distributed multicast multichannel paths

Telecommun Syst (2012) 50:55–70DOI 10.1007/s11235-010-9385-0

Distributed multicast multichannel paths

Ayaz Isazadeh · Mohsen Heydarian

Published online: 4 August 2010© Springer Science+Business Media, LLC 2010

Abstract Supporting multimedia applications inQoS-aware multicast deployment has become an importantresearch dimension in recent years. Future communicationnetworks will face an increase in traffic driven by multi-media applications with stringent requirements in the fol-lowing important functions: (1) nodes and links used dis-tributing, (2) packets duplication distributing, (3) QoS sup-porting, (4) multichannel routing. For improving these fourfunctions, in this paper we propose a new polynomial timealgorithm, named Nodes Links Distributed-Multicast Mul-tichannel Routing (NLD-MMR), based on the Constraint-Based Routing (CBR) and Linear Programming (LP). Thenew algorithm by constructing Distributed Multicast Mul-tichannel Paths (DMMCP) can distribute or compact bothpaths and traffic. Our simulation study shows that the pro-posed algorithm, as compared to other available algorithms,performs well and constructs a new generation of optimalpaths with the best cost and efficiency.

Keywords Mathematical modelling · Multicastmultichannel path · Nodes and links used distributing ·Packets duplication distributing · Quality of Services (QoS)

1 Introduction

Advances in routing protocols and switching technologiesare driven by three new inventions in mathematical mod-eling, communications technologies and new requirements

A. Isazadeh (�) · M. HeydarianDepartment of Computer Science, Tabriz University, Tabriz, Irane-mail: [email protected]: www.isazadeh.net

M. Heydariane-mail: [email protected]

in applications. The explosive growth of the Internet andbandwidth-intensive applications, such as multimedia con-ferencing and web applications, require high-bandwidthtransport networks whose capacity is much beyond whatcurrent high-speed networks, such as asynchronous trans-fer mode (ATM) network, can provide. While the need ofcommunication channels for high-bandwidth, low latency,reasonable packet loss rate, packet duplication and trafficdistribution has been on the rise, a continuous demand fornetworks of high capacities at low costs is seen now. Toachieve these, in this paper we provide a new routing al-gorithm which supports the four following important func-tions:

1. Nodes and links used distributing: in a transmission ses-sion, we can define two types of traffic distribution: in-creasing and decreasing the number of nodes-links usedwhich we call them as high level distribution (traffic ex-pansion) and low level distribution (traffic compression),respectively. By increasing and decreasing the number ofnodes-links used we can compact or expand traffic, re-spectively. Consequently, if it is necessary, by using thesetwo types of distribution we can manage traffic, band-width and congestion preferably.

2. Packets duplication distributing: this means that in thesubnet, every node can duplicate and dispatch arrivalpackets instead of other nodes. It is important that we de-termine which nodes instead of which other nodes mustduplicate and dispatch arrival packets and also how manypackets must be duplicated.

3. QoS supporting: Delay, jitter, bandwidth, and loss rateare common parameters, widely used to describe QoSrequirements. Multimedia applications such as Internettelephony or digital video conferencing are very sensi-tive to these parameters.

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56 A. Isazadeh, M. Heydarian

4. Multichannel routing: this means that transmitting a mes-sage using all available links. Multichannel routing canimprove network efficiency and traffic management.

Traditional algorithms such as trees and unicast algorithms[4, 6, 13, 15, 17] cannot support above functions and somenew available algorithms such as Optimal Multicast Multi-channel Routing (OMMR) [7] cannot support function (1)and (2). In addition, some new distribution algorithms suchas Distributed Optimal Multicast Multichannel Routing(DOMMR) [8] does not support function (1).

In this paper, we propose a new distributed algorithm forsolving the problem of multicast multichannel routing sup-porting four above functions. This algorithm is fully distrib-uted and can generate within acceptable time and messagecomplexities a multicast routing path, which not only sat-isfies the required multicast multichannel distributed QoSconstraints but also has a sub-optimal network cost. The re-sults of the simulations show that the multicast multichannelrouting path generated by our algorithm has better perfor-mance than the previous well-known results.

The rest of this paper is organized as follows. In Sect. 2,we present the problem, some terminologies and somemathematical definitions such as multicast multichannelpath, multicast multichannel routing and etc., which arenecessary to construct the proposed algorithm, Nodes-Links Distributed-Optimal Multicast Multichannel Routing(NLD-MMR). In Sect. 3 we present a brief survey of someexisting multicast algorithms such as DPST [6], OMMR[7] and (DOMMR) [8]. Section 4 explains terminologies,concepts and steps of NLD-MMR algorithm. This sectionalso includes theorems and mathematical formulations ofthe proposed algorithm. Section 5 shows the sub-optimalityof the previously published algorithms using mathematicaland network tools such as QSB, MATLAB, and OpNeT. Toachieve this, in this section, some sample networks, com-putations, and simulation results will be presented. Finally,Sect. 6 concludes the paper with a summary of the resultsand some important future topics of research.

2 The problem

End-to-end delay and bandwidth consumption are two im-portant multicast QoS parameters in data communicationsproblems to guarantee that the messages transmitted by thesource node can reach the destination node(s) within a cer-tain amount of time. To support such problems, efficientQoS multicast algorithm needs to be developed. In this sec-tion, at first, we describe our problem, finding distributedmulticast multichannel path (DMMCP), and then in the nextsections we propose a fast and efficient QoS multicast algo-rithm, nodes links distributed-multicast multichannel rout-ing algorithm (NLD-MMR), which can find a DMMCP

achieving smaller QoS multicast delay variation under themulticast multichannel end-to-end delay constraint.

2.1 Preliminaries

We use graphs to show the topology of the sample net-works. Graphs in this paper are considered to be directedand weighted including loops. A computer network is mod-eled by N = (V ,E,b,p, q), where G = (V ,E) is a directedsimple graph with vertex set V , edge set E, and integerweighting functions b(•), p(•) and q(•): b(u, v) ≥ 0 andp(u, v) > 0 are the bandwidth and propagation delay for a(directed) edge e = (u, v) ∈ E, and q(u) ≥ 0 is the queuingdelay at a vertex u ∈ V . Let s be a source vertex and t adestination vertex, unicast algorithms are interested in rout-ing a message of size σ from s to t as fast as possible whilesatisfying certain delay constraints.

Definition 2.1 A v1 − vk path in graph G can be shown asπ = 〈v1, v2, . . . , vk〉 and is called as a single path. For eachsingle path π the path delay is:

D(π) =∑

1<j≤k

p(vj−1, vj ) + q(vj ). (1)

The maximum usable bandwidth of a path π is

B(π) = min1<j≤k

b(vj−1, vj ). (2)

The time required to route a massage of σ > 0 along path π

using a bandwidth of b ≤ B(π) is

T (π,b,σ ) = D(π) +⌈

σ

b

⌉− 1. (3)

If we are seeking the fastest transmission of a messagefrom s to t using a single path, we have a unicast singlechannel routing problem (Fig. 1(a)). If we are seeking thefastest transmission of a message from s to t using all avail-able links, we have a unicast multichannel routing problem(Fig. 1(b)). Ford and Fulkerson [4, 5] and Xue [16] have for-mally formulated and studied unicast multichannel routingproblem. Also, the tree routing has been shown in Fig. 1(c).Finally, If we are seeking the fastest transmission of a mes-sage from s to more than one destination t using all avail-able links, we have a multicast multichannel routing prob-lem (Fig. 1(d)).

