IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 12, DECEMBER 2014 2121

Self-Similar Traffic End-to-End Delay MinimizationMultipath Routing Algorithm

Donghyuk Han and Jong-Moon Chung, Senior Member, IEEE

Abstract—In this letter, a multipath routing algorithm to reduceend-to-end delay and loss rate is proposed. The proposed mul-tipath routing algorithm, self-similar delay minimization (SDM),estimates network delay of realistic network traffic by analyz-ing self-similar parameters based on fractional Brownian motion(fBm) traffic models, computes the optimal number of paths foraverage delay minimization, and also derives an optimal trafficdistribution ratio for multipath routing using a cooperative gamealgorithm. Simulation results show that the average end-to-enddelay and loss rate performance can be significantly improvedwhen using SDM compared to using the average delay minimi-sation (ADM) or maximum delay minimisation (MDM) multipathrouting algorithms.

Index Terms—Fractional Brownian motion (fBm), multipathrouting, self-similar.

I. INTRODUCTION

MULTIPATH routing has been proposed as an efficientmechanism to use network resources by providing mul-

tiple paths between the source and destination. For examplein [1], two multipath routing algorithms, average delay min-imisation (ADM) and maximum delay minimisation (MDM)are introduced. ADM and MDM find the optimal proportionof traffic load for multiple routing paths, using minimizationof average delay or maximum delay, respectively, based on theM/M/1 queueing model for delay characteristic estimation. Amajor problem of existing multipath routing algorithms is thelack of consideration of realistic traffic characteristics. Due tothe self-similar nature of multiplexed network traffic [2]–[4],the average delay is much larger than the delay predicted bytraditional queueing analysis. As a result, network performance(e.g., delay and loss rate) degrades significantly when an inad-equate traffic model is applied as the network control scheme[2]. To cope with this problem, this paper proposes a multi-path routing algorithm called self-similar delay minimization(SDM). SDM takes the self-similar characteristic of networktraffic into account using the fractional Brownian motion (fBm)based traffic estimates and provides reliable end-to-end Qualityof Service (QoS) support by adjusting the number of routingpaths and the traffic distribution ratio. In addition, SDM adoptsdisjoint multipath routing to minimize the interference among

Manuscript received May 1, 2014; revised September 23, 2014; acceptedSeptember 23, 2014. Date of publication October 13, 2014; date of currentversion December 8, 2014. This work was supported by the Ministry of Science,ICT and Future Planning (MSIP) under the Information Technology ResearchCenter (ITRC) support program (NIPA-2014-H0301-14-1015) supervised bythe National IT Industry Promotion Agency (NIPA), and the ICT R&D pro-gram of MSIP/IITP [13-911-05-002, Access Network Control Techniques forVarious IoT Services], Republic of Korea. The associate editor coordinating thereview of this paper and approving it for publication was I.-R. Chen.

The authors are with the School of Electrical and Electronic Engineering,Yonsei University, Seoul 120-749, Korea (e-mail: jmc@yonsei.ac.kr).

Digital Object Identifier 10.1109/LCOMM.2014.2362747

multipaths and avoid shared bottleneck problems. By virtue ofits disjoint nature, SDM fully utilizes network resources andenables easier congestion control [5].

II. NETWORK MODEL

In this letter, the network is modeled as a graph G = (V,E),where V is the set of nodes and E is the set of links. Eachlink (u, v) from node u to v has a non-negative cost. For aconnection request from the source node to the destinationnode, P f = {P1, · · · , Pn} is the set of feasible routing paths,where the ith path Pi is the set of independent links from sourceto destination. The network model assumes that self-similartraffic with an average of λ bps is flowing into the network. Theinflow traffic is modeled as A(t) = λt+

√aλBH(t), where

A(t) represents the number of bits in the time interval [0, t),a is the variance coefficient, and BH(t) is the fBm process,in which H is the Hurst parameter [2]. The characteristics ofself-similar fBm traffic models are analyzed in [6], where theoverflow probability εa can be obtained from the supremumof the server workload, 2σ−2

i {(Cu,v − λ)/H}2H(1−H)2H−2,and the decay rate b2−2H

u,v , in which Cu,v bps is the capacity oflink (u, v) and σi is the standard deviation of the traffic on theith path. The corresponding statistical bound of the steady statebacklog queue length on link (u, v) of the ith routing path bu,vcan be modeled as in (1), where ri is the traffic distribution ratioon the ith path.

bu,v=

(− 1

2σ2i ln εa

(Cu,v−riλ

H

)2H(1

1−H

)2−2H) 1

2H−2

(1)

III. MULTIPATH ROUTING USING THE FBM PROCESS

A. Traffic Distribution of SDM

In the multipath routing algorithm, the target traffic flow isassumed to be distributed by the ratio r = {r1, r2, . . . , rn} overn possible routes, resulting in an average traffic inflow rateof riλ for the ith path. The proposed SDM algorithm aims todetermine the distribution ratio so that the end-to-end networkdelay can be minimized. SDM aims to minimize the averagepath delay, which is achieved by minimizing the sum of pathdelays when n is predefined. Minimizing the average delayis very effective because it enables formulation into a convexoptimization problem [7].

