An economic framework for routing and channel allocation in cognitive wireless mesh networks

188 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014

An Economic Framework for Routing and ChannelAllocation in Cognitive Wireless Mesh Networks

Reza Mossanen Amini and Zbigniew Dziong, Senior Member, IEEE

Abstract—We consider wireless mesh networks with cognitiveability of the wireless routers’ radios. The cognitive ability is acost efficient manner to increase available bandwidth but requiresan adaptive bandwidth management mechanism to deal withdynamics of primary users’ activities. In this paper, we investigatethe joint channel allocation and routing in cognitive wireless meshnetworks including the channel reuse opportunities in order toimprove the network performance. In particular we propose aneconomic framework for adaptation and control of the networkresources with the goal of network profit maximization. Theeconomic framework is based on the notion of state dependentnode shadow price that is derived from Markov decision theory.The node shadow prices are used as routing metrics while theiraverage values are used to allocate the channels among thedifferent nodes. Simulation results illustrate the network profitmaximization and effectiveness of the proposed channel alloca-tion scheme that is integrated with a channel reuse algorithm.

Index Terms—Cognitive radio, channel allocation, routing,channel reuse, economic model, Markov decision process, wire-less mesh network.

I. INTRODUCTION

EVER increasing bandwidth demands from users of wire-less networks force the network designers to consider

new network paradigms. The multi-hop wireless mesh net-works with cognitive ability have been identified as a validnetwork paradigm capable of providing significant increase ofspectrum usage efficiency [1], [2], [6]. In this environment,a channel of the primary user (channel owner) is used inan opportunistic manner by a secondary user (without accesslicense) to establish its communication. Still the primaryuser, PU, has priority so the transmission of the secondaryuser, SU, should not interfere with the transmission of PU.These constraints combined with the time varying PU chan-nels availability require new approaches for dynamic channelallocations, CAC and Routing and channel reuse in secondarynetworks. While in the literature there are some works thataddress particular issues (e.g. [4], [11],[18], [19]) they do notprovide a consistent framework that would cover all of theseissues. In contrast we propose a novel homogeneous economicframework that integrates the mentioned issues and thereforesimplifies the management and provides performance gains.

In order to describe in more details the proposed frameworkand compare it with related works, we first present theconsidered system and the basic assumptions. The addressed

Manuscript received August 15, 2013; revised October 26, 2013.The authors are with the Electrical Engineering Department, École de

Technologie Supérieure, University of Quebec, Montreal, Canada (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TNSM.2013.120413.120533

Fig. 1. Considered network architecture.

secondary network consists of wireless mesh routers with cog-nitive ability that allows to use opportunistically primary userspectrum, without harming the primary user transmissions,and to prevent the interference between the secondary userconnections. The available cognitive channels are dynamicallyallocated to the routers via a bandwidth management algorithmthat takes into account the traffic distribution and channelavailability. In this work we assume that the MAC layer usesthe TDMA/FDMA access scheme that facilitates schedulingand channel diversity. Moreover we assume that local connec-tions between the clients and the routers use separate radiosand frequencies. In this architecture, illustrated in Fig. 1, wealso assume that the channel availability information, whichis required during channel allocation and routing process,is available from the secondary network database fed by asignalling protocol.

The database indicates whether channels are busy or idleand the type of occupation (primary or secondary user).The proposed approach considers service demands on theconnection level. In this case each connection is allocatedan effective bandwidth that provides the required QoS onthe packet level but the packet level is not modelled in thispaper. If sufficient bandwidth is not available for a connectiondemand, the call admission control (CAC) rejects the demand.

The goal of this work is to develop a joint channel allo-cation and routing framework which provides network profitmaximization. The profit is defined as a difference betweenthe rewards received for serving the connections and thephysical cost of the network resources. To achieve this goalwe propose an approach where the dynamic channel allocationand the CAC and routing algorithms are integrated based ona Markov decision process framework. The basic concept ofthis framework is a state dependent node shadow price which

1932-4537/13/$31.00 c© 2013 IEEE

AMINI and DZIONG: AN ECONOMIC FRAMEWORK FOR ROUTING AND CHANNEL ALLOCATION IN COGNITIVE WIRELESS MESH NETWORKS 189

is a dynamic cost of accepting a connection in a node. Thenthe routing selects a path with minimum sum of the nodeshadow prices and this sum should be smaller than the rewardfrom the connection, otherwise the connection is rejected. Itis important to note that the average value of the node shadowprice corresponds to the node average revenue sensitivity tothe number of node channels. Therefore this metric is used toadapt the channel allocation between the wireless mesh routersin order to maximize the network profit. Resource utilizationoptimization is further improved via channel reuse algorithmthat is integrated with capacity adaptation procedure, whichincreases the network total capacity for the given total numberof available channels. The proposed approach is based on amix of distributed (between nodes) and centralized algorithmsthat can be implemented in centralized or decentralized modes.

In the literature there are several works related to therouting and channel allocation in wireless mesh networks andsome of them treat these issues in cognitive environment.Concerning routing, different routing metrics are proposedin the literature for multi-hop multichannel wireless networkincluding hop-count, expected transmission count ETX [3],weighted cumulative expected transmission time WCETT [4].The implementation of these metrics in the cognitive envi-ronments is more complex and the works presented in theliterature for cognitive wireless mesh networks (CWMNs)performing packet routing (e.g. [11], [18]) do not consider thepreemptive characteristic of the primary user while this is animportant feature that is taken into account in our approach. Todo that we apply a framework of reward maximization routingbased on Markov Decision theory, that was formulated forwired network in [5]. In our approach we extend the conceptof link shadow price into the node shadow price that includesthe preemptive mode of operation reflecting the primary users’priority. At the same time our routing metric provides loadbalancing.

Concerning channel allocation algorithms for wireless meshnetworks (WMNs) there are many works that propose heuristicchannel allocation algorithms (e.g. [9], [11]) but they cannotguarantee optimality of the solution or in some cases thealgorithms are very complex and not practical (e.g. [18]). Forexample [9] takes into account channel interference cost andchannel diversity factor to design a channel cost metric CCM,but this approach does not guarantee absence of interferenceand the channel allocation may be not optimal, as the CCMmetric is local. In our approach both of these limitations areovercome. The approach from [11] also does not provideguarantees for the co-channel interference absence. Whilealgorithm presented in [18] avoids the interference problem, itapplies INLP modified to ILP followed by heuristic process,which makes it difficult to be used in real time joint channelallocation and routing operations. In contrast our frameworkis decentralized which allows for real time execution.

Concerning the channel allocation adaptation for profit max-imization, the most relevant works were developed for overlaywired networks with adaptive online bandwidth provisioning:Duan in [12] and by Tran in [13], [14]. There are three maindifferences of our approach when compared with the mostrelevant works of [13], [14]. First we use the concept ofnode shadow price instead of the link shadow price concept.

Second, in our model the node shadow prices are calculatedtaking into account preemption from the primary users. Thethird difference comes from the fact that the resources inwireless network are limited and therefore in our model theadaptation of channel allocation to nodes is done exclusivelyby reallocating (borrowing) the channels between the nodesforming the network while in [13], [14] the needed bandwidthis leased on demand from external resource providers (ISPs)based on SLAs (Service Layer Agreements).

Concerning the algorithms for the channel reuse the mostrelevant are approaches developed for ad hoc wireless net-works. In [19] and [20] the algorithms are based on exchangeof control packets (RTS/CTS/RES) at each time a communi-cation link is established. In our approach the channel reusealgorithm is integrated in the capacity adaptation procedurewhich is performed periodically and the channels are assignedto different nodes by updating distributed database.

