensed busy due to a successful transmission. We define s i = P {T S = i}, i =1, 2,, as the probability distribution of T S. And T C is a random variable representing amount of time slots that channel is scanned busy due to collision. We define c i = P {T C = i}, i =1, 2,, as the probability distribution of T C.

5 The MAC layer packet service time distributions of DCF 505 The MAC layer service time T B has a discrete probability of b i = P {T B = i}, i =1, 2,. Obviously, the probability distribution b i,i =1, 2,, depends on the transmission rate, the length of the packet, and the specific medium access methods of Basic access method and RTS/CTS access method[5,6]. To find the PGF(Probability Generating Function of T B, let Y p be a general discrete random variable of packet sizes and let a i = P {Y p = i}, i=1, 2,,L p, where L p is defined by the maximum packet length. We assume that T S, T C, T B and Y p are integer times of slot time δ and that these random variables are independently identically distributed for all n active stations. Further we define C(z, S(z and B(z respectively by C(z = c i z i, S(z = s i z i, B(z = b i z i. (1 i=1 i=1 3.2 The processes of collision and successful transmission We first consider Basic access method. As shown in Figure 1, the period of successful transmission T S consists of DATA, ACK, SIFS and DIFS intervals. Let l 1 = (ACK+SIFS+DIFS/δ, where x is the smallest integer that exceeds x. Then we have the PGF of T S as follows L p S(z = a i z i+l1, (2 i=1 where a i is the probability mass function for the packet size. The period of collision T C also consists of DATA, ACK, SIFS and DIFS intervals. In Basic access method, T C is determined by the longest one of the collided packets. Assume that the probability of three or more packets simultaneously colliding is neglected and let Y 1 and Y 2 be two random variables of packet sizes engaged in collision, then its probability distribution can be calculated by the following equation P {T C = l 1 + i} = P {Y 1 = i, Y 2 i} + P {Y 2 = i, Y 1 i} P {Y 1 = i, Y 2 = i}. Thus we can obtain the PGF of the random variable T C as L p C(z = (2a i F Y (i (a i 2 z i+l1, (3 i=1 where F Y (i is the cumulative distribution function of a i,i=1, 2,,L p. Let us now consider the case that RTS/CTS access method is used. As shown in Figure 2, the period of successful transmission T S consists of RTS, CTS, DATA, ACK, 3 SIFSs and DIFS. Let l 2 = (RTS + CTS + ACK + 3SIFS + DIFS/δ, then we have the PGF of T S as L p S(z = a i z i+l2. (4 i=1 i=1

6 506 Dong Hwan Han and Chul Geun Park Similarly, to find the PGF of the collision period T C, let l 3 = (RTS + SIFS + CTS + DIFS/δ, then we have the PGF of T C as C(z =z l3. (5 3.3 Packet transmission probability and Markov chain The backoff process decreases its counter by one for every idle slot detected. Assume that there are n active stations in wireless LAN and that packet arrival processes at all stations are independent, identically distributed. Then we have p c =1 (1 τ n 1, (6 where τ is the packet transmission probability that the station transmits in a random time slot given that the station has packets to transmit. Let p s be the probability that there is one successful transmission among other (n 1 stations in the considered slot given that the tagged station does not transmit. Then, by the equation (6, we have ] p s =(n 1 [(1 p c n 2 n 1 + pc 1. (7 To find the packet transmission probability, we let W 0 = CW min and m be the maximum backoff stage such that 2 m W 0 = CW max and let W i =2 i W 0, where i(0 i m is called backoff stage. Let s(t, b(t and k(t bethe stochastic processes representing the backoff stage, the backoff counter and the frozen period of the station at a time slot t respectively. Then the 3-dimensional stochastic process {(s(t,b(t,k(t} forms a discrete time MC(Markov Chain depicted in Figure 3. At each transmission, the backoff counter is uniformly chosen in the range (0,W i, where i is the current backoff stage, i.e., the number of transmission failed for the considered packet. We denote the one-step transition probability of {(s(t,b(t,k(t} by P {(i 1,j 1,k 1 (i 0,j 0,k 0 } = P {(s(t +1=i 1,b(t +1=j 1,k(t +1=k 1 (s(t =i 0,b(t =j 0,k(t =k 0 }. Then the one-step transition probabilities for the backoff stages and backoff counters at the original state of the frozen period are summarized as follows, for i =0, 1,,m, P {(i, j, 0 (i, j +1, 0} =1 p c, 0 j W i 2, P {(i, j, 0 (i 1, 0, 0} = p c /W i,i 0, 0 j W i 1, { (1 p c /W 0, i m, 0 j W i 1, P {(0,j,0 (i, 0, 0} = 1/W 0, i = m, 0 j W i 1.

