Stability and Delay Analysis of an Adaptive Channel-Aware Random Access Wireless Network

Abstract

In this work, we consider an asymmetric two-user random access wireless network with interacting nodes, time-varying links and multipacket reception capabilities. The users are equipped with infinite capacity buffers where they store arriving packets that will be transmitted to a destination node. Moreover, each user employs a general transmission control protocol under which, it adapts its transmission probability based both on the state of the other user, and on the channel state information according to a Gilbert-Elliot model. We study a two-dimensional discrete time Markov chain, investigate its stability condition, and show that its steady state performance is expressed in terms of a solution of a Riemann-Hilbert boundary value problem. Moreover, for the symmetrical system, we provide closed form expressions for the average delay at each user node. Numerical results are obtained and show insights in the system performance.

Appendices

A Appendix

Regarding the derivation of (2), the queue evolution in (1) implies,

$$\begin{aligned} \begin{array}{l} E(x^{N_{1,n+1}}y^{N_{2,n+1}})=D(x,y)\left( P(N_{1,n}=N_{2,n}=0)\right. \\ \left. +E(x^{N_{1,n}}1_{\{N_{1,n}>0,N_{2,n}=0\}})[1+s_{G}^{(1)}q_{G1}^{*}\tilde{f}_{G,1/\{G,1\}}(\frac{1}{x}-1)]\right. \\ \left. +E(y^{N_{2,n}}1_{\{N_{1,n}=0,N_{2,n}>0\}})[1+s_{G}^{(2)}q_{G2}^{*}\tilde{f}_{G,2/\{G,2\}}(\frac{1}{y}-1)\right. \\ \left. +E(x^{N_{1,n}}y^{N_{2,n}}1_{\{N_{1,n}>0,N_{2,n}>0\}})[s_{G}^{(1)}q_{G1}(s_{G}^{(2)}\bar{q}_{G2}+s_{B}^{(2)}\bar{q}_{B2})\right. \\ \left. \times (1+f_{G,1/\{G,1\}}(\frac{1}{x}-1)) +s_{G}^{(2)}q_{G2}(s_{G}^{(1)}\bar{q}_{G1}+s_{B}^{(1)}\bar{q}_{B1})(1+f_{G,2/\{G,2\}}(\frac{1}{y}-1))]\right. \\ \left. +s_{G}^{(1)}q_{G1}s_{G}^{(2)}q_{G2}(1+f_{G,1/\{G,1;G,2\}}(\frac{1}{x}-1)+f_{G,2/\{G,1;G,2\}}(\frac{1}{y}-1))\right. \\ \left. +s_{G}^{(1)}q_{G1}s_{B}^{(2)}q_{B2}(1+f_{G,1/\{G,1;B,2\}}(\frac{1}{x}-1))+s_{B}^{(1)}q_{B1}s_{G}^{(2)}q_{G2}\right. \\ \left. \times (1+f_{G,2/\{B,1;G,2\}}(\frac{1}{y}-1))+(s_{G}^{(1)}\bar{q}_{G1}+s_{B}^{(1)}\bar{q}_{B1})(s_{G}^{(2)}\bar{q}_{G2}+s_{B}^{(2)}\bar{q}_{B2})\right. \\ \left. +s_{B}^{(1)}q_{B1}s_{B}^{(2)}q_{B2}+s_{B}^{(2)}q_{B2}(s_{G}^{(1)}\bar{q}_{G1}+s_{B}^{(1)}q_{B1})+s_{B}^{(1)}\bar{q}_{B1}(s_{G}^{(2)}\bar{q}_{G2}+s_{B}^{(2)}\bar{q}_{B2})\right) , \end{array} \end{aligned}$$

where $1_{\{A\}}$ denotes the indicator function of the event A. Note that

$$\begin{aligned} \begin{array}{c} H(x,0)-H(0,0)=\lim _{n\rightarrow \infty }E(x^{N_{1,n}}1_{\{N_{1,n}>0,N_{2,n}=0\}}),\\ H(0,y)-H(0,0)=\lim _{n\rightarrow \infty }E(y^{N_{2,n}}1_{\{N_{1,n}=0,N_{2,n}>0\}}),\\ H(x,y)-H(x,0)-H(0,y)+H(0,0)=\lim _{n\rightarrow \infty }E(x^{N_{1,n}}y^{N_{2,n}}1_{\{N_{1,n}>0,N_{2,n}>0\}}). \end{array} \end{aligned}$$

