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Performance Analysis of Network Coding Based Two-Way Relay Wireless Networks Deploying IEEE 802.11

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Abstract

In this paper, we investigate the performance analysis of the IEEE 802.11 DCF protocol at the data link layer. We analyze the impact of network coding in saturated and non-saturated traffic conditions. The cross-layer analytical framework is presented in analyzing the performance of the encode-and-forward (EF) relaying wireless networks. This situation is employed at the physical layer under the conditions of non-saturated traffic and finite-length queue at the data link layer. First, a model of a two-hop EF relaying wireless channel is proposed as an equivalent extend multi-dimensional Markovian state transition model in queuing analysis. Then, the performance in terms of queuing delay, throughput and packet loss rate are derived. We provide closed-form expressions for the delay and throughput of two-hop unbalanced bidirectional traffic cases both with and without network coding. We consider the buffers on nodes are unsaturated. The analytical results are mainly derived by solving queuing systems for the buffer behavior at the relay node. To overcome the hidden node problem in multi hop wireless networks, we develop a useful mathematical model. Both models have been evaluated through simulations and simulation results show good agreement with the analytical results.

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References

  1. Ishmael, J., Bury, S., Pezaros, D., & Race, N. (2008). Deploying rural community wireless mesh networks. IEEE Internet Computing, 12(4), 22–29.

    Article  Google Scholar 

  2. Soldani, D., & Dixit, S. (2008). Wireless relays for broadband access [radio communications series]. IEEE Communications Magazine, 46(3), 58–66.

    Article  Google Scholar 

  3. Eugster, P., Guerraoui, R., Kermarrec, A., & Massoulié, L. (2004). Epidemic information dissemination in distributed systems. Computer, 37(5), 60–67.

    Article  Google Scholar 

  4. Ieee std 802.11-2007, part 11. wireless lan medium access control (mac) and physical layer (phy) specifications. IEEE Std. 2007 2007; :Part 11. doi:10.1109/IEEESTD.2007.92296.

  5. Bianchi, G. (2000). Performance analysis of the ieee 802.11 distributed coordination function. IEEE Journal on Selected Areas in Communications, 18(3), 535–547.

    Article  Google Scholar 

  6. Robinson, J. W., & Randhawa, T. S. (2004). Saturation throughput analysis of ieee 802.11 e enhanced distributed coordination function. IEEE Journal on Selected Areas in Communications, 22(5), 917–928.

    Article  Google Scholar 

  7. Duffy, K., Malone, D., & Leith, D. (2005). Modeling the 802.11 distributed coordination function in non-saturated conditions. IEEE Communications Letters, 9(8), 715–717.

    Article  Google Scholar 

  8. Ni, Q., Li, T., Turletti, T., & Xiao, Y. (2005). Saturation throughput analysis of error-prone 802.11 wireless networks. Wireless Communications and Mobile Computing, 5(8), 945–956.

    Article  Google Scholar 

  9. Fragouli, C., & Widmer, J. (2008). Efficient broadcasting using network coding. IEEE/ACM Transactions on Networking, 16(2), 450–463.

    Article  Google Scholar 

  10. Wu, H., Peng, Y., Long, K., Cheng, S., & Ma, J. (2002) Performance of reliable transport protocol over ieee 802.11 wireless lan: analysis and enhancement. In INFOCOM 2002. Twenty-first annual joint conference of the IEEE computer and communications societies. proceedings. IEEE, vol. 2. IEEE, 2002, pp. 599–607.

  11. Chatzimisios, P., Boucouvalas, A., & Vitsas, V. (2003). Ieee 802.11 packet delay-a finite retry limit analy- sis. In Global telecommunications conference, 2003. GLOBECOM’03. IEEE, vol. 2. IEEE, pp. 950–954.

  12. Babich, F., & Comisso, M. (2009). Throughput and delay analysis of 802.11-based wireless networks using smart and directional antennas. IEEE Transactions on Communications, 57(5), 1413–1423.

    Article  Google Scholar 

  13. Zhang, L., Yantai, S., Oliver, Y., & Guanghong, W. (2006). Study of medium access delay in ieee 802.11 wireless networks. IEICE transactions on Communications, 89(4), 1284–1293.

