As discussed in Section 4.3.1, the overall system time consists of the waiting time and the extended data delivery time. Let E[Tpb] and E[Tsb] be the average data delivery time for the probability- and sensing-based spectrum decision methods, respectively. Furthermore, denote E[Wpb] and E[Wsb] as the average waiting time for the probability- and sensing-based spectrum decision methods, respectively. Then, we can have
E[Spb] = E[Wpb] + E[Tpb] , (4.8) and
E[Ssb] = E[Wsb] + E[Tsb] . (4.9) In the following, we will investigate how to obtain the average extended data delivery time and the average waiting time.
4.4.1 Extended Data Delivery Time
First, we investigate the effects of multiple interruptions on the extended data delivery time. Within the transmission period of a secondary connection, it is likely to have multiple spectrum handoffs due to the interruptions from the primary users. The spectrum handoff procedure helps the secondary users vacate the occupied channel and then resume the unfinished transmission
when this channel becomes idle. Clearly, multiple spectrum handoffs will increase the extended data delivery time and degrade the QoS for the latency-sensitive traffic of the secondary users [84].
Based on the PRP M/G/1 queueing model, we can derive the extended data delivery time of the secondary connections as follows. Let N(k) be the total number of interruptions for a secondary connection at channel k. Fur-thermore, denote Yp(k) as the duration from the time instant that channel k is occupied by the primary connections until the time instant that the high-priority queue becomes empty. This duration is called the busy period resulting from transmissions of multiple primary connections at channel k.
When a secondary connection is interrupted by primary users, it must stop transmitting on the current operating channel until all the primary connec-tions in the high-priority queue have been served. In this case, the secondary connections of channel k must wait for the duration of E[Yp(k)] on average after the interruption event occurs. Denote eXs as the actual service time of the secondary connections when the effects of sensing errors are considered2 and T(k) as the extended data delivery time of the secondary connections at channel k. We can have
E[T(k)] = E[ eXs] + E[N(k)]E[Yp(k)] . (4.10) Let eXp(k) be the actual service time of the primary connections at channel k when the effects of sensing errors are considered. One can obtain E[N(k)] =
2Although this chapter assumes that all M channels have the same data transmission rate (or equivalently service rate), the proposed model can be applied to the CR network where all channels have different data rates. In the CR network with heterogeneous data rates, the secondary connections at different channels have different average service time.
Hence, they will have different average actual service time. In this case, the notation eXs
in (4.10) should be replaced by the notation eXs(k), which is the actual service time of the secondary connections at channel k. More discussions had been shown in [79].
λ(k)p E[ eXs] and E[Yp(k)] = E[ eXp(k)]
1−λ(k)p E[ eXp(k)] according to to [85]. Note that E[ eXs] and E[ eXp(k)] will be derived in Section 4.5.
Finally, the average extended data delivery time for the probability- and sensing-based channel selection methods can be expressed as follows:
E[Tpb] = For various channel selection algorithms, we use different methods to eval-uate the corresponding distribution probability vectors p. For the probability-based scheme, the distribution probability vector ppbcan be designed by solv-ing the Overall System Time Minimization Problem for Probability-based Channel Selection Scheme in (4.2). For the sensing-Probability-based scheme, the distribution probability vector psb is determined inherently based on the given traffic patterns. Intuitively, a channel with larger idle probability will be selected more frequently through spectrum sensing. How to derive psb from the given traffic parameters will be discussed in Appendix A.
4.4.2 Waiting Time
Next, we focus on the derivations of the average waiting time for the probability-based and sensing-probability-based channel selection schemes.
