Tree-based multicast service provisioning with maximum immunity in cognitive radio networks

Abstract

With the emergence of cognitive radios, the open spectrum access paradigm that is proposed to overcome the shortcomings of the traditional spectrum allocation paradigm became viable. Researchers have been accumulating efforts to verify the validity of existing networking schemes and protocols as well as designing new ones for this new paradigm. As link failures, due to the activity of primary users, are common in cognitive radio networks, it is essential not to overlook their effect when developing networking solutions including those for multicasting. In this paper, we study the problem of provisioning a multicast session via a tree that has the maximum achievable immunity against any single spectrum failure. The immunity is measured as the minimum percentage of multicast receivers that remain connected with the root of the multicast tree upon the occurrence of a single spectrum failure. The problem is formulated an Integer Linear Program. We also propose a suboptimal algorithm to solve the problem and compare its performance to the optimal performance obtained using the ILP. Results proved the good performance of our algorithm which was always within $\approx 10 %$ from the optimal solution. Experiments were also conducted to understand how different network parameters affect the achievable immunity.

Keywords

Multicasting fault tolerance spectrum failure Cognitive Radio Networks

1. Introduction

Recent advancements in transceiver design enabled software-based reconfiguration of radio devices, which emerged as the software-defined radio technology (also referred to as cognitive radio) [30,34]. Facilitated by this new technology, and motivated by the significant spectrum crowdedness as well as underutilization, the Open Spectrum Access (OSA) paradigm was proposed [2,46]. This paradigm created a new type of wireless networks called cognitive radio networks (CRNs) in which unlicensed wireless users (also referred to secondary users (SUs)) may use any part of the spectrum as long as they do not cause any harm (in terms of interference) to licensed wireless users (also referred to as primary users (PUs)). In other words, the OSA paradigm opens the whole spectrum to any kind of wireless users provided that they do not interrupt or negatively affect the operation of PUs. This paradigm creates a number of challenges which distinguishes cognitive radio networks from traditional multi-channel wireless networks [1,3,6,13,18,21,24,31]. These challenges include:

Heterogeneous channel availability: depending on the spatial distribution of active PUs, different SUs may observe different sets of available frequency channels.

Time-varying channel availability: depending on the temporal channel occupation distribution of PUs in its vicinity as well as their possible mobility, the set of channels available for an SU to use may vary over time.

Time-varying network topology: the variability of channel availability, due to the dynamic activity of PUs, imposes variability in network topology. As a pair of connected SUs become disconnected if they no longer share a common channel, some nodes may lose connectivity with the rest of the network causing fragmentation.

Broadcast deformation: in CRNs, an SU may not be able to broadcast a packet to all of its neighboring SUs in one transmission because they do not all share a common frequency channel to use.

Frequent and correlated link failures: when a PU starts using its frequency channel, all SUs which can interfere with its transmission must vacate the channel. This means that all the links between this set of SUs which operate on the same channel have failed. Thus, traditional protection techniques that assume independent link failures are not applicable to CRNs. Furthermore, depending on the frequency of PUs activity, such failures may occur frequently.

The aforementioned challenges motivated a vast volume of research on different aspects of CRNs. Being an essential mode of communication in computer networks, data multicast has been one of these aspects [7,9,23,35,36,38,40,41,44]. In this paper, we are interested in studying the effect of spectrum failures (the appearance of a PU on its channel of operation is referred to as a spectrum failure in this paper) on multicast communication in CRNs. We aim at minimizing the damage that any spectrum failure may create to the multicast tree, where this damage is measured as the number of multicast members that get disconnected from the multicast source upon such a failure. Depending on their geographical locations, transmission power levels, and channels of operation, different PUs may cause different levels of damage to an existing multicast tree. Furthermore, the way the multicast tree is constructed will affect its immunity to spectrum failures (see Section 3.1 for an example).

1.1. Paper contributions

The contributions of this work are as follows:

Formalizing the problem of multicast tree construction with maximum immunity against spectrum failures, and presenting an immunity metric as the ratio of the multicast members that remain connected after any failure to the total number of members. We call this problem “Multicast Tree Construction with Maximum Immunity (MTMI)”, and we define it formally in Section 4.

Developing an Integer Linear Program (ILP) formulation [26] for the MTMI in order to find the optimal multicast tree (the one withe the maximum immunity).

As ILPs are not guaranteed to give a solution in polynomial time, we propose a suboptimal algorithm with excellent accuracy. The accuracy of the proposed algorithm is evaluated against the optimal solution obtained by solving the ILP using IBM ILOG CPLEX Optimizer [27].

1.2. Paper organization

The rest of this paper is organized as follows. We review the work related to multicast routing and recovery in cognitive radio networks in Section 2. In Section 3, we outline some preliminaries regarding the network model and the spectrum failure model. The motivation and problem statement are presented in Section 4. The MTMI problem is formulated as an Integer Linear Program in Section 5. Section 6 presents the details of the proposed sub-optimal algorithm to solve the MTMI problem. Results and performance evaluation are discussed in Section 7. We finally conclude the paper in Section 8.

2. Literature review

In this section, we review related works on multicast communication in cognitive radio networks.

