Cross-Layer Design of 2D Queuing Model for Multi-hop Wireless Networks

Abstract

Based on cross-layer design, a modified 2-dimensional queuing model (2DQM) is proposed in this paper to tackle the problem of end-to-end quality of service (QoS) metric calculation. This model exploits the traffic arrival process, multi-rate transmission in the physical layer and error recovery technology with the protocol of truncated automatic repeat request in the data link layer. Based on this model, QoS metrics of wireless links can be evaluated hop by hop. The model can be used in more realistic scenarios of multi-hop wireless networks, although the computational complexity of 2DQM is slightly higher compared with existing 1-dimensional queuing model. Simulation results indicate that the proposed model can estimate the end-to-end packet loss-rate and average delay more accurately than existing models, and a model based QoS routing algorithm can find routes with better QoS performance (with lower end-to-end packet loss-rate and delay).

Appendices

Appendix 1: Derivation of the Average Number of Transmissions per Data Packet

Let the maximum number of retransmissions be $N_r^{\max }$, then the average number of retransmissions per data packet can be expressed by the expectation of the number of transmissions, i.e. $\bar{{N}}=E(N).$ So we have,

$$\begin{aligned} \bar{{N}}(p_0 ,N_r^{\max } )=(1-p_0 )+2p_0 (1-p_0 )+3p_0^2 (1-p_0 )+\cdots +(N_r^{\max } +1)p_0^{N_r^{\max }}\qquad \end{aligned}$$

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or,

$$\begin{aligned} \bar{{N}}(p_0 ,N_r^{\max } )=1+p_0 +p_0^2 +\cdots +p_0^{N_r^{\max } } =\frac{1-p_0^{N_r^{\max } +1} }{1-p_0 } \end{aligned}$$

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Appendix 2: Computation of Channel State Transition Probability

We derive the channel state transition probability $P_{d|c}^{(k)}$ in Sect. 4. Suppose there are $N+1$ states (i.e. State $0, 1, {\ldots }, N$) in the channel, and the state transition only happens between adjacent states, while the transition probability between non-adjacent states being zero (e.g. slow fading channel), i.e.

$$\begin{aligned} P_{m,n} =0,\quad |m-n|\ge 2 \end{aligned}$$

Then, the transition probability between adjacent states will be [28],

$$\begin{aligned} \left\{ {\begin{array}{l@{\quad }l} P_{n,n+1} =\frac{N_{n+1} T_f }{\Pr (n)},&{} \textit{if}~ n=0,\ldots ,N-1 \\ P_{n,n-1} =\frac{N_n T_f }{\Pr (n)},&{} \textit{if}~ n=1,\ldots ,N \\ \end{array}} \right. \end{aligned}$$

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Where, $N_{n}$ is the crossing rate of states under TM $n$, which can be estimated by [29] (for Nakagami-m fading channels)

$$\begin{aligned} N_n =\sqrt{2\pi \frac{m\gamma _n }{\bar{{\gamma }}}}\frac{f_d }{\Gamma (m)}\left( \frac{m\gamma _n }{\bar{{\gamma }}}\right) ^{m-1}\exp \left( -\frac{m\gamma _n }{\bar{{\gamma }}}\right) \end{aligned}$$

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Where, $f_{d}$ is the Doppler frequency due to movement, $\gamma _n$ the SINR threshold of TM $n$, $\bar{{\gamma }}$ the average SINR ($\overline{\textit{SINR}} )$. Then the probability of channel staying at state $n$ in the next time frame is

$$\begin{aligned} P_{n,n} =\left\{ {\begin{array}{l@{\quad }l} 1-P_{n,n+1} -P_{n,n-1} ,&{}\textit{if}~ 0<n<N \\ 1-P_{0,1} ,&{} \textit{if}~ n=0 \\ 1-P_{N,N-1} ,&{} \textit{if}~ n=N \\ \end{array}} \right. \end{aligned}$$

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$P_{d|c}^{(k)}$ can now be computed using a FSMC channel model represented by a $(N+1)\times (N+1)$ state transition matrix below,

$$\begin{aligned} P_c =\left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} P_{0,0 }&{} P_{0,1}&{}\ldots &{} 0 \\ P_{1,0 }&{} P_{1,1}&{} P_{1,2}&{} \vdots \\ 0 &{}\ddots &{} \ddots &{} 0 \\ \vdots &{} P_{N-1,N-2}&{} P_{N-1,N-1} &{}P_{N-1,N} \\ 0&{} \ldots &{} P_{N,N-1} &{}P_{N,N} \\ \end{array}} \right] \end{aligned}$$

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