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2884 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 8, AUGUST 2012
OPERA: Optimal Routing Metric for
Cognitive Radio Ad Hoc Networks
Marcello Caleffi, Member, IEEE, Ian F. Akyildiz, Fellow, IEEE, and Luigi Paura, Member, IEEE
Abstract—Two main issues affect the existing routing metrics
for cognitive radio ad hoc networks: i) they are often based on
heuristics, and thus they have not been proved to be optimal; ii)
they do not account for the route diversity effects, and thus
they are not able to measure the actual cost of a route. In
this paper, an optimal routing metric for cognitive radio ad
hoc networks, referred to as OPERA, is proposed. OPERA
is designed to achieve two features: i) Optimality:OPERAis
optimal when combined with both Dijkstra and Bellman-Ford
based routing protocols; ii) Accuracy: OPERA exploits the route
diversity provided by the intermediate nodes to measure the
actual end-to-end delay, by taking explicitly into account the
unique characteristics of cognitive radio networks. A closed-
form expression of the proposed routing metric is analytically
derived for both static and mobile networks, and its optimality is
proved rigorously. Performance evaluation is conducted through
simulations, and the results reveal the benefits of adopting the
proposed routing metric for cognitive radio ad hoc networks.
Index Terms—Cognitive radio, routing metric, optimality,
route diversity, mobility.
I. INTRODUCTION
COGNITIVE Radio (CR) paradigm has been proposed as
a viable solution to counteract both spectrum inefficiency
and spectrum scarcity problems. The CR paradigm exploits
the concept of spectrum hole: a CR user is allowed to use
a spectrum band licensed to a primary user (PU) when it
is temporarily unused. The CR paradigm can be applied to
ad hoc scenarios, and the resulting networks, referred to as
CR Ad Hoc Networks (CRAHNs) [1], are composed by CR
users that exploit the spectrum holes for establishing multi-hop
communications in a peer-to-peer fashion. To fully unleash the
potentials of such networks, new challenges must be addressed
and solved at the network layer. In particular, effective routing
Manuscript received August 5, 2011; revised March 5, 2012; accepted
April 16, 2012. The associate editor coordinating the review of this paper
and approving it for publication was T. Hou.
M. Caleffi was with the Broadband Wireless Networking Laboratory,
School of Electrical and Computer Engineering, Georgia Institute of Tech-
nology, USA. He is now with the Dept. of Biomedical, Electronics and
Telecommunications Engineering, University of Naples Federico II, Italy (e-
mail: marcello.caleffi@unina.it).
I. F. Akyildiz is with the Broadband Wireless Networking Laboratory,
School of Electrical and Computer Engineering, Georgia Institute of Tech-
nology, USA (e-mail: ian.akyildiz@ee.gatech.edu).
L. Paura is with the Department of Biomedical, Electronics and Telecom-
munications Engineering, University of Naples Federico II, Italy (e-mail:
paura@unina.it).
This work was supported by the Italian National Program under Grants
PON01-00744 ”DRIVE-IN2: DRIVEr monitoring: technologies, methodolo-
gies, and IN-vehicle INnovative systems for a safe and ecocompatible driving”
and PON01-01936 ”HABITAT: HarBour traffIc opTimizAtion sysTem”,and
by the U.S. National Science Foundation under Grant ECCS-0900930.
Digital Object Identifier 10.1109/TWC.2012.061912.111479
metrics able to account for the distinguishable properties of
the CR paradigm are needed [2].
The routing metrics for CRAHNs can be classified in two
classes: a) the metrics originally proposed for multi-channel
environments and then adapted to CR networks; b) the metrics
specifically designed for CR networks. All the routing metrics
in the first class, due to their origin, cannot fully account
for the spectrum dynamics introduced by the PU activity.
Indeed, in a CR network the routes of the CR users are highly
affected by the spectrum dynamics, therefore these metrics fail
in discovering the optimal route. The routing metrics in the
second class, namely, the metrics explicitly designed for CR
networks, either they are not optimal or they are not able
to measure the actual path cost. We describe both the two
drawbacks in the following.
A. Metric Optimality
A routing metric is a function that assigns a weight (i.e., a
cost) to any given path [3]. A routing metric is defined optimal
when there exists an efficient shortest-path algorithm based on
such a metric that always discovers the lowest-weight route
between any pair of nodes in any connected network.
We explain the concept of optimality with an example of a
metric that is not optimal when combined with Belmann-Ford
(or Dijkstra) shortest-path algorithm: the Weighted Cumulative
Expected Transmission Time (WCETT), proposed in [4] to
account for the multi-channel inter-flow interference. For a
path p, WCETT is defined as:
WCETT(p)=(1−β)
link l∈p
Delay(l)+βmax
channel jXj
where βis a tunable parameter, Delay(l)denotes the trans-
mission delay for link l,andXjcaptures the inter-flow
interference by counting the number of times that channel jis
used along the path. Since WCETT metric is not optimal, there
is no efficient shortest-path algorithm able to always discover
the lowest-WCETT route.
Fig. 1 shows a simple network in which the Belmann-Ford
algorithm based on WCETT fails in discovering the optimal
route for β>4/9. More in detail, if we assume β=2/3, node
u1announces the route u1-u2-ud(with a cost 0.4/3+2/3)as
the optimal route to reach ud, since it is preferred to the route
u1-ud(with a cost 0.476/3+2/3). Hence, node usdoes not
have any chance to check the weight of the route u1-ud,and
it incorrectly sets the sub-optimal route us-u3-u4-u5-ud(with
acost1.7/3+2/3) as the route to ud, since it is preferred
to the route us-u1-u2-ud(with a cost 0.9/3+4/3). However,
the route us-u1-ud(with a cost 0.976/3+2/3) is the actual
1536-1276/12$31.00 c
2012 IEEE
CALEFFI et al.: OPERA: OPTIMAL ROUTING METRIC FOR COGNITIVE RADIO AD HOC NETWORKS 2885
Fig. 1. Metric Optimality: WCETT metric fails in finding the shortest
path between usand udwhenever β>4/9, since node ussets us-u3-
u4-u5-udas the optimal route, although us-u1-udis the actual optimal
one.
Fig. 2. Metric Accuracy: a metric that neglects the route diversity measures
a delay equal to 4.5for the route us-u1-u2-u4-ud, although the actual delay
for the combined route us-u1-(u2when vlis not active, u3otherwise)-u4-
udis 4.2.
optimal route. A similar issue occurs when Dijkstra algorithm
is applied.
The lack of the optimality property is not trivial: regard-
less of the adopted routing procedure, the packets can be
routed either through sub-optimal routes, wasting the network
resources, or even worse through route loops, causing un-
reachable destinations. Clearly, these issues become more
severe in CR networks.
B. Metric Accuracy
A routing metric is defined accurate when it always mea-
sures the actual route cost between every pair of nodes in any
connected network.
Most of the metrics explicitly designed for cognitive radio
networks measure the cost of a route only with reference to
the routing opportunities available when the route is not dis-
connected by the PU activity. Thus, they neglect the additional
routing opportunities provided by the route diversity, failing
so to measure the actual quality of a route.