2.1. Multicast routing in cognitive radio networks

Due to their special properties (reviewed in part in Section 1) which are enforced by the opportunistic nature of spectrum availability, CRNs needed new mechanisms and protocols to provision the multicast service different from the ones used in traditional wireless networks. A multipath on-demand multicast routing protocol for CRNs was presented in [5]. The protocol is based on the reinforcement learning method of artificial intelligence, and it aims at improving both throughput and end-to-end delay. Results proved the superiority of the proposed protocol over existing state-of-the-art protocols especially under high PU activity.

An assisted multicast scheduling strategy for a single cell in cognitive radio mesh networks is proposed in [8]. In this strategy, client nodes assist the base station in delivering the multicast data over orthogonal channels to reduce the total multicast time. Network coding is also utilized for further reduction in the overall multicasting time. The use of network coding for multicasting in CRNs has also been proposed in [37]. The aim of this work was to apply network coding, considering both channel uncertainty and node mobility, to minimize the total transmission cost (amount of transmitted data). In [19], random network coding-based multicasting approach is proposed and the objective was again to minimize the total number of transmissions.

Using a dynamic programming approach, an on-demand multicast routing protocol that considers channel assignment, channels heterogeneity, and channel-switching delay is proposed in [7]. The protocol aims at reducing end-to-end delay and enhancing throughput. The authors in [29] proposed an optimization method for multicast scheduling in CRNs using network coding. Several factors where considered in this optimization problem including PUs protection and fairness. Two methods were proposed: a greedy method which is based on a centralized optimization, and an online method that is based on stochastic optimization. In [42], an optimization model that jointly finds multicast routes and channels assignment is formulated with the objective of increasing multicast throughput in multi-hop CRNs. The application of Non-orthogonal multiple access (NOMA) to multicast in CRNs is investigated in [32]. Via SU scheduling, significant performance gains have been achieved by using cooperative NOMA. Other NOMA-inspired multicasting approaches for CRNs are proposed in [15,20].

The authors of [38] addressed the problem of constructing energy efficient multicast trees in CRNs, and proposed a low-complexity approximation algorithm with performance guarantees. A multicast tree construction method is proposed in [45] so that the overall bandwidth utilized by the multicast tree is minimized and the predetermined QoS requirements are met. The authors proposed two methods to solve this problem based on minimal spanning tree, such that time slots are assigned to tree links in a way that reduces bandwidth consumption. In [4], a minimum spanning tree-based multicast routing protocol is proposed. The objective in this work was to increase the overall throughput and enhance the packet delivery rate.

The problem of joint power and rate allocation to maximize the average sum multicast rate in CRNs while maintaining some QoS guarantees (in terms of service outage probability) is studied in [47]. In [39], the multichannel multicast problem was studied. The paper introduced the concept of local multicast to tackle the hidden channel problem, and presented an channel assignment algorithm for this purpose which considers both partially overlapped channels and orthogonal channels. In [14], the authors considered a wireless mesh network of multi-radio nodes which are equipped with directional antennas. The objective was to select the best multicast/broadcast tree, channels, and antenna beams to keep the interference minimized.

In [16], the problem of optimizing the multicast group power allocation to minimize the group outage probability is studied. This optimization is investigated under PU QoS guarantees and average transmission power constraints. The problem of energy efficient multicasting in CRNs using omni-directional and directional antennas is studied in [11]. The objective of this study was to build a multicast tree and to schedule the transmissions along this tree so that the total energy consumption is minimized. Designing a protocol to route video multicast data has been the focus of [33]. The protocol is proposed for a single-hop infrastructure-based CRN. In [28], a multi-objective optimization model is proposed to constrcut multicast routing trees for a given session so that the worst end-to-end delay is mininmized, the multicast rate is maximized, and the number of transmission links used is minimized.

For further review of existing work on multicast in CRNs, we refer the reader to the comprehensive survey on this topic presented in [36]. It also highlights open research problems and directions in this field. We further refer the reader to the work presented in [22] which discusses the research challenges related to multicast routing, spectrum allocation, cross-layer multicast scheduling, and multicasting with QoS guarantees in cognitive radio ad hoc networks.

2.2. Multicast recovery in cognitive radio networks

In general, two approaches are used to recover from failures in wireless networks, namely protection (also known as proactive) and restoration (also known as reactive) recovery. In the protection approach, backup resources are reserved a priori to be used for recovery in case of failure. The restoration approach, on the other hand, restores the functioning status of the provisioned service after the occurrence of the failure. Therefore, no resources are reserved a priori but instead new resources are looked for and allocated after the failure. After a deep literature review, we could identify only a few research articles, [10,12,17,25,40,43], that address the problem of recovering multicast service from spectrum failures in CRNs. There articles are reviewed below.

In [12], the authors use the protection approach to route multicast traffic based on a multilayer hyper-graph model. Primary and backup paths which are failure-disjoint are selected for each multicast receiver. On the other hand, the work in [25] uses the restoration approach. A local multicast-route recovery process is launched upon the occurrence of a spectrum failure in which the route is fixed around the failure instead of end-to-end recovery. In [43], the problem of robust multicast in mobile ad hoc cognitive radio networks is studied. A semi-mesh topology is used to provision the multicast session where two paths are maintained between the source and each multicast receiver. The two paths are not necessarily link disjoint, and therefore they may fail together. If such a case occurs, a restoration process is launched.