    Article  Google Scholar 

  14. Xiao, Y. (2005). Performance analysis of priority schemes for ieee 802.11 and ieee 802.11 e wireless lans. IEEE Transactions on Wireless Communications, 4(4), 1506–1515.

    Article  Google Scholar 

  15. Gupta, V., Gong, M., Dharmaraja, S., & Williamson, C. (2010). Analytical modeling of bidirectional multi-channel ieee 802.11 mac protocols. International Journal of Communication Systems, 24(5), 647–665.

    Google Scholar 

  16. Kao, H., Wu, P., & Lee, C. (2011). Analysis and enhancement of multi-channel mac protocol for ad hoc networks. International Journal of Communication Systems, 24(3), 310–324.

    Article  Google Scholar 

  17. Rui, X., Hou, J., & Zhou, L. (2012). Decode-and-forward with full-duplex relaying. International Journal of Communication Systems, 25(2), 270–275.

    Article  Google Scholar 

  18. Liu, W., Jin, H., Wang, X., & Guizani, M. (2011). A novel ieee 802.11-based mac protocol supporting cooperative communications. International Journal of Communication Systems, 24(11), 1480–1495.

    Article  Google Scholar 

  19. Ke, C., Wei, C., Lin, K., & Ding, J. (2011). A smart exponential-threshold-linear backoff mechanism for ieee 802.11 wlans. International Journal of Communication Systems, 24(8), 1033–1048.

    Article  Google Scholar 

  20. Engelstad, P., Osterbo, O. (2006). Analysis of the total delay of ieee 802.11 e edca and 802.11 dcf. In Communications, 2006. ICC’06. IEEE International Conference on, vol. 2. IEEE, pp. 552–559.

  21. Malone, D., Duffy, K., & Leith, D. (2007). Modeling the 802.11 distributed coordination function in nonsaturated heterogeneous condition, IEEE/ACM Transactions on Networking, 15(1), 159–172.

    Article  Google Scholar 

  22. Zhai, H., Kwon, Y., & Fang, Y. (2004). Performance analysis of ieee 802.11 mac protocols in wireless lans. Wireless Communications and Mobile Computing, 4(8), 917–931.

    Article  Google Scholar 

  23. Tickoo, O., & Sikdar, B. (2008). Modeling queueing and channel access delay in unsaturated ieee 802.11 random access mac based wireless networks. IEEE/ACM Transactions on Networking (TON), 16(4), 878–891.

    Article  Google Scholar 

  24. Felemban, E., & Ekici, E. (2011). Single hop ieee 802.11 dcf analysis revisited: Accurate modeling of channel access delay and throughput for saturated and unsaturated traffic cases. IEEE Transactions on Wireless Communications, 10(10), 3256–3266.

    Article  Google Scholar 

  25. Foh, C., Zukerman, M., & Tantra, J. (2007). A markovian framework for performance evaluation of ieee 802.11. IEEE Transactions on Wireless Communications, 6(4), 1265–1276.

    Article  Google Scholar 

  26. Liu, J., Goeckel, D., & Towsley, D. (2007). Bounds on the gain of network coding and broadcasting in wireless networks. In INFOCOM, 2007. 26th IEEE international conference on computer communications, IEEE. IEEE 2007, pp. 724–732.

  27. Chaporkar, P., & Proutiere, A. (2007) Adaptive network coding and scheduling for maximizing throughput in wireless networks. In Proceedings of the 13th annual ACM international conference on Mobile computing and networking, ACM, 2007, pp. 135–146.

  28. Chen, W., Letaief, K., & Cao, Z. (2007). Opportunistic network coding for wireless networks. In Communications 2007. ICC’07. IEEE international conference on, IEEE, 2007, pp. 4634–4639.

  29. Asterjadhi, A., Fasolo, E., Rossi, M., Widmer, J., & Zorzi, M. (2010). Toward network coding-based protocols for data broadcasting in wireless ad hoc networks. IEEE Transactions on Wireless Communications, 9(2), 662–673.