The remainder of this paper is organized as follows. InSection II we present the main issues (CAC and Routingpolicy, capacity adaptation, channel reuse) and used assump-tions. In Section III we formulate the economic frameworkbased on Markov decision process and node shadow pricethat integrates CAC & Routing policy and capacity adaptation.The node shadow price evaluation model for the consideredpreemptive system is derived in Section IV. Section V presentsthe capacity adaptation procedure without channel reuse whilethe case with channel reuse is described in Section VI. Sec-tion VII contains the performance evaluation of the proposedmodels based on simulation results. Finally, the conclusionsare presented in Section VIII.

II. MAIN ISSUES AND ASSUMPTIONS

A. Connection Level Modeling

We assume that the considered network should supportmultimedia and emergency services which require end-to-end QoS. To facilitate this task we apply connection levelbandwidth management where an effective bandwidth is al-located to connections in order to guarantee QoS in packetlevel (e.g. [21], [22]). This is consistent with current approachin wired networks that is based on MPLS (e.g.[23], [24]).Consequently, in this paper, we model only the connectionlevel.

B. Coexistence with the Primary Users

The considered cognitive radio of a secondary user usesopportunistically the licensed channels of the primary users,PUs, that are not used currently by PUs, and vacates themwhen PUs revisits this portion of the spectrum. The structureof channels accessible by a secondary user, SU, node is shownin Fig. 2. From the perspective of SU network we refer to thesechannels as cognitive channels. Management of the cognitivechannels requires two mechanisms. The first should indicatewhen a PU channel is freed and when it becomes occupiedagain by PU. In general this information can be obtainedby using a sensing mechanism at SU nodes (e.g. [31]–[34])or from a public database (e.g. [35]–[39]). In this paper weassume that this information is available. The second neededmechanism is related to connection preemption. Namely, when


Fig. 2. Structure of SU node channels.

the primary user appears in its licensed channel, the secondaryuser connection using this channel must be preempted andeither removed or moved to another idle channel. We assumethat switching connection to another idle channel is done fastenough to not interrupt the connection continuity and thatthe database updating is assured via a signaling protocol (notcovered in this paper).

To facilitate the presentation, in this paper we assumethat the connections of secondary users are homogeneous i.e.they have the same bandwidth requirements and the samemean service times (the same assumption is used in [14]).Nevertheless, this is not a limitation of the proposed approachas, analogously to the system analysed in [5], it can beextended to heterogeneous cases with some approximationsreducing the implementation complexity. We also assume thatthe primary user channel can carry k secondary connectionswhich gives dp = kds, where dp represents the PU channelbandwidth and ds is the bandwidth used by a SU connection,the same assumption are used in [28] and [29].

It should be mentioned that in general each secondary nodecan have also access to its own exclusive licensed channelsthat can be used to establish a reliable signalling links orto serve a portion of the secondary network users. Althoughour approach also covers such architecture, in this paper weconsider the system with cognitive channels only to simplifythe presentation as the challenges and novelty are related tomanagement of cognitive channels.

C. CAC and Routing Policy

The CAC and routing mechanism should provide a routewith the required bandwidth to each connection demand. Wealso assume that the routing should take into account thevariability of number of available channels due to primaryuser arrivals or channel/radio failures in order to optimizebandwidth utilization. In particular the preemptive aspect ofprimary user should be considered. Channel condition dete-rioration or Channel/radio failure cause preemption like PUarrival to its channel. If there is an ongoing communicationon the target channel the effect is the same. We achievethese objectives by designing an appropriate routing metricpresented in Subsection IV.B.

D. Capacity Adaptation

The fixed channel allocation is not efficient for temporal andspatial fluctuations in load distribution. To cope with this issueone can consider two possible channel allocation adaptationschemes: short term and long term. In the short term adap-tation scheme the total number of available channels in the

network is constant and the adaptation scheme reallocates thechannels to different wireless routers according to the needs.In the long term capacity adaptation scheme the total numberof available channels is adjusted to the long term changesin the demand. In this paper we address only the short termadaptation that is described in Section V and in Section VI.We also assume that there is always sufficient number of radiointerfaces when channels are reallocated.

E. Co-channel Interference Constraint and Channel Reuse

The neighboring radios operating on the same channel causethe co-channel interference when they transmit simultaneouslyand therefore there should be sufficient distance between a pairof nodes using the same channel. In our work we assume thatthe co-channel interference is acceptable when this distance isat least three hops. This assumption is used in channel reusescheme that is described in Section VI.

In conclusion of this section we want to underlined thatin this section as well as in the following sections we useseveral simplifying assumptions to facilitate the presentationof the proposed methodology but they are selected in the waythat does not affect significantly the main objective of thepaper which is presentation of the proposed approach and itsperformance.

III. ECONOMIC FRAMEWORK

In this section we introduce a model for capacity adap-tation and CAC and Routing that is based on an economicframework. First let us introduce the applied notation. Ourcognitive wireless mesh network is defined by set of nodesS =

{s1, s2, s3, . . . , s|S|

}. A secondary user connection of

class j, j = 1, 2, . . . , |J |, is characterized by its Origin-Destination (OD) pair, bandwidth requirement dj , Poissonconnection arrival process with rate λj , exponentially dis-tributed connection service time with mean μ−1j , connectionreward parameter rj , and a set of alternative paths Wj . It isassumed that each admitted connection brings to the networka reward at the rate of qj = rj · μj and therefore the rewardparameter can be interpreted as average revenue from class jconnection. Then the network average profit is expressed as:

P = R−∑

s∈S CS , (1)

where R represents the average reward from accepted connec-tions and CS represents node s channels’ physical cost ratedefined as: CS = cNS , where c is the channel bandwidth unitprice and NS is node s capacity.

Since under our assumptions of fixed number of channelsthe sum

∑s∈S CS is constant and does not depend on CAC

and routing and capacity adaptation, the network average profitmaximization corresponds to maximization of the averagereward from accepted connections defined as:

R (π) =∑

j∈J λjrj , (2)

where λj is the rate of class-j accepted connections andπ denotes the CAC and Routing policy. Note that in thisformulation the reward parameter can be used to achieve threedifferent goals. First, when the reward parameter corresponds


to the price paid for service, the network control maximizesthe revenue and profit from the network. Second, when thereward parameters are equal each other, the network controlmaximizes the network traffic defined as the number of ac-cepted connections. Third, when the reward parameter of oneconnection class is increased (decreased) the priority of thisclass is increased (decreased) and consequently its blockingprobability is decreased (increased) so the reward parametercan be used to control the distribution of blocking probabilitiesamong the connection classes.

Let us consider first maximization of (2) with respect toCAC and Routing policy π. Since under our assumptions thenetwork connection process constitutes a Markov process, itcan be easily shown that the optimal CAC & Routing policythat maximizes the network average reward is connectionstate dependent and can be found by applying one of thealgorithms from Markov decision process (MDP) theory. Un-fortunately in the exact MDP model, the network connectionstate and policy spaces can be very large even for moderate-sized networks. Therefore in our approach, similarly to theframework described in [5] which is also used in [13], [14][30],the network Markov process is decomposed into a set ofnode Markov processes, assumed to be independent, driven byPoisson node connection arrival processes with average rateλsj (π). This implies that a connection established on a path k

consisting of∣∣Sk

∣∣ nodes is decomposed into∣∣Sk

∣∣ independentnode connections characterized by the same mean service timeas the original connection. Then the network reward processis decomposed into a set of separable node reward processesby defining node connection reward parameters rsj with theobvious condition:

rj =∑

s∈Skrsj (π) . (3)

Concerning the division rule that distributes the connectionreward parameter among the nodes we use a simple equaldistribution rule, rsj =

rj|Sk| . This follows from numerical

results presented in [5] for wired networks that indicatedthat the network reward is not sensitive to this rule withinreasonable range. Then the CAC and Routing policy is definedby routing decision that selects a path that maximizes the pathnet-gain defined as

gmax = maxk∈Wj

[rj −

∑s∈Sk

psj (x, π)], (4)

where x is the node s state which represents the numberof SU connections and the number of active PUs in node sand psj (x, π) denotes node state dependent shadow price thatexpresses the expected loss of future revenue from connectionsrejected due to acceptance of the current connection demand.Obviously the new connection is rejected if the path net-gain isnegative. Since the node shadow price calculation is performedin periodic intervals, based on measurement based estimationsof λs

j (x, π), the described approach corresponds to the policyiteration algorithm that under stationary conditions convergesto the optimal policy defined by the node shadow pricevalues. Concerning models for calculation of shadow prices,some iterative models are presented in [5]. Unfortunatelythese models do not apply to our model with preemptiveMarkov process. Therefore we have developed exact and

Fig. 3. Economic framework.

approximate models for such a system that are presented inSubsections IV.B and IV.C.