In [40], the authors proposed a multicast routing approach which they call HyTopCast (short for Hybrid Topology Multicast). HyTopCast aims at establishing more stable multicast trees as well as providing a multicast route repair service. It does not take into account, however, the possibility of correlated failures in the constructed tree. A robust multicasting algorithm for CRNs is proposed in [10] based on the protection approach. The purpose of this work is protecting multicast sessions from a single spectrum failure at a time with the minimum protection cost, while at the same time maximizing the number of protected multicast trees that can be simultaneously provisioned.

In [17], the multicast problem in CRNs is studied with the objective of minimizing the outage probability of the multicast group by designing a power allocation policy. A group is considered to be in an outage when the aggregate rate of all the members of the multicast session drops below some predefined target rate.

Our work is different from the works reviewed above in the sense that we do not propose neither a proactive nor a reactive recovery technique. Instead, we propose a method to construct a multicast tree that has the best immunity against any single spectrum failure, i.e., we minimize the worst effect of any single spectrum failure on the constructed tree. This proposal can be used for both protection and restoration as follows:

If restoration is used, our approach ensures that the maximum number of multicast receivers that may get disconnected, and therefore need to be restored, is minimized.

If protection is used, our approach can be exploited to construct both the primary and the backup trees.

Table 1 compares between the existing works on reliable multicast routing in CRNs. We compare between them in terms of the reliability assurance method, the number of multicast sessions considered, and the metrics used to evaluate the solution or the joint objectives considered with the routing problem. Below, we explain what each of the reliability assurance methods mentioned in Table 1 means:

Protection: reserving enough resources to make sure any failure cannot interrupt service.

Restoration: handling the failure after it occurs in the most efficient way without reserving resources a priori.

Robust provisioning: multicast routes with the highest possible level of reliability are constructed, without reserving enough backup resources to achieve protection or presenting a restoration technique.

Table 1
The comparison between existing protection/restoration works on multicast in cognitive radio networks

Approach Reactive/Proactive Number of Sessions Metrics and joint objectives

[40] Restoration (Stable routes and fast route repair) Single End-to-End delay and packet delivery ratio

[12] Protection Single End-to-End delay

[25] Restoration Single End-to-End delay

[43] Hybrid (Robust provisioning via shared backup resources, in addition to restoration) Single Reducing the complexity and computation load on the sender of the multicast session

[10] Protection Multiple Minimizing the provisioning cost (based on channel leasing cost), and increasing the number of sessions that can be accommodated.

[17] Robust Provisioning Multiple Outage probability and power consumption

Approach	Reactive/Proactive	Number of Sessions	Metrics and joint objectives
[40]	Restoration (Stable routes and fast route repair)	Single	End-to-End delay and packet delivery ratio
[12]	Protection	Single	End-to-End delay
[25]	Restoration	Single	End-to-End delay
[43]	Hybrid (Robust provisioning via shared backup resources, in addition to restoration)	Single	Reducing the complexity and computation load on the sender of the multicast session
[10]	Protection	Multiple	Minimizing the provisioning cost (based on channel leasing cost), and increasing the number of sessions that can be accommodated.
[17]	Robust Provisioning	Multiple	Outage probability and power consumption

3. Preliminaries

In this section, we outline some preliminary models that we use later in the paper.

3.1. Network model

We consider a CRN that consists of a set of SUs denoted as $S$ , and a set of PUs denoted as $P$ . SUs opportunistically utilize K licensed spectrum channels, where each PU $p \in P$ operates on one of these channels denoted as $k_{p}$ . By utilizing spectrum sensing, each SU observes a set of available (unused by PUs) spectrum channels. For an SU i, this set is denoted as $C_{i}$ . Different SUs may observe different sets of available channels depending on the presence and activity of PUs in their vicinities. Furthermore, a link exists between a pair of SUs i and j, making them neighbors, if the following conditions are satisfied:

They are within the transmission range of each other.

They share at least one common channel (i.e., $C_{i} \cap C_{j} \neq \emptyset$ ).

We denote the set of SUs which are neighbors to SU i as

S_{i}

. We also use

E

to denote the set of links between all SUs.

Finally, a multicast session exists in the network where $s \in S$ is the source of this session and $R \in S ∖ {s}$ is the set of its multicast receivers.