We illustrate this issue with the example in Fig. 2, where
three routes exist between the source usand the destination
ud:i)the1st route is composed by the intermediate nodes u1-
u2-u4, and it is affected by the activity of PU vlwith activity
probability pl=(1−¯pl)=0.2and with mean activity time
Tl=2; ii) the 2nd route is composed by the intermediate nodes
u1-u3-u4; iii) the 3rd route is composed by the intermediate
nodes u5-u6-u7.
If we neglect the route diversity, we have the following
delays for each route: i) 1st route1: delay =4.5; ii) 2nd route:
delay =5; iii) 3rd route: delay =4.4. As a consequence,
a metric that does not account for the route diversity would
(incorrectly) announce the 3rd route as the lowest cost route,
1The delay of the 1st route is the sum of two terms: i) the delay introduced
by links us-u1and u4-ud, which is always equal to 2; ii) the delay introduced
by links u1-u2and u2-u4, which depends on vlactivity and grows linearly
with Tlfor each unsuccessful transmission. As a consequence, the delay of
the 1st route is equal to:
delay =2+¯pl2+pl¯pl(2 + Tl)+p2
l¯pl(2 + 2Tl)+... =
=2+2¯pl
∞
n=0
pn
l+Tl¯pl
∞
n=0
pn
ln=2+2+Tl
pl
1−pl
and uswill forward the packets through the intermediate node
u5, regardless from the adopted routing procedure.
Differently, if we consider the effects of the route diversity
in measuring the delay of the 1st route, we have also to
consider the additional routing opportunities provided by the
intermediate node u3when vlis active. Thus, we have that the
delay of the route u1-(u2when vlis not active, u3otherwise)-
u4,is
24.2. As a consequence, a metric that accounts for the
route diversity would (correctly) announce such a route as the
lowest cost route, and uswould forward the packets through
the intermediate node u1, i.e., through the combined route
with the actual lowest cost.
The lack of the accuracy property is not trivial in CR
networks: if the metric overestimates the route cost, the
packets can be routed through sub-optimal routes, wasting the
network resources.
In this paper, we propose a novel CR routing metric, called
Optimal Primary-aware routE quAlity (OPERA), with the
objective to overcome both the issues mentioned above: un
-optimality and un-accuracy.
More specifically, OPERA is designed to achieve two dis-
tinguishable features: i) Optimality: OPERA is optimal when
combined with both Dijkstra and Bellman-Ford based routing
protocols; ii) Accuracy: OPERA exploits the route diversity
provided by the intermediate nodes to measure the actual end-
to-end delay of a route, by taking explicitly into account the
unique characteristics of cognitive radio networks.
We analytically derive closed-form expressions of OPERA
metric for static and mobile networks, and the optimality
of OPERA is analytically proved by means of the routing
algebra theory. Performance evaluation is conducted through
simulations, and the results reveal the benefits of adopting the
proposed routing metric for cognitive radio ad hoc networks.
The rest of the paper is organized as follows. Sec. II
discusses related work. Sec. III describes the network model,
2The delay of the combined route is the sum of three terms: i) the delay
introduced by links us-u1and u4-ud, which is always equal to 2; ii) with
probability ¯pl, the delay introduced by links u1-u2and u2-u3, which is equal
to 2; iii) with probability pl, the delay introduced by links u1-u3and u3-u4,
which is equal to 3. As a consequence, if we neglect the additional delay
introduced by packet queueing at u2, the delay of the route is equal to:
delay =2+¯pl2+pl3=4.2.
2886 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 8, AUGUST 2012
whereas in Sec. IV the closed-form expression of OPERA is
derived for both static and mobile networks. In Sec. V we
prove the optimality of OPERA, and in Sec. VI we evaluate
its performance. Finally, conclusions are drawn in Sec. VII
and some proofs are gathered in the appendix.
II. RELATED WORK
As mentioned in Sec. I, the routing metrics for CRAHNs
can be classified as follows: i) the metrics originally proposed
for multi-channel environments and then adapted to CR net-
works; ii) the metrics specifically designed for CR networks.
With reference to the metrics belonging to the first class, the
works [5], [6], [7] propose to measure the quality of a route
in terms of delay. The expressions of the metrics account for
different delay components related to multi-channel environ-
ments, such as the channel switching delay or the link switch-
ing delay. However, none of them accounts for the spectrum
dynamics, and thus they fail in describing the characteristics
of a CR environment. Differently, OPERA is designed by
explicitly accounting for the spectrum dynamics related to the
PU activity and for the sensing process characteristics.
With reference to the metrics specifically designed for CR
networks, the authors in [8] propose a metric that combines the
link stability and the switching delays. However, it is possible
to prove that the proposed metric is not optimal since it is
not isotonic [9]. The paper [10] introduces a CR metric based
on the route stability. The stability of a route is defined as
a function of the route maintenance cost, which is measured
in terms of channel switching and link switching delays. The
paper [11] proposes a routing metric and a routing protocol
to achieve high throughput efficiency, by allowing the CR
users to opportunistically transmit according to the concept
of spectrum utility. The authors consider several problems,
such as spectrum and power allocation and distributed routing,
and the proposed algorithm is proved to be computationally
efficient and with bounded BER guarantees. In [12], the
authors perform an asymptotic analysis of the capacity and
delay for cognitive radio networks, by comparing shortest-path
routing, multi-path routing and network coding techniques.
However, they do not provide any heuristic or theory for
measuring the quality of a route. In [13], the authors pro-
pose an opportunistic path metric that accounts for the route
diversity effects. However, neither they provide a closed form
expression or they prove the optimality of their metric. In [14]
the authors propose a novel CR metric based on path stability
and availability over time, while in [15], the authors propose
a routing metric that aims to minimize the interference caused
by the CR users to the PUs. Unlike OPERA, none of the cited
works jointly addresses the optimality and accuracy issues.
III. NETWORK MODEL
We model the cognitive network with a direct temporal
graph:
G(t)=(U, E (t)) (1)
in which the spectrum is organized in Ndistinct bands
(channels), a vertex ui∈Udenotes a CR user, and an
edge eij (t)∈E(t)denotes the presence of at least one
communication link em
ij (t)from CR user uito CR user uj
at time tthrough the spectrum band m:
eij (t)=1⇐⇒ ∃ m∈{1,...,N}:em
ij (t)=1 (2)
Each link em
ij (t)is characterized by the link throughput
ψm
ij (t), which measures the amount of information for unit
time that uican successfully transmits to ujthrough the m-
th channel at time t. The link throughput jointly accounts for
several factors, such as the channel bandwidth, the channel
spectral efficiency, the wireless propagation conditions, the
channel contention delays, the node queueing delays, etc.. The
expected value of ψm
ij (t)at time tis denoted with ¯
ψm
ij (t),and
the CR users are able to estimate the expected values, i.e.,
through the past channel-throughput history.
