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Semi-Markov Process based Model for Performance Analysis of Wireless LANs Murali Krishna Kadiyala, Dipti Shikha, Ravi Pendse, and Neeraj Jaggi Department of Electrical Engineering and Computer Science Wichita State University, Wichita, KS 67260. Email:{mxkadiyala, dspathak, ravi.pendse, neeraj.jaggi}@wichita.edu
A BSTRACT In this paper, we propose a new semi-Markov process based model to compute the network parameters such as saturation throughput, for the IEEE 802.11 Distributed Coordination Function (DCF) employing the Binary Exponential Backoff (BEB). The backoff stages of BEB and their backoff intervals are modeled as the states of semi-Markov process and their state holding-times, respectively. The proposed model is simpler than Bianchi’s two-dimensional Markov chain based model, with the number of states in the proposed model being of the order O(m), where m is the number of backoff stages in the BEB, compared with the Bianchi’s model where number of states is of the order O(2m ). Using the proposed semi-Markov process model, we compute the parameters of interest in wireless LANs, such as conditional collision probability, packet transmission probability, and saturation throughput. We show that the proposed model is quite accurate in computing these parameters of interest. Moreover, we show that the computation time with the proposed model is approximately one-tenth of that with Bianchi’s model, using Matlab simulations. Thus, the proposed model achieves accurate results with less complexity and computation time, and is suitable to be used for performance evaluation of complex protocols such as IEEE 802.11e. K EYWORDS Distributed Coordination Function (DCF), Semi-Markov Process, Mean State Holding-Time. I. I NTRODUCTION The two-dimensional Markov chain [1] is used for analyzing IEEE 802.11 distributed coordination function (DCF) medium access control (MAC) protocol. The two dimensions of this Markov chain represent the backoff stages of Binary Exponential Backoff (BEB) mechanism and their backoff counters. The state space |S| of the two-dimensional Markov chain is given by [1] |S| =
m X i=0
2i CWmin = CWmin (2m+1 − 1)
(1)
where i represents the backoff stage of BEB mechanism, m represents the highest backoff stage, and CWmin represents the minimum contention window. For Frequency Hop Spread Spectrum (FHSS) physical layer specifications, the state space |S| of the Markov chain is |S| =
16 × (27 − 1) = 2032
(2)
The state space of the two-dimensional Markov chain is very large of the order O(2m ). We propose a new model with reduced state space which computes (without compromising the accuracy) the wireless network parameters such as conditional collision probability, node’s packet transmission probability, and saturation throughput in the wireless LAN. In this paper, we represent the BEB mechanism of the IEEE 802.11 DCF protocol as a semi-Markov process by modeling its backoff stages and backoff interval as the states of the semi-Markov process and their state holding times, respectively. The proposed model has a reduced number of states of the order O(m) compared to that of twodimensional Markov chain [1] of order O(2m ). Thus we provide a simpler approach for computing the parameters of interest such as conditional collision probability, node’s packet transmission probability, and saturation throughput. Additionally, the computation time for obtaining the stationary probabilities of the semi-Markov process is approximately one-tenth of that incurred by the Bianchi’s model. Thus, the semi-Markov process provides the accuracy close to Bianchi’s model with lower complexity and less computation time, and thus is more suitable for analyzing complex protocols such as IEEE 802.11e with EDCA [2]. This model finds a significant application in pervasive wireless LANs [3], [4], such as ad hoc networks or sensor networks, employed for mission-critical applications in which every node is required to assess the conditional collision probability and saturation throughput of the network before transmitting a packet. This model assumes that the network nodes are aware of network size [5], [6]. The remainder of this paper is divided into four sections. Section II describes related work. Sections III presents the proposed model with (m+1)-state semi-Markov Process. Section IV presents the evaluation of our model. Section V discusses the conclusions.
II. R ELATED W ORK
III. M ODELING THE BEB M ECHANISM U SING THE S EMI -M ARKOV P ROCESS
This section discusses performance analysis of DCF protocol and the MAC queuing models proposed in the literature. These research works are based on the two-dimensional Markov chain proposed in [1]. Bianchi [1] analyzed the saturation throughput of IEEE 802.11 DCF protocol using a two-dimensional Markov chain. The backoff stages and the backoff counters represent two dimensions of the Markov chain. The author’s analysis was primarily based on two parameters namely the packet transmission probability (τ ) and conditional collision probability (p). The effect of these parameters on protocol’s saturation throughput was analyzed. Shabdiz and Subramaniam [7] analyzed the saturation throughput of IEEE 802.11 DCF MAC for finite loads based on the two-dimensional Markov chain proposed in [1]. Kwak et al. [8] analyzed the stability and performance of IEEE 802.11 DCF protocol’s BEB mechanism under saturation condition. Chen and Li [9] analyzed the performance of a new packet transmission mechanism, which is the hybrid of basic access and RTS/CTS mechanism. The authors evaluated the saturation throughput of this mechanism for variable packet lengths. Foh and Tantra [10] further extended the model in [1]. The authors assumed that the channel access probability and station collision probability depend on the status of communication channel, and proposed another analytical model. Li and Battiti [11] analyzed the IEEE 802.11 DCF protocol and proposed various modifications to support differentiated services. The prior research on the IEEE 802.11 DCF protocol focused on its performance analysis, and the enhancements to the protocol and its backoff mechanism. These research works were based on the two-dimensional Markov chain introduced in [1]. The requirement and significance of reducing the state space of this Markov chain was not realized in the past. This paper represents the BEB mechanism of the IEEE 802.11 DCF protocol as a semi-Markov process. The proposed model reduces the order of state space of the system from O(2m ) to O(m), which significantly reduces the complexity of the analysis, and also provides a simpler approach for computing the essential parameters in wireless networks. The MATLAB evaluation of the semi-Markov process indicates that the proposed model requires approximately one-tenth the computation time required by the two-dimensional Markov chain. Thus, the proposed model is more suitable if the nodes in the network need to perform these computations in a distributed manner. Also, the proposed model would be suitable towards analyzing more sophisticated protocols such as IEEE 802.11e, due to its simplicity and reduced complexity. The following section presents the details of the proposed model.