Now, we recall some notations which have been pre-sented before [7]:

1. Unicasting Node (UcN): This is an intermediate nodethat must receive packets and only forward them to thenext neighbor nodes. In other words, it does not dupli-cate any arrival packets and does not use any duplicationcapacities.

Distributed multicast multichannel paths 57

Fig. 1 Some sample networkswith a source node s anddestination nodes t1, t2 and t3;(a) a single or unicast path〈s, x, y, t1〉; (b) a unicastmultichannel path including twosingle paths 〈s, x, y, t1〉 and〈s, x, t1〉; (c) a multicast treeincluding three single paths〈s, x, y, t1〉, 〈s, x, y, t2〉, and〈s, x, t3〉; (d) a multicastmultichannel path including sixsingle paths s − t1 and s − t2

2. Multicasting Node (McN): This is an intermediate nodethat receives arrival packets and duplicates some of themand sends duplicated packets to next neighbor nodes.Note that each duplicated packet belongs to a destina-tion node and must be forwarded to that node. The set ofthe multicasting nodes in a network, is shown by “Multi-cast”.

3. Duplication: Each packet that is dispatched by a sourcenode, must be multiplied by multicasting nodes in accor-dance with the number of destination nodes. The maxi-mum capacity of duplication in a multicasting node v ∈V is shown by the “Capacity(v)”. Also, dup(v) shows theamount of data duplication in multicasting node v whichmust be done by v until the required packets for the des-tination nodes are produced.

4. Duplication Delay: This is the time units required for theduplication of arrival packets in a multicasting node v ∈V . It is shown by c(v) ≥ 0. If v ∈ V be a unicasting nodeor dup(v) = 0, then c(v) = 0.

5. Multicast Path Delay: For each path π = 〈v1, . . . , vk〉 thepath delay is: D(π) = ∑

1<j≤k(p(vj−1, vj ) + q(vj )) +∑vj ∈Multicast c(vj ).

The types of the paths which we use for describing the NLD-MMR problem, are categorized as the following [8]:

1. Unicast Single-Channel Path (USCP): as mentionedin Definition 2.1, it is a single channel path v1–vk =〈v1, v2, . . . , vk〉. Figure 1(a) shows an USCP.

2. Unicast Multi-Channel Path (UMCP): it is a group ofUSCPs (more than one USCP) which is rooted in asource node s and have the same destination node. TheseUSCPs may have some shared links or nodes. Note thatan UMCP includes at least one loop. Figure 1(b) showsan UMCP.

3. Tree: it is a graph (or group of USCPs) rooted in a sourcenode s and ended to destination nodes t1, t2, . . . , tn whichdoes not include any loops. Note that each USCP used

by tree, can has some shared links or nodes with otherUSCPs, but can not form any loop in the tree. Figure 1(c)shows a tree.

4. Multicast Multi-Channel Path (MMCP): because thereare n destination nodes t1, t2, . . . , tn in a multicast ses-sion, therefore corresponding to each ti we have anUMCP. Thus, we have n UMCPs, which is called anMMCP. In fact, an MMCP is a group of UMCPs (morethan one UMCP), which is rooted in the source node s

and each UMCP will be ended to one different ti . Fig-ure 1(d) shows a MMCP. We can transform a tree to anMMCP by adding one or more than one loops to it: infact, an MMCP is a tree including some loops but cannotbe exactly equal to a tree. An MMCP has an importantproperty: if one unite of data be sent out from s, eachdestination node ti , must receive just one copy of it.

Figure 1 shows the kinds of paths which are considered inthis paper. As an aid to understand concepts of transmissionand delay, we consider a sample network shown in Fig. 1(a).There are different ways to transmit 9 units of data, (σ = 9),through single path P = 〈s, x, y, t1〉 from s to t1 illustratedin Fig. 1(a). One of them is the quickest way which is as fol-lows. Let us assume that each node has a buffer space largeenough to store excess data. We assume that data transmis-sion starts from node s at time 0. Considering edge (x, y),the maximum available bandwidth along path P is 5. Attime unit 2, 5 units of data arrives at node x, leaving 4 unitsof data at the node s. Since there is a queuing delay of 3 atnode x, no data can go out of node x until time unit 5. Attime unit 5, 5 unit of data leaves node x along edge (x, y).Note that at time unit 1, the other 4 units of data can go outof node s and follow 5 unites of data node-by-node. At timeunit 6, 5 unit of data arrives at node y and at time unit 8leaves node y. Since there is a transmission delay of 1 atedge (y, t1) and there is a queuing delay of 2 at node t1, the5 units of data are finally received by destination node t1 at


time unit 11. The other 4 units of data are received by nodet1 at time unit 12 and transmission will be completed in 12time units. As mentioned for a single path, the transmissionof a message can be done for a tree such as Fig. 1(c). Notethat in a tree, single paths may diverge at a node and datamust be duplicated. The duplication may produce delay andthis delay must be computed. Duplication capacity and du-plication delay will be discussed with Sects. 3 and 4.

2.2 Our approach and definitions

Now, we need to define some new notations as the follow-ing:

1. Incapable Node: the multicast node which its duplicationcapacity is not equal to zero and can duplicate packets,but data transmission algorithm does not allow it or can-not use it to duplicate packets.

2. Capable Node: a former unicast or incapable multicastnode which NLD-MMR algorithm select it to duplicatepackets instead of later multicast nodes.

3. MulticastNew: it is the set which contains all capablenodes. We use n(MulticastNew) to show the number ofcapable nodes.

4. DMMCP: it is an Distributed MMCP. In other words, ifwe increase the number of nodes and links used in anMMCP, we will obtain a DMMCP. An DMMCP can beproduced only by our algorithm and contains two typesmulticasting and capable nodes, but MMCP only con-tains multicasting nodes.

As mentioned before, the available algorithms such as Trees,OMMR and DOMMR do not support two following func-tions:

1. Link Used Distributing: in a multicast session includinga source node and some destination nodes, an DMMCPcan be routed for transmitting packets from the sourcenode to the destination nodes. An DMMCP is a new typeof MMCP which can increase the number of nodes andlinks used as compared to MMCPs. Note that, increasingthe number of nodes used in an DMMCP routed causesincreasing the number of links used and this causes trafficdistribution.

2. QoS supporting: the DMMCP routed must reduce totalconsumption of bandwidth and end-to-end delay whenthe used algorithm tries to compact or expand the traffic.