Probability-based Channel Selection Scheme
For the probability-based channel selection scheme, a secondary connection selects its operating channel based on the predetermined probability. Then, it is directly connected to the low-priority queue of the selected channel. It
cannot be served until all the primary and the secondary connections in the high-priority queue and the present low-priority queue of the selected channel have been served. Hence, the waiting time is the required duration from the time instant that a secondary connection arrives at the low-priority queue of the selected channel until the time instant that the selected channel becomes idle. That is, the waiting time is the duration spent in the waiting queue by a secondary connection. Hence, E[Wpb] can be expressed as follows:
E[Wpb] = XM
k=1
p(k)pb E[Wpb(k)] , (4.13)
where Wpb(k) is the waiting time of the secondary connections at channel k for the probability-based channel selection scheme. Applying the PRP M/G/1 queueing theory [86], one can obtain
E[Wpb(k)] = E[R(k)]
(1 − ρ(k)p )(1 − ρ(k)p − ρ(k)s ) , (4.14) where ρ(k)p and ρ(k)s are the busy probabilities resulting from the primary and the secondary connections at channel k when sensing errors are considered, respectively. Hence, we can have ρ(k)p = λ(k)p E[ eXp(k)] and ρ(k)s = λ(k)s E[ eXs].
Furthermore, E[R(k)] is the average remaining time to complete the service of the connection being served at channel k. Referring to [86], we have
E[R(k)] = 1
2λ(k)p E[( eXp(k))2] + 1
2p(k)pb λsE[( eXs)2] . (4.15) Then, substituting (4.14) and (4.15) into (4.13), we can obtain the closed-from expression for E[Wpb].
Finally, substituting (4.11) and (4.13) into (4.8), we can obtain the rela-tionship between the average overall system time and the distribution prob-ability vector ppb for the probability-based channel selection scheme. Then, the optimal distribution probability vector p∗ can be determined by solving
the Overall System Time Minimization Problem for Probability-based Channel Selection in (4.2).
Sensing-based Channel Selection Scheme
The waiting time Wsbfor the sensing-based channel selection method consists of the total sensing time and the queueing time (denoted by Wsb0 ). Let τ be the sensing time for scanning one candidate channel. Hence, nτ is the total sensing time for scanning all the n candidate channels. After wideband sensing, the secondary user can decide channel availability and then transmits data at one of the idle channels. Moreover, if the idle channel cannot be found, the secondary user cannot transmit immediately. In this case, the secondary user’s connection will be put into the low-priority queue of the randomly selected channel. Hence, we can have
E[Wsb] = nτ + Pr(E) × 0 + Pr(Ec) × E[Wsb0 ] , (4.16) where E is the event that at least one idle channel can be found after sensing, and Ec is the compliment of E.
Next, the closed-form expressions for Pr(E) and Pr(Ec) can be derived by the following two observations. First, a channel is called actual idle if and only if (1) this channel is not occupied by the primary connections and (2) the low-priority queue of this channel is empty. Note that the second condition should be contained because the FCFS scheduling discipline is adopted. Secondly, an actual idle channel is assessed as idle through spectrum sensing if and only if false alarm does not occur. Hence, we can have
Pr(E) = Xn k=1
[Pr(E|k channels are actually idle) × Pr(k channels are actually idle)]
=
where ρ(k) = ρ(k)p + ρ(k)s and PF is the false alarm probability. On the other hand, Ec is the compliment of E. That is,
Pr(Ec) = 1 − Pr(E) . (4.18)
Moreover, when all channels are assessed as busy, each channel is selected by the secondary users with probability 1/n. Hence, in this case, one can derive the average queueing time based on the PRP M/G/1 queueing theory as follows [86] :
Finally, substituting (4.12) and (4.16) into (4.9), we can obtain the relation-ship between the average overall system time and the number of candidate channels n for the sensing-based channel selection scheme.
Determining the optimal number of candidate channels (denoted by n∗) is the key issue for sensing-based spectrum decision scheme. Intuitively, a small number of candidate channels can reduce the total sensing time nτ in (4.16). However, it is harder to find one idle channel from fewer candidate channels, resulting in a larger value of Pr(Ec) in (4.16) and thus increasing the overall system time. The optimal number of candidate channels n∗can be determined by solving the Overall System Time Minimization Problem for Sensing-based Channel Selection in (4.6).