The semi-Markov processes is the generalization of Markov chain that augments the specification of the process by including a state holding-time. The future state of a semiMarkov process depends on the current state and its state holding-time. The holding time of state i is the amount of time that passes before making a state transition from i. Discrete time Markov chains have state holding-times that are equal to a unit time and that are independent of the next state transition. The sample paths for semi-Markov processes are timed sequences of state transitions. If the process is viewed at the times of state transition, then the sample paths are identical to those of a Markov chain. Such a process is known as embedded Markov chain. The transition probability Pii is zero in embedded Markov chains [12]. In this section, the BEB mechanism is modeled using the semi-Markov process. In subsection III A, we construct (m+1)-state Markov chain to describe the backoff stages of BEB mechanism. Because the backoff interval associated with different backoff stages of BEB mechanism are not equal, this discrete-time Markov chain with a unit state holding time for all its states cannot completely describe the BEB mechanism. So, we construct an embedded Markov chain in section II B that allows different state holding-times for its states. However, this embedded Markov chain does not include self-loops (transition from state i to itself). In subsection III C, we model the backoff intervals of backoff stages of BEB mechanism as state holding-times of the semiMarkov process which allows self-loops and also different state holding-time for its states. Also, we compute the stationary probabilities for the semi-Markov process. Subsection III D computes the parameters of interest (including saturation throughput) based upon the proposed model. A. Construction of (m+1)-state Markov Chain The (m+1)-states of the Markov chain in Figure 1 represent the backoff stages of the BEB mechanism of IEEE 802.11 DCF MAC protocol. The node performs its first packet transmission while it is in state zero. If the transmission is successful, it loops back to the same state and initiates the next packet transmission. In the event of collision, the node proceeds for a retransmission and enters the backoff stage one represented by state one. If the packet collides again, the node transitions to the next backoff stage represented by state two for the retransmission. These state transitions continue until the packet is successfully transmitted, or the node reaches the highest backoff stage m. When the node reaches the backoff stage m, it returns to the same backoff stage for retransmissions after collisions. Once the packet transmission count reaches the threshold value, the packet is dropped and a new packet is served. In summary, any state i of the semi-Markov process represents the ith backoff stage of the BEB mechanism. The transition
(1-p)
p
0
p
p
1
p
p
p
2
(m-1)
of a packet transmitted by each station, regardless of the number of retransmissions already suffered [1]. B. Construction of (m+1)-state Embedded MC
m
Next, we transform the above Markov chain to an embedded Markov chain (with Pii = 0 ∀ i), as shown in Figure 2. The element Pije of state transition probability matrix [P]e of an embedded Markov chain is [12]
(1-p)
(1-p)
Pije
(1-p)
= =
(1-p)
Figure 1. (m+1)-state Markov Chain with state transition probabilities Pij representing the transition from state i to j.
from a lower state i to a higher state (i + 1) indicates an unsuccessful transmission. The transition from any state i to state 0 indicates a successful transmission. The loopback transitions are possible only for states 0 and m. The state transitions of the (m+1)-state Markov chain are represented by state transition probability matrix [P] given by [1], [12] (1 − p) p 0 · · · 0 (1 − p) 0 p · · · 0 (1 − p) 0 0 · · · 0 (3) [P] = · · · · · (1 − p) 0 0 · · · p (1 − p) 0 0 · · · p where P(i−1)i , 0 ≤ i < m, is the probability of transition from state (i − 1) to state i, and is equal to conditional collision probability p defined as [1] p = 1 − (1 − τ )(N −1)
(4)
where τ is node’s packet transmission probability, and N is the number of nodes in the saturated LAN. The key assumption considered here regarding p is the same as the assumption of constant and independent collision probability p
1
0
1
p
2
(m-1)
m
(1-p)
(1-p)
(1-p)
e [P] =
0 1 0 (1 − p) 0 p (1 − p) 0 0 · · · (1 − p) 0 0 1 0 0
··· ··· ··· · ··· ···
0 0 0 · p 0
(7)
j6=i
Solving these simultaneous equations, we obtain the stationary probabilities of the embedded Markov chain as Πe0
=
Πei
=
(1 − p) (2 − p − pm ) (1 − p)pi−1 , ∀ i ∈ (2, m) (2 − p − pm )
Πe1 =
(9) (10)
which constitute the stationary probability vector Πe represented as [Πe0 Πe1 Πe2 . . . Πem ]. C. Stationary Probabilities of Semi-Markov Process Stationary probability vector Πs of the semi-Markov process is represented as [Πs0 Πs1 . . . Πsm ] [12] where Πsi
=
Πei × E[Hi ] , 0≤i≤m e j=0 {Πj × E[Hj ]}
Pm
(11)
The state holding-time Hi is defined as the time-period for which the node remains in a particular state i before transition to another state [12]. In this section, we model the backoff interval of backoff stage i as the state holdingtime for state i in the Markov chain. The state holding-time for state i is a random variable selected uniformly within the range (0, 2i CWmin ), for 0 ≤ i ≤ m. The expected value of state holding-time E[Hi ] for state i of the semi-Markov process is given by E[Hi ]
e = 0. (m+1)-state embedded Markov Chain with Pii
(6)
The stationary probability Πei of state i of the embedded Markov chain is given by [12] X e Πei = Πej Pji , ∀ i ∈ (0, m) (8)
1
Figure 2.