To achieve this, we try to define a new problem which con-sider these two functions and its solution supports them. In-formally, the new problem is: we are seeking a fastest trans-mission of the message from a source node s to a groupof destination nodes using all available links by support-ing QoS and traffic distribution. Traffic distribution includestwo major tasks: (1) reducing total consumption of band-width and end-to-end delay, (2) increasing the number of

nodes and links used. We call this problem as Nodes LinksDistributed-multicast multichannel routing problem (NLD-MMR).

2.3 Formal definition of the problem

Now, we present the NLD-MMR formally:

Definition 2.2 Let n be the number of destination nodes ti ,i = 1,2, . . . , n. Suppose from source node s to each destina-tion ti there are ki single path (USCP) πij , j = 1,2, . . . , ki .Furthermore, σ > 0 be the size of the message to be trans-mitted from s to each destination ti and a positive integer τ

be the number of time unites consumed in this transmission.The nodes links distributed-multicast multichannel routingdecision problem asks the existence of a decomposition

σ = DUPi + σi =∑

1≤j≤ki

σij (σij ≥ 0, 1 ≤ i ≤ n) (4)

where all parts σij must be transmitted along paths πij

(j = 1,2, . . . , ki) from s to ti . Note that for each ti , thepart σi is not necessary to be duplicated and will be sent byfirst multicasting node on the πij , but DUPi must be dupli-cated by multicasting nodes inside the network and transmit-ted along all paths πij (j = 1,2, . . . , ki). It means that for adestination node ti , the message σ must be transmitted us-ing an UMCP; for all destination nodes ti (i = 1,2, . . . , n),n × σ units of data must be transmitted using an DMMCP.It can be verified that

∑1≤i≤n DUPi = (n − 1) × σ . Also,

the bandwidth of edge (u, v) ∈ πij must be decomposed as

b(u, v) =∑

1≤i≤n,1≤j≤ki

bij (u, v) (bij (u, v) ≥ 0). (5)

In (5) if (u, v) does not belong to πij , then bij (u, v) = 0.Furthermore, for each multicasting node v ∈ πij , dupij (v)

is computed so that the following result is obtained:

max1≤j≤ki ,1≤i≤n

T (πij ,Bij (πij ), σij ) ≤ τ. (6)

Note that this transmission must be only done by an MMCPor an DMMCP, and any trees can not be used. Also the num-ber of nodes and links used must be maximized.

3 Related work

Over the recent few years, the design of distributed routingalgorithms has gained increasing importance in multicastmultichannel QoS applications. In the following, we give abrief literature review on some available algorithms.


3.1 Trees

Trees such as Minimum Spanning Tree (MST) [11] and Dy-namic Priority Spanning Tree (DPST) [6] are widely usedfor routing in communication networks. DPST algorithmuses the source routing strategy to establish the tree of pathsfrom the source to the multicasting destinations [6]. The al-gorithm allows for transfer of messages from the source tothe multicasting destination by using routing and messageduplication in network switches. The multicast tree is rootedin the source and all branches terminate at destinations. Thepacket are duplicated only when two branches of the pathdiverge. If a single link leads to multiple destinations, onlya single copy of packet traverses the link; if necessary thepacket will be duplicated later. DPST is a heuristic multicastrouting algorithm that can find the minimum cost multicastpath in a polynomial time, using at most O(n3) iterations,where n is the number of nodes in network. DPST supportsmulticast service and duplicates messages by using interme-diate nodes but does not support multichannel links and traf-fic distribution and also does not guarantee QoS parameterssuch as delay and bandwidth.

3.2 Optimal multicast multichannel routing

OMMR algorithm [7] gives solution for the following prob-lem: Let σ > 0 be the size of the message to be transmit-ted from source node s to destination nodes t1, t2, . . . , tn,

whereas each s − ti path i = 1,2, . . . , n, is a multichannelpath, then determine minimum number of the time units inthis transmission. OMMR solves a sequence of multicast dy-namic flow problems (M-MDF(τ )) to find the smallest in-teger τ which guarantees YES answer to the multichannelrouting decision problem. M-MDF(τ ) transmits data froma source node to more than one destination nodes. OMMRsupports QoS by decreasing end-to-end delay and employsmultichannel paths by using M-MDF(τ ) formulation, butdoes not guarantee traffic distribution optimally. M-MDF(τ )is as following:

(1) M-MDF(τ ): min∑

(u,v)∈E d(u, v)f (u, v)

+ ∑v∈V c(v)dup(v) − (τ + 1)nφ

s.t.(2)

∑(s,v)∈E f (s, v) − φ = 0,

(3)∑

v∈E f (v, ti) − φ = 0 (i = 1,2, . . . , n),(4)

∑(u,v)∈E f (u, v) + dup(v) − ∑

(v,u)∈E f (v,u) = 0,(u �= s, t),

(5)∑

f (s, v) = ∑f (u, ti), v ∈ V, {(s, v), (u, ti)} ∈ πij ,

j ∈ {1,2, . . . , ki},(6) (m − 1)

∑f (u, v) ≥ dup(v), v ∈ Multicast, m is the

number of output links of node v,(7)

∑(u,v)∈E f (u, v) ≥ f (v,u), v ∈ V ,

(8) 0 ≤ f (u, v) ≤ b(u, v), (u, v) ∈ E,(9) 0≤ dup(v) ≤ Capacity(v), (v) ∈ V .

Let (φ,f,dup) be an optimal solution to the above LP prob-lem. Then the flow f can be decomposed into a set of s − ti

arc-chain flows i = 1,2, . . . , n; for each malticasting nodev ∈ V , dup(v) can be distributed along v − ti ∈ πij paths;and the maximum amount of data from s to all of the des-tination nodes ti in τ time units is: χ(τ) = (τ + 1)nφ −∑

(u,v)∈E d(u, v)f (u, v) − ∑v∈V c(v)dup(v).

OMMR supports multichannel and multicast QoS rout-ing but it cannot support nodes-links used distributing andpackets duplication distributing.

3.3 Distributed-Optimal multicast multichannel routingalgorithm

Distributed-Optimal multicast multichannel routing algo-rithm (DOMMR) [8] gives solution for the following prob-lem: we are seeking a fastest transmission of the messagefrom a source node s to a group of destination nodes usingall available links by supporting QoS and traffic distribu-tion. Traffic distribution includes two major tasks: reducingtotal consumption of bandwidth and increasing the numberof link used.

Because M-MDF(τ ) cannot distribute traffic optimally,DOMMR presents DM-MDF(τ ), Distributed M-MDF(τ ),which is constructed by adding some new conditions to M-MDF(τ ), and it is called as DM-MDF(τ ). DM-MDF(τ )

produces different optimal solutions such as (φ,f,dup)

which only cover Multicast Multichannel Paths (MMCPs)and do not cover trees. An MMCP includes a tree contain-ing some loops which starts from source node and termi-nates in destinations nodes. DOMMR solves a sequence ofdistributed multicast dynamic flow problems (DM-MDF(τ ))to find the smallest integer τ which guarantees YES answerto the distributed multicast multichannel routing problem.DOMMR supports packets duplication distributing, but can-not support nodes and links used distribution.