3.2. Spectrum failure model

Two types of failures are usually of concern when it comes to computer networks; node failures and link failures. In wireless networks, a link exists between a pair of nodes if they are within the transmission range of each other and they share a common wireless channel to communicate over. As channel availability is a function of PU activity in CRNs, link failures are often frequent (contrary to node failures). Furthermore, link failures in CRNs are usually correlated. To illustrate this correlation, consider the example in Fig. 1. The example shows a network of eight SUs ${s_{1}, \dots, s_{8}}$ and one PU $p_{1}$ . Links between SUs are labeled with the frequency channels used for communication between the two ends of the link. The primary user operates on channel 1, which is represented in the figure with the orange color. When $p_{1}$ is inactive (i.e, does not use channel 1) as shown in Fig. 1.a, it is safe for the CRN to use channel 1 on links ${(s_{1}, s_{2}), (s_{6}, s_{7}), (s_{8}, s_{3}), (s_{8}, s_{5})}$ . However, when $p_{1}$ starts using its channel (that is channel 1) as shown in Fig. 1.b, nodes $s_{1}$ , $s_{6}$ , $s_{7}$ , and $s_{8}$ (which can observe the presence of $p_{1}$ ) must backoff and stop using the channel. This means that the links ${(s_{1}, s_{2}), (s_{6}, s_{7}), (s_{8}, s_{3}), (s_{8}, s_{5})}$ are all broken. This example shows that a correlation exists between link failures in CRNs. Therefore, the design of algorithms that aim at providing CRNs with fault tolerance against link failures must take into account the correlation between such failures. In this paper, we are concerned with link failures only.

Fig. 1.

An example to illustrate the correlation between links failures in CRNs. The network consists of eight SUs ${S_{1}, \dots, S_{8}}$ and one PU $P_{1}$ operating on frequency channel 1 (represented by the orange color).

From now on, we will use the term “spectrum failure” to refer to the appearance1

In the form a wireless transmission.

of any PU

p \in P

on its channel of operation

k_{p}

. This term is defined formally below.

Definition 1.

Spectrum Failure: is the event of an idle PU $p \in P$ starting to use its channel of operation $k_{p}$ .

Keep in mind that a spectrum failure might cause one or more links in the network (or none) to fail as the example above demonstrated.

4. Motivation and problem statement

In this section, we first give an example to illustrate the motivation behind this work in Section 4.1. Then, in Section 4.2, we present the problem statement and formally define the “Multicast Tree Construction with Maximum Immunity (MTMI)” problem.

4.1. Motivation

We start this section by giving a motivational toy example (shown in Fig. 2) to explain the importance of channel assignment in reducing the effect of a spectrum failure. The example includes one PU $p_{1}$ which operates on channel 1 and a multicast session rooted at SU s, four multicast receivers $r_{1}$ through $r_{4}$ (in black) and two relaying SUs (in white). Improper channel assignment may cause the entire multicast tree to break apart upon a spectrum failure as shown in Fig. 2.a (links are marked with the channels assigned to them). The channel assignment in Fig. 2.b exposes only two links to the potential spectrum failure of $p_{1}$ . This results in disconnecting $75 %$ of the multicast members when the spectrum failure takes place. The channel assignment in Fig. 2.c causes only one member of the multicast group to be disconnected upon the occurrence of the failure. Finally, the channel assignment in Fig. 2.d makes the tree completely immune to the spectrum failure of $p_{1}$ .

Fig. 2.

A toy example to explain how the damage that a spectrum failure might cause to the multicast tree depends on the channel assignment. Black circles represents SUs which are members of the multicast session, while white circles represent SUs that are just relay nodes.

4.2. Problem statement

Before giving the formal definition of the problem, we need to give two preliminary definitions.

Definition 2.
Multicast Tree: for a given multicast session defined by the tuple $⟨ s, R ⟩$ , any tree that satisfies the following conditions is called a multicast tree of $⟨ s, R ⟩$ :
s is the root of the tree.

The tree spans all the nodes in $R$ .

Any link $(i, j)$ on the tree is part of a path that connects s to one of the nodes in $R$ (otherwise the link is not needed).

Each link $(i, j)$ on the tree is assigned a channel from the set $C_{i} \cap C_{j}$ .

Definition 3.
Immunity Level: for a given multicast tree T, its immunity level, denoted as $I (T)$ , is the minimum ratio of the number of multicast receivers that remain connected with the source to the size of the multicast session among all possible spectrum failures. In other words, $\begin{matrix} (1) & I (T) = min_{p \in P} \frac{| R | - γ_{p}}{| R |} \times 100 % \end{matrix}$ where $γ_{p}$ is the number of multicast receivers that get disconnected from the source s upon a spectrum failure caused by PU p on channel $k_{p}$ .

We are now ready to formally define the problem of constructing a multicast tree with maximum immunity against spectrum failures, abbreviated as the MTMI problem. Definition 4.
Multicast Tree construction with Maximum Immunity (MTMI): given a CRN defined by the tuple $⟨ S, P, C_{i} \forall i \in S, S_{i} \forall i \in S, k_{p} \forall p \in P ⟩$ and a multicast session defined by the tuple $⟨ s, R ⟩$ , find a multicast tree T for $⟨ s, R ⟩$ with maximum immunity level $I (T)$ .

5. Integer linear programming formulation

In this section, we propose an integer linear programming (ILP) formulation to solve the MTMI problem. Before presenting the ILP formulation, we need to present some notations:

$e_{i, j}^{k}$ is a binary variable indicating whether link $(i, j)$ is assigned channel k (if it equals 1) or not (if it equals 0).

$γ_{p, i}$ is a binary variable to indicate whether PU p breaks the flow coming into SU i (by breaking the link carying that flow).