Concerning adaptive capacity allocation, it can be shown,based on [5], that the node average reward sensitivity to itscapacity can be approximated by the node average shadowprice of a connection class with unit bandwidth requirement.

ps (Ns) = Rs (Ns)−Rs (NS − 1) ∼= ∂Rs

∂Ns, (5)

where ps (Ns) and Rs (Ns) represent average shadow priceand average reward for node s with capacity Ns. Based onthis property the optimality condition for the network profitmaximization can be easily derived for two cases, without andwith channel reuse, that are presented in Sections V and VI,respectively. Note that under the assumption of fixed numberof available channels in the network, capacity adaptation isrealized by channel assignment which is based on channelborrowing/lending mechanism that in our approach is executedperiodically at the end of each sampling interval.

It is important to underline that the presented economicmodel integrates the CAC and Routing function with capacityadaptation procedure in the same economical framework. Thisintegration is realized by common use of the node shadowprice concept as illustrated in Fig. 3. Obviously for stabilityreason the capacity adaptation period is longer than the CACand routing adaptation period. It should be also mentionedthat the proposed framework can be extended further tocontrol the connection rejection rates by adapting periodicallythe connection reward parameters. Increasing the connectionreward parameter causes the increase of connection acceptancerate and vice versa. Such a mechanism can be used to provideaccess fairness. Obviously this should be done in a cycle thatis longer that the capacity adaptation cycle, as illustrated inFig. 3, although this issue is not treated in this paper. Themulti-timescale aspect of the framework presented in Fig. 3is consistent with the multi-time scale traffic managementapproach for MPLS network presented in [40].

Implementation of the presented approach can be realized inseveral ways. Concerning calculation of the shadow prices andtheir averages it is important that both of these metrics canbe calculated locally in the respective nodes based on local


measurements of traffic distribution. The CAC and Routingalgorithm could use a centralized server to calculate optimalroutes but a decentralized (among the nodes) implementa-tion seems to be better although it requires distribution ofthe metrics to all nodes so each node can calculate theoptimal routes when needed. Such decentralized approachis analogous to OSPF routing algorithm implementation inautonomous Internet systems that is decentralized although ituses Dijkstra’s algorithm that requires all network link states.Similarly, the adaptive capacity allocation can be implementedin centralized or decentralized way depending whether thenetwork operator wants to have a centralized database in aserver or a decentralized database with a complete image ineach node. Each of these solutions has well known advantagesand disadvantages related to reliability, signaling traffic, anddatabase synchronization problems. In this paper we do notaddress these issues.

IV. PREEMPTIVE NODE SHADOW PRICE EVALUATION

In this section we present algorithms related to the nodeshadow price calculations. First we present the required nodetraffic parameters estimation. Then the exact and approximatealgorithms for calculation of preemptive node shadow pricesare presented.

A. Estimation of Node Offered Traffic Parameters

The traffic estimation is based on measured statistics. In thissubsection we describe the estimation methods for primaryand secondary traffic parameters. In order to reduce thecomplexity, in the following the different SU homogenousconnection classes that are routed through a given node s areaggregated into one class and referred by index 1 and the PUclass is referred by index 2. Thus the node state is expressedby vector x = [xl] containing two components, where x1

represents the number of SU connections carried by the nodes and x2 is the number of active PUs in node s.

1) Estimation of Primary User Node Arrival and ServiceRates: We assume that each primary user channel occupationfollows on-off distribution with the non-active period averageduration equal to 1

λc2

and the average service time equal to 1μs2

.We also assume that both periods are exponentially distributed.Then, when there are x2 active primary users, node s totalprimary user arrival rate follows binomial distribution definedas:

λs2 (x2) = (N c

s − x2)λc2, (6)

where N cs is the number of cognitive channels allocated to

node s and λc2 is one primary user arrival rate. We can also

define average number channels occupied by primary users atnode s as:

ρs2 = N cs ∗

1μs2

1λc2+ 1

μs2

(7)

By transforming this equation we arrive at:

λc2 =

(ρs2μ

s2

N cs − ρs2

), (8)

Note that the values of average number of channels occupiedby primary users ρs2 and primary user average service time 1

μs2

can be easily estimated in real system using an exponentialsmoothing method. Then using these estimates in (8) weobtain an estimate of one primary user arrival rate λc

2.2) Estimation of Secondary User Node Arrival and Service

Rates: The secondary user mean service time, 1μs1

, is estimatedby an exponential smoothing average. To estimate secondaryuser class j connection arrival rate in node s, λs

j (π), we useestimates of average arrival rate of class j connections, λj (π),and average rate of connections accepted on each alternativepath, λ

k

j . These estimates are based on statistics measured

during sampling intervals. Once λj (π) and λk

j are estimatedwe use load sharing formula to calculate offered rate of classj connections on path k:

λkj (π) = λj

λk

j (π)∑k∈Wj

λk

j (π), (9)

where Wj is a set of alternative paths. Then the path classarrival rate obtained in (9), allows to estimate the node offeredrate for each connection class from:

λsj (π) = λk

j (π)∏◦∈Sk\{s}

(1− b◦j (π)

), (10)

where b◦j (π) is the class j rejection probability calculated fornode ◦. Equation (10) can be solved iteratively via Erlang’sfixed point method, using substitutions of node class blockingvalues. The iterations stop when the node class blockings donot change. Concerning calculation of b◦j (π) it is obtained bysumming all rejection state probabilities that are obtained bysolving the set of linear equations (20) for the state probabilityvector Q.

B. Calculation of Preemptive Node Shadow Price

As explained in Subsection II.B, the secondary users canbe preempted when primary user becomes active. This pre-emption possibility is taken into account in the correspondingexample of continuous Markov process transition diagrampresented in Fig. 4 for a node with 3 cognitive channels,where each of them can accommodate 2 SU connections. Inthis model each state is represented by a vector x = [x1, x2],where x1 is the number of secondary user connections andx2 is the number of primary users connections. In general,the PU required bandwidth is d2 = k ∗ d1; k ≥ 1 and inFig. 4 we have k = 2 and d1 = 1 representing SU requiredbandwidth. As defined in previous subsection, N c

s representsthe total number of cognitive channels allocated to node sgiving the node capacity of Ns = N c

s ∗ k bandwidth units.If ((x1 +(x2 +1) ∗ d2)−Ns) is positive then it represents

the number of SUs connections that have to be removed in thecase of PU arrival. Such preemptions are indicated by diagonaltransitions in Fig. 4.

The possible state changes are denoted as follows:

- x→ x+ δ1; admission of SU connection,- x→ x+δ2; admission of PU connection, no pre-emption,- x → x + Δ; admission of PU connection, with pre-

emption,- x→ x− δ1; departure of SU connection,- x→ x− δ2; departure of PU connection.


Fig. 4. Preemptive Markov process.