4 The new algorithm

Trees and OMMR algorithms can not support any typesof distribution and also, DOMMR algorithm only supportspackets duplication distributing. In this section we presenta new algorithm, Nodes Links Distributed Multicast Multi-channel Routing (NLD-MMR) Algorithm, which supportspackets duplication, nodes and links used distribution. Wewill show that in the one hand, NLD-MMR increase thenumber of nodes and links used in a DMMCP routed andon the other hand, it reduces the total consumption of band-width and total time of message transmission. In this way, itcan increase traffic distribution and network efficiency.


4.1 Nodes links distributed multicast MDF(τ )

NLD-MMR algorithm provides an iterative solution basedon the constraint-based routing. Linear programing formu-lation is one of the most important of approaches whichare used by constraint-based routing. OMMR and DOMMRpresent M-MDF(τ ) and DM-MDF(τ ) models based on thelinear programming, respectively, for formulating itemssuch as: propagation delay, bandwidth, total time of mes-sage transmission, switching delay and switches’ packetsduplication capacities, but does not model nodes used distri-bution.

For removing this weakness, we present a new DM-MDF(τ ), Nodes Links Distributed Multicast MDF(τ ), whichis constructed by adding some new conditions to DM-MDF(τ ), and we call it as NLDM-MDF(τ ).

Theorem 4.1 The Nodes Links Distributed multicast maxi-mal dynamic flow problem or briefly NLDM-MDF(τ ) for τ

periods can be computed by solving the following minimumcost static flow problem:

(1) NLDM-MDF(τ ): min∑

(u,v)∈E d(u, v)f (u, v)

+ ∑v∈V c(v)dup(v) − (τ + 1)nφ

s.t.(2)

∑(s,v)∈E f (s, v) − φ = 0,

(3)∑

v∈E f (v, ti) − φ = 0 (i = 1,2, . . . , n),(4)

∑(u,v)∈E f (u, v) + dup(v) − ∑

(v,u)∈E f (v,u) = 0,(u �= s, t) and v ∈ Multicast ∪ Unicast ∪ MulticastNew,

(5)∑

f (s, v) = ∑f (u, ti), v ∈ V, {(s, v), (u, ti)} ∈ πij ,

j ∈ {1,2, . . . , ki},(6) (m − 1)

∑v∈Multicast f (u, v) ≥ dup(v), m is the num-

ber of output links of the node v OR (n − 1)∑v∈MulticastNew f (u, v) ≥ dup(v), n is the number of

destination nodes ti ,(7)

∑(u,v)∈E f (u, v) ≥ f (v,u), v ∈ Multicast ∪ Unicast,

(8)∑n

i=1∑ki

j=1 xj (v, ti)+∑N(Multicast)i=1

∑INP(vi )j=1 yj (u, vi)

− (n + N(Multicast)) > 0, vi ∈ Multicast; INP(vi) isthe number of input flows in each multicasting nodevi ∈ Multicast; N(Multicast) is the size of the setMulticast,

(9) 0 < f (v, ti) ≤ xj (v, ti)b(v, ti), and 0 < f (u, vi) ≤yj (u, vi)b(u, vi) where xj (v, ti) = Sign(f (v, ti)) ∈{0,1}, j = 1,2, . . . , ki and yj (u, vi) = Sign(f (u, vi))

∈ {0,1} j = 1,2, . . . , INPvi, vi ∈ Multicast,

(10) 0 ≤ f (u, v) ≤ X(u,v)b(u, v), where X(u,vi) =Sign(f (u, v)) ∈ {0,1}, and v ∈ V − {Multicast ∪{t1, t2, . . . , tn}},

(11)∑

i

∑j dupk

ij = dupk , that dupk = dup(vk), vk ∈Multicast, k = 1,2, . . . , n(Multicast), dupk

ij is dis-patched to πij by node vk ∈ πij ,

(12)∑

j

∑vk∈πij

dupkij ≤ φ, that πij = USCPij , USCPij ∈

UMCPi , UMCPi ∈ MMCP,

(13)∑

k dupk = (n − 1)φ, that k = 1,2, . . . , n(Multicast),(14) 0 < dup(v) ≤ Y(v)Capacity(v), where Y(v) ∈ (0,1]

and v ∈ Multicast ∪ Unica is arbitrarily selected node,(15) 0 ≤ dup(v) ≤ Capacity(v), (v) ∈ Multicast,(16) 0 ≤ f (u, v) ≤ b(u, v), (u, v) ∈ E.

Let (φ,f,dup) be an optimal solution to the above LP prob-lem. Then the flow f can be decomposed into a set of s − ti

arc-chain flows i = 1,2, . . . , n; for each malticasting nodev ∈ V , dup(v) can be distributed along v − ti ∈ πij paths;and the maximum amount of data from s to all of the desti-nation nodes ti in τ time units is:

χ(τ) = (τ + 1)nφ −∑

(u,v)∈E

d(u, v)f (u, v)

−∑

v∈V

c(v)dup(v). (7)

NLDM-MDF(τ ) produces different optimal solutionssuch as (φ,f,dup) which only cover MMCPs and DMM-CPs and do not cover trees. There is always an optimalsolution to NLDM-MDF(τ ), which can be decomposedinto a set of s − ti arc chain flows. The decision vari-ables of NLDM-MDF(τ ) are: f (u, v), ∀(u, v) ∈ E; dup(v),∀v ∈ Multicast; φ, xj , yj , Y1, Y2 and X(u,v). Condition(13) enables NLDM-MDF(τ ) to increase the number ofnodes and links used. It shows that an unicasting node v canbe transformed to a multicasting node (or a capable node)and duplicates packets and we call it as New Node. A mul-ticasting node can duplicate packets by itself or receive theduplicated packets from a former New Node which has du-plicated and dispatched these packets. In the other words,a New Node is the multicasting node which can duplicatepackets instead of other multicasting nodes. Note that M-MDF(τ ) and DM-MDF(τ ) cannot produce New Nodes, butNLDM-MDF(τ ) can do it. Condition (10) is based on thetree properties of graph theory and causes NLDM-MDF(τ )constructs only DMMCPs. As condition (12) shows, a flowf (u, v) can be zero or positive; therefore, its sign is ‘0’ or‘+’. In contrast to DM-MDF(τ ), NLDM-MDF(τ ) in the op-timal solution (φ,f,dup), by using variables xj , yj , and X

can decide a flow f (u, v) be zero or not. We can arbitrarilydecide that variables xj , yj , and X be zero or not. If oneof these variables be zero, its corresponding USCP will beomitted, therefore the UMCP which contains this USCP,will be changed or compressed. Consequently DMMCPsolution will be changed or compressed. This means thatNLDM-MDF(τ ) as compared to DM-MDF(τ ), can increaseor decrease the number of nodes and links used, and thisbalances traffic optimally, and increases traffic distribution.NLDM-MDF(τ ) distributes and balances traffic using theconditions (10), (11), (12) and (13).