The CR users, equipped with a single radio interface,
are assumed heterogeneous, namely, they can have different
transmission ranges. Also the PUs, denoted with vl∈V,
are assumed heterogeneous, namely, they can have different
transmission ranges, activity probabilities pl, and mean activity
times. The CR user time is organized into fixed-sized slots
of duration T, as shown in Fig. 33. Each time slot Tis
further organized in a sensing period Ts, which measures the
portion of time slot assigned to spectrum sensing, and in a
transmission period Ttx, which measures the portion of time
slot devoted to unlicensed access to the licensed spectrum for
packet transmission.
In the following we give some definitions adopted through
the paper.
Definition 1. (Interfering Set) The interfering set Vm
i(t)⊆V
is the set of PUs that can prevent CR user uieither to receive
or transmit through the m-th channel at time t4:
Vm
i(t)={vl∈V:||ui−vl||(t)<min {Rui,R
vl}} (3)
where || · ||(t)denotes the Euclidean distance at time t,and
Ruiand Rvldenote the interference range of uiand vl,
respectively. Similarly, we denote with Vm
ij (t)=Vm
i(t)∪
Vm
j(t)the set of PUs whose activities can interfere with the
communications through link em
ij at time t.
Definition 2. (Throughput Sequence) The throughput se-
quence (τ1
ij (t),τ2
ij (t),...,τN
ij (t)), with τm
ij (t)∈{1,...,N},
is the sequence of channels ordered according to the decreas-
ing expected link throughput ¯
ψm
ij (t)at time t:
¯
ψτm
ij (t)
ij (t)≥¯
ψτm+1
ij (t)
ij (t)(4)
Definition 3. (Link Availability Probability) The link avail-
ability probability pm
ij (t)is the probability of link em
ij being
not affected by PU activity at time t5, i.e., the probability that
3In the following, we assume for the sake of simplicity that the slot periods
of the CR users are synchronized. Nevertheless, the results presented through
the paper can be easily extended to the case of asynchronous slot periods.
4The rationale for adopting a symmetric interference model is that, in
a multi-hop environment, a CR user is expected to forward the traffic
received by other CR users. Thus, both the transmission and the reception
functionalities are required.
5We note that the time dependence of pm
ij (t)is due to the PU and/or CR
user mobility, which can affect Vm
ij (t)in time.
CALEFFI et al.: OPERA: OPTIMAL ROUTING METRIC FOR COGNITIVE RADIO AD HOC NETWORKS 2887
Fig. 3. Link Delay for packet transmission due to the presence of PU
activity.
Fig. 4. Difference between Block Probability and Conditional Block
Probability.
no PU belonging to the interfering set Vm
ij (t)is active at time
t:
pm
ij (t)
=P(Em
ij (t)=1)=
vl∈Vm
ij (t)
(1 −pl)(5)
where Em
ij (t)is the random process that models the PU
activity on link em
ij , namely, Em
ij (t)=1denotes the event
’link em
ij is not affected by PU activity at time t’, and
¯pm
ij (t)
=1−pm
ij (t).
Definition 4. (Block Probability) Given a CR user uiand
an ordered set C={c1,...,c
n}of neighbors of ui,theblock
probability ¯zm
iCn(t)is the probability of the links em
ic1,...,e
m
icn
being affected by PU activity at time t:
¯zm
iCn(t)
=PEm
ic1(t)=0,...,Em
icn(t)=0
(6)
Remark. (Independent PU Activity Condition) We observe
that, if the random variables {Em
icj(t)}n
j=1 modeling the PU
activity at time tare independent each other, we have that:
¯zm
iCn(t)=
n
j=1
¯pm
icj(t)(7)
Definition 5. (Conditional Block Probability) Given a CR
user uiand an ordered set C={c1,...,c
n}of neighbors of
ui,theconditional block probability ¯zm
i\Cn(t)is the probability
of the links em
ic1,...,e
m
icn−1being affected by PU activity at
time t, given that the link em
icnis not affected by PU activity
at time t:
¯zm
i\Cn(t)
=PEm
ic1(t)=0,...,Em
icn−1(t)=0|Em
icn(t)=1
(8)
Remark. Fig. 4 highlights the difference between the block
probability and the conditional block probability,which
concerns link em
icn. In fact, the first probability measures the
probability of all the nlinks (em
icnincluded) being affected by
PU activity. Differently, the conditional probability measures
the probability of the first n−1links being affected by PU
activity, given that the n-th link em
icnis not affected by PU
activity.
In the following, we consider two scenarios: i) static net-
works; ii) mobile networks. In static networks, both the CR
users and the PUs are static; as a consequence, the parameters
em
ij ,Em
ij ,¯
ψm
ij ,pm
ij ,¯zm
iCn,and¯zm
i\Cndo not depend on the time.
In mobile networks, the CR users and/or the PUs are mobile.
IV. OPERA: OPTIMAL ROUTING METRIC
In this section we define OPERA and we analytically
derive its expression, first with reference to static net-
works (Sec. IV-A), then with reference to mobile networks
(Sec. IV-B).
A. OPERA for Static Networks
In this sub-section, we derive the expression of OPERA
with reference to static networks (Theorem 1). To this aim,
we first present Lemma 1.
Lemma 1. The expected link delay lij experienced by a packet
sent by uito the neighbor ujis given by:
lij =1
1−qN
ij ⎛
⎝
N
m=1
qm−1
ij pτm
ij
ij
L
¯
ψτm
ij
ij
+qN
ij T⎞
⎠(9)
where Lis the packet length, τm
ij is defined in (4),pτm
ij
ij is
defined in (5), and qm
ij is given by:
qm
ij =⎧
⎪
⎨
⎪
⎩
1if m=0
m
n=1
¯pτn
ij
ij otherwise (10)
Proof: See Appendix A.
Remark. The expected link delay (9) allows us to estimate
the delay for a packet sent over a link by accounting for two
main factors that affect the transmission of a packet over a
CR link: i) the PU characteristics, via the probabilities pτm
ij
ij
and qm
ij ; ii) the sensing process characteristics, via the time
parameter T. More in detail, the expected link delay is function
of the weighted sum of two delay terms: i) the delay for packet
transmission, which depends on the inverse of the channel
capacity ¯
ψτm
ij
ij ; ii) the delay introduced by PU activity, which
depends on the time parameter T. The delay introduced by
PU activity is weighted by the probability qN
ij , which is the
probability of at least one PU being active on each available
channel.
2888 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 8, AUGUST 2012
Remark. By ordering the channels according to the decreas-
ing expected link throughput ¯
ψm
ij ,theexpected link delay
expression (9) computes the actual minimum expected delay,
as proved in [16].
By means of Lemma 1, we can now derive the expression
of OPERA for static networks.
Theorem 1. (OPERA for static networks) The expected
end-to-end delay Dik (C)for the source-destination (ui,u
k)
through the ordered set C={cj}n
j=1 of neighbors of uiis
given by:
Dik(C)= 1
1−¯qN
icn⎛
⎝
n
j=1
N
m=1
¯qm−1
icjpτm
icj
icj⎛
⎝
L
¯
ψτm
icj
icj
+Dcjk⎞
⎠+
+¯qN
icnT(11)
where:
¯q˜m
icj=
˜m
m=1
¯zτm
icj
iCj¯zτ˜m+1
icj
i\Cj
N
m=˜m+2
¯zτm
icj
iCj−1(12)
and where Lis the packet length, τm
icjis defined in (4),pτm
icj
icj
is defined in (5),¯zτm
icj
iCjis defined in (6),¯zτm
icj
i\Cjis defined in
(8),Dcjkis the minimum expected end-to-end delay for the
couple source destination (cj,u
k).