7 The MAC layer packet service time distributions of DCF 507 Figure 3. State transition diagram of the discrete time Markov chain The one-step transition probabilities of the frozen period at the backoff stages i and the backoff counter j are given by, for i =0, 1,,m, P {(i, j, k 1 (i, j, k} =1, 0 j W i 1, 0 <k L, P {(i, j, k (i, j, 0} = p c q k, 1 j W i 1, 0 <h 1 <k L, where L = h 1 + L p and in case of Basic access method, h 1 is equal to l 1 and in case of RTS/CTS access method, h 1 is equal to l 2. So we have s k = a k h1, c k =2a k h1 F Y (k h 1 a 2 k h 1, k = h 1 +1,,L for Basic method and s k =

8 508 Dong Hwan Han and Chul Geun Park a k h1, c l3 = 1 for RTS/CTS method by the definitions of T S and T C. Hence q k, k = h 1,,L is a distribution of the discrete random variable representing sum of packet size and some inter-frame spaces as follows q k = p s s k + p c p s c k. (8 p c p c Let b i,j,k = lim P {s(t =i, b(t =j, k(t =k}, 0 i m, 0 j W i 1, t 0 k L be the stationary distribution of the MC. Then, in steady state, we can derive following relations through the state transition diagram. First, we have for i =0, 1,,m, j =1,,W i 1, p c b i,j,0, 1 k h 1, b i,j,k = L p c b i,j,0 q l, h 1 +1 k L. By the recursive relation, we have b i,0,0 =(p c i b 0,0,0, 1 i m, b i,0,0 =(1 p c b i,1,0 + p c b,0,0, 1 i m, W i b i,j,0 =(1 p c b i,j+1,0 + p c b,0,0 + b i,j,1 W i = W i j b i,0,0, 1 i m, 1 j W i 2. (1 p c W i By the normalization condition, we have 1= b i,j,0 + l=k L b i,j,k + b i,0,0. For simplicity, let η =1+p c L kq k, then we can refer to Appendix A.1 for k=h 1+1 finding b 0,0,0 as follows [ ηw0 (1 (2p c m+1 η(1 pm+1 c b 0,0,0 = 2(1 p c (1 2p c 2(1 p c ] 1 pm+1 c. 1 p c When the backoff counter reaches zero, a station will attempt to transmit packet regardless of backoff stage. So we can find the probability τ that the station which has a packet in its buffer transmits in a randomly chosen time slot. Thus we can refer to Appendix A.2 for finding τ by [ ηw0 (1 (2p c m+1 ] 1 τ = 2(1 2p c (1 p m+1 c η 2(1 p c The PGF of MAC layer packet service time In this subsection, we obtain the PGF of the MAC layer service time. In the backoff process, if the medium is idle, the backoff counter has the probability k=1

9 The MAC layer packet service time distributions of DCF p c to decrement by 1 during a time slot and the probability p c to stay at the original state (i, j, 0 during the frozen period. Moreover the frozen period has the probability p s to stay at the original state during T S and has the probability p c p s to stay at the original state during T C. Let H b (z be the PGF of time interval needed to decrement the backoff counter by 1 and H f (z be the PGF of a frozen period. Then we have H b (z =p c zh f (z+(1 p c z, L H f (z = q k z k H b (z, (9 k=h 1+1 where q k, k = h 1 +1,,L, is well described in (8. By some manipulations of (9, we have (1 p c z H b (z =. L 1 p c z q k z k k=h 1+1 By using the above equation, we can obtain the PGF of the MAC layer service time T B, denoted as B(z in (1, which can be represented by S(z, C(z and H b (z as follows m 1 B(z =(1 p c S(z (p c C(z i H i (z+(p c C(z m S(zH m (z, (10 where S(z and C(z are given in the equations (2-(5 and H i (z is defined by H i (z = i j=0 2 j W 0 1 k=0 1 2 j W 0 (H b (z k. Finally, by the equation (10, we obtain the system throughput S at the saturation state as (1 p m+1 c ia i i=1 S = B, (1 where p m+1 c is the packet discarding probability due to transmission failures, {a i } is the probability distribution of the packet size and L p is the maximum packet length. L p 4. Numerical results In this section, we present some numerical results to show the property of the discrete probability distribution of B(z for the MAC layer packet service time. We use the system parameters for FHSS(Frequency Hopping Spread Spectrum PHY-specification and DCF access method, which are summarized in Table 1.