    Article  Google Scholar 

  30. Fasolo, E., Rossi, M., Widmer, J., & Zorzi, M. (2007). On mac scheduling and packet combination strategies for practical random network coding. In Communications, 2007. ICC’07. IEEE international conference on, IEEE, 2007, pp. 3582–3589.

  31. Sagduyu, Y., & Ephremides, A. (2008). Cross-layer optimization of mac and network coding in wireless queueing tandem networks. IEEE Transactions on Information Theory, 54(2), 554–571.

    Article  MathSciNet  Google Scholar 

  32. Le, J., Lui, J., & Chiu, D. (2008). How many packets can we encode? An analysis of practical wireless network coding. In INFOCOM 2008. The 27th conference on computer communications. IEEE, IEEE, pp. 371–375.

  33. Argyriou, A. (2009). Wireless network coding with improved opportunistic listening. IEEE Transactions on Wireless Communications, 8(4), 2014–2023.

    Article  Google Scholar 

  34. Hsu, Y., Abedini, N., Ramasamy, S., Gautam, N., Sprintson, A., Shakkottai, S., et al. (2011). Opportunities for network coding: To wait or not to wait. Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on, IEEE, pp. 791–795.

  35. Cloud, J., Zeger, L., & Médard, M. (2012). Mac centered cooperation synergistic design of network coding, multi-packet reception, and improved fairness to increase network throughput. IEEE Journal on Selected Areas in Communications, 30(2), 341–349.

    Article  Google Scholar 

  36. Hasegawa, J., Yomo, H., Kondo, Y., Davis, P., Suzuki, R., Obana, S., et al. (2009). Bidirectional packet aggregation and coding for voip transmission in wireless multi-hop networks. In Communications, 2009. ICC’09. IEEE international conference on, IEEE, pp. 1–6.

  37. Fouli, K., Casse, J., Sergeev, I., Médard, M., & Maier, M. (2012). Broadcasting xors: On the application of network coding in access point-to-multipoint networks. Multiple Access Communications. Springer, 25–36.

  38. Hsu, Y. P., & Sprintson, A. (2012). Opportunistic network coding: Competitive analysis. In International Symposium on Network Coding (NetCod), 2012, IEEE, pp. 191–196.

  39. Antonopoulos, A., Verikoukis, C., Skianis, C., & Akan, O. B. (2012). Energy efficient network coding-based mac for cooperative arq wireless networks. Ad Hoc Networks.

  40. Katti, S., Rahul, H., Hu, W., Katabi, D., Médard, M., & Crowcroft, J. (2008). Xors in the air: practical wireless network coding. IEEE/ACM Transactions on Networking (TON), 16(3), 497–510.

    Google Scholar 

  41. Hirantha Sithira Abeysekera, B., Matsuda, T., & Takine, T. (2008). Dynamic contention window control mechanism to achieve fairness between uplink and downlink flows in ieee 802.11 wireless lans. IEEE Transactions on Wireless Communications, 7(9), 3517–3525.

    Google Scholar 

  42. Chatzimisios, P., Boucouvalas, A., & Vitsas, V. (2004). Effectiveness of rts/cts handshake in ieee 802.11 a wireless lans. Electronics Letters, 40(14):915–916.

    Google Scholar 

Download references

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Correspondence to Karim Faez.

Appendices

Appendix 1: Proof of Lemma 1

Proof

The sum of steady-state probabilities \(Q(0), Q(1*)\) and \(Q(2*)\) and the ratio of the steady-state probabilities \(Q(1*)\) to \(Q(2*)\) are proportional:

$$\begin{aligned} Q(0) + Q(1*) + Q(2*) = 1, \frac{Q(1*)}{Q(2*)} = \frac{\lambda _1}{\lambda _2}\frac{\mu _2}{\mu _1} = \frac{\rho _1}{\rho _2}. \end{aligned}$$
(44)

where \(Q(v*) = \sum _{\fancyscript{V}_1^n\in \{1,2\}^n} Q(v\fancyscript{V}_1^n)\)

By solving the equations in Eq. 44,

$$\begin{aligned} Q(v*) = \frac{\rho _v}{\rho _1 + \rho _2}(1-Q(0)). \end{aligned}$$
(45)