Proof: See Appendix B.
Remark. Similarly to the expected link delay (9), the ex-
pression (11) of OPERA explicitly accounts for two main
factors that affect the transmission of a packet over a CR
link: i) the PU characteristics, via the probabilities pτm
icj
icj,
¯q˜m
icj,¯zτm
icj
iCj,and¯zτm
icj
i\Cj; ii) the sensing process characteristics,
via the time parameter T. Unlike (9), the delay for packet
transmission in OPERA is defined as the inverse of the
channel capacities ¯
ψτm
icj
icjtoward forwarder cjplus the expected
end-to-end delay Dcjkfrom cjto the final destination uk.
The delay introduced by PU activity (T) is weighted by the
probability ¯qN
icnthat represents the probability that all the
available channels towards all the forwarders c1,...,c
nare
affected by PU activity. Clearly, the expected end-to-end delay
depends on the set of forwarders C; with Theorem 3 we give
the optimal choice for the set of forwarders C.
B. OPERA for Mobile Networks
In this sub-section, we derive (Theorem 2) the expression of
OPERA with reference to mobile networks. Since the proof of
Theorem 2 requires three definitions, Definition 6-8, and one
lemma, Lemma 2, we first present these intermediate results.
In this scenario, the expected end-to-end delay can change
dramatically due to the effect of the PU and/or CR user
mobility. Therefore, OPERA measures the expected end-to-
end delay by explicitly accounting for the effect of the relative
movement between two CR users (13) and between a CR
user and a PU (14). Moreover, OPERA measures the delay
with reference to the time interval δ, which represents the
route update period. In such a way, OPERA can be efficiently
adopted by any routing protocol based on periodic route
updates.
Definition 6. (Expected Link Utilization) The expected link
utilization factor ρij (t, n)measures the fraction of the time
interval [t+nT,t+(n+1)T)during which uiand ujare
able to communicate, measured at time t:
ρij (t, n)=⎧
⎨
⎩
1if tij (t, n)=T
tij (t, n)/T if 0≤tij (t, n)<T
0otherwise
(13)
where tij (t, n)is the expected cumulative contact time period
between uiand ujin time interval [t+nT,t+(n+1)T),
measured at time t.
Definition 7. (Expected Link Interference) The expected
link interference factor σl
ij (t, n)measures the fraction of the
time interval [t+nT,t+(n+1)T)during which the activity
of PU vlcan interfere with the communications between ui
and uj, measured at time t:
σl
ij (t, n)=⎧
⎨
⎩
1if tl
ij (t, n)=T
tl
ij (t, n)/T if 0≤tl
ij (t, n)<T
0otherwise
(14)
where tl
ij (t, n)is the expected cumulative contact time period
between vland either uior ujin time interval [t+nT,t+
(n+1)T), measured at time t.
Definition 8. (Link Availability Probability) The link avail-
ability probability pm
ij (t, n)between uiand ujis the proba-
bility of link em
ij being: i) connected, ii) not affected by PU
activity, in the time interval [t+nT,t+(n+1)T), measured
at time t:
pm
ij (t, n)
=ρij (t, n)
vl∈Vm
ij (t,n)1−σl
ij (t, n)¯pl(15)
where Vm
ij (t, n)is the interfering set for link em
i,j in the time
interval [t+nT,t+(n+1)T), measured at time t.
Remark. The expression (15) of the link availability proba-
bility allows us to account for the dynamics introduced in the
link connectivity by: i) the relative movement between two
CR users, through the expected link utilization factor ρij (t, n)
(13); ii) the relative movement between a CR user and a PU,
through the expected link interference factor σl
ij (t, n)(14).
By means of Definition 8, we derive now the expression
of the expected link delay for mobile networks in the time
interval δ.
Lemma 2. The expected link delay lij (t, δ)experienced by
apacketsentbyuito the neighbor ujin the time interval
[t, t +δ)measured at time tis shown in (16),whereLis the
packet length, τm
ij (t)is defined in (4),pτm
ij (t)
ij (t, e)is defined
in (15),edenotes the number of transmission attempts failed
due to the presence of PU activity on each available channel,
and qm
ij (t, e)is given by:
qm
ij (t, e)=⎧
⎪
⎨
⎪
⎩
1if m=0
m
n=1
¯pτm
ij (t)
ij (t, e)otherwise (17)
CALEFFI et al.: OPERA: OPTIMAL ROUTING METRIC FOR COGNITIVE RADIO AD HOC NETWORKS 2889
lij (t, δ)=
δ
T
e=0 ⎛
⎝
e
r=1
qN
ij (t, r)
N
m=1
qm−1
ij (t, e)pτm
ij (t)
ij (t, e)L
¯
ψτm
ij (t)
ij (t)⎞
⎠+
δ
T
e=1 e
r=1
qN
ij (t, r)eT 1−qN
ij (t, e)(16)
Proof: It is straightforward to prove the theorem by
following the reasoning adopted to prove Lemma 1.
Remark. Similarly to static networks (see (9)), the expression
of the expected link delay for mobile networks (16) accounts
for both the PU characteristics and the sensing process char-
acteristics. Unlike (9), expression (16) accounts also for the
dynamics introduced in the link delay by the movement of the
CR users and/or the PUs. Moreover, expression (16) depends
on the time interval tand on the time horizon δ,which
can be interpreted as the route update period. The shorter is
δ, the more accurate is the estimation of the expected link
delay provided by (16), but the higher is the routing overhead
generated by the adopted routing protocol.
By means of Lemma 2, we can now derive the expression
of OPERA for mobile networks.
Theorem 2. (OPERA for mobile networks) The expected end-
to-end delay Dik(t, δ, C)for the source-destination (ui,u
k)
through the ordered set C={cj}n
j=1 of neighbors of uiin
the time interval [t, t +δ)measured at time tis shown in (18)
reported at the top of the next page, where
¯q˜m
icj(t, e)=
˜m
m=1
¯zτm
icj(t)
iCj(t, e)¯zτ˜m+1
icj(t)
i\Cj(t, e)·
·
N
m=˜m+2
¯zτm
icj(t)
iCj−1(t, e)(19)
¯zm
iCn(t, e)
=PEm
ic1(t, e)=0,...,Em
icn(t, e)=0
(20)
¯zm
i\Cn(t, e)
=
PEm
ic1(t, e)=0,...,Em
icn−1(t, e)=0|Em
icn(t, e)=1
(21)
and where Lis the packet length, τm
icj(t)is defined in (4),
pτm
icj(t)
icj(t, e)is defined in (15),Em
icj(t, e)denotes the random
process that models the PU activity on link em
ij in the time
interval [t+eT , t +(e+1)T), and edenotes the number of
transmission attempts failed due to the presence of PU activity
on each available channel towards each forwarder cj.