(5)
which results in
p
p
0 f or i = j Pij f or i = 6 j (1 − Pii )
= =
2i CWmin , 2 2i−1 CWmin
0≤i≤m (12)
As the node visits state 0 and state m successively after successful transmission in backoff stage 0 and collision in backoff stage m, respectively, the expected number of 1 , consecutive visits to states 0 and m equal p1 and (1−p) respectively. Hence, the expected value of state holding-time for states 0 and m are 20 CWmin 1 CWmin E[H0 ] = × = (13) 2 p 2p 2m CWmin 1 2m−1 CWmin E[Hm ] = × = (14) 2 (1 − p) (1 − p) Pm Using (11) and i=0 Πsi = 1, the stationary probabilities of the semi-Markov process are given by Πs0
1 2p
=
CWmin 2p
+A+
2m−1 CW
S=
Ptr Ps
min
A=
{Πej ∗ 2j−1 CWmin }
(15)
j=1
Πs1
=
Πsi
=
Πsm
=
1 CWmin 2p
2m−1 CWmin (1−p) i−1
+A+ (2p)
2m−1 CWmin (1−p)
CWmin 2p
+A+
CWmin 2p
(2p)m−1 (1−p) m−1 CW min + A + 2 (1−p)
i ∈ (1, m − 1) Stationary probability Πsi of semi-Markov process represents the fraction of time spent by a node in backoff stage i. D. Node Packet Transmission Probability and Saturation Throughput of the Channel Next, we use the stationary probability distribution of the semi-Markov process, and the state holding-times to compute the saturation throughput in the network. The packet transmission probability τ is calculated as follows. If the system is in state i, the node transmits once after an expected time interval E[Hi ], for 0 < i < m. In state 0, the node transmits once after an expected time interval of E[H0 ]/(1/p). Similarly, the average time interval before transmission is computed for state m. Thus, τ can be expressed as =
=
=
Πs0 ( p1 )
1 ) Πsm ( 1−p Πs1 Πs2 + + ... + E[H0 ] E[H1 ] E[H2 ] E[Hm ] n Pm−2 j pm−1 o 1−p 1 + m j=0 (p) + 1−p 2−p−p p n o P m−1 m−2 (1−p)CWmin 1 j + (2p) + (2p) m j=0 2−p−p 2p 1−p n Pm−2 j pm−1 o 1 j=0 (p) + 1−p p + n o (16) Pm−2 m−1 1 CWmin 2p + j=0 (2p)j + (2p) 1−p
+
(17)
1 − (1 − τ )N (18) (N −1) N τ (1 − τ ) (19) = 1 − (1 − τ )N = DIF S + RT S + CT S + Hdr + E[P ] + ACK =
+
(1−p)
Tcrts m−1 X
Ps Ptr E[P ] (1 − Ptr )ρ + Ptr Ps Ts + Ptr (1 − Ps )Tc
where Ptr is the probability of a node transmitting a packet, Ps is the probability of successful transmission, ρ is the idle slot time, Ts is the average time spent in a successful transmission, and Tc is the average time spent in collisions. These parameters are defined as [1]
Tsrts
where
τ
The saturation throughput S of the channel is [1]
3SIF S + 4δ
= DIF S + RT S + δ
(20) (21)
In the above computation, we will use the τ as given by (16). DIF S represents the distributed inter-space frame period, RT S is the time spent for RTS transmission, CT S is the time spent for CTS transmission, Hdr is the sum of MAC and physical layer headers, E[P ] is the average payload, ACK is the time spent for ACK transmission, SIF S is the short inter-space frame period, and δ is the propagation delay. IV. T HE E VALUATION OF P ROPOSED M ODEL Matlab evaluation was carried out: (a) to compute the essential network parameters (τS , pS , and SS ) of wireless networks using the proposed model, and (b) to evaluate the computation time for the proposed semi-Markov process. The results were validated by comparing them with those from Bianchi’s model [1]. τB denotes the value of τ obtained using Bianchi’s analysis, and τS denotes the value of τ obtained using the proposed model (eq.(16)). For (a): The conditional collision probability (pS ) and packet transmission probability (τS ) were computed using (4) and (16) for a fixed number of nodes, using fixed point iteration method. These outputs were further used for computing the saturation throughput of the channel using (17) through (21). Table I presents various parameters and their values used for computing saturation throughput (SS ). These computations were performed for different values of N for fixed values of m = 3 and CWmin = 32, and the results obtained were compared with those from Bianchi’s model (τB , pB , and SB ). The outputs obtained for both models are presented in Table II. Furthermore, the saturation throughputs SB and SS are compared for the variations in CWmin (from 8 through 1024), and these results are presented in Table III. We observe that the saturation throughput obtained using the proposed model is fairly accurate compared with that obtained using the Bianchi’s model for all values of N and CWmin .
For (b): We compare the computation time needed to solve for the saturation throughput under both models for various values of N presented in Table II. The time includes the time needed to solve for the stationary probability distribution, and fixed point equations for τ and p. These results are presented in Table IV. Our simulations show that the proposed model requires one-tenth of the time required for computing the stationary probabilities using Bianchi’s model. Table I FHSS S YSTEM PARAMETERS U SED F OR C OMPUTING S ATURATION T HROUGHPUT [1] P arameter E[P ] M AC header P HY header ACK RT S CT S Channel Bit Rate P ropagation Delay Slot T ime SIF S DIF S
V alue 8184 bits 272 bits 128 bits 112 bits + P HY header 160 bits + P HY header 112 bits + P HY header 1 M bits/s 1 µs 50 µs 28 µs 128 µs
Table II C OMPARISON OF (τB , pB , SB ) AND (τS , pS , SS ) FOR VARIATIONS IN N ; FOR CONSTANT m AND CWmin OF 3 AND 32, RESPECTIVELY. N 2 3 5 10 15 20 30 40 50 60 70 80 90 100 150
(τB , pB , SB ) (0.0570, 0.0570, 0.8189) (0.0538, 0.1046, 0.8279) (0.0481, 0.1794, 0.8343) (0.0387, 0.2986, 0.8371) (0.0329, 0.3750, 0.8367) (0.0291, 0.4302, 0.8356) (0.0242, 0.5078, 0.8329) (0.0212, 0.5644, 0.8300) (0.0190, 0.6104, 0.8269) (0.0174, 0.6467, 0.8240) (0.0162, 0.6772, 0.8209) (0.0153, 0.7012, 0.8182) (0.0144, 0.7266, 0.8148) (0.0137, 0.7470, 0.8116) (0.0115, 0.8233, 0.7944)
(τS , pS , SS ) (0.0586, 0.0586, 0.8198) (0.0551, 0.1071, 0.8284) (0.0491, 0.1823, 0.8345) (0.0392, 0.3024, 0.8371) (0.0333, 0.3770, 0.8366) (0.0293, 0.4328, 0.8355) (0.0243, 0.5116, 0.8327) (0.0213, 0.5665, 0.8299) (0.0191, 0.6123, 0.8268) (0.0175, 0.6483, 0.8238) (0.0162, 0.6788, 0.8208) (0.0152, 0.7050, 0.8177) (0.0145, 0.7256, 0.8149) (0.0137, 0.7485, 0.8113) (0.0115, 0.8244, 0.7941)
Table III C OMPARISON OF (τB , pB , SB ) AND (τS , pS , SS ) FOR VARIATIONS IN CWmin ; FOR CONSTANT m AND N OF 3 AND 100, RESPECTIVELY. CWmin 8 16 32 64 128 256 512 1024
(τB , pB , SB ) (0.0331, 0.9647, 0.6496) (0.0205, 0.8720, 0.7749) (0.0137, 0.7470, 0.8116) (0.0095, 0.6134, 0.8267) (0.0066, 0.4761, 0.8339) (0.0044, 0.3558, 0.8362) (0.0028, 0.2367, 0.8340) (0.0016, 0.1459, 0.8253)
(τS , pS , SS ) (0.0336, 0.9662, 0.6436) (0.0206, 0.8738, 0.7740) (0.0137, 0.7485, 0.8113) (0.0095, 0.6142, 0.8266) (0.0066, 0.4765, 0.8339) (0.0044, 0.3563, 0.8362) (0.0028, 0.2368, 0.8340) (0.0016, 0.1460, 0.8253)