4.2 Nodes links distributed multicast multichannel routingalgorithm

We now present NLD-MMR algorithm, solving the NodesLinks Distributed Multicast Multichannel Routing prob-lem. NLD-MMR algorithm solves a sequence of NLDM-MDF(τ ) problems to find the smallest τ which guaranteesYES answer to the nodes links distributed multicast multi-channel routing decision problem. Also, NLD-MMR algo-rithm supports multicasting service and recognizes all opti-mal and usable multicast paths which start from s and ter-minate at destinations ti , i = 1, 2, . . . , n, (n is the number ofdestination nodes). NLD-MMR can present those solutionswhich are established by trees, OMMR and DOMMR, butthey can not present those solutions which are constructedby NLD-MMR. Also, NLD-MMR only presents solutionswhich are as an DMMCP, but available algorithms some-times construct trees and MMCPs instead of DMMCPs.

NLD-MMR algorithm:

(1) Let α := 0 and β := σ +D(π∗). // initialization. π∗ andD(π∗) have been computed before by signaling proto-cols and can be employed by NLD-MMR.

(2) If β = α + 1 goto Step-5.(3) Let γ := �(α + β)/2� and solve the γ -periods nodes

links distributed multicast maximal dynamic flow prob-lem. If the solution is infeasible, terminate problem, elselet χ(γ ) be the maximum amount of data that can betransmitted from s to all of the ti in γ units of time. //Solving a nodes links distributed multicast maximal dy-namic flow problem.

(4) If �(χ(γ ))/n� ≥ σ then let β := γ else let α := γ endif;goto Step-2. // A cycle for solving a sequence of NLDM-MDF(τ ) problems to find the smallest integer τ whichguarantees an YES answer.

(5) Let τ := β and solve the minimum cost flow problemNLDM-MDF(τ ) to obtain flow value φ and edge flowassignments f . Let i := 0. // Computing optimal solu-tion (φ,f,dup).

(6) Let i = i + 1, j = 0; if i > n goto Step-11, else gotoStep-7; endif. // Counting the number of UMCPs bycounter ‘i’ and USCPs by counter ‘j ’.

(7) Let j = j+1; Compute an s − ti path πij in G(V,E,

b, d) to maximize B(πij ) = min(u,v)∈πijf (u, v);

if πij is a marked path then let j = j − 1, ki = j ; gotoStep-6. // Rejecting a marked USCP.else if B(πij ) = 0 then let j = j − 1, π = πij ; gotoStep-9. // Rejecting a vanished USCP.endif; // Computing USCPs and counting the number ofthem.

(8) Choose the node vm ∈ Multicast on the path πij whichis the closest multicasting node to ti and dup(vm) > 0.(a): if vm �= s then

Compute the USCP 〈vm − ti〉 = 〈s − ti〉 − 〈s − vm〉 andcall it π , (π ∈ πij ). Compute less =min(u,v)∈π {dup(vm), f (u, v)}.if less = 0 then goto Step-9. endifLet dup(vm) = dup(vm) − less // Updating duplicationcapacity on node vm.Let dupij (vm) = less // Assigning consumed duplicationcapacity to node vm.for each e ∈ E doif e ∈ π thenLet bij (e) = bij (e) + less // Updating bandwidth usedon edge e.Let f (e) := f (e) − less // Updating edge flow assignment

on edge e.

elseLet bij (e) = 0endif; endfor; endif; goto Step-8(b): if vm = s thenCompute less = min(u,v)∈πij

f (u, v).if less = 0 then let π = πij goto Step-9. endif;for each e ∈ E doif e ∈ πij thenLet bij (e) = bij (e) + lessLet f (e) := f (e) − lesselseLet bij (e) = 0endif; endfor; endif.

(9) Let D(π) = c(v) + ∑(u,w)∈π d(u,w). B∗(πij ) =

max(u,v)∈πijf (u, v), j = 1,2, . . . , ki . // Computing

transmission delay of path π . Notice that v is a mul-ticasting node and it is the closest node to ti anddupij (v) > 0. The location of the edge (u, v) is afterthe multicasting node v on the πij .

(10) Mark πij ; goto Step-7.(11) Compute a positive integer decomposition σ . σ =

DUPi + σi = ∑1≤j≤ki

σij (σij ≥ 0 and i ∈ {1, . . . , n})such that (τ − D(πij ))B

∗(πij ) ≤ σij ≤ (τ − D(πij )

+ 1)B∗(πij ).

The goal of Step-1 through Step-5 of NLD-MMR algorithmis to compute τ , the minimum number of time unites re-quired to transmit σ units of data from s to destination nodesti , i = 1,2, . . . , n. The goal of Step-6 through Step-10 ofNLD-MMR is to compute an optimal positive integer de-composition of σ as in (8), a corresponding decompositionof the bandwidth for every edge (u, v) ∈ E as in (9), andall πij path (i = 1,2, . . . , n, j = 1,2, . . . , ki ), and dupij (v)

for each multicasting node v ∈ πij , (i = 1,2, . . . , n, j =1,2, . . . , ki ) such that max1≤j≤ki ,1≤i≤nT (πij , Bij (πij ),σij ) ≤ τ . Step-11 computes one optimal assignment.


5 Computational result and metrics analyzing

In this section, we present our computational result and per-formance comparison of NLD-MMR algorithm with DPST,OMMR and DOMMR algorithms. For making comparisonsbetween our algorithm and above algorithms, we will usesome samples of network to show the steps of NLD-MMRalgorithm, and to show the sub-optimality of existing algo-rithms.

Computations are done by mathematical and computa-tional tools such as WinQSB, MATLAB6. The WinQSB isused for solving all kinds of LPF, Non-LPF and networkfollow problems. The MATLAB6 is an integrated techni-cal computing environment that combines numeric compu-tation, advanced graphics and visualization, and a high-levelprogramming language. The MATLAB6 includes hundredsfunctions for: Data analysis and visualization, numeric com-putation, modeling, simulation, prototyping, programmingand application development. We suppose that the buffersize of the each node is unlimited.

5.1 Computational metrics

The following metrics have been presented before [7, 8]:

1. Total Consumption of Bandwidth (TCoB): For each algo-rithm, TCoB is equal to

∑(u,v)∈E f (u, v) where f (u, v)

is amount of bandwidth of (u, v) which is used by thealgorithm.

2. Total Time of Message Transmission (τ ): It is the timerequired to transmit a message of size σ ≥ 0 from thesource node to one or more than one destination nodesusing an algorithm.

3. Data Rate (DR): For each algorithm is equal to στ

.4. Total consumption of duplication capacity (dup): Sup-

pose an algorithm uses a set A of nodes to route amessage from a source node to a group of destinationnodes. For this algorithm we have; dup = ∑

dup(v),∀v ∈ Multicast, and v ∈ A. If node v does not duplicatepackets, then dup(v) = 0.

5. Input flow φ: It is input flow which is loaded by source s.In fact, node s presents a constant load φ of data unitsper time unit destined all destinations.