In the following we present an approach to calculate thenode shadow prices for the presented preemptive model. Theapproach is based on the value iteration algorithm and webegin with introduction of notation used in the development.

Let λs1 (π) and λs

2 (x2) represent SU and PU node s ar-rival rates, respectively. λs

1 (π) is the aggregation of all SUconnection classes arrival rates offered to node s

λs1 (π) =

∑j∈Js

λsj (π) . (11)

The departure rates of secondary user and that of primary userare represented by μ1 and μ2 respectively.

The rate of node s reward in state x, is given by

q (x) = rs1 (π) · x1 · μ1, (12)

where rs1 (π) is the aggregated reward parameter for node sdefined as:

rs1 (π) =

∑j∈Ji

rsj (π)λs

j (π)∑j∈Ji

λs

j (π), (13)

whereλs

j (π) denotes the average rate of class j connectionsaccepted in node s. Since the aggregated reward process isstatistically close to the original process we assume that

psj (x, π)∼= psi (x, π) j ∈ Ji, (14)

where Ji denotes a set of connection classes aggregated intoone class (i = 1) at the node level.

Using the above notation the value iteration equations aredefined as (15) at the top of the next page.

Once the value iteration algorithm is converged, the nodenet-gains corresponding to acceptance of a secondary userconnection are given by

gs1 (x, π) = limn→∞ [V s

n (x+ δ1, π)− V sn (x, π)] , (16)

where n is the iteration index. The net gain expresses theincrease in the future node reward caused by acceptance of asecondary user connection in state x. Then the node shadowprice for state x is defined as a difference between the nodeaggregated reward parameter and the node net-gain in state x,which is expressed by:

ps1 (x, π) = rs1 (π)− gs1 (x, π) . (17)

As indicated in Section III, the shadow price expresses theexpected loss of future revenue from SU connections rejecteddue to acceptance of the current SU connection demand and

as a consequence they allow us to calculate the path net-gainfrom carrying class j connection on path k during the routingprocess

gkj (y, π) = rj −∑

s∈Skps1 (x, π) , (18)

where y = {xs}, denotes the network state in the decomposedmodel.

In our approach described in Section III, the node averageshadow price for SU connections is used for node capacityadaptation. This average shadow price represents the expectedrevenue loss when the node capacity is reduced by 1 and isdefined as

ps1 (π) =∑

x∈Xs−xsb

Q1 (x) ps1 (x, π) , (19)

where Q1 (x) denotes the probability that class i = 1,connection is accepted in state x and Xs

b is a set of states inwhich class 1 connections are blocked (cannot be accepted).To find Q1 (x) probabilities we need to calculate first the stateprobabilities in our preemptive loss system from the followingset of equations: {

ΛT ·QT = 0,∑x∈Xs Q (x) = 1

}, (20)

where Λ is the transition-rate matrix and Q is the stateprobability vector. Then Q1 (x) probabilities are found bynormalizing the state probabilities to exclude the blockingstates:

Q1 (x) = Q (x) /∑

x∈Xs−xsb

Q (x) . (21)

C. Approximation of the Preemptive Model

As explained in Subsection B, the state probabilities arerequired for evaluation of connection acceptance probabilitiesfor admissible states which are used for calculation of nodeaverage shadow price value from (19). When the numberof states is significant, calculation of the state probabilitiesfrom (20) is time and memory consuming. To overcome theselimitations we propose an approximate method for shadowprice calculation in the considered system. It is based on ourhypothesis that the SU shadow prices in the preemptive systemwill be similar to the SU shadow prices in a non-preemptivesystem if some significant reward parameter r2 is allocated tothe PU connections in order to reflect their priority. To find thisvalue, we minimize the sum of squared error of approximateshadow prices over r2 values

min∀r2

{∑x∈Zs

(psn (x, π)− psn (x, π))2}= min∀r2

∑x∈Zs

e2x,

(22)where psn (x, π) and psn (x, π) represent the approximated andexact state dependent node shadow price for SU connection,respectively, and Zs is the set of states where SU connectionsare admissible in node s.

To find the minimum of the sum of squared errors, we useNewton method with the steps defined as

rn+12 = rn2 +Δrn2 , (23)

where

Δrn2 =−f (rn2 )

(rn2 − rn−12

)f (rn2 )− f

(rn−12

) , (24)


V sn (x, π) =

{q (x) · τ + λs

1 (π) τ[V sn−1 (x+ δ1, π)− V s

n−1 (x, π)]

+λs2 (x2) τ

[V sn−1 (x+ δ2, π)− V s

n−1 (x, π)]

+λs2 (x2) τ

[V sn−1 (x+Δ, π)− V s

n−1 (x, π)]+∑2

l=1xlμlτ

[V sn−1 (x− δl, π)− V s

n−1 (x, π)]+ V s

n−1 (x, π)}, x ∈ Xs (15)

TABLE INODE’S TRAFFIC PARAMETER

Arrival rate of class1 : 1.8 Service rate of class 1: 1Node arrival rate of class2 : 0.18 Service rate of class2: 0.2

6 :)slennahc-bus(yticapac edoNReward parameter of class1: 5Bandwidth requirement of class2: 2

where f (rn2 ) =∑

x∈Zs e2x, represents the sum of squarederrors. The iterations terminate once new value of f (rn2 ) isgreater than previous one and then the previous r2 value isselected.

To assess the accuracy of the proposed approximation weuse scenario from Fig. 4 with parameters given in Table I.

Fig. 5 shows the sum of squared errors as a functionof r2. For this scenario the applied Newton algorithm findsthe minimum sum of squared errors for r2 = 17.115 in 6iterations.

Once the state dependent shadow price values are approxi-mated, (19) can be used to evaluate the node average shadowprice value. The state probability for non preemptive losssystem can be calculated via classical product form equationand then connection acceptance probability for admissiblestate, Q1 (x), used in (19), can be easily evaluated. The stateprobability calculation is also required for evaluation of nodeclass j rejection probability which is used in (10), to estimateλsj (π).

V. CAPACITY ADAPTATION WITHOUT CHANNEL REUSE

The main objective of channel allocation adaptation ismaximization of the network profit which is also equivalentto the average reward maximization in the considered caseof fixed number of channels. In this section we considerthe network operation mode in which the channels are notreused. We first deduce the optimality condition for profitmaximization in Subsection A. Then the capacity adaptationprocedure is introduced in Subsection B. This procedure usesone of two algorithms for the required capacity calculationpresented in Subsections C and D.

A. Optimality Conditions (No Reuse Case)

In the decomposed model, the average reward from thenetwork, defined by (2), can be expressed as the sum of meanvalue of reward from all nodes forming the network i.e.

R =∑

s∈S Rs, (25)

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

The value of R2LS

E

Fig. 5. Minimization of approximation error.

where Rs represents the mean value of reward from node s.In the considered system the channel allocation operation iscarried out via lending/borrowing mechanism as the number ofavailable channel in the network is constant. We assume thatany given node can borrow the channels from any other nodeto increase its bandwidth. Then it is obvious that to maximizethe network profit, the derivative of the average reward fromthe network with respect to the bandwidth of node i shouldbe equal to zero. If node i borrows one channel from node jthis condition can be expressed as:

∂R

∂Nij=

∂∑

s∈S Rs

∂Nij=

∂Ri

∂Nij+

∂Rj

∂Nij+

∑◦∈S/{i,j}

∂R◦∂Nij

= 0,

(26)where ∂Nij denotes change of node i and node j capacitieswhen node i borrows channel from node j. We assume thatsensitivity of node reward to node capacity for the nodes thatare directly concerned e.g., (i, j) is dominant and thereforewe use the following approximation: ∂R◦

∂Nij= 0. Then the

condition (26) can be approximated by:

∂R

∂Nij

∼= ∂Ri

∂Ni− ∂Rj

∂Nj= 0. (27)

According to (5), we know that the node average rewardsensitivity to its capacity can be approximated by the nodeaverage shadow price of a connection class with unit band-width. Therefore we have

∂R

∂Nij

∼= pi (Ni)− pj (Nj) = 0. (28)

Therefore the optimality condition corresponds to equalizationof all node average shadow prices:

p1 (N1) = p2 (N2) = p3 (N3) . . . = ps (Ns) . (29)


Since the number of available channels in the network isassumed to be constant, this optimality condition can beachieved via periodic channel lending/borrowing between thenodes. To do that we propose an iterative capacity adaptationprocedure described in Subsection B. This procedure usesanalytical or heuristic algorithm to calculate required nodecapacities based on the optimality condition in each iteration.The analytical and heuristic algorithms are described in Sub-sections C and D, respectively.