Table IV C OMPARISON OF C OMPUTATIONAL T IME FOR B IANCHI ’ S MODEL (CB ) AND THE PROPOSED MODEL (CS ) N 2 3 5 10 15 20 30 40 50 60 70 80 90 100 150
CB (in ms) 1.3ms 0.77390ms 0.78945ms 0.83832ms 0.82677ms 0.80233ms 0.79434ms 0.79567ms 0.79389ms 0.80811ms 0.79922ms 0.77657ms 0.80278ms 0.79789ms 0.82188ms
CS (in ms) 0.10174ms 0.081299ms 0.082632ms 0.079522ms 0.099070ms 0.098626ms 0.080855ms 0.082632ms 0.095960ms 0.082188ms 0.081299ms 0.081744ms 0.084409ms 0.082181ms 0.083077ms
A. Results and Discussion This subsection presents the analysis of results presented in Tables II and IV. 1) Output Parameters: The simulation results presented in Table II show that the outputs of the proposed model are close to those of Bianchi’s model with a maximum difference of 0.1 percent for the saturation throughput S. An increase in N results in an increase in p and a decrease in τ for both models. For N = 2, the values of p and τ are equal according to (4). With an increase in N , the saturation throughput increased until the channel throughput reached its maximum value. With the further increase in the number of network nodes, the saturation throughput reduces and reaches zero as N → ∞. This pattern is visible in the outputs of the proposed model as also seen from the Bianchi’s model from Table II. The saturation throughput reached its peak very close to 84 percent at N = 10 in both cases validating the accuracy of the proposed model. Figure 3 presents the variations of p, τ , and S with respect to N for both Bianchi’s model and the proposed model, and also with respect to CWmin for a constant N of 100. 2) Computation Time: The Matlab implementation of the two models shows that the proposed model requires one-tenth of the time required for implementing (computing the stationary probabilities of) Bianchi’s model. The computation time is obtained using the Matlab commands tic and toc. Table IV presents the computation time for the two models for variations in N . The computation time is almost constant for both models for variations in N .
0.85
0.07
1
0.88
0.9
0.06
0.6
0.5
Bianchi‘s Model Semi−Markov Process
0.4
0.3
Bianchi‘s Model Semi−Markov Process Model 0.05
0.04
0.03
0.8
0.84 Saturation Throughput (S)
0.7
Saturation Throughput (S)
Packet Transmission Probability (tau)
Conditional Collision Probability (p)
Bianchi‘s Model Semi−Markov Process Model
0.86
0.8
0.82
0.8
0.78
0.75
Bianchi‘s Model Semi−Markov Process Model 0.7
0.76
0.2
0.74
0.65
0.02 0.1
0.72
20
40
60
80
100
120
Number of Nodes (N)
140
0.01
20
40
60 80 100 Number of Nodes (N)
120
140
0.7
20
40
60 80 100 Number of Nodes (N)
120
140
0
100
200
300
400
500
600
700
800
900
1024
Minimum Contention Window (CWmin)
Figure 3. (a) Variations in packet transmission probability with number of network nodes. (b) Variations of condition collision probability with number of network nodes. (c) Variations of saturation throughput with number of nodes. (d) Variations of saturation throughput with minimum contention window.
V. C ONCLUSIONS In this paper, we model the Binary Exponential Backoff (BEB) mechanism of the IEEE 802.11 DCF protocol using the semi-Markov process. Various backoff stages of the BEB mechanism and their backoff counters were modeled as the state of the semi-Markov process and their state holding times, thus reducing the two-dimensional Markov chain to a one-dimensional process. Unlike the two-dimensional Markov chain with an order of O(2m ) states, the semiMarkov process has very few states of order O(m). The proposed model presents a new and simpler approach to compute the essential parameters such as conditional collision probability, packet transmission probability, and saturation throughput without compromising the accuracy. Matlab evaluation of the proposed semi-Markov process shows that it requires one-tenth of the time required for computing the same network parameters of interest using Bianchi’s twodimensional Markov chain model. Thus, the proposed model achieves accurate results close to those of Bianchi’s model with less complexity and computation time, and hence is more suitable towards analyzing sophisticated protocols such as IEEE 802.11e with EDCA. R EFERENCES [1] Giuseppe Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination function,” IEEE Journal on Selected Areas in Communications, vol. 18, no. 3, pp. 535–547, March 2000. [2] I. Inan, F. Keceli, and E. Ayanoglu, “Analysis of the 802.11e enhanced distributed channel access function,” IEEE Transactions on Communications, vol. 57, pp. 1753–1764, June 2009. [3] D. Estrin, D. Culler, K. Pister, and G. Sukhatme, “Connecting the physical world with pervasive networks,” IEEE Pervasive Computing, 2002.
[4] S. Tompros, N. Mouratidis, M. Caragiozidis, H. Hrasnica, and A. Gavras, “A pervasive network architecture featuring intelligent energy management of households,” Proceedings of the 1st international conference on PErvasive Technologies Related to Assistive Environments, 2008. [5] N. G. Neumann, “Two algorithms for leader election and network size estimation in mobile ad hoc networks,” Master’s thesis, Texas A and M University, Dec 2004. [6] R. Ali, S. Lor, and M. Rio, “Two algorithms for network size estimation for master/slave ad hoc networks,” IEEE 3rd International Symposium on Advanced Networks and Telecommunication Systems (ANTS), pp. 1–3, Dec 2009. [7] F. Alizadeh-Shabdiz and S. Subramaniam, “A finite load analytical model for the IEEE 802.11 distributed coordination function mac,” in Proc. WiOpt’03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, France, March 2003. [8] B.-J. Kwak, N.-O. Song, and L. E. Miller, “Analysis of the stability and performance of exponential backoff,” in Proc. IEEE Wireless Communication and Networking Conference (WCNC’03), New Orleans, LA, March 2003. [9] H. Chen and Y. Li, “Performance model of IEEE 802.11 dcf with variable packet length,” IEEE Communications Letters, vol. 8, no. 3, pp. 186–188, March 2004. [10] C. H. Foh and J. W. Tantra, “Comments on IEEE 802.11 saturation throughput analysis with freezing of backoff counters,” IEEE Communication Letters, vol. 9, no. 2, pp. 130– 132, February 2005. [11] B. Li and R. Battiti, “Performance analysis of an enhanced IEEE 802.11 distributed coordination function supporting service differentiation,” in Proc. International Workshop on Quality of Future Internet Services (QoFIS’03), October 2003, pp. 152–161. [12] R. Nelson, Probability, Stochastic Processes and Queuing Theory: Mathematics of Computer Performance and Analysis, ser. Springer-Verlag, New York, 1995.