6. Number of Links Used (NoLU): Is the number of links ofthe network which are used by an algorithm to transmit amessage from the source node to destination nodes.

7. Efficiency (): It is clear that if an algorithm as comparedto some other algorithms, transmits a specific message ofsize σ > 0 in minimum time and bandwidth, it is moreefficient than other algorithms. In fact, the less is, themore efficient algorithm is. It can be mathematically de-scribed as:

= τ × TCoB

σ.

We use the following metrics to evaluate and compare ofperformance and traffic distribution of the algorithms:

1. Number of Multicasting Nodes Used (NoNU): Is thenumber of multicasting nodes of the network which areused by an algorithm to transmit a message from thesource node to destination nodes.

2. Distribution Coefficients (DC) and (DC′): For each al-gorithm we define DC = TCoB

NoLU and DC′ = TCoBNoLU+NoNU .

DC and DC′ say that an algorithm distributes traffic suc-cessfully, if it minimizes TCoB and maximizes NoLUand NoNU. In other words, traffic distribution can be in-creased, if the algorithm decrease the value of DC.

3. Duplication Distributing Coefficient (DDC): For each al-gorithm we define DDC = dup

NoNU . DDC says that an algo-rithm distributes switches’ duplication capacity success-fully, if it minimizes dup and maximizes NoNU. In otherwords, duplication distributing can be increased, if thealgorithm decrease the value of DDC.

5.2 Manual example

We now consider simple network illustrated in Fig. 2(a).This figure consists 7 nodes and 7 directed links. The nodes is a source node, whereas two nodes T1 and T2 are desti-nation nodes. In this network, against two nodes v1 and v4,channels are branch by two nodes v2 and v3.

By considering that which algorithm be used, the inter-mediate vertexes v1, v2, v3 and v4 can be unicasting ormulticasting vertexes. We assume that amount of message

Fig. 2 A sample networks witha source node s and twodestination nodes T1 and T2


Table 1 Comparing of DPST, OMMR, DOMMR and DPST algorithms based on the distribution metrics according to Fig. 3

Algorithm TCoB τ DR NoNU dup φ NoLU DDC DC′

DPST 28 18 3.33 1 4 5 6 8.4 5 4

OMMR 25 12 5 1 5 5 5 5 5 4.16

DOMMR 26 12 5 2 5 5 7 5.2 2.5 2.88

NLD-MMR 26.5 12 5 3 5 5 7 5.3 1.66 2.65

Fig. 3 Applying DPST, OMMR, D-OMMR and NLD-MMR algo-rithms to Fig. 1. (a) DPST: a traditional tree, (b) OMMR: an optimalQoS tree, (c) D-OMMR: an optimal MMCP, (d) NLD-MMR: an opti-mal DMMCP

to be transmitted from s to each destination t1 and t2 isσ = 30. To simplify notation, we will assume e1 = (s, v1),e2 = (v1, v2), e3 = (v2, v4), e4 = (v4, t1), e5 = (v2, v3),e6 = (v3, t1), e7 = (v3, t2). In Fig. 2(a), each edge ei ∈ E,i = 1,2, . . . ,7, and also each node v ∈ V , has been labeledby (b, d) and [Capacity(v), c(v)] respectively.

Using WinQSB software tool, we can apply DPST algo-rithm to Fig. 2 and obtain the tree illustrated in Fig. 3(a) con-taining the solution: f1 = 5, f2 = 5, f3 = 5, f4 = 5, f5 = 4,f6 = 0, f7 = 4, dup1 = 0, dup2 = 4, dup3 = 0, τ = 18,φ = 5. This tree is not optimal and transmits σ = 30 unitesof data from s to t1 and t2 at least within τ = 18 time unites.

Figure 3(a) shows that only node v2 is a multicasting nodeand can duplicate arrival packets.

By applying OMMR algorithm to Fig. 2, we can obtainanother tree which has been illustrated in Fig. 3(b) con-taining the optimal solution (φ,f,dup) as: f1 = 5, f2 = 5,f3 = 0, f4 = 0, f5 = 5, f6 = 5, f7 = 5, dup1 = 0, dup2 = 0,dup3 = 5, τ = 12, φ = 5. Because this tree is optimally ob-tained, we call it as Optimal tree. This optimal tree differswith minimum spanning tree, but supports QoS parametersand transmits σ = 30 unites of data from s to t1 and t2 atmost within τ = 12 time unites. Optimal tree contains mul-ticast node v3 and as compared to DPST, reduces total timeof message transmission.

By applying DOMMR algorithm to Fig. 2, we can in-crease the number of nodes and link used as shown inFig. 3(c). This means that DOMMR can distribute traf-fic, while this distribution is arisen by increasing the num-ber of links used. DOMMR produces the optimal solution(φ,f,dup) as: f1 = 5, f2 = 5, f3 = 1, f4 = 1, f5 = 5,f6 = 4, f7 = 5, dup1 = 0, dup2 = 1, dup3 = 4, τ = 12,φ = 5. This optimal solution contains two multicast nodesv2 and v3.

Now, we apply algorithm NLD-MMR to Fig. 2 and pro-duce Fig. 3(d) containing the optimal solution (φ,f,dup)

as: f1 = 5, f2 = 5.5, f3 = 1, f4 = 1, f5 = 5, f6 = 4, f7 = 5,dup1 = 0.5, dup2 = 0.5, dup3 = 4, τ = 12, φ = 5. This op-timal solution shows that node v1 can duplicate 0.5 unit ofdata per time unit instead of the node v2 and this means thatNLD-MMR can transform unicasting node v1 to a multi-casting node. As shown in this figure, NDL-MMR increasesNoLU, NoNU and traffic distribution.

Table 1 presents the metrics numerical result for DPST,OMMR, DOMMR and NLD-MMR algorithms. Table 1shows that NLD-MMR increases traffic distribution muchmore than the other algorithms and has acceptable networkefficiency.

5.3 Changing unicasting nodes to multicasting nodes forincreasing network efficiency

Now we consider sample network shown in Fig. 4. In thisnetwork, OMMR algorithm cannot formulate the flows fi

and duplication capacities dupi , because OMMR only candetermine node v3 as a multicasting node while v3 is not


Fig. 4 A sample networks witha source node s and threedestination nodes t1, t2 and t3

a multicasting node. In fact node v3 cannot duplicate anypackets, because its duplication capacity, [Capacity = 0,1],is equal to zero and is not capable for duplicating packets.Note that, two conditions (6) and (7) in M-MDF(τ ) causethat OMMR considers two nodes v1 and v2 as incapableunicasting nodes. Consequently, two nodes v1 and v2 can-not duplicate any packets by themselves or instead of inca-pable multicasting node v3. This means that OMMR cannottransmit any data across the network.

