Opportunistic Channel Scheduling for Ad Hoc Networks with Queue Stability

Appendix: Proof of theorem 1

We first prove part 1). From Lyapunov algorithm in Table 1, we can obtain the optimal stopping time Nl∗ at frame l by calculating the following equation:

Obviously, any other strategy leads to a smaller value than Lyapunov algorithm, including the i.i.d algorithm. Therefore, we have

where Niid means an i.i.d algorithm. Before further proof, we give Lemma 2.

Lemma 2: When the constraints in eq. (5) are feasible and bound requirements of {yNl,TNl,Bk,Nl} in Section 3 hold, then for any η>0 and ε>0, there is an i.i.d algorithm satisfying

Proof: Denote Ξ as the set that contains all the values of {E[yNiid],(E[Bk,Niid])k=1K,E[TNiid]} obtained by i.i.d algorithms. For any i.i.d algorithm, we can express it as the linear mixture of two other i.i.d algorithms, thus the set Ξ is convex. From the bound constraints Ymin≤E[yNl|Q(l)]≤Ymax and Tmin≤E[TNl|Q(l)]≤Tmax in Section 3, we can conclude the set is also bounded.

We know that the optimal stopping rule Nl∗ which may not be an i.i.d algorithm is determined by the queue information Q(l). It is noted that Nl∗ can also be regarded as one of the i.i.d algorithms based on the unique conditional distribution which is corresponding to the queue information Q(l). Therefore, all the stopping rules belong to the set Ξ.

From the statement above, Nl∗ at frame l can be replaced with some i.i.d algorithm Niid, thus the summation 1L{∑l=0L−1E[yNl*],∑l=0L−1E[TNl*],∑l=0L−1E[Bk,Nl*]} can be regarded as the summation of different i.i.d algorithms. Since the set Ξ is convex and bounded, the linear combination of different i.i.d algorithms also forms an i.i.d algorithm. For any i.i.d algorithm in eq. (30) which attains near to the optimal value λˆ∗, the proof is similar with the optimal solution Nl∗, thus the proof is completed. □

Substituting eq. (30) into eq. (29), we have

Adding −E[VyNl|Q(l)] on both sides of eq. (10), and implementing the Lyapunov algorithm, we have

Substituting eq. (31) into eq. (32), we have

Taking expectation on both sides, and using the law of iterated expectation, we have

where the law of iterated expectation for any random variables X and Y is given by E{X}=E[E{X|Y}]. Summing eq. (34) from l=0 to l=L−1, then dividing L on both sides and taking the limitation L→∞, we have the following equation when η→0 and ε→0:

Since E[F(0)]=0 and E[F(L)]≥0, rearranging the terms in eq. (35), the proof of part 1) is completed.

To prove 2), we have the following formulation by using the bound constraints Ymin≤E[yNl|Q(l)]≤Ymax and Tmin≤E[TNl|Q(l)]≤Tmax in Section 3 to eq. (33):

Taking expectations of eq. (36) and summing from l=0 to l=L−1, then dividing L on both sides and taking the limitation L→∞, we have eq. (12) when η→0. □