A BSTRACT In this paper, we propose a new semi-Markov process based model to compute the network parameters such as saturation throughput, for the IEEE 802.11 Distributed Coordination Function (DCF) employing the Binary Exponential Backoff (BEB). The backoff stages of BEB and their backoff intervals are modeled as the states of semi-Markov process and their state holding-times, respectively. The proposed model is simpler than Bianchi’s two-dimensional Markov chain based model, with the number of states in the proposed model being of the order O(m), where m is the number of backoff stages in the BEB, compared with the Bianchi’s model where number of states is of the order O(2m ). Using the proposed semi-Markov process model, we compute the parameters of interest in wireless LANs, such as conditional collision probability, packet transmission probability, and saturation throughput. We show that the proposed model is quite accurate in computing these parameters of interest. Moreover, we show that the computation time with the proposed model is approximately one-tenth of that with Bianchi’s model, using Matlab simulations. Thus, the proposed model achieves accurate results with less complexity and computation time, and is suitable to be used for performance evaluation of complex protocols such as IEEE 802.11e. K EYWORDS Distributed Coordination Function (DCF), Semi-Markov Process, Mean State Holding-Time. I. I NTRODUCTION The two-dimensional Markov chain [1] is used for analyzing IEEE 802.11 distributed coordination function (DCF) medium access control (MAC) protocol. The two dimensions of this Markov chain represent the backoff stages of Binary Exponential Backoff (BEB) mechanism and their backoff counters. The state space |S| of the two-dimensional Markov chain is given by [1] |S| =
m X i=0
2i CWmin = CWmin (2m+1 − 1)
(1)
where i represents the backoff stage of BEB mechanism, m represents the highest backoff stage, and CWmin represents the minimum contention window. For Frequency Hop Spread Spectrum (FHSS) physical layer specifications, the state space |S| of the Markov chain is |S| =
16 × (27 − 1) = 2032
(2)
The state space of the two-dimensional Markov chain is very large of the order O(2m ). We propose a new model with reduced state space which computes (without compromising the accuracy) the wireless network parameters such as conditional collision probability, node’s packet transmission probability, and saturation throughput in the wireless LAN. In this paper, we represent the BEB mechanism of the IEEE 802.11 DCF protocol as a semi-Markov process by modeling its backoff stages and backoff interval as the states of the semi-Markov process and their state holding times, respectively. The proposed model has a reduced number of states of the order O(m) compared to that of twodimensional Markov chain [1] of order O(2m ). Thus we provide a simpler approach for computing the parameters of interest such as conditional collision probability, node’s packet transmission probability, and saturation throughput. Additionally, the computation time for obtaining the stationary probabilities of the semi-Markov process is approximately one-tenth of that incurred by the Bianchi’s model. Thus, the semi-Markov process provides the accuracy close to Bianchi’s model with lower complexity and less computation time, and thus is more suitable for analyzing complex protocols such as IEEE 802.11e with EDCA [2]. This model finds a significant application in pervasive wireless LANs [3], [4], such as ad hoc networks or sensor networks, employed for mission-critical applications in which every node is required to assess the conditional collision probability and saturation throughput of the network before transmitting a packet. This model assumes that the network nodes are aware of network size [5], [6]. The remainder of this paper is divided into four sections. Section II describes related work. Sections III presents the proposed model with (m+1)-state semi-Markov Process. Section IV presents the evaluation of our model. Section V discusses the conclusions.
II. R ELATED W ORK
III. M ODELING THE BEB M ECHANISM U SING THE S EMI -M ARKOV P ROCESS
This section discusses performance analysis of DCF protocol and the MAC queuing models proposed in the literature. These research works are based on the two-dimensional Markov chain proposed in [1]. Bianchi [1] analyzed the saturation throughput of IEEE 802.11 DCF protocol using a two-dimensional Markov chain. The backoff stages and the backoff counters represent two dimensions of the Markov chain. The author’s analysis was primarily based on two parameters namely the packet transmission probability (τ ) and conditional collision probability (p). The effect of these parameters on protocol’s saturation throughput was analyzed. Shabdiz and Subramaniam [7] analyzed the saturation throughput of IEEE 802.11 DCF MAC for finite loads based on the two-dimensional Markov chain proposed in [1]. Kwak et al. [8] analyzed the stability and performance of IEEE 802.11 DCF protocol’s BEB mechanism under saturation condition. Chen and Li [9] analyzed the performance of a new packet transmission mechanism, which is the hybrid of basic access and RTS/CTS mechanism. The authors evaluated the saturation throughput of this mechanism for variable packet lengths. Foh and Tantra [10] further extended the model in [1]. The authors assumed that the channel access probability and station collision probability depend on the status of communication channel, and proposed another analytical model. Li and Battiti [11] analyzed the IEEE 802.11 DCF protocol and proposed various modifications to support differentiated services. The prior research on the IEEE 802.11 DCF protocol focused on its performance analysis, and the enhancements to the protocol and its backoff mechanism. These research works were based on the two-dimensional Markov chain introduced in [1]. The requirement and significance of reducing the state space of this Markov chain was not realized in the past. This paper represents the BEB mechanism of the IEEE 802.11 DCF protocol as a semi-Markov process. The proposed model reduces the order of state space of the system from O(2m ) to O(m), which significantly reduces the complexity of the analysis, and also provides a simpler approach for computing the essential parameters in wireless networks. The MATLAB evaluation of the semi-Markov process indicates that the proposed model requires approximately one-tenth the computation time required by the two-dimensional Markov chain. Thus, the proposed model is more suitable if the nodes in the network need to perform these computations in a distributed manner. Also, the proposed model would be suitable towards analyzing more sophisticated protocols such as IEEE 802.11e, due to its simplicity and reduced complexity. The following section presents the details of the proposed model.