$γ_{p}$ is an integer variable that counts the number of multicast receivers that get disconnected from the source s upon a spectrum failure caused by PU p (i.e., $γ_{p} = \sum_{i \in R} γ_{p, i}$ ).

$Ψ_{(i, j)}^{p}$ is a binary indicator of effect of PU p on link $(i, j)$ . In other words, $Ψ_{(i, j)}^{p} = 1$ if a spectrum failure caused by PU p on channel $k_{p}$ breaks link $(i, j)$ assuming the link operates on channel $k_{p}$ . It equals 0 otherwise.

Before presenting the details of the proposed ILP, we need to explain the network flow model we used to facilitate the tree construction requirement for the ILP. In graph theory, a flow network is a directed graph in which each link may carry an amount of flow that does not exceed its capacity, and each node can be either a supply node, a demand node, or a relay node. A supply node is one that has flow to supply to others, a demand node is one that demands flow to consume, and a relay node is one that relays the flow it receives to others. In general types of flow problems, any node i of any of the three types above must satisfy the following condition: $\begin{matrix} Incoming flow to i + Supply flow of i = Demand flow of i + Outgoing flow from i \end{matrix}$

The most common type of flow problems is the min-cost flow in which all the supply from supply nodes must be delivered to all demand nodes with minimum cost, where the cost is usually assigned to links per unit of flow. In our case, the source of the multicast session s is a supply node of $| R |$ units of flow. The multicast receivers in $R$ , on the other hand, are all demand nodes each of which demanding one unit of flow. The rest SUs are all relay nodes that must relay all of their incoming flow. This model gives each link a capacity of $| R |$ . Our task then becomes to deliver all the traffic from the single supply node to all demand nodes with the maximum revenue measured as the immunity level of resulting multicast tree. An important thing to point out here is that a general solution (the induced graph which contains only links that carry flow, their end-nodes, and the amount of flow they carry) of a flow problem is not necessarily a tree. Therefore, we need to add special conditions to enforce the solution to be a tree. In our case, the following two conditions are sufficient:

The only supply node s receives no flow (i.e., no incoming links in the resulting solution).

Each other SU node may receive flow from no more than one SU and via a single link (i.e., at most one incoming link in the resulting solution).

These two conditions guarantee that the solution will always be a tree (the proof of this is trivial and therefore omitted). We are now ready to present the ILP formulation. Apparently, the objective function is to maximize

I

\begin{array}{l} Maximize I \end{array}

Subject to the following constraints:

Tree constraints. The source node s must have no incoming links (constraint (2)), while other SUs may have no more than one incoming link (constraint (3)). $\begin{array}{l} (2) & \sum_{j \in S_{s}} \sum_{k \in C_{s} \cap C_{j}} e_{j s}^{k} = 0 \\ (3) & \sum_{j \in S_{i}} \sum_{k \in C_{i} \cap C_{j}} e_{j i}^{k} ⩽ 1, i \in S ∖ {s} \end{array}$

Network flow constraints. This set of constraints govern the flow. Constraint (4) guarantees that no flow passes through link $(i, j)$ unless a channel (from the set $C_{i} \cap C_{j}$ ) is assigned to that link, and that the maximum amount of flow to carry via the link is $| R |$ . Constraint (5) ensures that the flow coming out of node s equals the size of the multicast session. Constraint (6), on the other hand, ensures the following:

A multicast receiver node must receive exactly one unit of flow from its neighbors.

The source node s does not receive any flow from its neighbors.

Any other node must forward all the flow it receives (flow conservation, i.e., $flow-in = flow-out$ ).

The

d_{i}

used in this constraint is defined as follows:

\begin{matrix} d_{i} = \{\begin{matrix} - | R | & if i = s \\ 1 & if i \in R \\ 0 & otherwise \end{matrix} \end{matrix}

Finally, constraint (7) prevents channel assignment for a link unless it caries a flow. This is needed for evaluating the cost of spectrum failures in the rest of this ILP.

\begin{array}{l} (4) & f_{i j} ⩽ | R | \cdot \sum_{k \in C_{i} \cap C_{j}} e_{i j}^{k}, \forall i \in S, j \in S_{i} \\ (5) & \sum_{j \in S_{s}} f_{s j} = | R | \\ (6) & \sum_{j \in S_{i}} f_{j i} - \sum_{j \in S_{i}} f_{i j} = d_{i}, \forall i \in S \\ (7) & \sum_{k \in C_{i} \cap C_{j}} e_{j i}^{k} ⩽ f_{j i} \forall i \in S, j \in S_{i} \end{array}