The arrival rate \(\lambda _R\) in relay node \(\mathbf{R}\) and the departure rate \(\mu _R\) from relay node \(\mathbf{R}\) are balanced in steady-state. They are expressed as:

$$\begin{aligned}&\displaystyle \lambda _R = (\lambda _{0,1} + \lambda _{0,2})Q(0) + (\lambda _1 + \lambda _2)(1-Q(0)) = (\lambda _1 + \lambda _2)\bigg [\frac{1-\tau _R(1-Q(0))}{1-\tau _R}\bigg ], \nonumber \\&\displaystyle \mu _R = \mu _1Q(1*) + \mu _2Q(2*) = (\lambda _1 + \lambda _2)\frac{(1-Q(0))}{\rho _1 + \rho _2}, \nonumber \\&\displaystyle \mu _R = \lambda _R \Rightarrow Q(0) = \frac{(1-\tau _R)(1-\rho _1 - \rho _2)}{(1-\tau _R) + \tau _R(\rho _1 + \rho _2)}. \end{aligned}$$
(46)

We can calculate the relation between \(Q(1\fancyscript{V}_1^n)\) and \(Q(2\fancyscript{V}_1^n)\) as Eq. and applying the detailed balance equations in ascending order of queue state length gives these results:

$$\begin{aligned} \frac{Q(1\fancyscript{V}_1^n)}{Q(2\fancyscript{V}_1^n)}&= \frac{\rho _1}{\rho _2}, \nonumber \\ (\lambda _{0,1} + \lambda _{0,2})Q(0)&= \mu _1Q(1) + \mu _2Q(2),\nonumber \\ (\lambda _1 + \lambda _2)Q(\fancyscript{V}_1^n)&= \mu _1Q(1\fancyscript{V}_1^n) + \mu _2Q(2\fancyscript{V}_1^n). \end{aligned}$$
(47)

By applying these equations, the following Lemma is obtained.\(\square \)

Appendix 2: Proof of Lemma 2

Proof

From Eq. 7, the steady-state probability \(P_v(0)\) can be expressed as:

$$\begin{aligned} P_v(0) = Q(0) + \sum _{n_{\bar{v}}=1}^\infty \frac{\rho _{\bar{v}}^{n_{\bar{v}}}Q(0)}{(1-\tau _R)} = Q(0)\bigg [ \frac{1-\tau _R(1-\rho _{\bar{v}})}{(1-\tau _R)(1-\rho _{\bar{v}})}\bigg ]. \end{aligned}$$
(48)

From Eq. 7, the steady-state probabilities \(P(n)\) after some algebra can be expanded as:

$$\begin{aligned} P(n) =\sum _{\fancyscript{V}_1^n\in \{1,2\}^n} Q(\fancyscript{V}_1^n) = \frac{(\rho _1 + \rho _2)^n}{(1-\tau _R)}P(0); n>0 , P(0) = Q(0). \end{aligned}$$
(49)

\(\square \)

Appendix 3: Proof of Lemma 3

Proof

It is assumed that the steady-state probability \(P(0,0)\) is positive, i.e. both virtual queues are non-saturated. Figure 13 illustrates the Markov chain with respect to the number of packets in virtual queue \(v\) at relay node \(\mathbf{R}\). The state transition probability from states 0 to 1 is equal to:

$$\begin{aligned} \lambda _{1,v} = \lambda _v\bigg ( 1-\frac{P(0,0)}{P_v(0)}\bigg ) + \lambda _{0,v}\frac{P(0,0)}{P_v(0)}. \end{aligned}$$
(50)

The detailed balance equations are obtained as follows:

$$\begin{aligned} P_v(1)&= \rho _vP_v(0) + \frac{\rho _v\tau _R}{1-\tau _R}P(0,0), \nonumber \\ P_v(n+1)&= \rho _vP_v(n); n\ge 1. \end{aligned}$$
(51)

Summing all the steady-state probabilities \(P_v(n)\) , the normalized condition and some algebra enable us to obtain Lemma 3 as follows: \(\sum _{n=0}^\infty P_v(n) = 1 \Rightarrow \frac{P_v(0)(1-\tau _R)+\rho _v\tau _RP(0,0)}{(1-\rho _v)(1-\tau _R)} = 1\).\(\square \)