B. Capacity Adaptation Procedure

The calculation of required node capacities and corre-sponding channel assignments is performed by executing aniterative algorithm with iteration index n. The current networkcapacities are used as initial capacities N0

s for n = 0. Hereare the algorithm steps:

1) Use analytical, RC-A, or heuristic, RC-H, algorithm,described in the following Subsections B and C, tocalculate new required node capacities, Nn+1

s , based onthe optimality conditions.

2) Set n← n+1 and carry out the channel assignment toreach the new required capacities Nn

s by using channelborrowing/lending operations,.

3) Estimate new node average shadow prices ps (Nns )∀s ∈

S and stop the algorithm if the relative error of shadowprice equalization objective is equal or less than athreshold:

max p (Nn)−min p (Nn)

max p (Nn)≤ η, (30)

where max max p (Nn) and min p (Nn) represent max-imum and minimum node average shadow price, re-spectively, and η is a specified precision threshold (inour numerical experiments η = 0.06 following [30]).Otherwise go to Step 1.

When the node capacities are adjusted by the adaptationalgorithm, this event induces a corresponding change in thenode carried traffic. Then, estimated secondary user arrivalrates λs

1 =∑

j∈Js λsj are then used to recalculate the SU

rejection probabilities bsj , for each node. This rejection prob-ability allows us to recalculate the admitted rate of each SUclass in each node

λs

j = λsj

(1− bsj

)(31)

These values are used in (13), to recalculate the node aggregatereward parameter.

It should be mentioned that to transfer a channel from alending node to a borrower node a channel management isneeded. Specifically, if all channels are occupied and thereare some SU connections then a SU connection is removedto provide a channel to the borrower node. If all channels areoccupied by PUs, a channel is transferred but it cannot be usedby the borrower node until the channel is released by PU.

C. Required Capacity Analytical Calculations RC-A

First we approximate the derivative of node s averageshadow price with respect to the node capacity in iteration

n by:

p′s (Nns )∼=

(ps (N

ns )− ps

(Nn−1

s

))(Nn

s −Nn−1s

) . (32)

Then we have a set of |S| non-linear equations to solve,{ps

(Nn+1

s

)− pj(Nn+1

j

)= 0 ∀j ∈ S, j �= s,∑

s∈S δs = 0,

}(33)

where δs represents the change in capacity value of node s.By substituting each function in (33), with its Taylor series

and keeping only the linear terms we arrive at:⎧⎨⎩

ps (Nns ) + δsp

′s (N

ns )− pj

(Nn

j

)− δjp′j

(Nn

j

)= 0

∀j ∈ S, j �= s,∑s∈S δs = 0.

⎫⎬⎭(34)

Then by rearranging (34) we obtain:⎧⎨⎩

δsp′s (N

ns )− δjp

′j

(Nn

j

)= pj

(Nn

j

)− ps (Nns )

∀j ∈ S, j �= s,∑s∈S δs = 0.

⎫⎬⎭ (35)

The matrix form of the above set of equations is representedby:

Jδ = v, (36)

where J is a Jacobian matrix, and v is vector of the differ-ence of average shadow prices. Then solution to this matrixequation can be expressed by:

δ = J−1v (37)

Then the new capacity is calculated from

Nn+1s = nint (Nn

s + αn · δns ) ∀s ∈ S, (38)

where αn is a dumping factor used to avoid oscillations andto speed up the convergence toward the root. The dumpingfactor is a function of the oscillation range measured by themean, m of absolute value of delta vector δ. If this mean ishigh, m ≥ 1 the dumping factor is inversely proportional tom, otherwise it is equal to 1.{

αn = 1m m ≥ 1

αn = 1 m < 1

}with m =

1

|S|∑

s∈S |δns | . (39)

D. Required Capacity Heuristic Calculations RC-H

The concept behind the heuristic algorithm is that the nodewith smallest average shadow price value lends one channel tothe node with greatest average shadow price value. Then thesetwo nodes are excluded from the test in the current iteration

and the same operation is performed for other nodes.The following steps describe the proposed heuristic algo-

rithm:1) Calculate weighted average of all node average shadow

prices:

p =

∑s∈S ps (Ns)λ

s

1∑s∈S λ

s

1

, (40)

where λs

1 =∑

j∈Js λs

j , is the aggregated secondary useradmitted connection rates in node s.

2) Construct list L1 that contains newly calculated nodeaverage shadow prices, ps (Ns) sorted in ascendingorder.


3) Browse list L1 starting with the first element andif(max p (N) > p and ps (Ns) < p

), where max p (N)

represents maximum node average shadow price valuein L1, then the node s becomes lender and the nodewith maximum average shadow price value becomesborrower. The capacity of lender node is reduced by onewhile that of borrower is increased by the same quantityand then the max p (N) value is removed from the list.The test is repeated for the next element in L1 until thelist L1 is exhausted.

The new capacities Nn+1s resulting from this procedure are

returned to the capacity adaptation procedure without the useof a damping factor since here the capacity changes are limitedby the fact that a node can borrow only one channel.

VI. CAPACITY ADAPTATION WITH CHANNEL REUSE

The principle objective of the channel reuse scheme is toincrease the network total capacity expressed as a sum of nodecapacities, NT =

∑∀s∈S Ns when compared with the total

number of channels that can be available in the network, Nu.Consequently we have the following relation

Nu ≤ NT =∑∀s∈S Ns. (41)

The principal constraint in the reuse is the co-channel in-terference which can be avoided only if there is sufficient dis-tance between the nodes using the same channel. We assumethat a channel can be reused only in the distance greater thantwo hops which we call the reuse distance. In the followingwe first present two equivalent algorithms (analytical andheuristic) for channel reuse allocations in Subsection A. Thenthe possible channel borrowing/lending operations in networkswith reuse are explained in Subsection B. In Subsection Cwe deduce the network profit optimality condition for thereuse case. Then the capacity adaptation procedure using themechanisms describe in Subsections A and B is presented inSubsection D.

A. Channel Reuse Maximization

In the proposed channel reuse allocation algorithm we firstmaximize channel reuse by the node with maximum averageshadow price value. Then this process is repeated for othernodes in decreasing order of shadow price values. In the fol-lowing we present two algorithms realizing the same objective.The first is based on the binary integer programming, BIP. Thesecond is a heuristic.

1. BIP based channel reuse maximization, CRM-BIP

1) Calculate all node average shadow prices ps (Ns) andtheir weighted average value p (40). Then construct twolists of nodes, L1 and L2. The first list L1 containsthe nodes with average shadow prices smaller thanthe network average: ps (Ns) < p ∀s ∈ S sorted inascending order of the average shadow price values.The second list L2 contains the remaining nodes thatare sorted in descending order of the average shadowprice values.

2) Exit the algorithm if all nodes in list L1 have alreadybeen considered for channel reuse.

3) Select the first node from L1 whose channels have notbeen yet considered for channel reuse. Then add all itschannels to channel-list V.

4) To maximize the channels reuse carry out the BIP codedescribed below and then go to step 1.