Spectrum failure constraints. This set of constraints accounts for the cost of spectrum failures. The binary variable $γ_{p, i}$ indicates, if it equals 1, that the spectrum failure caused by PU p affects the flow (i.e., breaks the links carrying that flow) coming into SU i. Constraint (8) ensures that $γ_{p, i}$ is equal to zero if SU i does not receive any flow. Constraint (9), on the other hand, forces $γ_{p, i}$ to be equal to 1 if SU i receives flow via a link that can be affected by a spectrum failure caused by PU p. Constraint (10) propagates the effect of the spectrum failure caused by PU p on the flow coming into SU j to all the members of the subtree rooted at j. The total number of multicast receivers that get isolated from the source upon a spectrum failure cause by PU p (denoted $γ_{p}$ ) is calculated in constraint (11). Finally, the immunity of the achieved multicast tree is accounted for in constraint (12). $\begin{array}{l} (8) & γ_{p, i} ⩽ \sum_{j \in S_{i}} f_{j i}, \forall p \in P, i \in S \\ (9) & γ_{p, i} ⩾ \sum_{j \in S_{i}, k_{p} \in C_{i} \cap C_{j}} e_{j i}^{k_{p}} \cdot Ψ_{(j, i)}^{p}, \forall p \in P, i \in S \\ (10) & γ_{p, i} ⩾ γ_{p, j} + \sum_{k \in C_{i} \cap C_{j}} e_{j i}^{k} - 1, \forall p \in P, j \in S, i \in S_{j} \\ (11) & γ_{p} = \sum_{i \in R} γ_{p, i}, \forall p \in P \\ (12) & I ⩽ 1 - \frac{1}{R} \cdot γ_{p} \forall p \in P \end{array}$

Figure 3 explains constraint (10). When a PU p affects the flow incoming to a node j in the resulting tree, constraint 3 propagates this effect to all neighbors in the subtree rooted at j. Then, those neighbors propagate in a similar manner to all their neighbors downstream in the tree. This method helps us account for the loss of connectivity with mutlicast receivers down in the tree due to a failure caused by a PU upstream although this PU does not directly affect those receivers. Before discussing the explanation in Fig. 3, we would like to reassert the two conditions enforced on the solution by constraints (4) and (7) which are:

No flow $f_{i j}$ may pass through link $(i, j)$ unless there is a channel assigned to that link, i.e., unless $\sum_{k \in C_{i} \cap C_{j}} e_{i j}^{k} = 1$ .

Fig. 3.

Explanation of constraint (10) of the proposed ILP formulation.

No channel may be assigned to link $(i, j)$ unless the link carries flow, i.e., $\sum_{k \in C_{i} \cap C_{j}} e_{i j}^{k} ⩽ f_{i j}$ .

Given the two assertions above, for a pair of SUs j and i where

i \in S_{j}

and a PU p, there are four possible scenarios summarized in the tabular in Fig. 3.

PU p does not affect the flow incoming to j (i.e., $γ_{p, j} = 0$ ) and no flow is going from j to i (or equivalently, no channel is assigned to link $(j, i)$ ). Then, the lower bound on $γ_{p, j}$ is set to −1, and as we are minimizing the sum of γ’s over $R$ , it will be set to 0 in the optimal solution (unless of course it receives flow from another node u which is affected by p). In other words, there was no effect on j to propagate to i.

PU p does not affect the flow incoming to j (i.e., $γ_{p, j} = 0$ ) and there is flow going from j to i (or equivalently, there is a channel to link $(j, i)$ ). Then, the lower bound on $γ_{p, j}$ is set to 0, and again as we are minimizing the sum of γ’s over $R$ , it will be set to 0 in the optimal solution unless i receives flow from another node u which is affected by p. In this case as well, there was no effect on j to propagate to i.

PU p affects the flow incoming to j (i.e., $γ_{p, j} = 1$ ) and no flow is going from j to i (or equivalently, no channel is assigned to link $(j, i)$ ). Then, the lower bound on $γ_{p, j}$ is set to 0, and as we are minimizing the sum of γ’s over $R$ , it will be set to 0 in the optimal solution (unless it receives flow from another node u which is affected by p). In this case, the effect is not propagated from j to i because no flow is passing in this direction (i.e., this link is not part of the tree).

PU p affects the flow incoming to j (i.e., $γ_{p, j} = 1$ ) and there is flow is going from j to i (or equivalently, a channel is assigned to link $(j, i)$ ). Then, the lower bound on $γ_{p, j}$ is set to 1 ( $γ_{p, i} ⩾ 1$ ). This way, the effect of p on j is propagated to i.

To give an example, consider the tree in Fig. 2. PU $p_{1}$ affects the relay node to the left (call it $e_{l}$ ), but it does not directly affect the multicast receivers $r_{2}$ and $r_{3}$ . Constraint (8) will force $γ_{p, e_{l}}$ to be equal to 1. As the links between $e_{l}$ and both $r_{2}$ and $r_{3}$ carry flow from $e_{l}$ , constraint (10) will force (as explained above) the following: $γ_{p, r_{2}} ⩾ 1$ (i.e., $γ_{p, r_{2}} = 1$ as it is a binary variable) and $γ_{p, r_{3}} ⩾ 1$ (i.e., $γ_{p, r_{3}} = 1$ as it is a binary variable). This way, the effect of $p_{1}$ on $e_{l}$ is propagated to the subtree rooted at $e_{l}$ .

6. Tree construction and channel assignment algorithm

In this section, we propose a greedy algorithm to solve the MTMI problem. We refer to this algorithm throughout the paper as the G-MTMI algorithm, and it is outline in Algorithm 2. The G-MTMI algorithm exploits another algorithm that we call the “Minimum Cost Route (MCR)” which is outlined in Algorithm 1. The MCR algorithm finds the best route from the source of multicast session s to one multicast receiver from an input set $\tilde{R}$ . We explain both algorithms next.

