Greedy Target Channel Selection Strategy

In each spectrum handoff, the greedy target channel selection strategy is sug-gested to select the channel with the shortest expected handoff delay. Some permutations of the target channel sequences will never occur when this shortest-handoff-delay strategy is adopted. Here, we give such an example.

Consider a secondary user’s connection whose default channel is channel 1 (Ch1). In a two-channel system, it can select either channel 1 or channel 2 (Ch2) for its target channel when an interruption event occurs. Now, we assume that the average busy period of Ch1 (denoted by E[Yp⁽¹⁾]) is shorter than the sum of channel switching time (denoted by t_s) and the average waiting time of Ch2 (denoted by E[Ws⁽²⁾]). Hence, when the first inter-ruption occurs and the greedy target channel selection strategy is adopted, the secondary user selects Ch1 as its target channel for spectrum handoff.

The similar argument can be held again for all the upcoming interruptions.

That is, the target channel sequence will be (Ch1, Ch1, Ch1, Ch1, Ch1, · · · ).

In this case, some permutations of the target channel sequences, such as

Figure 6.2: Six kinds of candidate sequences for the Cumulative Hand-off Delay Minimization Problem when the greedy shortest-handHand-off-delay target channel selection strategy is adopted.

(Ch1, Ch2, Ch2, Ch2, Ch2, · · · ) or (Ch1, Ch2, Ch1, Ch1, Ch1, · · · ), will never occur. In Theorem 4, we prove that only six permutations of the target chan-nel sequences are required to be compared when the greedy shortest-handoff-delay target channel selection strategy is employed.

Theorem 4. The shortest-handoff-delay target channel selection strategy only requires to compare six permutations of the target channel sequences, as shown in Fig. 6.2.

Proof. Consider a secondary user’s connection whose default channel is α (α ∈ Ω). If the strategy is to select a channel with the shortest handoff de-lay when an interruption event occurs, the secondary user will compare the expected handoff delay of staying on the same channel and that of changing

to a new channel. Now, we discuss what conditions will cause target chan-nel sequences not to be considered by the greedy target chanchan-nel selection strategy.

(1) At the first interruption: The secondary user can select channel α or channel k (k ∈ Ω/{α}) for the target channel. If staying on channel α, the expected delay for the non-hopping spectrum handoff equals the average busy period of the primary users’ connections at channel α (denoted by E[Yp^(α)]). If changing its operating channel to channel k, the secondary user will experience the delay of the hopping spectrum handoff, which is equal to the sum of the channel switching time (denoted by ts) and the average waiting time on channel k (denoted by E[Ws^(k)]). On the one hand, if the following condition (C1) is satisfied,

(C1) : E[Y_p^(α)] ≤ min

∀k∈Ω/{α}{E[W_s^(k)] + t_s} ,

channel α is the first element in the target channel sequence. This implies that the interrupted secondary user must wait until all the primary users’

connections leave channel α. When the traffic statistics of all channels are stationary, the interrupted secondary user will always stay on channel α because (C1) always can be satisfied for all the upcoming interruptions.

That is, the target channel sequence becomes (α, α, α, α, α, α, · · · ), as shown in Fig. 6.2. On the other hand, if the condition

(C2) :

is satisfied, the first element in the target channel sequence is channel β.

Clearly, (C2) is not sufficient to determine the remaining elements in the target channel sequence.

(2) At the second interruption: When (C2) is satisfied, the secondary user will encounter one of the following three conditions at the second inter-ruption. Denote E[Yp^(β)], E[Ws^(γ)]+t_s, and E[Ws^(α)]+t_sas the average handoff delay for staying on channel β, that for changing to channel γ (γ 6= α and β), and that for switching back to channel α, respectively.

• First, we consider the case

(C3) : E[Y_p^(β)] ≤ min

∀k∈Ω/{β}{E[W_s^(k)] + t_s} .

Similar to (C1), the interrupted secondary user prefers staying on chan-nel β when (C3) is satisfied. Thus, (C2) and (C3) lead to the target channel sequence (β, β, β, · · · ).

• Next, we consider the condition (C4) : E[W_s^(α)] + t_s < min{ min

∀k∈Ω/{α,β}{E[W_s^(k)] + t_s}, E[Y_p^(β)]} . When (C4) is satisfied, the interrupted secondary user will switch back to channel α. After that, the third interruption event may occur. If so, this interrupted secondary user will switch back to channel β due to (C2). Hence, (C2) and (C4) yields the target channel sequence (β, α, β, α, β, α, · · · ).

• When (C3) and (C4) are not satisfied, it implies that there exists channel γ (γ 6= α) such that

Then, the second element in the target channel sequence is channel γ. Since (C2) and (C5) are not sufficient to determine the remaining

elements in the target channel sequence, the third interruption event will be considered.

(3) At the third interruption: In this case, we need to compare the expected handoff delay of staying on channel γ and that of switching back to channels α and β, i.e., E[Yp^(γ)], E[Ws^(α)] + t_s, and E[Ws^(β)] + t_s, respectively.

Given (C2) and (C5), three different situations exist.

• First of all, if the condition

(C6) : E[Y_p^(γ)] ≤ min

∀k∈Ω/{γ}{E[W_s^(k)] + t_s}

is satisfied, the interrupted secondary user prefers staying on channel γ.

Hence, (C2), (C5), and (C6) result in the the target channel sequence (β, γ, γ, γ, · · · ).

• Furthermore, if the condition

(C7) : E[W_s^(α)] + t_s< min{ min

∀k∈Ω/{α,γ}{E[W_s^(k)] + t_s}, E[Y_p^(γ)]}

is satisfied, the interrupted secondary user switches back to channel α.

Then, (C2) and (C5) will make this secondary user switches to channel β and channel γ at the fourth and fifth interruptions, respectively.

Thus, when (C2), (C5) and (C7) are satisfied, the target channel sequence becomes (β, γ, α, β, γ, α, β, γ, α, · · · ).

• Similarly, when

(C8) : E[W_s^(β)] + t_s< min{ min

∀k∈Ω/{β,γ}{E[W_s^(k)] + t_s}, E[Y_p^(γ)]}

is satisfied, one can show that the target channel sequence is (β, γ, β, γ, β, γ, · · · ).

Now, we will prove that it is not necessary to consider other channels except for channels α, β, and γ when the third interruption event occurs.

Assume that there exists channel ξ = arg min

∀k∈Ω/{α,β,γ} However, from (C2), we know that

E[W_s^(β)] + ts< E[W_s^(k)] + ts, ∀ k 6= α, β . (6.17) Since (6.16) yields E[Ws^(ξ)] < E[Ws^(β)], but (6.17) yields E[Ws^(β)] < E[Ws^(ξ)], it leads to a contradiction and proves that no other channels need to be considered further.

From the above discussions, (C1)-(C8) have considered all the condi-tions between E[Ws^(k)] and E[Yp^(k)]. Hence, we can conclude that the greedy shortest-handoff-delay target channel selection strategy results in only six permutations of the target channel sequences that are needed to be further compared in the Cumulative Handoff Delay Minimization Problem, are shown in Fig. 6.2.

Based on Theorem 4, the transmitter and receiver need to consider only three channels for spectrum handoff as long as the greedy shortest-handoff-delay target channel selection strategy is considered. Thus, it relieves channel consensus issue in CR networks. Theorem 4 can be extended to other greedy strategies for the target channel selection based on various criteria, such as the longest expected remaining idle period.