BIP code: Maximize the objective function defined as:

max∑

ch∈V CT y(ch), subject to: (42)∑∀o∈S,d(s,o)≤2 yo ≤ 1 ∀s ∈ L2, (43)

ys = 0 ∀s ∈ IRs0 , ∀s0 (44)

where:- y denotes a column vector with the same length as L2.

In particular each component of y is an indicator variabledesignating a node in L2 with the same index, which is1 when channel ch is reused by the corresponding nodeand 0 otherwise.

- C designates a column vector with 1 in all positions.- IRso refers to the interference region of node so i.e.IRso = {s|s ∈ S, d (so, s) ≤ 2, s �= so}, where so is thenode which already uses channel ch before executingchannel reuse algorithm and d (so, s), is a distance func-tion. The constraint (43) does not allow node s channelreuse in node s interference region, and the constraint (44)sets the indicator variables to zero, which corresponds tothe nodes residing in the interference region of the nodeso. This constraint is dynamically created by the code foreach channel to be reused.

2. Heuristic channel reuse maximization, CRM-HThe heuristic algorithm uses the same first three steps as

the BIP algorithm and then the following steps are executed:4) For each channel, ch ∈ V , execute the steps 5.1 and 5.2,

Once all channels in V are checked for reusing then go to step1

- 5.1 For l = 1, . . . , |L2|:Create a channel ch reuse configuration by checking allnodes reuse possibility in the following order: s =l, . . . , |L2| , 1, . . . , l − 1. Record the channel reuse configu-ration.

- 5.2 From the set of reuse configurations obtained in Step5.1 select the configuration with maximum channel chreuse. If there are several configurations with the samemaximum channel reuse then select the configurationwith the highest average shadow price of node l in step5.1.

B. Channel Borrowing/Lending Operations

Channel reuse maximization performed by CRM-BIP orCRM-H does not necessary achieve the minimization of p,which is the capacity adaptation objective. To obtain thisobjective some channel reallocations using lending/borrowingoperations may be required. In this subsection we presentpossible lending/borrowing operations in the reuse scheme thatare used in capacity adaptation procedure for channel reusepresented in the following subsection.

In channel reuse mode, we can try first to use the direct oneto one borrowing operation that was used in ‘no reuse mode’


Fig. 6. (n to one) lending operations.

but even if such operation is not available due to interferencecaused by another node using the considered channel in theinterference region of the borrower, the borrowing is stillpossible by using indirect one to one or n to one lend-ing/borrowing operations. The mentioned interference regionincludes the set of all nodes residing within two hops distancefrom node i, that is IRi = {s|s ∈ S, d (i, s) ≤ 2, s �= i},where d (i, s), is the distance function. In the following weexplain the new lending/borrowing concepts.

The indirect one to one lending/borrowing operation con-sists of using intermediate nodes in such a way that the lendernode lends a channel to next intermediate node and in turn theintermediate node lends a different channel to the next oneuntil the borrower node is reached. Obviously the co-channelinterference also has to be avoided in each of the intermediatenodes. To do that, we construct a tree based structure inwhich the borrower node is the root of the tree. Then, lendingborrowing chains between a lender and the borrower node aresearched. If there are several alternative chains connecting alender to desired borrower node, we choose the shortest chainfor the ease of management. The selected chain is traversedin reverse direction to perform lending/borrowing operationsto provide a channel to the desired borrower node.

In n to one lending/borrowing operation, the borrower nodeborrows the same channel from n > 1 nodes that are located

within the borrower’s interference region. The two to one andthree to one operations are illustrated in Fig. 6, where the limitof the interference region, IRi, of borrower node i is repre-sented by a circle. The set of nodes {j, k, l} ⊂ IRi representsthe lender nodes that use the same channel simultaneously.It is assumed that the reuse distance between lending nodesis respected. In this situation one to one borrowing operationcannot be used since it would create a co-channel interference.This can be avoided if the channel is borrowed from k andj for two to one operation and k, l and j for three to oneoperation. Note that in n to one operation each lender nodeloses one channel while the borrower node obtain only onechannel thus the network total capacity NT , is reduced byn− 1.

C. Optimality Conditions (Reuse Case)

From optimization viewpoint, channel reuse introduces twoimportant changes when compared with no-reuse case. Firstthe co-channel interference constraints are added to the prob-lem. Second, the network total capacity, NT =

∑∀s∈S Ns,

is not constant since it depends on the reuse pattern. Con-sequently, the solution that provides maximization of theaverage reward from the network may be achieved for channelallocation that does not correspond to the average shadowprice equalization required in no reuse case (29). This canbe simply illustrated in a scenario where some nodes havepossibility to reuse more channels than others due to theirprivileged location. For example, the nodes on the edge of thenetwork usually can reuse more channels due to the fact thatthey have less interfering neighbours. This may result in loweraverage shadow price for a node in such privileged locationand due to the interference constraints there may be nopossibility of moving channels from this node to other nodeswith higher average shadow price in order to achieve shadowprice equalization. Consequently in the following we presentmore general optimality conditions that can be also appliedto networks with channel reuse. The first optimality conditionis related to the sensitivity of network average reward to anychannel borrowing/lending operation. This sensitivity can bedefined, by extending definitions included in (27) and (28), asfollows:

∂R

∂Ni←j

∼= ∂Ri

∂Ni−∑

j∈J∂Rj

∂Nj

∼= pi (Ni)−∑

j∈J pj (Nj) ,

(45)where J denotes set of n nodes from which a channel ismoved to node i in the considered n to one operation, andNi←j denotes changes in node capacities corresponding tothis operation.

Then this optimality condition is defined as follows:OC1 – Based on (45), under optimal channel allocation any

possible channel borrowing/lending operation cannot providefurther increase of average network reward. Therefore, thefollowing condition has to be valid:

∂R

∂Ni←j= pi (Ni)−

∑j∈J pj (Nj) ≤ 0, (46)

for all possible channel borrowing/lending operations i← J .Note that this optimality condition corresponds to minimiza-tion of average of all node average shadow prices, p.

The second optimality condition is related to reuse oppor-tunities:

OC2 – Under optimal channel allocation, any node cannotincrease its capacity by a new channel reuse due to theco-channel interference constraints. Otherwise the additionalchannel from new channel reuse would decrease the node’saverage shadow price and therefore the average reward fromthe network would be increased.

D. Capacity Adaptation Procedure with Channel Reuse

In the following we present the capacity adaptation proce-dure for the networks with channel reuse:

1) Calculate new node channel allocations with reuse us-ing the selected channel reuse maximization algorithm(CRM-BIP or CRM-H). This gives new node capacitiesNs. Then execute iteratively steps 2 to 4.

2) Optimize the channel assignment by using channel bor-rowing/lending operations in the following sequence ofoperation classes: 1 → 1, 2 → 1, 3 → 1, . . . ,M , where


Fig. 7. Network topology.

M denotes maximum of number of nodes using thesame channel in the interference region of any node.In each operation class the sequence of the borrow-ing/lending operations is defined by performing first theoperation that provides maximum positive increase ofthe average network reward defined by

max∂R

∂Ni←j= max

{pi (Ni)−

∑j∈J pj (Nj) > 0

},

(47)These operations are performed until no positive in-crease in R can be found.

3) Verify if the channel lending/borrowing operations exe-cuted in Step 2 created new reuse opportunities for somechannels. If this is the case, use heuristic algorithm CR-H to allocate these channels to the eligible nodes withhighest values of their average shadow prices in orderto achieve our objective of the average network shadowprice minimization.

4) If channel allocations are the same as in the previousiteration stop the algorithm since both optimality con-ditions, OC1 and OC2, are met, otherwise go to Step2.