Proof: It is straightforward to prove the theorem by
following the same reasoning adopted to prove Theorem 1.
Remark. Differently from the expression of OPERA derived
for static networks (11), the expression (18) depends on the
time interval tand on the time horizon δ, which can be
interpreted as the route update period. The shorter is δ,the
more accurate is the estimation of the expected end-to-end
delay provided by (18), but the higher is routing overhead
generated by the adopted routing protocol.
V. O P E R A O PTIMALITY
In this section, we first define the conditions for route metric
optimality with a preliminary lemma in Sec. V-A. Then, in
Sec. V-B we prove (Theorem 3) that OPERA metric is optimal
when combined with both Dijkstra and Bellman-Ford based
routing protocols.
A. Preliminaries
For the source-destination (ui,u
k), each forwarder can
route the packets through a set of different candidate routes.
Such a set can be regarded as a rooted tree, called CR-
tree, where all the leaves coincide with the destination uk.
Accordingly, we use the following definition and lemma based
on theorems (1) and (4) in [3] to prove the optimality of
OPERA.
Definition 9. (Optimality) A routing protocol is optimal if it
always routes packets along the path with the minimum route
cost between every pair of nodes in any connected network.
Lemma 3. (Conditions for Optimality) Both a Dijkstra-based
and a Bellman-Ford-based CR routing protocols are optimal
when adopted by a CR algebra with the following properties:
relay beneficial condition, strictly preference preservation, and
relay order optimality.
In the following, we first define the concept of routing
algebra, and then we define the three properties listed by
Lemma 3.
Definition 10. (CR algebra)ACR algebra is a 4-tuple:
CR =(P,⊕,L,≤)(22)
where Pis the set of CR-trees, ⊕is the operator that
concatenates two CR-trees in a single CR-tree, L:P→R
is a function that assigns a cost to a CR tree, and ≤is the
ordering relation over the CR-trees in terms of costs.
We use the following notations. We denote with Pik(C)∈P
the CR-tree for the source-destination pair (ui,u
k)through
the forwarder set C. The cost of the CR-tree is denoted with
L(Pik (C)). We say that Pik (C)is strictly preferred to Pik (˜
C)
if L(Pik (C)) <L(Pik(˜
C)).
Definition 11. (Relay Beneficial Condition) The algebra CR
is relay-conditionally-beneficial if, ∀ui,u
k∈Uand ∀c˜n∈C,
it results:
L(Pik(c˜n)) <L(Pik (˜
C)) ⇐⇒ L (Pik (˜
C⊕c˜n)) <L(Pik (˜
C))
(23)
where Cdenotes the set of forwarders and ˜
C⊂C. Intuitively,
the relay beneficial conditional property requires that a neigh-
bor cjis selected as forwarder if and only if it can decrease
the delay from uito uk.
2890 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 8, AUGUST 2012
Dik(t, δ, C)=
δ
T
e=0 ⎛
⎝
e
r=1
¯qN
icn(t, r)
n
j=1
N
m=1 ⎛
⎝¯qm−1
icj(t, e)pτm
icj(t)
icj(t, e)⎛
⎝
L
¯
ψτm
icj(t)
icj(t)
+Dcjk(t, δ)⎞
⎠⎞
⎠⎞
⎠
+
δ
T
e=1 e
r=1
¯qN
icn(t, r)eT 1−¯qN
icn(t, e)(18)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10−5
10−4
10−3
10−2
10−1
100
101
102
103
PU Activity Probability
Expected Delay [s]
Theoretical Link Delay: N=5
Experimental Link Delay: N=5
Theoretical End−to−End Delay: N=5
Experimental End−to−End Delay: N=5
Theoretical End−to−End Delay: N=3
Experimental End−to−End Delay: N=3
Fig. 5. OPERA validation: theoretical delays vs experimental delays
for expressions (9) and (11). Logarithmic scale for axis y.
0 200 400 600 800 1000
0
100
200
300
400
500
600
700
800
900
1000
X Position [m]
Y Position [m]
PU
CR users
Un−Effective Algorithm:
Sub−Optimal Route 1.36s
OPERA: Optimal Route 0.85s
Fig. 6. Considered topology: the figure shows two different routes
and the respective delays between the same couple source-destintion,
the route singled out by OPERA (black line) and the route singled out
by the un-effective algorithm.
Definition 12. (Strictly preference preservation) The alge-
bra CR is strictly preference-preservable if, ∀ui,u
k∈Uand
∀c˜n∈C, it results:
L(Pik (c˜n)) <L(Pik(˜
C)) ⇐⇒ L (Pik(c˜n)) <L(Pik (˜
C⊕c˜n))
(24)
where Cdenotes the set of forwarders and ˜
C⊂C. Intuitively,
a CR algebra is strictly-preference preservable if the concate-
nation operation preserves the preference.
Definition 13. (Relay order optimality) The algebra CR is
relay-order-optimal if, ∀ui,u
k∈U, it results:
L(Pik (C)) ≤L(Pik (˜
C)) (25)
where Cis the set of forwarders ordered according to the
preferences associated with the corresponding CR-trees and ˜
C
is any permutation of the set C. Intuitively, a CR algebra is
relay-order-optimal if it exists a unique criterion to order the
set of forwarders.
B. Optimality
Theorem 3. Any Dijkstra-based or Bellman-Ford-based rout-
ing protocol combined with OPERA is optimal when the
independent PU activity condition (7) holds, and when the
set of forwarders Cfor an arbitrary couple source-destination
(ui,u
k)is given by:
Cik ={cj∈N
i:Wik (cj)<Wik(cj+1 )and
Dik (cj)<Dik({c1,...,c
j−1,c
j})}(26)
where
Wik(ucj)= 1
1−qN
icj
N
m=1
qm−1
icjpτm
icj
icj⎛
⎝
L
¯
ψτm
icj
icj
+Dcjk⎞
⎠(27)
and where Lis the packet length, τm
ij is defined in (4),pτm
ij
ij
is defined in (5),qm
ij is defined in (10), and Dik is defined in
(11).
Proof: The proof follows from Lemmas 4-6.
Lemma 4. OPERA is relay-conditionally-beneficial.
Proof: See Appendix C.
Lemma 5. OPERA is strictly preference-preservable.
Proof: See Appendix D.
Lemma 6. OPERA is relay order optimal.
Proof: See Appendix E.
VI. PERFORMANCE EVA L UA T I O N
In this section, we evaluate the performance of OPERA in
terms of optimality and accuracy.
More in detail, first we validate the theoretical results de-
rived in Sec. IV, i.e., the expressions of the expected link delay
(9) and expected end-to-end delay (11). Then, we validate the
optimality property of OPERA derived in Sec. V (Theorem 3),
by comparing the performance of OPERA with that provided
by the exhaustive search of the minimum end-to-end delay.