Fig. 13
figure 13

The Markov chain with respect to the number of packets in virtual queue \(v\) at relay node \(\mathbf{R}\) in the NC-CSMA/CA protocol

Appendix 4: Proof of Lemma 4

Proof

Based on Fig. 5, we can express the detailed balance equation as follows:

$$\begin{aligned} P(1,0)&= \frac{\rho _1}{1-\tau _R}P(0,0), P(0,1) = \frac{\rho _2}{1-\tau _R}P(0,0), \nonumber \\ P(n_1+1,n_2)&= \rho _1P(n_1,n_2), P(n_1,n_2+1) = \rho _2P(n_1,n_2), \end{aligned}$$
(52)

for any \((n_1,n_2)\ne (0,0)\). The above detailed balance equations provide:

$$\begin{aligned} P(n_1,n_2) = \frac{\rho _1^{n_1}\rho _2^{n_2}}{1-\tau _R}P(0,0). \end{aligned}$$
(53)

for any \((n_1,n_2)\ne (0,0)\).

Summing all the steady-state probabilities \(P(n_1,n_2)\), which are functions of \(P(0,0)\), and the normalized condition enable us to obtain

$$\begin{aligned} 1&= \sum \limits _{n_1=0}^\infty \sum \limits _{n_2=0}^\infty P(n_1,n_2) = \frac{P(0,0)}{1-\tau _R} (\sum \limits _{n_1=0}^\infty \sum _{n_2=0}^\infty \rho _1^{n_1}\rho _2^{n_2}-\tau _R) \nonumber \\&= \frac{1-\tau _R(1-\rho _1)(1-\rho _2)}{(1-\tau _R)(1-\rho _1)(1-\rho _2)}P(0,0). \end{aligned}$$
(54)

and then an approximate expression of \(P(0,0)\) is derived as

$$\begin{aligned} P(0,0) = \frac{(1-\tau _R)(1-\rho _1)(1-\rho _2)}{1-\tau _R(1-\rho _1)(1-\rho _2)} \end{aligned}$$

Appendix 5: Proof of Proposition 1

Proof

First, we note the following relations:

$$\begin{aligned} \pi _{i,0}&= P_{eq}\pi _{i-1,0}=P_{eq}^i\pi _{0,0}, 1\le i\le m.\end{aligned}$$
(55)
$$\begin{aligned} \pi _{i,k}&= \frac{W_i-k}{P_dW_i}P_{eq}^i\pi _{0,0}, 0\le i\le m, 1\le k\le W_i-1. \end{aligned}$$
(56)

The stationary probability to be in state \(\pi _I\) can be evaluated as follows:

$$\begin{aligned} \pi _I=\pi _I(1-q)+\pi _{m,0}(1-q)P_{eq}+(1-q)(1-P_{eq})\sum _{i=0}^m\pi _{i,0} \Rightarrow \pi _I= \frac{1-q}{q}\pi _{0,0}.\nonumber \\ \end{aligned}$$
(57)

Employing the normalization condition, after some mathematical manipulations, and remembering the relation \(\sum _{i=0}^m\pi _{i,0} = \pi _{0,0}\frac{1-P_{eq}^{m+1}}{1-P_{eq}}\), it is possible to obtain:

$$\begin{aligned}&\displaystyle \sum \limits _{i=0}^m\sum \limits _{k=0}^{W_i-1}\pi _{i,k} +\pi _I = 1,\nonumber \\&\displaystyle \sum \limits _{k=1}^{W_i-1}\pi _{i,k} = \frac{P_{eq}^i\pi _{0,0}}{2P_d}(W_i-1),\nonumber \\&\displaystyle \sum \limits _{i=0}^m\frac{P_{eq}^i\pi _{0,0}}{2P_d}(2^iW-1) =\frac{\pi _{0,0}}{2P_d} \left( \frac{W(1-(2P_{eq})^{m+1})}{1-2P_{eq}} - \frac{(1-P_{eq}^{m+1})}{1-P_{eq}}\right) .\end{aligned}$$
(58)
$$\begin{aligned}&\displaystyle \sum \limits _{i=0}^m\sum \limits _{k=0}^{W_i-1}\pi _{i,k} = \sum \limits _{i=0}^m\sum \limits _{k=1}^{W_i-1}\pi _{i,k}+\sum \limits _{i=0}^m\pi _{i,0} = \nonumber \\&\displaystyle \frac{\pi _{0,0}}{2P_d} \left( \frac{W(1-(2P_{eq})^{m+1})}{1-2P_{eq}} + \frac{(2P_d-1)(1-P_{eq}^{m+1})}{1-P_{eq}}\right) . \end{aligned}$$
(59)