VII. NUMERICAL RESULTS

We illustrate the performance of the proposed algorithmsusing a specific network scenario that was chosen to stress theadaptive algorithm and to allow easy understanding of trafficand performance changes. It has 18 nodes and its topologyillustrated in Fig. 7. The principal network parameters arepresented in Table II. There are 140 available cognitivechannels which are initially distributed among the networknodes in such a way that 14 nodes obtain the capacity of 8channels and the remaining 4 nodes obtain the capacity of 7channels. Each channel is subject to primary user arrival andwe assume that each secondary user connection occupies onechannel.

The PU channel occupation follows binomial distributionwith the busy period probability of 0.2, and the mean service

TABLE IINETWORK PARAMETERS

Number of available channels 140

Number of offered classes 306

SU mono hop connection traffic, λ 0.6,1.8

SU multi hops connection traffic, λ 0.15

Number of alternative paths Reward parameter, R SU mean service time, μ 1 PU mean service time, μ 5 SU required bandwidth, d 1

PU required bandwidth, d , 1

Simulation duration T 1550

time of μ−12 = 5, which is 5 times larger than the secondaryuser’s mean service time (the same assumptions are used in[7], [8]). There are 306 SU directional connection classes,each characterized by its unique O-D pair, and each class hasat most 5 alternative paths.

The SU traffic it is divided into two levels. A low leveltraffic is offered in one half of the network while the otherhalf receives a high level of traffic. The two halves areindicated in Fig. 7 by the vertical dashed line that separatesthe two regions. The traffic levels are reversed gradually forboth halves during 50 time units starting at time 750 whilethe simulation lasts 1550n time units. Two cases of networkoperation are considered. In the first case the channels are notreused while the second case employs channel reuse. It shouldbe mentioned that other scenarios were also tested but sincethe results were consistent only one representative scenario ispresented in the paper due the space limitations.

The event driven simulation model executes the CAC androuting algorithm and the capacity adaptation proceduresproposed in this paper. Measurements of parameters are doneat the end of each sampling interval of 5 time units thatcontains 50 samples. Then the node state dependent shadowprices are calculated using value iteration algorithm and thesevalues are used for routing process, while their averages areused for capacity adaptation and channel reuse operations. Theresults are presented for heuristic algorithms RC-H and CRM-H that calculate the required capacity in the no-reuse case andmaximize the channel reuse in the reuse case, respectively.Nevertheless, it was tested that the corresponding analyticalalgorithms RC-A and CRM-BIP give almost the same results.This is illustrated in Subsection E for no reuse case.

A. Equalization of Average Shadow Prices and Minimizationof Average Value of Average Shadow Prices

The capacity adaptation optimality condition (29) for nochannel reuse mode requires equalization of all node averageshadow prices. In Fig. 8, we present the average shadow pricesin no reuse case for selected nodes 1, 2, 3, 4, 7, 8. The nodes3, 4, 8 receive the high traffic level until time 750 and thenodes 1, 2, 7 receive the low traffic during the same period oftime. After time 750 the traffic level is reversed gradually fortwo node groups as explained before. Note that the average


0 200 400 600 800 1000 1200 1400 1600

0

0.5

1

1.5

2

2.5

3

3.5

4

Simulation duration

Nod

e A

vera

ge S

hado

w P

rice

afte

r cap

acity

ada

ptat

ion

Node 1Node 2Node 3Node 4Node 7Node 8

Fig. 8. Node average shadow price for no reuse case.

0 200 400 600 800 1000 1200 1400 1600

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Simulation Duration

Nod

e A

vera

ge S

hado

w P

rice

afte

r cap

acity

ada

ptat

ion

Node 3

Node 4

Node 8

Node 1Node 7

Node 2

Fig. 9. Node average shadow price for reuse case.

shadow price values follow each other closely although theyare not equalized exactly due to the integer value of nodecapacity and stochastic variations of traffic. Fig. 9 shows theaverage shadow prices of the selected nodes for the reusecase in which we minimize the average value of node averageshadow prices according to the optimality conditions OC1 andOC2. Note that due to the channel reuse the average shadowprices for all nodes are significantly lower compared to theno reuse case. Moreover, the average shadow prices for nodeswith low level of traffic are significantly lower than the onesfor nodes with high level of traffic. This follows from the factthat the channels from nodes with low shadow prices cannotbe borrowed to the nodes with high shadow prices due to thereuse constraints.

In Fig. 10, we compare the value of p for both cases. Notethat the value of p is significantly lower than in no reusecase. Figs. 11 and 12, show the average shadow prices for theselected nodes in both cases but with the capacity adaptationperformed only until time 750. For the no reuse case, shown inFig. 11, the gap between average shadow prices is becomingvery large starting from time 750. In particular the averageshadow price values of the nodes which receive the high trafficlevel after time 750 are increased due to lack of requiredcapacity which leads to 26% increase in average rejection rate

0 200 400 600 800 1000 1200 1400 1600

0

0.5

1

1.5

2

2.5

3

3.5

Simulation duration

Ave

rage

of a

ll no

de a

vera

ge s

hado

w p

rices

No Reuse case

Channel Reuse case

Fig. 10. Average of node average shadow prices for no reuse and reusecases.

0 200 400 600 800 1000 1200 1400 1600

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Simulation duration

Nod

e A

vera

ge S

hado

w P

rice

Node1

Node2

Node7

Node4

Node3

Node8

Fig. 11. Node average shadow price, the capacities are not adapted after750ts (no reuse case).

in Period 2, as shown later in Subsection VII.C. On the otherhand the average shadow price values of the nodes whichreceive less traffic after time 750 are low due to capacityexcess. In the reuse case, shown in Fig. 12, the average shadowprices are increased (nodes 1, 2, 8) or decreased (node 7)or about the same (nodes 3, 4) when compared with theadaptation case. This not optimal capacity allocation leads to22% increase in average rejection rate in Period 2, as shownlater in Subsection VII.C.

B. Convergence

As indicated before the offered traffic levels are stationaryin the intervals (0, 750) and (800, 1550). Fig. 13 showsthe trajectory of capacity allocation in the no reuse case forthe selected nodes (1, 2, 3, 4, 7, 8). The figure indicatesthat the system converges from the initial allocated capacityto a relatively stable state in twenty two sampling intervals(110 time units). After the convergence period, the capacityallocation fluctuations are due to stochastic variation of PUand SU arrivals until time 750, where the traffic levels arereversed gradually during 50 time units. The system againconverges to a relatively stable state in thirty eight samplingintervals at time 940. Fig. 14, shows the trajectory of capacityallocation in the reuse case for the same selected nodes. Thefigure indicates similar convergence characteristics to the noreuse case but the capacity changes between two periods are


0 200 400 600 800 1000 1200 1400 1600

0

0.5

1

1.5

2

2.5

Simulation duration

Nod

e av

erag

e sh

adow

pric

e

Node 1Node 2

Node 8Node 7

Node 4 Node 3

Fig. 12. Node average shadow price, the capacities are not adapted after750ts (reuse case).

0 200 400 600 800 1000 1200 1400 1600

0

5

10

15

20

25

Simulation duration

Nod

e C

apac

ity a

dapt

atio

n

Node 4

Node 3

Node 1

Node 2

Node 7 Node 8

Fig. 13. Capacity allocation for nodes 3,4,8 and 1,2,7(no reuse case).

less pronounced due to excess capacity in the nodes with lowtraffic level in Period 1.

Table III provides the percentage of channel lend-ing/borrowing operations used in this simulation. We observethat direct one to one channel lending/borrowing operationconstitutes the majority of used operations but the number oftwo to one operations is also significant at 30%.

