Based on the proposed PRP M/G/1 queuing network model, we can eval-uate many performance metrics of the secondary connections with various target channel sequences. In this chapter, we focus on the analysis of the extended data delivery time, which is an important performance measure for the latency-sensitive traffic of the secondary connections.
A secondary connection may encounter many interruptions during its transmission period. Without loss of generality, we consider a secondary
High-priority
Figure 5.2: The PRP M/G/1 queueing network model with three channels where λ(k)p , λ(k)s , and ωn(k) are the arrival rates of the primary connections, the secondary connections, and the type-n secondary connections (n ≥ 1) at channel k. Note that ω0(k) = λ(k)s . Furthermore, fp(k)(x) and fi(k)(φ) are the pdfs of Xp(k) and Φ(k)i , respectively.
connection whose default channel is channel η in the following discussions.
Let N be the total number of interruptions of this secondary connection.
Then, the average extended data delivery time of this secondary connection can be expressed as considered secondary connection can be divided into many segments due to multiple interruptions as discussed in Fig. 5.1. Hence, the extended data delivery time of this secondary connection consists of the original service time and the cumulative delay resulting from multiple handoffs. Let Di be the handoff delay of the considered secondary connection for the ithinterruption.
When N = n, we have Di = 0 if i ≥ n+1. Then, the conditional expectation of the extended data delivery time of the considered secondary connection given the event N = n can be derived as
E[T |N = n] = E[Xs(η)] + Xn
i=1
E[Di] . (5.2)
Next, we investigate how to derive the value of Pr(N = n) of (5.1).
For the considered secondary connection, denote s0,η and si,η as its default channel and its target channel at the ith interruption, respectively. Thus, we have s0,η = η and this secondary connection’s target channel sequence can be expressed as (s1,η, s2,η, s3,η, · · · ). Let p(si i,η) be the probability that the considered secondary connection is interrupted again at channel si,η when it has experienced i interruption. Then, the probability that the considered secondary connection is interrupted exactly n times can be expressed as
Pr(N = n) = (1 − p(snn,η))
n−1Y
i=0
p(si i,η) . (5.3)
Finally, substituting (5.2) an (5.3) into (5.1) yields
where the values of E[Di] and p(k)i can be obtained from the Propositions 1 and 2, respectively.
Proof. The handoff delay E[Di] depends on which channel is selected for the target channel at the ith interruption. For the secondary connection with (i − 1) interruptions, its current operating channel is si−1,η. When it is interrupted again, its new operating channel is si,η. When si−1,η = si,η, it means that the considered secondary connection will stay on the current channel. When si−1,η 6= si,η, it represents that the considered secondary connection will change its operating channel to another channel. Both cases are discussed as follows.
(1) Staying case: When the considered secondary connection stays on its current operating channel si,η = k, it cannot be resumed until all the
high-priority primary connections of channel k finish their transmissions.
Hence, the handoff delay is the busy period resulting from multiple primary connections of channel k (denoted by Yp(k)) as discussed in Section 5.2.2.
That is, we can have E[Di] = E[Yp(k)].
The value of E[Yp(k)] can be derived as follows. Denote Ipas the idle period resulting from the primary connections. This idle period is the duration from the termination of the busy period to the arrival of the next primary connection. Because of the memoryless property, the idle period follows the exponential distribution with rate λ(k)p . Hence, we have
E[Ip(k)] = 1 λ(k)p
. (5.8)
Next, according to the definition of the utilization factor at channel k, we have
ρ(k)p = λ(k)p E[Xp(k)] . (5.9) Because ρ(k)p is also the busy probability resulting from the primary connec-tions of channel k, we have
ρ(k)p = E[Yp(k)]
E[Yp(k)] + E[Ip(k)] . (5.10) Then, substituting (5.8) and (5.9) into (5.10), we can obtain (5.6).
(2) Changing case: In this case, the considered secondary connection will change to channel si,η = k0. After switching channel from channel k to k0, it must wait in the low-priority queue of channel k0 until all the traffic in the high-priority and the present low-priority queues of channel k0 are served as discussed in Section 5.2.2. Denote Ws(k0) as this waiting time for the secondary connections at channel k03. Hence, we have E[Di] = E[Ws(k0)] + ts.