Similar to OMMR, by the same reasons, DOMMR can-not transmit any data across the network either. By apply-ing two algorithms OMMR and DOMMR to Fig. 4, we willobtain the optimal solution (φ,f,dup) which is the sameas zero: f1 = 0, f2 = 0, f3 = 0, f4 = 0, f5 = 0, f6 = 0,f7 = 0, f8 = 0, dup1 = 0, dup2 = 0, dup3 =, τ = 0, φ = 0.The optimal solution (φ,f,dup) shows that all of the vari-ables fi , dupi and φ are equal to zero and no data can betransmitted by this solution. Note that there is one importantdifference between two algorithms OMMR and DOMMRand it is that either algorithm produces an optimal solution,but DOMMR’s solution distributes traffic and as comparedto OMMR’s solution contains more nodes and links used.

For transmitting data across the Fig. 4, we can employNLD-MMR algorithm, because it can transform incapableunicasting nodes to capable multicasting node and can dis-tribute nodes and links used optimally. By applying NLD-MMR to the network we can transform one or both of nodesv1 and v2 to multicasting node as shown in Fig. 5. Therefore,three different optimal solutions will be obtained as the fol-lowing:

State 1. Transforming v1 to a capable multicasting node:Using conditions (11) through (14) of NLDM-MDF(τ ) wecan decide which one of two incapable nodes v1 and v2 mustbe changed to a capable multicasting node. In this case weselect only node v1 and obtain optimal solution (φ,f,dup)

shown in Fig. 5(a) as: f1 = 3.66, f2 = 0, f3 = 5, f4 = 6,f5 = 5, f6 = 3.66, f7 = 3.66, f8 = 3.66, dup1 = 7.33,dup2 = 0, dup3 = 0, τ = 16, φ = 3.66.

State 2. Transforming v2 to a capable multicasting node:by changing coefficients and values in conditions (11)

Fig. 5 Applying NLD-MMR to Fig. 4 and transforming unicast nodesto multicasting nodes

through (14) we can select only node v2 as capable multicas-ting node and obtain optimal solution (φ,f,dup) shown inFig. 5(b) as: f1 = 2, f2 = 0, f3 = 2, f4 = 0, f5 = 6, f6 = 2,f7 = 2, f8 = 2, dup1 = 0, dup2 = 4, dup3 = 0, dup2

11 = 2,dup2

21 = 2, dup231 = 0, τ = 26, φ = 2.

State 3. Transforming v1 and v2 to a capable multicastingnode: by enforcing condition (14) for two nodes v1 and v2

we can transform them to two capable multicasting nodesand obtain optimal solution (φ,f,dup) shown in Fig. 5(b)as: f1 = 4, f2 = 0, f3 = 2, f4 = 6, f5 = 6, f6 = 4, f7 = 4,f8 = 4, dup1 = 4, dup2 = 4, dup3 = 0, dup2

21 = 2, dup231 =

2, dup112 = 2, dup1

22 = 2, τ = 15, φ = 4.Table 2 shows values of computational metrics for three

states. This table shows that NLD-MMR algorithm presentsvarious optimal solutions for data transmission and estab-


Table 2 Applying NLD-MMR algorithm to Fig. 4 and obtaining Metrics computational result for three different states shown in Fig. 5

Algorithm TCoB τ DR NoNU dup φ NoLU DDC DC′

State 1 30.64 16 2.50 1 7.33 3.66 7 12.256 7.33 3.83

State 2 16 26 1.54 1 4 2 6 10.400 4 2.28

State 3 30 15 2.67 2 8 4 7 11.250 4 3.33

Fig. 6 A sample networkcontaining 60 nodes and 120edges

lishes different changeable configurations for using networkresources.

5.4 Data transmission sessions and algorithms behavior

In this section we consider network topology shownin Fig. 6. This figure contains 60 nodes and 120 edges. Weassign one of these values {6,7,8} as available bandwidthto each edge randomly, and suppose each node’s duplica-tion capacity is equal to 8 data units per time unit. Also,queuing delay of each node is the same as one of these val-ues {1,2,3} and propagation delay of each edge belongs to{2,3,4}.

5.4.1 Synchronous sessions and network services

Now, we consider some synchronous sessions which everysession contains a source node and five destinations nodessupposing that these source and destination nodes are ran-domly selected. It is important that these sessions requiredata transmission in the same time and we want to knowthat how MST, OMMR, DOMMR and NLD-MMR algo-rithms serve these sessions. These algorithms serve as thefollowing:

• MST: Constructs traditional trees.• OMMR: Constructs trees, optimal trees (Op-Tree) and

MMCPs.• DOMMR: Constructs MMCPs and distributed MMCPs.

• NLD-MMR: Constructs MMCPs and DMMCPs.

Because all of the solutions which are produced byMST can be produced by OMMR, we survey only OMMR,DOMMR and NLD-MMR algorithms. We have appliedeach algorithm to Fig. 6 separately and obtain the pre-sented result in Fig. 7. This figure shows that two algorithmsOMMR and DOMMR can only serve four synchronous datatransmission sessions and cannot serve other sessions suchas sessions 5, 6, and 7, But NLD-MMR can serve seven syn-chronous sessions as shown in Fig. 7. Three columns DR,DC, and of Fig. 7 have been illustrated in Figs. 8, 9, 10respectively, and these figures show that NLD-MR is moreefficient than the other algorithms.

5.4.2 Periodic sessions and algorithms’ behavior

It is known that a session can be served and terminated byan algorithm and then, a new session will be started. Sup-pose each session contains at most four or five destinationnodes. Let Algorithms OMMR, DOMMR and NLD-MMRcan transmit data within 200 time units. It is important thatwe know how many data units can each algorithm trans-mit within 200 time units? We apply these algorithms toFig. 6 and obtain result shown in Fig. 11. Two Fig. 11(a) andFig. 11(b) show that NLD-MMR as compared to OMMRand DOMMR increases the number of data units deliveryand sessions.


Fig. 7 Comparison ofdistribution metrics for threealgorithms OMMR, DOMMRand NLD-MMR

Fig. 8 Comparison of DR for three algorithms OMMR, DOMMR andNLD-MMR

Fig. 9 Comparison of DC for three algorithms OMMR, DOMMR andNLD-MMR

5.5 Simulation

Now, we simulate the implementation of OMMR, DOMMRand NLD-OMMR algorithms using the OpNet network sim-ulation toolkit. Our implementation builds on a Campus

Fig. 10 Comparison of network efficiency for three algorithmsOMMR, DOMMR and NLD-MMR

Network provided by OpNet, to generate an Ethernet do-main and create suitable test networks. The geographicalsize of the our Campus Network is determined by X span(10 Km) and Y span (10 Km).

