A Simple but Effective Collision and Error Aware Adaptive Back-Off Mechanism to Improve the Performance of IEEE 802.11 DCF in Error-Prone Environment

Abstract

In a realistic wireless network, all the unsuccessful transmission attempts are not due to packet collisions only. But the transmission errors also played an important role because of the non-ideal channel conditions. In addition, non-saturation condition for a contending station is also a realistic phenomenon. It is now very established fact that performance of the distributed coordination function (DCF) significantly degrades in non-ideal situations due to the traditionally fixed behavior of its binary exponential back-off (BEB) mechanism. To enhance the performance of the DCF, many adaptive back-off mechanisms have been suggested. However, these mechanisms still suffer due to inefficient adaptation procedure followed by them. In order to overcome such inefficiencies, in this paper, we propose an effective contention window (CW) and contention slot selection (CSS) mechanism, called Collision and Error Aware Adaptive Back-off (CEAAB), to improve the performance of IEEE 802.11 DCF. In the proposed CEAAB mechanism, CW and CSS strategies differ from the BEB back-off mechanism to adopt the network situation. A Markov chain based analytical model is developed for the proposed CEAAB mechanism to derive the performance metrics-throughput and delay. Simulation results illustrate that the CEAAB mechanism outperforms the earlier proposed mechanisms.

Appendix

1.1 Derivation of E[BT _wf ]

Equation (44) can be re-written as follows:

$$ \begin{aligned} E\left[ {BT_{wf} } \right] &= \sum\limits_{k = 1}^{{W_{0} - 1}} {k \times \pi_{0,k} } + \sum\limits_{k = 1}^{{W_{i} - 1}} {k \times \pi_{0,k} } + \sum\limits_{k = 1}^{{W_{m} - 1}} {k \times \pi_{0,k} } \\ &= {\rm M} + {\rm N} + {\rm O} \\ \end{aligned} $$

(44a)

The values of M, N and O are derived as following:

$$ \begin{aligned} {\rm M} = & \sum\limits_{k = 1}^{{W_{0} - 1}} {k \cdot \pi_{0,k} } \\ = & \sum\limits_{k = 1}^{{W_{0} - 1}} {k \cdot \left[ {\frac{{\pi_{0,0} }}{{\left( {1 - P_{tr} } \right)}}\sum\limits_{j = k}^{{W_{0} - 1}} {\left\{ {\varPsi \cdot F_{0} \left( j \right) + P_{e} \left( {1 - P_{c} } \right) \cdot G_{0} \left( j \right)} \right\}} } \right]} \\ = & \frac{{\pi_{0,0} }}{{W_{0} \left( {1 - P_{tr} } \right)}}\left[ {\sum\limits_{k = 1}^{{W_{0} - 1}} {\left\{ {k \cdot \sum\limits_{j = k}^{{W_{0} - 1}} {\left\{ {\varPsi + P_{e} \left( {1 - P_{c} } \right)} \right\}} } \right\}} } \right] \\ = & \frac{{\pi_{0,0} }}{{6 \cdot \left( {1 - P_{tr} } \right)}}\left\{ {\left( {W_{0} } \right)^{2} - 1} \right\} \cdot \left\{ {\varPsi + \xi } \right\} \\ \end{aligned} $$

(44b)

The value of N can be derived as follows:

$$ \begin{aligned} {\rm N} = & \sum\limits_{i = 1}^{m - 1} {\sum\limits_{k = 1}^{{W_{i} - 1}} {k \cdot \pi_{i,k} } } \\ = & \sum\limits_{i = 1}^{m - 1} {\left[ {\frac{{\left( {P_{f} } \right)^{i - 1} \cdot \pi_{0,0} }}{{\left( {1 - P_{tr} } \right)}}\left\{ {\sum\limits_{k = 1}^{{W_{i} - 1}} {\sum\limits_{j = k}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {P_{c} \cdot F_{i} \left( j \right) + P_{e} \left( {1 - P_{c} } \right)P_{f} \cdot G_{i} \left( j \right)} \right\}} \right]} } } \right\}} \right]} \\ = & \sum\limits_{i = 1}^{m - 1} {\left[ {\frac{{\left( {P_{f} } \right)^{i - 1} \cdot \pi_{0,0} }}{{\left( {1 - P_{tr} } \right)}}\left\{ {\sum\limits_{k = 1}^{{W_{i} - 1}} {\left[ {kP_{c} \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {F_{i} \left( j \right)} } \right\} + kP_{e} \left( {1 - P_{c} } \right)P_{f} \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {G_{i} \left( j \right)} } \right\}} \right]} } \right\}} \right]} \\ = & \frac{{\pi_{0,0} }}{{\left( {1 - P_{tr} } \right)}}\sum\limits_{i = 1}^{m - 1} {\left[ {\left( {P_{f} } \right)^{i - 1} \left\{ {P_{c} \cdot \varPhi_{1} + P_{e} \left( {1 - P_{c} } \right)P_{f} \cdot \varPhi_{2} } \right\}} \right]} \\ \end{aligned} $$