10 510 Dong Hwan Han and Chul Geun Park For simplicity, the other parameters such as the number of active stations n = 10, the maximum backoff stage m = 5 and the initial value of a contention window W 0 = 32 are fixed. We assume that the propagation delay is neglected and the channel is error-free and all stations are awake all the time[3]. We also assume that three modes correspond to the most frequent packet sizes: 64 bytes(47%, 587 bytes(15% and 1518 bytes(28%. In addition, we consider the other packet sizes of 300 bytes(5%, 1300 bytes(5%[9,10]. Table 1. IEEE System parameters. Parameters Values Parameters Values PHY header 192 bits MAC header 224 bits RTS length 160bits+PHY header DIFS length 128µs ACK length 112bits+PHY header SIFS length 28µs CTS length 112bits+PHY header Unit time 50µs (FHSS Channel bit rate 2Mbps Table 2 shows how the packet size distributions influence the mean MAC layer packet service time of both Basic access and RTS/CTS access methods for four different cases of packet sizes: (i one packet of 1500 bytes, (ii one packet of 1000 bytes, (iii one packet of 624 bytes and (iv the considered five packets with mean 624 bytes when the collision probability varies. Table 2. The mean MAC layer service times( 5 ms Collision probabilities Basic access method RTS/CTS access method p c =0.05 p c =0.1 p c =0.2 p c =0.3 p c =0.4 case case case case case case case case From this table, we can see that in case (iv, the MAC service time of Basic access is shorter than that of RTS/CTS access when the collision probability is lower(p c =0.05 and vice versa, when the collision probability is higher(p c = 0.4. But in the other cases (i, (ii and (iii, this fact does not hold. Thus we conceive that Basic access has the better delay performance than RTS/CTS when the collision probability is low and the small sizes of packet are dominant. We also see that RTS/CTS access has the better delay performance than Basic access when the large sizes of packet are dominant. Figures 4 and 5 illustrate how the collision probability p c has influence on the probability distribution of the MAC layer packet service time in the two

11 The MAC layer packet service time distributions of DCF n=10, m=5, p c =0.05, Basic, 5 packets Real data Exponential Log normal Erlang(E Probability MAC service time [x5 ms] n=10, m=5, p c =0.05, RTS/CTS, 5 packets Real data Exponential Log normal Erlang(E Probability MAC service time [x5ms] Figure 4. PDFs of MAC layer service time(basic vs. RTS/CTS, p c =0.05 respective cases of Basic access method and RTS/CTS access method. In these Figures, we can see that the 2-stage Erlangian distribution is a good approximation to the real distribution of the MAC layer packet service time. In Figure 4, we choose a lower collision probability p c =0.05, which means that the considered station has the more successful packet transmission chance. We can see that the MAC service time of Basic access is shorter than that of RTS/CTS access. This is because that we have shorter mean length of T C and longer mean length of T S and that RTS/CTS access method has longer mean length of T S than Basic access method. In Figure 5, we choose a higher collision probability p c =0.4 which means that the considered station has the less successful packet transmission chance. We can see that the MAC service time of Basic access is longer than that of RTS/CTS access. This is because that we have longer mean length of T C and

12 512 Dong Hwan Han and Chul Geun Park n=10, m=5, p c =0.4, Basic, 5 packets Real data Exponential Log normal Erlang(E Probability MAC service time [x5 ms] n=10, m=5, p c =0.4, RTS/CTS, 5 packets Real data Exponential Log normal Erlang(E Probability MAC service time [x5ms] Figure 5. PDFs of MAC layer service time(basic vs. RTS/CTS, p c =0.4 shorter mean length of T S and that RTS/CTS access method has shorter mean length of T C than Basic access method. Figure 6 illustrates how the collision probability p c has influence on the system throughput in saturation condition. The left figure of Figure 6 shows the throughput of Basic access for four cases presented in Table 2, when the collision probability varies. In this figure, we see that the system has the same throughput performance for the same mean of the packet sizes, though the probability distributions of packet sizes are different. In the right figure of Figure 6, we can see that the throughput performance of RTS/CTS access is a little bit better than that of Basic access when the collision probability is low around p c =0.05 ad vice versa. 5. Conclusion

13 The MAC layer packet service time distributions of DCF n=10, m=5, Basic access case 1 case 2 case 3 case Throughput Collision probability (p c n=10, m=5, Basic and RTS/CTS Bas, case 3 RTS, case 3 Bas, case 4 RTS, case Throughput Collision probability (p c Figure 6. System throughput in saturation condition In this paper, we derived the probability generating function of the MAC layer packet service time when the arrival packet size of each station was a generally distributed random variable. We also presented the delay and the throughput analysis of the IEEE MAC protocol by investigating the MAC layer packet service time of wireless stations when arrival packet sizes have a general probability distribution function. By using the discrete probability distribution of the MAC layer service time, we analyze the mean packet delay and the system throughput of the IEEE MAC protocol. We take some numerical examples for the system throughput and the mean packet delay for three cases of only one packet size and one case of five modal packet sizes. We need a further study to investigate the effect of input traffic burstiness on the MAC layer service time and the queue dynamics of a mobile station in wireless LAN environment to the Internet.