This Ethernet implementation has three components, twoof which implement the profile configuration and Applica-tion configuration, while the third component is the topol-ogy of network. Application configuration specifies the dif-ferent tier names used in the network model and also speci-fies applications using available application types. The pro-file configuration can be used to create user profiles. Ourtest network has been shown in Fig. 12. This figure includes8 ethernet16-bridge nodes, 52 ethernet-station nodes whichhave been categorized in 6 subnet, 6 server node which everyserver node belongs to a subnet, 14 hub which have been cat-egorized in 6 subnet, 6 subnet, an Application Config nodeand a Profile Config node. Profile Config node consists fol-lowing applications: Data Base Access, File Transfer, VideoConferencing, Web and Email. All links are 10BaseT du-


Fig. 11 Comparison of data unit delivery and sessions for three algorithms OMMR, DOMMR and NLD-MMR

plex link to represents an Ethernet connection operating at10 mega bit per second (Mbps). The test network must trans-mit data from the source node to the destination nodes asshown in Fig. 12. Corresponding to each algorithm, we willproduce a scenario using OpNet. The Scenario of each algo-rithm is configured based on the solution of that algorithm.Note that each scenario implements the corresponding al-gorithm. Finally, by running OpNet simulation we can ob-tain the simulated results on the implementation of the algo-rithms. The results are presented as global statistics. We se-lect a global statistics Traffic received (packet/min) to com-pare the scenarios. Figure 13 shows these results.

In Fig. 13, the duration of simulation is 180 minutes. Inthese figure the horizontal axes present duration time andvertical axes indicate Ethernet traffic received (packet/min).This figure shows that the proposed algorithm (NLD-MMR)has the stable maximum traffic received as compared toother algorithms.

5.6 Network security

For a long time there has been a research interest to achieveand provide suitable levels of security in networks. Theselevels include flexibility, extensibility and stability in net-work elements containing resources, services and user datacommunications. These levels of security in computer andnetwork systems address three requirements [14]:

1. Secrecy: Requires that the information in computer sys-tem only be accessible for reading by authorized parties.

2. Integrity: Requires that computer system assets can bemodified only by authorized parties.

3. Availability: Requires that computer system assets areavailable to authorized parties.

For a data transmission from a source node to a destinationnode, as depicted in Fig. 14, the types of attacks or networkproblems such as disconnected links and etc., can threatentransmission security and are categorized as the following:Interruption, Interception, Modification, Fabrication.

Routing and switching are two important services whichcan be threatened by attacks and network problems such asflow measurement, traffic monitoring, penetrating to routedpaths, cracking message authentication, congestion, packetloss and etc. There are many algorithms and methods [1–3,9, 10, 12, 18] for reinforcing and increasing network securitycontaining nodes and links mechanism which can improvethe following security characteristics:

1. Using nodes having sufficient resources. This can guar-antee more stability and can reduce servicing weak-nesses.

2. Increasing relations between intermediate nodes andsharing their services and functions. This can reinforcerelations between nodes and consequently can overcomemore attacks.

3. Changing tasks and services of the nodes. Note that fixedand invariable tasks and services are more defencelessand fragile.

4. Changing and displacing path. This can change trafficand routes, consequently, can confuses attackers.

5. Differentiating services and resources. Because differen-tiated services and resources can limit and despair attack-ers.

We claim that our algorithm, NLD-MMR, as compare toother algorithms, DPST, OMMR and DOMMR, can supportabove characteristics properly and satisfactorily. This claim


Fig. 12 Simulating of three algorithms OMMR, DOMMR and NLD-MMR using network simulator OpNet

Fig. 13 Comparing of threealgorithms OMMR, DOMMRand NLD-MMR based on trafficreceived (packet/min) usingnetwork simulator OpNet


Fig. 14 Security threats

can be proved based on the following reasons for each char-acteristic:

• Characteristic 1: NLD-MMR uses capable nodes insteadof incapable nodes and this means that NLD-MMR algo-rithm uses nodes having sufficient resources.

• Characteristics 2 and 3: NLD-MMR distributes packetduplication tasks and traffic functions among nodes andlinks. For achieving this, NLD-MMR computes capacitiesand services of the nodes and links and also interchangesthese capacities and services between nodes and links.

• Characteristics 4 and 5: NLD-MMR changes paths andguarantees differentiated services. NLD-MMR decom-poses flows as separated sub-flows into virtual channelsand each flow can be treated and leaded by different andvarious services based on the QoS characteristics.

Note that the other algorithms, DPST, OMMR andDOMMR, cannot support all above characteristics and thismeans that NLD-MMR can provide more security for net-work and services.

6 Conclusion

In this paper we have presented a new algorithm Nodes-Links Distribution Multicast Multichannel Path (NLD-MMR) which increases network efficiency and traffic dis-tribution. The new algorithm obtains this by transformingunicasting nodes to multicasting nodes and increasing thenumber of nodes and links used. Our algorithm unlike theother available algorithms such as MST, OSPF, OMMR andDOMMR, can establish all types of the paths Trees, Opti-mal QoS Trees, MMCPs and DMMCPs. The simulation re-sult show that our algorithm is more efficient than the otheravailable algorithms.

Future work

1. Mathematical studding and presenting theorematic proofsfor relation among trees, optimal trees, MMCPs andDMMCPs. This study tries to transform linear program-ming model of DMMCP to a model in graph theory.

2. Dynamic NLD-MMR: It is important to consider thatduring a session, how NLD-MMR behaves when somedestination nodes be disconnected from the session orsome new destination nodes be added to the session.

3. Virtual channel NLD-MMR: We would like to investi-gate how NLD-MMR can support virtual channel. It isknown that virtual channels are widely used in new net-works and technology such as MPLS, ATM and etc.

4. Multi Source NLD-MMR: This future work will try toenable NLD-MMR to transmit data from more than onesource nodes to more than one destination nodes.

5. Sensor Network NLD-MMR: By changing and match-ing the linear programming model in NLD-MMR, it cansupport data communication in sensor networks.

6. Optimal Multicast Multichannel Switching: Switchinglike routing can be formulated by a new linear program-ming model which maximizes the number of switchedpackets in switch and decreases queuing delay in switchbuffers.

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Ayaz Isazadeh received the B.Sc.degree in Mathematics from Uni-versity of Tabriz in 1971, the M.S.E.degree in Electrical Engineeringand Computer Science from Prince-ton University in 1978, and thePh.D. degree in Computing and In-formation Science from Queen’sUniversity in 1996. Before return-ing to graduate studies in 1992, heworked for several years, as a Mem-ber of Technical Staff, at AT&TBell Laboratories on telecommuni-cation and manufacturing softwaresystems. Dr. Isazadeh is currently

an Associate Professor in the Department of Computer Science at Uni-versity of Tabriz and has been a founding member of this departmentsince 1999. His current research interests include Software Engineer-ing, Formal Methods, and Information and Communication Technolo-gies. He is a member of Mathematical Society of Iran, a member ofComputer Society of Iran, and had been a senior member of IEEE until2002!

Mohsen Heydarian received theB.Sc. degree in Applied Mathemat-ics from University of Tabriz in1999, the M.S.E. degree in AppliedMathematics (Numerical Analysis)from Tabriz University in 2002,and the Ph.D. degree in AppliedMathematics and Computer Sci-ence from Tabriz University in2010.Dr. Heydarian is currently an As-sistance Professor in the Depart-ment of Information Technologyand Computer Engineering atAzarbaijan University of Tarbiat

Moallem and has been a founding member of this department since2007. His current research interests include Communication Technolo-gies, Mathematical Optimization, and Information Technology.

distributed multicast multichannel paths

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