B Appendix

In the following, we proceed with the study of the location of the intersection points of $R(x,y)=0$, $A(x,y)=0$ (resp. B(x, y)). These points (if exist) are potential singularities for the functions H(x, 0), H(0, y), and thus, their investigation is crucial regarding the analytic continuation of H(x, 0), H(0, y) outside the unit disk. We only focus on the intersection points of $R(x,y)=0$, $A(x,y)=0$.

For and $R(x,y) = 0$, $y = Y_{\pm }(x)$, the resultant in y of the two polynomials R(x, y) and A(x, y) is $Res_{y}(R,A;x)=x(x-1)s_{G}^{(2)}q_{G2}\widehat{q}_{21}Z(x)$, where

$$\begin{aligned} \begin{array}{rl} Z(x)=&{}-\widehat{\lambda }_{1}(s_{G}^{(2)}q_{G2}\widehat{q}_{21}+(1+\widehat{\lambda }_{1})d_{1,2})x^{2}+x[(\widehat{\lambda }+\widehat{\lambda }_{1}\widehat{\lambda }_{2})d_{1,2}\\ {} &{}+(s_{G}^{(2)}q_{G2}\widehat{q}_{21}+d_{1,2})s_{G}^{(1)}q_{G1}^{*}\tilde{f}_{G1/\{G1\}}]-s_{G}^{(1)}q_{G1}^{*}\tilde{f}_{G1/\{G1\}}d_{1,2}. \end{array} \end{aligned}$$

Note also that $Z(0)>0$ since $d_{1,2}<0$, and $Z(1)>0$, due to the stability conditions (see Lemma 1). If $q_{G1}^{*}\le min\{1,\frac{s_{G}^{(2)}q_{G2}\widehat{q}_{21}+(1+\widehat{\lambda }_{2})s_{G}^{(1)}q_{G1}\widehat{q}_{12}}{(1+\widehat{\lambda }_{2})s_{G}^{(1)}\tilde{f}_{G1/\{G1\}}}\}$, then $\lim _{x\rightarrow \infty }Z(x)=-\infty$, and $Z(x)=0$ has two roots of opposite sign, say $x_{*}<0<1<x^{*}$. If $\frac{s_{G}^{(2)}q_{G2}\widehat{q}_{21}+(1+\widehat{\lambda }_{2})s_{G}^{(1)}q_{G1}\widehat{q}_{12}}{(1+\widehat{\lambda }_{2})s_{G}^{(1)}\tilde{f}_{G1/\{G1\}}}<\alpha _{1}^{*}\le 1$, then $\lim _{x\rightarrow \infty }Z(x)=+\infty$, and $Z(x)=0$ has two positive roots, say $1<\tilde{x}_{*}<x_{3}<x_{4}<\tilde{x}^{*}$ (due to the stability conditions). In the former case we have to check if $x^{*}\in S_{x}$, while in the latter case if $\tilde{x}_{*}\in S_{x}$. These zeros, if they lie in $S_{x}$ such that $|Y_{0}(x)|\le 1$, are poles of A(x, y). Denote from hereon $\bar{x}=x^{*}$, if $\alpha _{1}^{*}\le min\{1,\frac{s_{G}^{(2)}q_{G2}\widehat{q}_{21}+(1+\widehat{\lambda }_{2})s_{G}^{(1)}q_{G1}\widehat{q}_{12}}{(1+\widehat{\lambda }_{2})s_{G}^{(1)}\tilde{f}_{G1/\{G1\}}}\}$, and $\bar{x}=\tilde{x}_{*}$, if $\frac{s_{G}^{(2)}q_{G2}\widehat{q}_{21}+(1+\widehat{\lambda }_{2})s_{G}^{(1)}q_{G1}\widehat{q}_{12}}{(1+\widehat{\lambda }_{2})s_{G}^{(1)}\tilde{f}_{G1/\{G1\}}}<\alpha _{1}^{*}\le 1$.