Finally, we evaluate the benefits of the accuracy property by
CALEFFI et al.: OPERA: OPTIMAL ROUTING METRIC FOR COGNITIVE RADIO AD HOC NETWORKS 2891
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Exhaustive Search Delay [s]
OPERA Delay [s]
Fig. 7. OPERA optimality: OPERA delay vs Exhaustive Search delay
for the topology shown in Fig. 6. For each dot, coordinate (x)represents
the delay computed through exhaustive search, whereas coordinate (y)
represents the delay computed through OPERA expression.
0 0.2 0.4 0.6 0.8 1
10−2
10−1
100
101
PU Activity Probability
Delay [s]
Un−Effective Algorithm
OPERA
Fig. 8. OPERA accuracy: average delay vs PU activity probability
pl. Logarithmic scale for axis y.
comparing the performance of OPERA with that provided by
the algorithm that neglects the route diversity effects.
OPERA Validation
In Fig. 5, we validate the theoretical expressions of the
expected link delay (9) and expected end-to-end delay (11).
More in detail, we compare the theoretical delays with those
obtained through Montecarlo simulations as the PU activity
probability increases.
The adopted simulation set is as follows: the packet length is
L= 1500B, the expected link throughput is ¯
ψ= 54Mbps, the
number of channels Nis reported in the legend, each available
channel is affected by a PU with an activity provability pl∈
[0.05,0.95], and the sensing process is characterized by the
times T=2sandTs=0.001s. For the expected link delay,
we consider two neighbors, while for the expected end-to-end
delay, we consider a set Ccomposed by two forwarders, each
of them one hop away from the destination.
First, we note that there is a very good agreement be-
tween the theoretical and the experimental results for both
the expressions. Moreover, we observe that, for the lowest
values of probability pl, the delays are dominated by the
packet transmission delay L
¯
ψ. Differently, as probability pl
increases, the delays are dominated by the time period T.
This result agrees with the intuition: when the PU activity
is negligible, the path delay depends on the throughputs of
the links belonging to the path, while when the PU activity
is dominant, the delay depends on T. Finally, we observe
that the end-to-end delay decreases as the number of channels
increases. This result is reasonable, since, as the number of
channels increases, the probability ¯qN
icn(11) of unsuccessful
transmission due to the PU activity decreases. Thus, the delay
¯qN
icn
1−¯qN
icn
Tintroduced by the PU activity decreases as well.
OPERA Optimality
In this subsection, we validate the optimal property of
OPERA stated by Theorem 3. More in detail, for every couple
source-destination, we compare the delays computed with
expression (11) with those obtained by exhaustive search of
the minimum end-to-end-delay.
The adopted simulation set is as follows: the network
topology is shown in Fig. 6 and it is similar to the one used
in [15], with 64 CR users spread in a square region of side
1000m. The CR user transmission standard is IEEE 802.11g,
the packet length is L= 1500B, the expected link throughput
is ¯
ψm= 54Mbps, the transmission range is equal to 50m,
the number of channels is N=2, and the sensing process
is characterized by the times T=0.7sandTs=0.1sasin
[15]. The PU interference range Rvlis shown in Fig. 6, and
pl=0.5.
Fig. 7 presents the difference between the OPERA delays
and the exhaustive search delays by showing a dot for each
couple of CR users. The (x)coordinate of the dot represents
the delay computed through exhaustive search, whereas the (y)
coordinate of the dot represents the delay computed through
expression (11). Clearly, if y=x, then the two delays are
exactly the same, meaning that OPERA actually finds the
minimum end-to-end delay. Since Fig. 7 clearly shows that
for each couple of CR users we have y=x,thenOPERAis
optimal according to Definition 9. We further observe that the
delays shown in Fig. 7 have the same order of magnitude of
those shown in Fig. 5 for the same value of pl. The reason is
that, since pl=0.5and since the topology is relatively small,
the delay ¯qN
icn
1−¯qN
icn
Tintroduced by the PU activity dominates
the end-to-end delay.
OPERA Accuracy
In this subsection, we analyze the benefits provided by
the route diversity in terms of actual delay of a route. More
in detail, we compare the delays computed with expression
(11) with those obtained by the algorithm that neglects the
route diversity effects by recursively computing the path
delay through (9), called Un-Effective Algorithm. We consider
three scenarios: i) static networks, as probability plincreases;
ii) static networks, as time period Tincreases; iii) mobile
2892 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 8, AUGUST 2012
200 400 600 800 1000
0
0.5
1
1.5
CR user pair distance
Delay [s]
Un−Effective Algorithm: p=0.5
OPERA: p=0.5
Un−Effective Algorithm: p=0.3
OPERA: p=0.3
Fig. 9. OPERA accuracy: delay vs CR user pair distance for different
values of PU activity probability pl.
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Sensing Time Period T [s]
Average Delay [s]
Un−Effective Algorithm
OPERA
Fig. 10. OPERA accuracy: average delay vs time period T.
networks, as plincreases. For the static scenarios, the sim-
ulation set is the same adopted in previous subsection, with
pl∈[0.1,0.9] and with T∈[0.6,0.9]. For the mobile scenario,
we adopt the simulation set used to validate OPERA.
Fig. 8 presents both the mean and the variance values of
the delays for T=0.6, as probability plincreases. Each point
represents the average of the delays of all the couples source-
destination for a certain value of pl. First, we observe that the
higher is pl, the lower is the confidence of the mean values.
This result is reasonable, since the PU activity probability is
a measure of the ”noise” introduced in the delay estimation.
Moreover, we observe that for each value of pl, the difference
between the average delays computed by OPERA and those
computed by Un-Effective Algorithm is significant (axis yis
log-scale). This result highlights that, also for low number of
channels (N=2), it exists a significant route diversity that
must be exploited to effectively measure the route delays.
To better understand the effects of the route diversity on
the end-to-end delay, in Fig. 9 we report the delays for two
different values of plas the distance between the source and
destination increases. First, we observe that the difference
between the delays computed by OPERA and those computed
by Un-Effective Algorithm increases with the distance. This
result is reasonable, since the longer is the path, the more
numerous are the PUs affecting it, and the more significant
are the effects of the route diversity. Moreover, we observe
that the Un-Effective Algorithm measures delays that are in
average the 30% longer than those measured by OPERA.
As a consequence, a routing protocol based on Un-Effective
Algorithm forwards the packets on sub-optimal routes, as
showninFig.6.
Fig. 10 presents both the mean and the variance values of
the delays for pl=0.5, as the time period Tincreases. First,
we observe that, unlike the results of Fig. 8, Thas no impact
on the confidence of the mean values. This result confirms
the consideration made for Fig. 8, since in this scenario the
PU activity probability is fixed. Moreover, we observe that the
delays grow roughly linearly with T. This result confirms the
consideration made in the previous subsection: for pl=0.5
the delay ¯qN
icn
1−¯qN
icn
T, introduced by the PU activity, dominates
the end-to-end delay. Finally, we observe that, as Tgrows, the
difference between the delays computed by OPERA and those
computed by Un-Effective Algorithm increases as well. This
result highlights the importance of the route diversity effects
in CRAHNs.
In Fig. 11, we report the delays for two different values of
the time period Tas the distance between the source and the
destination increases. We observe that the difference between
the delays computed by OPERA and those computed by Un-
Effective Algorithm increases as the distance between the
nodes increases. This result confirms the consideration made
for Fig. 9: the longer is the path, the more significant are the
effects of the route diversity.