14 514 Dong Hwan Han and Chul Geun Park Appendix A.1 Derivation of b 0,0,0 By the normalization condition of the stationary probabilities {b i,j,k, 0 i m, 0 j W i 1, 0 k L}, we have 1= = = b i,j,0 + b i,j,0 + For simplicity, let η =1+p c 1=η = ηb 0,0,0 2(1 p c L b i,j,k + k=1 ( L b i,j,0 1+p c b i,j,0 + L k=h 1+1 b i,j,0 ( k=h 1+1 p c kq k + L k=h 1+1 b i,0,0 kq k + b i,0,0. kq k, then we have b i,0,0 m (W i p i c pi c + p i c b 0,0,0 = b 0,0,0 [ ηw0 (1 (2p c m+1 2(1 p c (1 2p c A.2 Derivation of τ η(1 pm+1 c b i,0,0 2(1 p c pm+1 c 1 p c Since τ is the probability that the station having a packet in its buffer transmits in a randomly chosen time slot, we have τ = b i,0,0 = p i cb 0,0,0 = 1 pm+1 c b 0,0,0. 1 p c From the result of Section A.1, we have τ = 1 [ pm+1 c ηw0 (1 (2p c m+1 1 p c 2(1 p c (1 2p c [ ηw0 (1 (2p c m+1 = 2(1 2p c (1 pc m+1 η 2(1 p c +1 η(1 pm+1 c ]. 2(1 p c pm+1 c 1 p c ] 1. ] 1

15 The MAC layer packet service time distributions of DCF 515 References 1. IEEE Standard Part II, Wireless LAN medium access control (MAC and physical layer(phy specifications ( M.S. Gast, Wireless networks, the definitive guides, O Reilly ( Y.T. Lee, D.H. Han and C.G. Park, Saturation throughput and dealy analysis of DCF in the IEEE wireless LAN, WSEAS Trans. on Comm. 4 (2005, P. Chatzimisios, V. Vitas and A.C. Boucouvalas, Throughput and delay analysis of IEEE protocol, IEEE INFOCOM 02, (2002, H. Zhai, Y.G. Kwon and Y. Fang, Performance analysis of IEEE MAC protocols in wireless LANs, Wireless Comm. and Mobile Computing 4 (2004, G. Bianchi, Performance analysis of the IEEE distributed coordination function, IEEE J. on Selected Areas in Comm. 18 (2000, P. Chatzimisios, A.C. Boucouvalas and V. Vitas, Packet dealy analysis of IEEE MAC protocol, Electronics Letters 39 (2003, H. Chen and Y. Li, Analytical analysis of hybrid access mechanism of IEEE DCF, IEICE Trans. on Comm. E87-B (2004, G. Kramer, B. Mukherjee, X. Dixit Y. Ye and R. Hirth, Supporting differentiated class of service in ethernet passive optical networks, J. of Optical Networking 1 (2002, C.G. Park, D.H. Han and B. Kim, Packet dealy analysis of dynamic bandwidth allocation scheme in an ethernet PON, Networking-ICN 05, LNCS 3420 (2005, Dong Hwan Han is Professor at the Department of Mathematics, Sunmoon University. He received B.S. in Mathematics from Sogang University, M.S. and Ph.D. in Applied Mathematics from Korea Advanced Institute of Science and Technology. In 1993, he joined the Communication Processing Research Laboratory at Eletronics and Telecommunications Research Institute(ETRI. His main areas of interests are queueing theory and Stochastic model in communication system. Department of Mathematics, Sunmoon University, Asan, Chungnam , Korea dhhan@sunmoon.ac.kr Chul Geun Park is Associate Professor at Department of Information and Communication Engineering, Sunmoon University. He received his B.S. degree in Mathematics from Pusan National University, Korea, in 1983, his M.S. degree in Applied Mathematics from Korea Advanced Institute Science and Technology (KAIST, in 1986 and Ph.D. degree in Applied Probability from KAIST in In 1986, he joined the Telecommunication Network Research Laboratory at Korea Telecom, Seoul and worked on traffic egineering and performance evaluation of Public Switched Telephone Networks. His current research intrests include traffic engineering, performance analysis of communication systems and networks by queueing approaches Department of Information and Communication Engineering, Sunmoon University, Asan, Chungnam , Korea cgpark@sunmoon.ac.kr