Algorithm 1:

Minimum Cost Route (MCR) algorithm

6.1. G-MTMI algorithm

The G-MTMI algorithm, outlined in Algorithm 2, takes the following inputs:

The network of SUs represented as an undirected graph $G (S, E)$ , where $S$ is the set of SUs and $E$ is the set of links between SUs (nodal connectivity). A link exists between a pair of SUs if they are within the transmission range of each other and they share at least one common frequency channel for data communication.

The set of PUs $P$ .

The set of multicast receivers $R$ .

The source of the multicast session s.

The channel availability $C_{u}$ for every SU u in $S$ .

As an output, the algorithm returns the constructed multicast tree T and its immunity level

I

. If it fails to construct such a tree, an empty tree and an immunity of −1 are returned.

For a set $\tilde{R}$ of multicast receivers, the G-MTMI algorithm calls the MCR algorithm $| \tilde{R} |$ times where in each call a route is established (by the MCR algorithm) from the source of the multicast session s to one of the nodes in $\tilde{R}$ , and this node is then removed from $\tilde{R}$ . If the MCR algorithm fails to find a route to any multicast receiver from $\tilde{R}$ , the G-MTMI algorithm terminates in failure.

6.2. Minimum Cost Route (MCR) algorithm

The MCR algorithm, outlined in Algorithm 1, is based on the well-known Dijkstra’s algorithm. It finds the minimum-cost path that connects the source s to a multicast receiver2

²
The node that is reached first.

from an input set

\tilde{R}

. The part of the algorithm from line 2 to line 29 is a modified version of the Dijkstra’s shortest path algorithm. Before illustrating the modifications made, we need to introduce the variable

ζ_{u, v, k}

which represents the cost of connecting a multicast receiver via a route that goes through the link

(u, v)

on channel k. This cost is the maximum total number of multicast receivers that get disconnected from the multicast source s upon a spectrum failure given that link

(u, v)

(on channel k) is added to the multicast tree under construction. Formally,

ζ_{u, v, k}

is defined as follows:

\begin{matrix} (13) & ζ_{u, v, k} = \{\begin{matrix} {max}_{p \in P_{(u, v, k)}} γ_{p} + 1, & | P_{(u, v, k)} | > 0 \\ 0, & otherwise \end{matrix} \end{matrix}

where

P_{(u, v, k)}

denotes the set of PUs that can break a link

(u, v)

on channel k, and is defined as follows:

\begin{matrix} (14) & P_{(u, v, k)} = {p : p \in P, k_{p} = k, Ψ_{(u, v)}^{p} = 1} \end{matrix}

Note that $γ_{p}$ is updated during the tree construction process for all PUs (line 36–39).

The modifications made over the original version of Dijkstra’s algorithm to make up the MCR algorithm are:

Because there are multiple channels that may be used on each link $(u, v)$ , all the channels that can be used on this link (i.e., all channels in $C_{u, v}$ ) must be considered when calculating the cost of using link $(u, v)$ as part of the route. The channel that gives the minimum cost ( $ζ_{u, v, k}$ ) is used as shown in line 19.

The cost of a route is the maximum of the individual costs of the links that constitute the route, and not the sum of those costs as it is in the original version of Dijkstra’s algorithm.

The algorithm terminates when reaching a node that is a member of the input set $\tilde{R}$ .

After finding the route with the minimum cost, the second part of the algorithm (from line 31 to line 42) performs the following updates for each link

(v, u)

and channel k along the route:

The set of channels to be used between nodes u and v on a link $(v, u)$ is set to ${k}$ (line 34).

All links incoming to node u except $(v, u)$ are removed from the graph G (line 35).

Algorithm 2:

Greedy MTMI algorithm (G-MTMI)

$γ_{p}$ is incremented for each PU p if it can break any link along the calculated route (line 36–39). To make sure that $γ_{p}$ is incremented no more than once per route for any PU p, p is removed from the temporary set $\tilde{P}$ upon incrementing as shown in line 38.

$\forall (u, v) \in G . E$ .

⁴

$\forall (u, v) \in G . E$ .

⁵

$\forall p \in E$ .

7. Performance evaluation

To evaluate the performance of the proposed G-MTMI algorithm, we used a network topology in which SUs are deployed on a squared-grid field of side length L. As Fig. 4 shows, the side length of each cell in the grid is r ( $r = L / \sqrt{| S |}$ ). Furthermore, each SUs has a transmission range of $\sqrt{2 r^{2}}$ , which means that each SU has a maximum of eight neighboring SUs. PUs, on the other hand, are deployed uniformly at random in the network field where each PU is assigned one channel of operation (selected at random) out of the total of K spectrum channels. As for the multicast group, node $s_{0}$ (i.e., the top-left-most node in the grid) is assumed to always be the source of the multicast session. The rest of the members (i.e., the multicast receivers) are selected at random from the remaining SUs.

Fig. 4.

The grid topology of SUs, $S_{t x} = \sqrt{2 r^{2}}$ is the transmission range of SUs.

Fig. 5.

The immunity versus PU activity range for a multicast group of size 8.

Fig. 6.