The semi-Markov processes is the generalization of Markov chain that augments the specification of the process by including a state holding-time. The future state of a semiMarkov process depends on the current state and its state holding-time. The holding time of state i is the amount of time that passes before making a state transition from i. Discrete time Markov chains have state holding-times that are equal to a unit time and that are independent of the next state transition. The sample paths for semi-Markov processes are timed sequences of state transitions. If the process is viewed at the times of state transition, then the sample paths are identical to those of a Markov chain. Such a process is known as embedded Markov chain. The transition probability Pii is zero in embedded Markov chains [12]. In this section, the BEB mechanism is modeled using the semi-Markov process. In subsection III A, we construct (m+1)-state Markov chain to describe the backoff stages of BEB mechanism. Because the backoff interval associated with different backoff stages of BEB mechanism are not equal, this discrete-time Markov chain with a unit state holding time for all its states cannot completely describe the BEB mechanism. So, we construct an embedded Markov chain in section II B that allows different state holding-times for its states. However, this embedded Markov chain does not include self-loops (transition from state i to itself). In subsection III C, we model the backoff intervals of backoff stages of BEB mechanism as state holding-times of the semiMarkov process which allows self-loops and also different state holding-time for its states. Also, we compute the stationary probabilities for the semi-Markov process. Subsection III D computes the parameters of interest (including saturation throughput) based upon the proposed model. A. Construction of (m+1)-state Markov Chain The (m+1)-states of the Markov chain in Figure 1 represent the backoff stages of the BEB mechanism of IEEE 802.11 DCF MAC protocol. The node performs its first packet transmission while it is in state zero. If the transmission is successful, it loops back to the same state and initiates the next packet transmission. In the event of collision, the node proceeds for a retransmission and enters the backoff stage one represented by state one. If the packet collides again, the node transitions to the next backoff stage represented by state two for the retransmission. These state transitions continue until the packet is successfully transmitted, or the node reaches the highest backoff stage m. When the node reaches the backoff stage m, it returns to the same backoff stage for retransmissions after collisions. Once the packet transmission count reaches the threshold value, the packet is dropped and a new packet is served. In summary, any state i of the semi-Markov process represents the ith backoff stage of the BEB mechanism. The transition
(1-p)
p
0
p
p
1
p
p
p
2
(m-1)
of a packet transmitted by each station, regardless of the number of retransmissions already suffered [1]. B. Construction of (m+1)-state Embedded MC
m
Next, we transform the above Markov chain to an embedded Markov chain (with Pii = 0 ∀ i), as shown in Figure 2. The element Pije of state transition probability matrix [P]e of an embedded Markov chain is [12]
(1-p)
(1-p)
Pije
(1-p)
= =
(1-p)
Figure 1. (m+1)-state Markov Chain with state transition probabilities Pij representing the transition from state i to j.
from a lower state i to a higher state (i + 1) indicates an unsuccessful transmission. The transition from any state i to state 0 indicates a successful transmission. The loopback transitions are possible only for states 0 and m. The state transitions of the (m+1)-state Markov chain are represented by state transition probability matrix [P] given by [1], [12] (1 − p) p 0 · · · 0 (1 − p) 0 p · · · 0 (1 − p) 0 0 · · · 0 (3) [P] = · · · · · (1 − p) 0 0 · · · p (1 − p) 0 0 · · · p where P(i−1)i , 0 ≤ i < m, is the probability of transition from state (i − 1) to state i, and is equal to conditional collision probability p defined as [1] p = 1 − (1 − τ )(N −1)
(4)
where τ is node’s packet transmission probability, and N is the number of nodes in the saturated LAN. The key assumption considered here regarding p is the same as the assumption of constant and independent collision probability p
1
0
1
p
2
(m-1)
m
(1-p)
(1-p)
(1-p)
e [P] =
0 1 0 (1 − p) 0 p (1 − p) 0 0 · · · (1 − p) 0 0 1 0 0
··· ··· ··· · ··· ···
0 0 0 · p 0
(7)
j6=i
Solving these simultaneous equations, we obtain the stationary probabilities of the embedded Markov chain as Πe0
=
Πei
=
(1 − p) (2 − p − pm ) (1 − p)pi−1 , ∀ i ∈ (2, m) (2 − p − pm )
Πe1 =
(9) (10)
which constitute the stationary probability vector Πe represented as [Πe0 Πe1 Πe2 . . . Πem ]. C. Stationary Probabilities of Semi-Markov Process Stationary probability vector Πs of the semi-Markov process is represented as [Πs0 Πs1 . . . Πsm ] [12] where Πsi
=
Πei × E[Hi ] , 0≤i≤m e j=0 {Πj × E[Hj ]}
Pm
(11)
The state holding-time Hi is defined as the time-period for which the node remains in a particular state i before transition to another state [12]. In this section, we model the backoff interval of backoff stage i as the state holdingtime for state i in the Markov chain. The state holding-time for state i is a random variable selected uniformly within the range (0, 2i CWmin ), for 0 ≤ i ≤ m. The expected value of state holding-time E[Hi ] for state i of the semi-Markov process is given by E[Hi ]
e = 0. (m+1)-state embedded Markov Chain with Pii
(6)
The stationary probability Πei of state i of the embedded Markov chain is given by [12] X e Πei = Πej Pji , ∀ i ∈ (0, m) (8)
1
Figure 2.