Finally, in Fig. 12 we report the delays obtained by (18)
as plincreases for three different values of the route update
period δnormalized to the sensing process parameter T.In
this experiment, we adopt the simulation set used to validate
OPERA, since it allows us to better analyze the impact of
the route update period on the delay estimation. First, we
observe that, surprisingly, the delays for mobile networks are
significantly lower than those obtained for static networks
(Fig. 5). The reason is that relation (18) accounts only for
the delays of the packets that can be delivered in the time
interval [t, t +δ), since packets with longer delays do not
contribute to the delay estimation. Moreover, we observe that
a similar effect is observable by comparing the delays for
different values of δ: the longer is the route update period,
the larger are the delays. Finally, we observe that for the
lowest values of pl, the delays are dominated by the packet
transmission delay L
¯
ψ,while,asplincreases, the delays are
dominated by the route update period δ. This result confirms
the considerations made previously.
VII. CONCLUSIONS
In this paper, an optimal routing metric for cognitive radio
ad hoc networks, OPERA, is proposed. OPERA has been
designed to achieve two features: i) Optimality:OPERAis
optimal when combined with both Dijkstra and Bellman-
Ford based routing protocols; ii) Accuracy: OPERA exploits
the route diversity provided by the intermediate nodes to
CALEFFI et al.: OPERA: OPTIMAL ROUTING METRIC FOR COGNITIVE RADIO AD HOC NETWORKS 2893
200 400 600 800 1000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
CR user pair distance
Delay [s]
Un−Effective Algorithm: T=0.8
OPERA: T=0.8
Un−Effective Algorithm: T=0.6
OPERA: T=0.6
Fig. 11. OPERA accuracy: delay vs CR user pair distance for different
values of time period T.
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
PU Activity Probability
Delay [s]
OPERA: = 4T
OPERA: = 8T
OPERA: = 16T
Fig. 12. Mobile networks: OPERA delay vs PU activity probability
plfor different values of the routing update period δnormalized to the
time period T.
measure the actual end-to-end delay, by taking explicitly into
account the unique characteristics of cognitive radio networks.
A closed-form expression of OPERA has been analytically
derived for both static and mobile networks, and its optimality
has been proved rigorously. The performance evaluation con-
firmed the benefits of adopting the proposed routing metric
for cognitive radio ad hoc networks.
ACKNOWLEDGEMENT
The authors would like to thank Dr. A.S. Cacciapuoti, M.
Pierobon, Dr. Z. Sun and P. Wang for their valuable feedbacks
and contributions to improve this work.
APPENDIX
A. Lemma 1
Proof: First, we observe that the additional delay for each
unsuccessful transmission is6T, as shown in Fig. 3. Therefore,
it is possible to express the expected link delay lij as:
lij =
+∞
e=0 qN
ij eN
m=1 ⎛
⎝qm−1
ij pτm
ij
ij
L
¯
ψτm
ij
ij ⎞
⎠+
+
+∞
e=1 qN
ij eeT
N
m=1 qm−1
ij pτm
ij
ij (28)
where edenotes the number of transmission attempts failed
due to the presence of PU activity on each available channel.
By using the notable relations ∞
n=0 nxn=x/(1 −x)2and
∞
n=0 xn=1/(1−x)if |x|<1, and by exploiting the relation
N
m=1 qm−1
ij pτm
ij
ij =1−qN
ij , we obtain (9).
6For the sake of simplicity, we have assumed that the arrival times of the
packets are synchronized with the slot period T. Nevertheless, it is easy to
derive (28) for the more general case of arrival times uniformly distributed
in the transmission period Ttx , by setting the delay introduced by the first
unsuccessful transmission equal to Ttx
2+Ts.
B. Theorem 1
Proof: Similarly to the proof of Lemma 1, it is possible
to express Dik(C)as in :
Dik (C)=
+∞
e=0 ¯qN
icnen
j=1
N
m=1
¯qm−1
icjpτm
icj
icj⎛
⎝
L
¯
ψτm
icj
icj
+Dcjk⎞
⎠+
+∞
e=1 ¯qN
icneeT
n
j=1
N
m=1
¯qm−1
icjpτm
icj
icj(29)
where edenotes the number of transmission attempts failed
due to the presence of PU activity on each available chan-
nel towards each available neighbor cj. By using the no-
table relations ∞
n=0 nxn=x/(1 −x)2and ∞
n=0 xn=
1/(1 −x)if |x|<1, and by exploiting the relation
n
j=1 N
m=1 ¯qm−1
icjpτm
icj
icj=1−qN
icn, we obtain (11).
C. Lemma 4
Proof: We consider the CR algebra CR (22), where the
path cost Lis defined according to OPERA metric (11), since
the proof for mobile networks follows accordingly. Since the
independent PU activity condition (7) holds, we have:
Dik (˜
C)= 1
1−qN
i˜
C˜
DC+qN
i˜
CT(30)
Dik(c˜n)= 1
1−qN
ic˜n˜
Dc˜n+qN
ic˜nT(31)
Dik(˜
C⊕c˜n)= 1
1−qN
i˜
CqN
ic˜n˜
DC+qN
i˜
C˜
Dc˜n+qN
i˜
CqN
ic˜nT(32)
where qN
icjis defined in (10) and qN
i˜
C=
ucj∈˜
C
qN
icj.
Case 1 (⇒): We prove the case with a reductio ad absurdum
by supposing that:
∃c˜n∈C
ik :Dik(c˜n)<Dik (˜
C)=⇒D
ik(˜
C⊕c˜n)>Dik (˜
C)
(33)
2894 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 8, AUGUST 2012
By substituting (30) and (32) in the right side of (33), we
obtain:
Dik (c˜n)>1
1−qN
i˜
C˜
DC+1−qN
i˜
CqN
ic˜n
1−qN
ic˜n
T(34)
and, since qN
i˜
C≤1and Dik(˜
C)>Dik (c˜n), (34) constitutes a
reductio ad absurdum.
Case 2 (⇐): Since, from (26), we have that ∀c˜n∈C
ik :
Dik(c˜n)<Dik (˜
C⊕c˜n), and since, for hypothesis Dik (˜
C⊕
c˜n)<Dik(˜
C), it results that ∀c˜n∈C
ik :Dik(c˜n)<Dik (˜
C).
D. Lemma 5
Proof: From (26), we have that:
∀c˜n∈C
ik :Dik (c˜n)<Dik(˜
C⊕c˜n)(35)
E. Lemma 6
Proof: We consider the CR algebra CR (22), where the
cost Lis defined according to OPERA metric (11), since the
proof for mobile networks follows accordingly. We prove the
lemma with a reductio ad absurdum by supposing that:
Wik(cj)<Wik (cj+1 )=⇒D
ik (C)>Dik (˜
C)(36)
where Wik(cj)is defined in (26), and C=
{c1,...,c
j,c
j+1,...,c
n}and ˜
C={c1,...,c
j+1,c
j,...,c
n}
are ordered sets of neighbors. Since we can re-write the left
side of (36) as Wik (cj)>Wik(cj+1 ),wehaveareductio ad
absurdum.