The immunity versus PU activity range for a multicast group of size 12.

7.1. The accuracy of the G-MTMI algorithm

In order to verify the accuracy of the G-MTMI algorithm, we performed four sets of experiments depicted in Figs 5, 6, 7, and 8 for a multicast group of size 8, 12, 16, and 20 respectively. In all of these figures, we compare the optimal immunity (obtained using the ILP in Section 5) to the immunity obtained using the G-MTMI algorithm. The protection range of PUs is varied from $4 r$ to $7 r$ on an increment of r (where r is the side length of each cell in the grid), and each point on the curves is the average of 200 randomly generated instances.

Fig. 7.

The immunity versus PU activity range for a multicast group of size 16.

Fig. 8.

The immunity versus PU activity range for a multicast group of size 20.

As the results show, the accuracy of the G-MTMI algorithm is within $\approx 10 %$ in all sets of experiments, where the accuracy is calculated as: $\begin{matrix} 100 % \times \frac{optimal immunity - G-MTMI immunity}{optimal immunity} . \end{matrix}$

7.2. Immunity versus network parameters

In this section, we study how the immunity of the constructed multicast tree varies with respect to network parameters like number of channels, number of PUs, geographical distribution of PUs in the network field, PU protection range, and the size of the multicast group. Instead of studying immunity against each of the different parameters concerning PUs, we define a new metric that we call the “reliability factor” which takes into consideration all of those aspects together. Given a SU s, its reliability factor on channel k, denoted $σ (s, k)$ , is defined as follows: $\begin{matrix} (15) & σ (s, k) = \{\begin{matrix} 0, & \exists p \in P_{active} : k_{p} = k and I_{(s, k)}^{p} = 1 \\ \frac{1}{1 + \sum_{p \in P_{idle} : k_{p} = k} I_{(s, k)}^{p}}, & otherwise \end{matrix} \end{matrix}$ where $I_{(s, k)}^{p}$ is a binary indicator that is equal to 1 if a spectrum failure caused by p on channel k prevents SU s from accessing channel k, and equals 0 otherwise.

The worst value of $σ (s, k)$ is 0 when there exists an active PU operating on channel k close enough to s preventing it from using the channel. The best value, on the other hand, is 1 when there is no PU (active or idle) that can interrupt SU s on channel k. The value of $σ (s, k)$ decreases with the increase of the number of idle PUs operating on channel k which can cause a spectrum failure to s on k.

The average reliability factor for SU s, denoted $σ (s)$ , is defined as follows: $\begin{matrix} (16) & σ (s) = \frac{\sum_{k \in C} σ (s, k)}{| C |} \end{matrix}$

Finally, the average reliability for the entire network, denoted as σ, is defined as follows: $\begin{matrix} (17) & σ = \frac{\sum_{s \in S} σ (s)}{| S |} \end{matrix}$

To study the relationship between σ and immunity, we varied the size of the multicast group from 8 to 20 in an increment of 4. For each group size, we varied the number channels from 4 to 10, the number of PUs from 10 to 40, and the PU protection range from $4 r$ to $10 r$ . The total number of network instances generated (at random) for each group size is 588000. For each network instance, σ is calculated according to equation (17). Then, instances are grouped in bins of width 0.01 according to the value of σ, and the immunity of all instances within the same bin is calculated. The averaged immunity over these bins is plotted in Fig. 9.

Fig. 9.

The immunity versus $\bar{σ}$ for different numbers of multicast receivers.

As Fig. 9 shows, immunity increases in an exponential pattern with the increase of σ. The pattern is the same regardless of the size of the multicast group. Furthermore, the achieved immunity is approximately the same, on the average sense, regardless of the size of the multicast group. This result suggests that the problem of calculating the achievable multicast immunity can be transformed into the problem of calculating σ. In other words, σ is a network property from which the immunity of multicast sessions in the network can be inferred. We are currently investigating the problem of developing distributed algorithms to calculate σ for any given CRN.

8. Conclusions

The problem of constructing multicast trees with maximum immunity against spectrum failures, abbreviated as the MTMI problem, is studied in this paper. The immunity of a multicast tree was defined as the minimum ratio of the number of multicast receivers that remain connected with the source of the multicast tree to the size of the multicast session among all possible spectrum failures. The MTMI problem was formulated as an Integer Linear Program (ILP) to find the optimal solution. Furthermore, a sub-optimal algorithm was also proposed. Results shows that the proposed algorithm is, on average, within $\approx 10 %$ of the optimal solution.

We have also defined the reliability metric σ and showed that the achievable immunity of a multicast session in a network is a function of σ. Therefore, evaluating σ can give a hint of how immune multicast sessions are going to be in a given CRN.

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Tree-based multicast service provisioning with maximum immunity in cognitive radio networks

Abstract

Keywords

1. Introduction

1.1. Paper contributions

1.2. Paper organization

2. Literature review

2.1. Multicast routing in cognitive radio networks

2.2. Multicast recovery in cognitive radio networks

3.1. Network model

3.2. Spectrum failure model

4.1. Motivation

6.2. Minimum Cost Route (MCR) algorithm

2 The node that is reached first.

References

²
The node that is reached first.