(5)
which results in
p
p
0 f or i = j Pij f or i = 6 j (1 − Pii )
= =
2i CWmin , 2 2i−1 CWmin
0≤i≤m (12)
As the node visits state 0 and state m successively after successful transmission in backoff stage 0 and collision in backoff stage m, respectively, the expected number of 1 , consecutive visits to states 0 and m equal p1 and (1−p) respectively. Hence, the expected value of state holding-time for states 0 and m are 20 CWmin 1 CWmin E[H0 ] = × = (13) 2 p 2p 2m CWmin 1 2m−1 CWmin E[Hm ] = × = (14) 2 (1 − p) (1 − p) Pm Using (11) and i=0 Πsi = 1, the stationary probabilities of the semi-Markov process are given by Πs0
1 2p
=
CWmin 2p
+A+
2m−1 CW
S=
Ptr Ps
min
A=
{Πej ∗ 2j−1 CWmin }
(15)
j=1
Πs1
=
Πsi
=
Πsm
=
1 CWmin 2p
2m−1 CWmin (1−p) i−1
+A+ (2p)
2m−1 CWmin (1−p)
CWmin 2p
+A+
CWmin 2p
(2p)m−1 (1−p) m−1 CW min + A + 2 (1−p)
i ∈ (1, m − 1) Stationary probability Πsi of semi-Markov process represents the fraction of time spent by a node in backoff stage i. D. Node Packet Transmission Probability and Saturation Throughput of the Channel Next, we use the stationary probability distribution of the semi-Markov process, and the state holding-times to compute the saturation throughput in the network. The packet transmission probability τ is calculated as follows. If the system is in state i, the node transmits once after an expected time interval E[Hi ], for 0 < i < m. In state 0, the node transmits once after an expected time interval of E[H0 ]/(1/p). Similarly, the average time interval before transmission is computed for state m. Thus, τ can be expressed as =
=
=
Πs0 ( p1 )
1 ) Πsm ( 1−p Πs1 Πs2 + + ... + E[H0 ] E[H1 ] E[H2 ] E[Hm ] n Pm−2 j pm−1 o 1−p 1 + m j=0 (p) + 1−p 2−p−p p n o P m−1 m−2 (1−p)CWmin 1 j + (2p) + (2p) m j=0 2−p−p 2p 1−p n Pm−2 j pm−1 o 1 j=0 (p) + 1−p p + n o (16) Pm−2 m−1 1 CWmin 2p + j=0 (2p)j + (2p) 1−p
+
(17)
1 − (1 − τ )N (18) (N −1) N τ (1 − τ ) (19) = 1 − (1 − τ )N = DIF S + RT S + CT S + Hdr + E[P ] + ACK =
+
(1−p)
Tcrts m−1 X
Ps Ptr E[P ] (1 − Ptr )ρ + Ptr Ps Ts + Ptr (1 − Ps )Tc
where Ptr is the probability of a node transmitting a packet, Ps is the probability of successful transmission, ρ is the idle slot time, Ts is the average time spent in a successful transmission, and Tc is the average time spent in collisions. These parameters are defined as [1]
Tsrts
where
τ
The saturation throughput S of the channel is [1]
3SIF S + 4δ
= DIF S + RT S + δ
(20) (21)
In the above computation, we will use the τ as given by (16). DIF S represents the distributed inter-space frame period, RT S is the time spent for RTS transmission, CT S is the time spent for CTS transmission, Hdr is the sum of MAC and physical layer headers, E[P ] is the average payload, ACK is the time spent for ACK transmission, SIF S is the short inter-space frame period, and δ is the propagation delay. IV. T HE E VALUATION OF P ROPOSED M ODEL Matlab evaluation was carried out: (a) to compute the essential network parameters (τS , pS , and SS ) of wireless networks using the proposed model, and (b) to evaluate the computation time for the proposed semi-Markov process. The results were validated by comparing them with those from Bianchi’s model [1]. τB denotes the value of τ obtained using Bianchi’s analysis, and τS denotes the value of τ obtained using the proposed model (eq.(16)). For (a): The conditional collision probability (pS ) and packet transmission probability (τS ) were computed using (4) and (16) for a fixed number of nodes, using fixed point iteration method. These outputs were further used for computing the saturation throughput of the channel using (17) through (21). Table I presents various parameters and their values used for computing saturation throughput (SS ). These computations were performed for different values of N for fixed values of m = 3 and CWmin = 32, and the results obtained were compared with those from Bianchi’s model (τB , pB , and SB ). The outputs obtained for both models are presented in Table II. Furthermore, the saturation throughputs SB and SS are compared for the variations in CWmin (from 8 through 1024), and these results are presented in Table III. We observe that the saturation throughput obtained using the proposed model is fairly accurate compared with that obtained using the Bianchi’s model for all values of N and CWmin .
For (b): We compare the computation time needed to solve for the saturation throughput under both models for various values of N presented in Table II. The time includes the time needed to solve for the stationary probability distribution, and fixed point equations for τ and p. These results are presented in Table IV. Our simulations show that the proposed model requires one-tenth of the time required for computing the stationary probabilities using Bianchi’s model. Table I FHSS S YSTEM PARAMETERS U SED F OR C OMPUTING S ATURATION T HROUGHPUT [1] P arameter E[P ] M AC header P HY header ACK RT S CT S Channel Bit Rate P ropagation Delay Slot T ime SIF S DIF S
V alue 8184 bits 272 bits 128 bits 112 bits + P HY header 160 bits + P HY header 112 bits + P HY header 1 M bits/s 1 µs 50 µs 28 µs 128 µs
Table II C OMPARISON OF (τB , pB , SB ) AND (τS , pS , SS ) FOR VARIATIONS IN N ; FOR CONSTANT m AND CWmin OF 3 AND 32, RESPECTIVELY. N 2 3 5 10 15 20 30 40 50 60 70 80 90 100 150
(τB , pB , SB ) (0.0570, 0.0570, 0.8189) (0.0538, 0.1046, 0.8279) (0.0481, 0.1794, 0.8343) (0.0387, 0.2986, 0.8371) (0.0329, 0.3750, 0.8367) (0.0291, 0.4302, 0.8356) (0.0242, 0.5078, 0.8329) (0.0212, 0.5644, 0.8300) (0.0190, 0.6104, 0.8269) (0.0174, 0.6467, 0.8240) (0.0162, 0.6772, 0.8209) (0.0153, 0.7012, 0.8182) (0.0144, 0.7266, 0.8148) (0.0137, 0.7470, 0.8116) (0.0115, 0.8233, 0.7944)
(τS , pS , SS ) (0.0586, 0.0586, 0.8198) (0.0551, 0.1071, 0.8284) (0.0491, 0.1823, 0.8345) (0.0392, 0.3024, 0.8371) (0.0333, 0.3770, 0.8366) (0.0293, 0.4328, 0.8355) (0.0243, 0.5116, 0.8327) (0.0213, 0.5665, 0.8299) (0.0191, 0.6123, 0.8268) (0.0175, 0.6483, 0.8238) (0.0162, 0.6788, 0.8208) (0.0152, 0.7050, 0.8177) (0.0145, 0.7256, 0.8149) (0.0137, 0.7485, 0.8113) (0.0115, 0.8244, 0.7941)
Table III C OMPARISON OF (τB , pB , SB ) AND (τS , pS , SS ) FOR VARIATIONS IN CWmin ; FOR CONSTANT m AND N OF 3 AND 100, RESPECTIVELY. CWmin 8 16 32 64 128 256 512 1024
(τB , pB , SB ) (0.0331, 0.9647, 0.6496) (0.0205, 0.8720, 0.7749) (0.0137, 0.7470, 0.8116) (0.0095, 0.6134, 0.8267) (0.0066, 0.4761, 0.8339) (0.0044, 0.3558, 0.8362) (0.0028, 0.2367, 0.8340) (0.0016, 0.1459, 0.8253)
(τS , pS , SS ) (0.0336, 0.9662, 0.6436) (0.0206, 0.8738, 0.7740) (0.0137, 0.7485, 0.8113) (0.0095, 0.6142, 0.8266) (0.0066, 0.4765, 0.8339) (0.0044, 0.3563, 0.8362) (0.0028, 0.2368, 0.8340) (0.0016, 0.1460, 0.8253)
Table IV C OMPARISON OF C OMPUTATIONAL T IME FOR B IANCHI ’ S MODEL (CB ) AND THE PROPOSED MODEL (CS ) N 2 3 5 10 15 20 30 40 50 60 70 80 90 100 150
CB (in ms) 1.3ms 0.77390ms 0.78945ms 0.83832ms 0.82677ms 0.80233ms 0.79434ms 0.79567ms 0.79389ms 0.80811ms 0.79922ms 0.77657ms 0.80278ms 0.79789ms 0.82188ms
CS (in ms) 0.10174ms 0.081299ms 0.082632ms 0.079522ms 0.099070ms 0.098626ms 0.080855ms 0.082632ms 0.095960ms 0.082188ms 0.081299ms 0.081744ms 0.084409ms 0.082181ms 0.083077ms
A. Results and Discussion This subsection presents the analysis of results presented in Tables II and IV. 1) Output Parameters: The simulation results presented in Table II show that the outputs of the proposed model are close to those of Bianchi’s model with a maximum difference of 0.1 percent for the saturation throughput S. An increase in N results in an increase in p and a decrease in τ for both models. For N = 2, the values of p and τ are equal according to (4). With an increase in N , the saturation throughput increased until the channel throughput reached its maximum value. With the further increase in the number of network nodes, the saturation throughput reduces and reaches zero as N → ∞. This pattern is visible in the outputs of the proposed model as also seen from the Bianchi’s model from Table II. The saturation throughput reached its peak very close to 84 percent at N = 10 in both cases validating the accuracy of the proposed model. Figure 3 presents the variations of p, τ , and S with respect to N for both Bianchi’s model and the proposed model, and also with respect to CWmin for a constant N of 100. 2) Computation Time: The Matlab implementation of the two models shows that the proposed model requires one-tenth of the time required for implementing (computing the stationary probabilities of) Bianchi’s model. The computation time is obtained using the Matlab commands tic and toc. Table IV presents the computation time for the two models for variations in N . The computation time is almost constant for both models for variations in N .