(44c)

where,

$$ \varPhi_{1} = \sum\limits_{k = 1}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {F_{i} \left( j \right)} } \right\}} \right]} $$

(44d)

$$ \varPhi_{2} = \sum\limits_{k = 1}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {G_{i} \left( j \right)} } \right\}} \right]} $$

(44e)

From Eq. (44d), we have:

$$ \begin{aligned} \varPhi_{1} = & \sum\limits_{k = 1}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {F_{i} \left( j \right)} } \right\}} \right]} \\ = & \sum\limits_{k = 1}^{{W_{i} /2}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {F_{i} \left( j \right)} } \right\}} \right]} + \sum\limits_{{k = \frac{{W_{i} }}{2} + 1}}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {F_{i} \left( j \right)} } \right\}} \right]} \\ = & \sum\limits_{k = 1}^{{\frac{{W_{i} }}{2}}} k - \frac{1}{{2^{i} W_{i} }} \cdot \sum\limits_{k = 1}^{{\frac{{W_{i} }}{2}}} {\left( {k^{2} } \right)} + \left( {1 - \frac{1}{{2^{i + 1} }}} \right) \cdot \varPhi_{11} - \frac{2}{{W_{i} }}\left( {1 - \frac{1}{{2^{i + 1} }}} \right)\varPhi_{12} \\ \end{aligned} $$

(44f)

where,

$$ \begin{aligned} \varPhi_{11} = & \sum\limits_{k = 1}^{{\frac{{W_{i} }}{2} - 1}} {\left( {\frac{{W_{i} }}{2} + k} \right)} \\ = & \frac{3}{8}W_{i} \left( {W_{i} - 2} \right) \\ \end{aligned} $$

(44g)

and

$$ \begin{aligned} \varPhi_{12} = & \sum\limits_{{k = \frac{{W_{i} }}{2} + 1}}^{{W_{i} - 1}} {k \cdot \left( {k - \frac{{W_{i} }}{2}} \right)} \\ = & \frac{1}{48}W_{i} \left( {W_{i} - 2} \right)\left( {5W_{i} - 2} \right) \\ \end{aligned} $$

(44h)

From Eq. (44f–44h), we get the value of Φ ₁ as given below:

$$ \varPhi_{1} = \frac{7}{24}\left( {W_{i} } \right)^{2} - \frac{1}{{2^{i + 3} }}\left( {W_{i} } \right)^{2} - \frac{1}{6} $$

(44i)

From Eq. (44e), we have:

$$ \begin{aligned} \varPhi_{2} = & \sum\limits_{k = 1}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {G_{i} \left( j \right)} } \right\}} \right]} \\ = & \sum\limits_{k = 1}^{{W_{i} /2}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {G_{i} \left( j \right)} } \right\}} \right]} + \sum\limits_{{k = \frac{{W_{i} }}{2} + 1}}^{{W_{i} - 1}} {\left[ {k \cdot \left\{ {1 - \sum\limits_{j = 0}^{k - 1} {G_{i} \left( j \right)} } \right\}} \right]} \\ = & \sum\limits_{k = 1}^{{\frac{{W_{i} }}{2}}} k - \frac{1}{{W_{i} }}\left( {2 - \frac{1}{{2^{i} }}} \right) \cdot \sum\limits_{k = 1}^{{\frac{{W_{i} }}{2}}} {\left( {k^{2} } \right)} + \frac{1}{{2^{i + 1} }} \cdot \varPhi_{11} - \frac{1}{{2^{i} W_{i} }}\varPhi_{12} \\ \end{aligned} $$

(44j)

From Eqs. (44g, 44h, 44j), we get the value of Φ ₂ as given below:

$$ \varPhi_{2} = \frac{1}{24}\left( {W_{i} } \right)^{2} + \frac{1}{{2^{i + 3} }}\left( {W_{i} } \right)^{2} - \frac{1}{6} $$

(44k)

Putting the values of Φ ₁ and Φ ₂ into Eq. (44c), we get the value of N as given below:

$$ \begin{aligned} {\rm N} = & \frac{{\pi_{0,0} }}{{\left( {1 - P_{tr} } \right)}}\sum\limits_{i = 1}^{m - 1} {\left( {P_{f} } \right)}^{i} \left[ {\left\{ {7\theta + \xi } \right\}\frac{{\left( {W_{i} } \right)^{2} }}{24} + \left\{ {\xi - \theta } \right\}\frac{{\left( {W_{i} } \right)^{2} }}{{2^{i + 3} }} - \frac{1}{6}\left\{ {\theta + \xi } \right\}} \right] \\ = & \frac{{\pi_{0,0} }}{{\left( {1 - P_{tr} } \right)}} \cdot \left[ \begin{gathered} \frac{1}{24}\left\{ {7\theta + \xi } \right\} \cdot \left( {W_{0} } \right)^{2} \cdot \frac{{1 - \left( {4P_{f} } \right)^{m - 1} }}{{1 - 4P_{f} }} \hfill \\ + \frac{1}{8}\left\{ {\xi - \theta } \right\} \cdot \left( {W_{0} } \right)^{2} \cdot \frac{{1 - \left( {2P_{f} } \right)^{m - 1} }}{{1 - 2P_{f} }} \hfill \\ - \frac{1}{6}\left\{ {\theta + \xi } \right\} \cdot \frac{{1 - \left( {P_{f} } \right)^{m - 1} }}{{1 - P_{f} }} \hfill \\ \end{gathered} \right] \\ \end{aligned} $$

(44l)

Finally, the value of O can be derived as follows:

$$ \begin{aligned} O &= \sum\limits_{k = 1}^{{W_{i} - 1}} {k \cdot \pi_{i,k} } \\ &= \frac{{P_{c} \left( {P_{f} } \right)^{m - 1} \cdot \pi_{0,0} }}{{\left( {1 - P_{eq} } \right)\left( {1 - P_{tr} } \right)}}\left\{ {\sum\limits_{k = 1}^{{W_{m} - 1}} {k \cdot \sum\limits_{j = k}^{{W_{m} - 1}} {\left[ {\left\{ {\left( {1 + P_{c} - P_{eq} } \right) \cdot F_{m} \left( j \right) + P_{e} \left( {1 - P_{c} } \right) \cdot G_{m} \left( j \right)} \right\}} \right]} } } \right\} \\ &= \frac{{P_{c} \left( {P_{f} } \right)^{m - 1} \cdot \pi_{0,0} }}{{\left( {1 - P_{eq} } \right)\left( {1 - P_{tr} } \right)}}\left\{ {\sum\limits_{k = 1}^{Wm - 1} {\left[ {k\left( {1 + P_{c} - P_{eq} } \right)\left\{ {1 - \sum\limits_{j = 0}^{k - 1} {F_{i} \left( j \right)} } \right\} + kP_{e} \left( {1 - P_{c} } \right)\left\{ {1 - \sum\limits_{j = 0}^{k - 1} {G_{i} \left( j \right)} } \right\}} \right]} } \right\} \\ &= \frac{{P_{c} \left( {P_{f} } \right)^{m - 1} \cdot \pi_{0,0} }}{{\left( {1 - P_{eq} } \right)\left( {1 - P_{tr} } \right)}}\left[ {\left( {1 + P_{c} - P_{eq} } \right)\varPhi_{3} + P_{e} \left( {1 - P_{c} } \right)\varPhi_{4} } \right] \\ \end{aligned} $$

(44m)

where,

$$ \begin{aligned} \varPhi_{3} = & \sum\limits_{k = 1}^{{W_{m} - 1}} {k \cdot \left[ {1 - \sum\limits_{j = 0}^{k - 1} {F_{m} \left( j \right)} } \right]} \\ = & \frac{7}{24}\left( {W_{m} } \right)^{2} - \frac{1}{{2^{m + 3} }}\left( {W_{m} } \right)^{2} - \frac{1}{6} \\ \end{aligned} $$

(44n)

and,

$$ \begin{aligned} \varPhi_{4} &= \sum\limits_{k = 1}^{{W_{m} - 1}} {k \cdot \left[ {1 - \sum\limits_{j = 0}^{k - 1} {G_{m} \left( j \right)} } \right]} \\ &= \frac{1}{24}\left( {W_{m} } \right)^{2} + \frac{1}{{2^{m + 3} }}\left( {W_{m} } \right)^{2} - \frac{1}{6} \hfill \\ \end{aligned} $$

(44o)

From Eqs. (44m–44o), we get the value of O as given below:

$$ O = \frac{{P{}_{c}\left( {P_{f} } \right)^{m - 1} \cdot \pi_{0,0} }}{{\left( {1 - P_{eq} } \right) \cdot \left( {1 - P_{tr} } \right)}} \cdot \left[ \begin{aligned}& \frac{{2^{2m} }}{24}\left\{ {7\theta + \xi } \right\} \cdot \left( {W_{0} } \right)^{2} \hfill \\ &+ \frac{{2^{m} }}{8}\left\{ {\xi - \theta } \right\} \cdot \left( {W_{0} } \right)^{2} \hfill \\ &- \frac{1}{6}\left\{ {\theta + \xi } \right\} \hfill \\ \end{aligned} \right] $$

(44p)

Putting the respective values of M, N and O from Eqs. (44b), (44l) and (44p) into the Eq. (44a), the required value of E[BT _wf ] can be given by Eq. (45).