The network average rejection rates BT are given in Ta-ble IV for both no reuse and reuse cases and the values areshown for two options with and without capacity adaptationin Period 2. The rejection rates are presented for two periods.The first period starts from time 5 to 750ts, and the secondperiod starts from 755 to the end of simulation time. The thirdcolumn presents the average rejection rates for total simulationduration. When there is no adaptation in Period 2, the rejectionrates are increased by 26 and 22 percent for the no reuse andreuse cases, respectively. Note that the total network averagerejection rate for the reuse case with adaptation in Period 2 isdecreased by around 91% when compared with the no reusecase.

C. Channel Reuse

We define the network channel reuse ratio as ratio of theaverage value of network total capacity, NT , to the totalnumber of available channels, Nu. In the tested scenario theaverage network channel reuse was 2.76 = 386/140.

0 200 400 600 800 1000 1200 1400 1600

0

5

10

15

20

25

30

35

40

Simulation duration

Nod

e C

apac

ity a

fter a

dapt

atio

n Node 4 Node 1Node 3

Node 2

Node 8 Node 7

Fig. 14. Capacity allocation for nodes 3,4,8 and 1,2,7(reuse case).

0 20 40 60 80 100 120 140

0123456789

10111213141516171818

Available Channels

Net

wor

k N

odes

Fig. 15. Channel reuse at time 1550ts with reuse ratio of 2.76.

Fig. 15, shows channel reuse at the end of simulation timewhere the network total capacity, NT is 386. The vertical axisrepresents the node indices while the horizontal axis representsthe 140 available channel indices. In the considered networktopology (Fig. 7) the same channel can be used by at mostthree nodes. This is illustrated in Fig. 15, where each verticalline represents a channel and each circle on the vertical linerepresents the node which uses this channel. For the clarity ofpresentation only every four channels are shown in Fig. 15.

D. Heuristic vs. Analytical Algorithms

In this subsection we compare performance of the analyt-ical RC-A and heuristic RC-H algorithms that calculate therequired capacity in the capacity adaptation procedure forno-reuse case. In particular, the network average rejectionrates over whole simulation are given in Table V for the twoalgorithms and the results are nearly identical.

VIII. CONCLUSION

We have proposed an economic framework that integratesCAC and Routing and channel allocation in cognitive wirelessmesh networks. This integration is realized through the useof novel node shadow price concept that takes into accountpreemption of secondary users by primary users and is basedon decomposed Markov decision process. This framework


TABLE IIIPERCENTAGE OF USED LENDER OPERATIONS

One to one Direct 61.92 One to one Indirect 7.43

56.03 eno ot owTThree to one 0

TABLE IVNETWORK AVERAGE REJECTION RATE

Period 1 Period 2 Total

Period 2 with adaptation, no reuse case

0.4036 0.4068 0.4053

Period 2 without adaptation, no reuse case

0.4036 0.5123 0.4604

Period 2 with adaptation Reuse case

0.0394 0.0301 0.0346

Period 2 without adaptation, Reuse case

0.0394 0.0368 0.0381

allows to derive the conditions for profit maximization for ‘nochannel reuse’ and ‘channel reuse’ cases. Using these optimal-ity conditions we propose the capacity adaptation procedurefor each case based on channel lending/borrowing strategybetween the nodes. The simulation results demonstrate thatthe system converges to a stable state within reasonableperiod of time. It is important to underline that the proposeddecomposed model allows for a decentralized implementationof channel allocation and routing algorithms. This featureresults in simpler protocols and faster execution times whencompared with other centralized algorithms proposed in theliterature.

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TABLE VNETWORK AVERAGE REJECTION RATE

Network average rejection rate over whole simulation period

Analytical algorithm 0.4049 Heuristic algorithm 0.4053

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Reza Mossanen Amini received his master’s de-gree in telematics and computer communicationsfrom University Paris VII - Diderot in 1993. Heis currently a Ph.D. candidate in telecommunica-tions and networking at the Department of Electri-cal Engineering, École de Technologie Supérieure,University of Quebec, Montreal, Canada. Prior tostarting his Ph.D. studies in 2007, he earned morethan 15 years of experience in scientific comput-ing and distributed multimedia systems design andnetwork programming. He also taught open system

architecture and network programming for four years at an academic level.

Zbigniew Dziong received his M.Sc. and Ph.D.degrees from Warsaw University of Technology,Poland, both in electrical engineering. After gradua-tion, he worked at Warsaw University of Technologyas an Assistant Professor. During this period, he wason sabbatical at the Centre National d’Etudes desTelecommunications, Paris, France, and at the De-partment of Communication Systems, Lund Instituteof Technology, Sweden. From 1987 to 1997, he waswith INRS-Telecommunications, Montreal, Canada,as a professor. From 1997 to 2003, he worked for

the Performance Analysis Department at Bell Labs, Lucent Technologies,Holmdel, New Jersey, USA. Since 2003, he has been with the École detechnologie supérieure (University of Quebec), Montreal, Canada, wherehe teaches on both the undergraduate and graduate levels as an associateprofessor.

Dr. Dziong is an expert in the field of performance, protocol, architecture,and resource management for data, wireless, and optical networks. He hasparticipated in research projects for many leading companies including BellLabs, Nortel, Ericsson, and France Telecom. His research achievements aredocumented in over 100 scientific publications and 17 patents and patentapplications. He won the prestigious STENTOR Research Award (1993,Canada) for collaborative research in the domain of resource management forbroadband networks. His monograph ATM Network Resource Management(McGraw Hill, 1997) has been used in several universities for graduatecourses. Currently, he is engaged in several research projects supported byindustry and government agencies.


TABLE VIGLOSSARY OF NOTATIONS USED IN THIS PAPER

Latin letters

SU class rejection probability for node SU class rejection probability for node network average rejection rate node channels’ physical cost rateSU required bandwidth referred by index 1PU required bandwidth referred by index 2required bandwidth of class j connection squared error of approximate shadow price in state Maximum net-gain over all feasible paths node net-gain in state , for SU connections aggregated into class 1 path net-gain from carrying class connection on path

Set of SU connection classes set of SU connection classes aggregated into one classnetwork total capacity average value of network total capacity total number of available channels in the network number of cognitive channels allocated to node Node capacity in iteration nnode shadow price in state , for SU connections aggregated into class 1 node shadow price in state , for SU class connection

approximated node shadow price for SU connection, in iteration n

average shadow price for node with capacity average shadow price for node with capacity , in

iteration n derivative of node average shadow price with respect

to the node capacity in iteration node average shadow price for SU connections

aggregated into class 1 maximum node average shadow price , in iteration n minimum node average shadow price, in iteration n

weighted average of all node average shadow prices network average profit reward rate of each admitted class j connection rate of node reward in state state probability vectorstate probabilityprobability that class 1 connection is accepted in state reward parameter of SU class j connection node aggregated reward parameter for SU connection classesreward parameter of node class j connection average reward from the network average reward for node with capacity set of nodes forming the network value function (Markov decision theory) set of alternative paths for class j connections node statenumber of SU connections carried by the nodenumber of active PUs in node set of node states set of node states in which class 1 connections are blocked

Greekletters

dumping factor used in iteration nvector of change in capacity value of all nodes change in capacity value of node precision threshold arrival rate of SU class j connection

rate of SU class-j accepted connectionsaggregation of all SU connection classes arrival rates offered to nodesecondary user class connection arrival rate offered to node

aggregated SU admitted connection rates in node average rate of class connections accepted in node

offered rate of class connections on path average rate of class connections accepted on path

one PU arrival rate 1 non-active period average duration for one PU

state dependent PU arrival rate when there are active PUs in node departure rate of SU referred by index 1

1

SU mean service time for node

mean service time of class j connection departure rate of PU referred by index 2

1 PU average service time for node

vector of the difference of node average shadow prices CAC & Routing policy average number of channels occupied by PUs in node transition-rate matrix

Education

An economic framework for routing and channel allocation in cognitive wireless mesh networks