## 2 Problem Formulation  
We consider a similar system model as we formulated in [12]. Specifically, we consider two cognitive radios in a cognitive radio network communicating with each other. They operate on the same frequency range which consists of \(N\) channels. During communications, the two radios exchange the channel state information so common vacant channels are known to both radios. Wedenote \(M\) current vacant channels (except the current operating channel) as \(\{c_{k}\in\{1,2,...,N\}\}_{k=1}^{M}\), \(M\leq N-1\). If a primary user appears on the current channel, the two cognitive radios would initiate spectrum handoff and switch to another vacant channel for data transmission. A proactive-sensing spectrum handoff is assumed where the target channels for spectrum handoff are ready before initiating the spectrum handoff process [16]. In this paper, \(\{c_{k}\in\{1,2,...,N\}\}_{k=1}^{M}\) are used as target channels for spectrum handoff. Before spectrum handoff, the visiting order of these target channels needs to be determined to optimize some objective. Denote the vector corresponding to the visiting order as \(\boldsymbol{v}=[v_{1},v_{2},...,v_{M}]\), then the process of spectrum handoff can be illustrated as in Figure 1. When handshake on a particular channel is successful, spectrum handoff is completed successfully and a new link is maintained on this channel. However, in a dynamic spectrum environment, as primary users may reoccupy the visiting target channel during spectrum handoff, handshake may still fail. When CRs try out all the channels and all handshakes fail, then spectrum handoff is failed.  
Denote \(\rho_{i}\) as the probability that the handshake on \(v_{i}\) is successful, then the probability of spectrum handoff failure is  
\[P_{\text{fail}}=\prod_{i=1}^{M}\left(1-\rho_{i}\right) \tag{1}\]  
An important metric for evaluating spectrum handoff is spectrum handoff delay. If spectrum handoff is successful, then the handoff delay is \(T\ell\), where \(\ell\)  is the number of handshakes till success. If spectrum handoff is failed, then some other rendezvous scheme needs to be performed to maintain communications. We assume the rendezvous time is \(\tau\). So the handoff delay when spectrum handoff is failed is \(MT+\tau\). The expected handoff delay is  
\[E[t]=\sum_{L=1}^{M}LT\Pr\{\ell=L\}+\left(MT+\tau\right)P_{\text{fail}}=Tp_{1 }+\sum_{i=2}^{M}iTp_{i}\prod_{k=1}^{i-1}\left(1-p_{k}\right)+\left(MT+\tau \right)\prod_{i=1}^{M}\left(1-p_{i}\right) \tag{2}\]  
Denote the set of vectors corresponding to all permutations of all elements in \(\{c_{k}\}_{k=1}^{M}\) as \(\Omega\), then in order to minimize spectrum handoff delay, the problem of target channel visiting order design for spectrum handoff can be represented as  
\[\boldsymbol{v}^{*}=\underset{\text{rel}\Omega}{\arg\min}\,E[t] \tag{3}\]  
Figure 1: Handshake process for spectrum handoff.  
Note that \(\rho_{i}\) depends on the specific distribution of channel \(v_{i}\) remaining vacant. In this paper, we consider five distributions: Uniform, Exponential [17], Generalized Pareto [18], Rayleigh [19], and Weibull [19]. As in [17], we do not consider the case where the channel state changes twice or more because the probability is too low within the relatively short duration. The probability density functions (PDFs) and the corresponding expressions of \(\rho_{i}\) are shown in Table 1. For uniform distribution, the probability that the handshake on \(v_{i}\) is failed equals to the probability that the state of channel \(v_{i}\) changes to occupied before the end of this handshake, which is  
\[1-\rho_{i}=\begin{cases}\int_{0}^{(i-1)T+T_{k}}1/\,b_{v_{i}}dx=\dfrac{(i-1)T+T_ {k}}{b_{v_{i}}},&(i-1)T+T<b_{v_{i}}\\ 1,&(i-1)T+T\geq b_{v_{i}}\end{cases} \tag{4}\]  
Equation (4) can be further simplified as  
\[1-\rho_{i}=\min\left\{\dfrac{(i-1)T+T_{k}}{b_{v_{i}}},1\right\} \tag{5}\]  
Other expressions of \(\rho_{i}\) can be obtained similarly.