REFERENCES
[1] I. F. Akyildiz, W.-Y. Lee, and K. R. Chowdhury, “Crahns: cognitive
radio ad hoc networks,” Elsevier Ad Hoc Netw. J., vol. 7, no. 5, pp.
810–836, 2009.
[2] M. Cesana, F. Cuomo, and E. Ekici, “Routing in cognitive radio
networks: challenges and solutions,” Elsevier Ad Hoc Netw. J.,vol.9,
no. 3, pp. 228–248, 2011.
[3] M. Lu and J. Wu, “Opportunistic routing algebra and its applications,”
in Proc. 2009 IEEE Int. Conf. Comput. Commun., pp. 2374–2382.
[4] B. Z. R. Draves and J. Padhye, “Routing in multi-radio, multi-hop
wireless mesh networks,” in Proc. 2004 Annual Int. Conf. Mobile
Comput. Netw., pp. 114–128.
[5] G. Cheng, W. Liu, Y. Li, and W. Cheng, “Spectrum aware on-demand
routing in cognitive radio networks,” in Proc. 2007 IEEE Int. Symp.
New Frontiers Dynamic Spectrum Access Netw., pp. 571–574.
[6] H. Ma, L. Zheng, X. Ma, and Y. Luo, “Spectrum aware routing for
multi-hop cognitive radio networks with a single transceiver,” in Proc.
2008 Int. Conf. Cognitive Radio Oriented Wireless Netw. Commun., pp.
1–6.
[7] Z. Yang, G. Cheng, W. Liu, W. Yuan, and W. Cheng, “Local coordination
based routing and spectrum assignment in multi-hop cognitive radio
networks,” Springer Mobile Netw. Appl., vol. 13, pp. 67–81, Apr. 2008.
[8] A. Sampath, L. Yang, L. Cao, H. Zheng, and B. Y. Zhao, “High
throughput spectrum-aware routing for cognitive radio based ad-hoc
networks,” in Proc. 2008 Int. Conf. Cognitive Radio Oriented Wireless
Netw. Commun.
[9] J. Sobrinho, “An algebraic theory of dynamic network routing,”
IEEE/ACM Trans. Netw., vol. 13, no. 5, pp. 1160–1173, Oct. 2005.
[10] I. Filippini, E. Ekici, and M. Cesana, “Minimum maintenance cost
routing in cognitive radio networks,” in Proc. 2009 IEEE Int. Conf.
Mobile Adhoc Sensor Syst., pp. 284–293.
[11] L. Ding, T. Melodia, S. Batalama, and M. J. Medley, “Rosa: distributed
joint routing and dynamic spectrum allocation in cognitive radio ad hoc
networks,” in Proc. 2009 ACM Int. Conf. Modeling, Analysis Simulation
Wireless Mobile Syst., pp. 13–20.
[12] Z. Wang, Y. Sagduyu, J. Li, and J. Zhang, “Capacity and delay scaling
laws for cognitive radio networks with routing and network coding,” in
Proc. 2010 Military Commun. Conf., pp. 1375–1380.
[13] S.-C. Lin and K.-C. Chen, “Spectrum aware opportunistic routing in
cognitive radio networks,” in IEEE 2010 Global Telecommun. Conf.,
pp. 1–6.
[14] F. Cuomo and A. Abbagnale, “Gymkhana: a connectivity-based routing
scheme for cognitive radio ad hoc networks,” in Proc. 2010 IEEE Int.
Conf. Comput. Commun. Workshops, pp. 1–5.
[15] K. R. Chowdhury and I. F. Akyildiz, “CRP: a routing protocol for
cognitive radio ad hoc networks,” IEEE J. Sel. Areas Commun.,pp.
794–804, Apr. 2011.
[16] A. S. Cacciapuoti, M. Caleffi, and L. Paura, “Optimal constrained
candidate selection for opportunistic routing,” in Proc. 2010 IEEE
Global Telecommun. Conf., Dec. 2010, pp. 1–5.
Marcello Caleffi received the Dr. Eng. degree
summa cum laude in Computer Science Engineering
from University of Lecce in 2005, and the Ph.D.
degree in Electronic and Telecommunications En-
gineering from University of Naples Federico II in
2009. Since 2008, he is with the Dept. of Biomed-
ical, Electronic and Telecommunications Engineer-
ing, University of Naples Federico II, as postdoctoral
research fellow. From 2010 to 2011, he was with
Broadband Wireless Networking Laboratory, Geor-
gia Institute of Technology, Atlanta, as well as with
the NaNoNetworking Center in Catalunya (N3Cat), Universitat Politcnica de
Catalunya (UPC), Barcelona, as a visiting researcher. His research interests
are in cognitive radio networks and human-enabled wireless networks.
Ian F. Akyildiz received the B.S., M.S., and Ph.D.
degrees in Computer Engineering from the Uni-
versity of Erlangen-N ¨urnberg, Germany, in 1978,
1981 and 1984, respectively. Currently, he is the
Ken Byers Chair Professor with the School of Elec-
trical and Computer Engineering, Georgia Institute
of Technology, Atlanta, the Director of Broadband
Wireless Networking Laboratory and Chair of the
Telecommunication Group at Georgia Tech. In June
2008, Dr. Akyildiz became an honorary professor
with the School of Electrical Engineering at Uni-
versitat Polit`ecnica de Catalunya (UPC) in Barcelona, Spain. He is also
the founding Director of the N3Cat (NaNoNetworking Center in Catalunya)
at UPC in Barcelona. He is the Editor-in-Chief of Computer Networks
(Elsevier) Journal, and the founding Editor-in-Chief of the Ad Hoc Networks
(Elsevier) Journal, the Physical Communication (Elsevier) Journal, and the
Nano Communication Networks (Elsevier) Journal. Dr. Akyildiz serves on
the advisory boards of several research centers, academic departments, high
tech companies, journals and conferences. He is an IEEE Fellow (1996) and
an ACM Fellow (1997). He received numerous awards from IEEE and ACM.
His research interests are in nano-networks, cognitive radio networks and
wireless sensor networks.
Luigi Paura received the Dr. Eng. degree summa
cum laude in Electronic Engineering in 1974 from
University of Napoli Federico II. From 1979 to
1984, he was with the Dept. of Biomedical, Elec-
tronic and Telecom. Engineering, University of
Naples Federico II, first as an Assistant Professor
and then as an Associate Professor. Since 1994, he
has been a Full Professor of Telecom.: first, with the
Dept. of Mathematics, University of Lecce, Italy;
then, with the Dept. of Information Engineering,
Second University of Naples; and, finally, from
1998 he has been with the Dept. of Biomedical, Electronic and Telecom.
Engineering, University of Naples Federico II. He also held teaching positions
at University of Salerno, at University of Sannio, and at University Parthenope
of Naples. In 1985-86 and 1991 he was a visiting researcher at Signal and
Image Processing Lab, University of California, Davis. His research interests
are mainly in digital communication systems and cognitive radio networks.