0.85
0.07
1
0.88
0.9
0.06
0.6
0.5
Bianchi‘s Model Semi−Markov Process
0.4
0.3
Bianchi‘s Model Semi−Markov Process Model 0.05
0.04
0.03
0.8
0.84 Saturation Throughput (S)
0.7
Saturation Throughput (S)
Packet Transmission Probability (tau)
Conditional Collision Probability (p)
Bianchi‘s Model Semi−Markov Process Model
0.86
0.8
0.82
0.8
0.78
0.75
Bianchi‘s Model Semi−Markov Process Model 0.7
0.76
0.2
0.74
0.65
0.02 0.1
0.72
20
40
60
80
100
120
Number of Nodes (N)
140
0.01
20
40
60 80 100 Number of Nodes (N)
120
140
0.7
20
40
60 80 100 Number of Nodes (N)
120
140
0
100
200
300
400
500
600
700
800
900
1024
Minimum Contention Window (CWmin)
Figure 3. (a) Variations in packet transmission probability with number of network nodes. (b) Variations of condition collision probability with number of network nodes. (c) Variations of saturation throughput with number of nodes. (d) Variations of saturation throughput with minimum contention window.
V. C ONCLUSIONS In this paper, we model the Binary Exponential Backoff (BEB) mechanism of the IEEE 802.11 DCF protocol using the semi-Markov process. Various backoff stages of the BEB mechanism and their backoff counters were modeled as the state of the semi-Markov process and their state holding times, thus reducing the two-dimensional Markov chain to a one-dimensional process. Unlike the two-dimensional Markov chain with an order of O(2m ) states, the semiMarkov process has very few states of order O(m). The proposed model presents a new and simpler approach to compute the essential parameters such as conditional collision probability, packet transmission probability, and saturation throughput without compromising the accuracy. Matlab evaluation of the proposed semi-Markov process shows that it requires one-tenth of the time required for computing the same network parameters of interest using Bianchi’s twodimensional Markov chain model. Thus, the proposed model achieves accurate results close to those of Bianchi’s model with less complexity and computation time, and hence is more suitable towards analyzing sophisticated protocols such as IEEE 802.11e with EDCA. R EFERENCES [1] Giuseppe Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination function,” IEEE Journal on Selected Areas in Communications, vol. 18, no. 3, pp. 535–547, March 2000. [2] I. Inan, F. Keceli, and E. Ayanoglu, “Analysis of the 802.11e enhanced distributed channel access function,” IEEE Transactions on Communications, vol. 57, pp. 1753–1764, June 2009. [3] D. Estrin, D. Culler, K. Pister, and G. Sukhatme, “Connecting the physical world with pervasive networks,” IEEE Pervasive Computing, 2002.
[4] S. Tompros, N. Mouratidis, M. Caragiozidis, H. Hrasnica, and A. Gavras, “A pervasive network architecture featuring intelligent energy management of households,” Proceedings of the 1st international conference on PErvasive Technologies Related to Assistive Environments, 2008. [5] N. G. Neumann, “Two algorithms for leader election and network size estimation in mobile ad hoc networks,” Master’s thesis, Texas A and M University, Dec 2004. [6] R. Ali, S. Lor, and M. Rio, “Two algorithms for network size estimation for master/slave ad hoc networks,” IEEE 3rd International Symposium on Advanced Networks and Telecommunication Systems (ANTS), pp. 1–3, Dec 2009. [7] F. Alizadeh-Shabdiz and S. Subramaniam, “A finite load analytical model for the IEEE 802.11 distributed coordination function mac,” in Proc. WiOpt’03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, France, March 2003. [8] B.-J. Kwak, N.-O. Song, and L. E. Miller, “Analysis of the stability and performance of exponential backoff,” in Proc. IEEE Wireless Communication and Networking Conference (WCNC’03), New Orleans, LA, March 2003. [9] H. Chen and Y. Li, “Performance model of IEEE 802.11 dcf with variable packet length,” IEEE Communications Letters, vol. 8, no. 3, pp. 186–188, March 2004. [10] C. H. Foh and J. W. Tantra, “Comments on IEEE 802.11 saturation throughput analysis with freezing of backoff counters,” IEEE Communication Letters, vol. 9, no. 2, pp. 130– 132, February 2005. [11] B. Li and R. Battiti, “Performance analysis of an enhanced IEEE 802.11 distributed coordination function supporting service differentiation,” in Proc. International Workshop on Quality of Future Internet Services (QoFIS’03), October 2003, pp. 152–161. [12] R. Nelson, Probability, Stochastic Processes and Queuing Theory: Mathematics of Computer Performance and Analysis, ser. Springer-Verlag, New York, 1995.
