Fig. 8

The optimal action \(a_{1}^{*}\) vs. queue state b ₁ and channel state g ₂ when b ₂=3 and g ₁=3 (left), and \(a_{2}^{*}\) vs. b ₂ and g ₁ when b ₁=2 and g ₂=1 (right). p ₁=0.1, p ₂=0.2, λ=0.05, τ=1, η=2, ξ _o =4 and β=0.95. Two channels are both Rayleigh fading with \(\overline {\gamma }_{1}=\overline {\gamma }_{2}=0\text {dB}\) and are both modeled by 8-state equiprobable FSMCs. In this case, \(\beta \leq \frac {P_{e}(g_{i})-P_{e}(g_{i}+1)} {\sum _{g'_{i}}P_{g_{i}g'_{i}} (P_{e}(g'_{i})-P_{e}(g'_{i}+1)) }\), and according to Corollary 4.16, Theorem 4.15 holds. Therefore, \(a_{1}^{*}\) and \(a_{2}^{*}\) are nondecreasing in (b ₁,g ₂) and (b ₂,g ₁), respectively