3A secondary connection needs to change its operating channel only when a primary connection appears. Because the arrivals of the primary connections follow Poisson
dis-The value of E[Ws(k0)] can be derived as follows. Let E[Q(kp 0)] be the average number of the primary connections which are waiting in the high-priority queue of channel k0 and E[Q(ki 0)] be the average number of the type-i secondary connections which are waiting in the low-priority queue of channel k0. Because the newly arriving secondary connections cannot be established until all the secondary connections in the low-priority queue and the primary connections in the high-priority queue have been served, the average waiting time of channel k0 is expressed as
E[Ws(k0)] = E[R(ks 0)]+E[Q(kp 0)]E[Xp(k0)]+
X∞ i=0
E[Q(ki 0)]E[Φ(ki 0)]+λ(kp0)E[Ws(k0)]E[Xp(k0)] , (5.11) where E[R(ks 0)] is the average residual effective service time of channel k0. That is, E[R(ks 0)] is the remaining time to complete the service of the con-nection being served at channel k0. This connection being served can be the primary connection or the type-i secondary connection. Furthermore, E[Q(kp 0)]E[Xp(k0)] andP∞
i=0E[Q(ki 0)]E[Φ(ki 0)] in (5.11) are the cumulative work-load resulting from the primary connections and the secondary connections in the present queues of channel k0, respectively. Moreover, the fourth term (λ(kp 0)E[Ws(k0)]E[Xp(k0)]) in (5.11) is the cumulative workload resulting from the arrivals of the primary connections during Ws(k0).
In (5.11), the closed-form expression for E[Φ(ki 0)] is derived in Appendix C. Next, we will derive E[R(ks 0)], E[Q(kp 0)], and E[Q(ki 0)]. Firstly, according to
tribution, the arrivals of the interrupted secondary connections at channel k0 also follow Poisson distribution. Applying the property of Poisson arrivals see time average (PASTA) on the arrivals of the interrupted secondary connections at channel k0 [97], all of them must spend time duration E[Ws(k0)] on average to wait for an idle channel k0. This waiting time is uncorrelated to the number of interruptions.
the definition of residual time in [98], we have
where ω(ki 0) is derived in Appendix B. Secondly, according to Little’s formula, it follows that
E[Q(kp 0)] = λ(kp 0)E[Wp(k0)] , (5.13) where E[Wp(k0)] is the average waiting time of the primary connections at channel k0. It is the duration from the time instant that a primary connection enters the high-priority queue of channel k0 until it gets a chance to transmit at channel k0. Hence, it follows that
E[Wp(k0)] = E[R(kp 0)] + E[Q(kp 0)]E[Xp(k0)] , (5.14) where E[R(kp 0)] is the average residual service time resulting from only the primary connections of channel k0 and E[Q(kp 0)]E[Xp(k0)] is the total workload of the primary connections in the present high-priority queue of channel k0. According to [98], we have E[R(kp 0)] = 12λ(kp 0)E[(Xp(k0))2]. Then, solving (5.13) Next, according to Little’s formula, we can obtain
E[Q(ki 0)] = ω(ki 0)E[Ws(k0)] . (5.17) Finally, substituting (5.12), (5.16), and (5.17) into (5.11), we can obtain (5.7).
Proposition 2.
Proof. The value of p(k)i can be evaluated as follows. Because the considered secondary connection will operate at channel si,η after ith interruption, we have p(k)i = 0 when k 6= si,η. Furthermore, for the case that k = si,η, we consider the time interval [0, t] at channel k. Total λ(k)p t primary connec-tions and ω(k)i t type-i secondary connections arrive at channel k during this interval. Hence, there are total ωi(k)tp(k)i type-i secondary connections will be interrupted on average during this interval. Furthermore, applying the property of Poisson arrivals see time average (PASTA) on the arrivals of the primary connections [97], we can obtain the probability of a primary connec-tion finding channel k being occupied by the type-i secondary connecconnec-tions is ρ(k)i . Thus, during this interval, the total λ(k)p tρ(k)i primary connections can see a busy channel being occupied by the type-i secondary connections.
For each primary connection, it can interrupt only one secondary connection when it arrives at a busy channel being occupied by the secondary connection because only one secondary user can transmit at any instant of time. Thus, the total number of the interrupted secondary connections at channel k is also λ(k)p tρ(k)i . Hence, we have ωi(k)tp(k)i = λ(k)p tρ(k)i . That is,
ρ(k)i = ωi(k) λ(k)p
p(k)i . (5.19)
Next, we consider a type-i secondary connection at channel k. Before the (i + 1)th interruption event occurs, its effective service time is E[Φ(k)i ]. Thus, from queueing theory, we can have
ρ(k)= ω(k)E[Φ(k)] . (5.20)
Comparing (5.19) and (5.20), we can obtain (5.18).