The normalization condition yields the following equation for computation of \(\pi _{0,0}\):

$$\begin{aligned}&\pi _{0,0} \nonumber \\&\quad =\frac{2qP_d(1-P_{eq})(1-2P_{eq})}{2P_d(1-q)(1-P_{eq})(1-2P_{eq})+qW(1-P_{eq})(1-(2P_{eq})^{m+1})+q(1-2P_{eq})(2P_d-1)\left( 1-P_{eq}^{m+1}\right) }.\nonumber \\ \end{aligned}$$
(60)

Equation 60 is then used to compute \(\tau \) , the probability that a station starts a transmission in a randomly chosen time slot. In fact, taking into account that a packet transmission occurs when the back-off counter reaches zero, we have:

$$\begin{aligned} \tau&= \sum \limits _{i=0}^m\pi _{i,0} = \pi _{0,0}\frac{1-P_{eq}^{m+1}}{1-P_{eq}} \nonumber \\&= \frac{2qP_d(1-P_{eq}^{m+1})(1-2P_{eq})}{2P_d(1-q)(1-P_{eq})(1-2P_{eq})+qW(1-P_{eq})(1-(2P_{eq})^{m+1})+q(1-2P_{eq})(2P_d-1)\left( 1-P_{eq}^{m+1}\right) }.\nonumber \\ \end{aligned}$$
(61)

\(\square \)

Appendix 6: Proof of Proposition 2

Proof

According to Fig. 7, there are three durations that the considered node spends at a particular back-off state, \(D_I\), \(D_S\) and \(D_C\). In the idle state, the considered node waits one time slot before decrementing the back-off counter. When the considered node enters the successful state we can compute the duration in this state as follows:

$$\begin{aligned} D_S= \frac{1}{1-p_{ss}}T_{suce}+ \frac{p_{si}}{1-p_{ss}}D_I + \frac{p_{sc}}{1-p_{ss}}D_S, \end{aligned}$$
(62)

where \(D_I=1\). Similarly, when the node enters a back-off state and finds the channel busy with a collision, this duration can be expressed as:

$$\begin{aligned} D_C= \frac{1}{1-p_{cc}}\overline{T_{coll}} + \frac{p_{cs}}{1-p_{cc}}D_S + \frac{p_{ci}}{1-p_{cc}}D_I. \end{aligned}$$
(63)

Let us consider the two cases in detail to calculate the average slot duration for each case:

  • Entering from a previous back-off state: The average slot duration in this case can be expressed using \(P_d\) as

    $$\begin{aligned} D_1 = \frac{1}{P_{d}}(p_{ei}D_I+p_{es}D_S+ p_{ec}D_C). \end{aligned}$$
    (64)
  • Entering from a transmission state: In this case we can compute the average slot duration as follows

    $$\begin{aligned} D_2 = \frac{\overline{CW}-1}{q\overline{CW}}(p_{ei}D_I+p_{es}D_S+ p_{ec}D_C). \end{aligned}$$
    (65)

Then we can compute the average slot duration as \(\fancyscript{D} = (1-\tau )D_1 + \tau D_2\). \(\square \)

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Mirrezaei, S.M., Faez, K. & Ghasemi, A. Performance Analysis of Network Coding Based Two-Way Relay Wireless Networks Deploying IEEE 802.11. Wireless Pers Commun 76, 41–76 (2014). https://doi.org/10.1007/s11277-013-1485-1

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