Minimizen∑

i=1

∑(u,v)∈Pi

bu,v (2)

1089-7798 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2122 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 12, DECEMBER 2014

Subject to∑

i∈{1,···,n}ri = 1, 0 ≤ ri ≤ 1,

and riλ ≤ Cu,v, ∀ i ∈ {1, 2, · · · , n} (3)

The problem of distributing traffic can be solved by treatingthe problem as a cooperative game, where each route is a gameplayer and a coalition is the combination of routes. As theproblem is mapped to a N -person game [8], the characteristicfunction v(S) for the coalition S is proposed to be

v(S) =

⎛⎜⎝∑

i∈S∑

(u,v)∈Piα(

Cu,v−riλH

) −2H2H−2

∑k∈A

∑(x,y)∈Pk

α(

Cx,y−rkλH

) −2H2H−2

⎞⎟⎠ (4)

whereA is the grand coalition set andα=(−2σ2 ln εa)1/(2H−2)/

(1−H), α > 0. The characteristic function represents theworth of coalition when the players in S cooperate with eachother regarding the expected sum of the steady state backlog.v(S) is proposed based on the self-similar traffic characteristicin (1). The initial value of ri is 1/n and σi = σ for ∀ i (i.e.,traffic is uniformly distributed), then the optimal distributionratio is obtained by the optimization technique which con-cerns the link capacity and the traffic characteristics (i.e., λand H). Applying these relations, (1) can be expressed asα−1((Cu,v − riλ)/H)2H/(2H−2). As (1) describes the statisti-cal queue length as well as the link delay, the reciprocal value

is applied as α((Cu,v − riλ)/H)−2H2H−2 in v(S) to represent the

profit in terms of delay. The path delay parameter of Pi isobtained by summing up all the link delay parameters in thepath (i.e.,

∑(u,v)∈Pi

α((Cu,v − riλ)/H)−2H/(2H−2)) which isdivided by the sum of delay parameters over all feasible pathsfor normalization. It needs to be checked if the proposedcharacteristic function v(S) meets the requirements for a char-acteristic function of a N -person game, which is presented inthe following Lemma 1.

Lemma 1: The proposed characteristic function v(S) of (4)meets the two requirements:

1) v(φ) = 0, where φ represents the null set.2) v(S1 ∪ S2) ≥ v(S1) + v(S2), if S1 ∩ S2 = φ �

Proof: Since v(φ) = 0 refers to no routes being selectedfor packet delivery, no capacity exists to calculate the

delay. Defining ψ(A) =∑

k∈A

(∑(x,y)∈Pk

α((Cx,y − rkλ)/

H)−2H2H−2

), v(φ) can be expressed as v(φ) = 0/ψ(A), which

results in 0. When the number of paths in the coalitionsare defined as m1 and m2, respectively for S1 and S2, thecharacteristic function of the coalition S1 ∪ S2 satisfies v(S1 ∪

S2)=∑

i∈S1

(∑(u,v)∈Pi

α(

Cu,v−λ/(m1+m2)H

) −2H2H−2

)/ψ(A)+∑

j∈S2

(∑(u′,v′)∈Pj

α(Cu′,v′−λ/(m1+m2)

H

) −2H2H−2

)/ψ(A), and

v(S1) + v(S2) can be obtained as v(S1) + v(S2) =∑i∈S1

(∑(u,v)∈Pi

α(

Cu,v − λ / m1

H

) −2H2H−2

)/ ψ(A )+∑

j∈S2

(∑(u′,v′)∈Pj

α(

Cu′,v′−λ/m2

H

) −2H2H−2

)/ψ(A). As it is

assumed that 0.5 ≤ H < 1 and riλ ≤ Cu,v , ∀ i ∈ {1, 2, · · · , n},

we obtain (Cu,v − λ/(m1 +m2)) ≥ (Cu,v − λ/m1), which

leads to∑

i∈S1

(∑(u,v)∈Pi

α(

Cu,v−λ/(m1+m2)H

) −2H2H−2

)≥∑

i∈S1

(∑(u,v)∈Pi

α(

Cu,v−λ/m1

H

) −2H2H−2

). As the same

relation applies to coalition S2, it can be concluded thatv(S1 ∪ S2) ≥ v(S1) + v(S2). �

To obtain the optimal distribution ratio of the traffic con-sidering each player’s profit, the Shapley value method isused. The Shapley value function ϕi(v) is the solution for aN -person cooperative game, which is a payoff correspondingto the contribution of the ith player.

ϕi(v) =∑

S⊂A,i∈A

(|S| − 1)! (n− |S|)!n!

(v(S)− v (S− {i}))

(5)

The SDM distribution ratio r∗ = {r∗1, . . . , r∗n} is obtained bycalculating the Shapley value of (5), that is, r∗i = ϕi(v). Usingthis method, the total expected steady state backlog of multiplerouting paths is minimized.

B. Number of Multipath Selection

In this section, the optimal number of paths n∗ is derived tominimize the end-to-end delay of multipath self-similar traffic.As mentioned in Section III-A, a link delay of self-similartraffic can be represented as α−1((Cu,v − riλ)/H)2H/(2H−2).When it is assumed that the traffic is distributed overn routing paths (where a uniform distribution ratio isapplied), the link delay over n paths can be represented asd(n) =

∑ni=1

∑(u,v)∈Pi

α−1HH/(1−H)(Cu,v − λ/n)H/(H−1).For an average path length l (represented in number ofhops) and an average link capacity C for the n routingpaths, the sum of average end-to-end link delays d(n) canbe approximated by the sum of average link delays d(n) asd(n) = nlα−1HH/(1−H)(C − λ/n)H/(H−1). Determinationof the number of paths must precede the Shapley valueoptimization process, which considers major influencefactors of end-to-end link delay. The simplified model d(n)enables formulation of a convex optimization problem, wherethe degree of margin between d(n) and d(n) is analyzedin Lemma 2.

Lemma 2: The difference between d(n) and d(n) is boundedby a finite upper bound. �

Proof: Based on l and C, as H/(H − 1) < 0,d(n) ≤ nlα−1HH/(1−H)(Cmin − λ/n)H/(H−1) and d(n) ≥nlα−1HH/(1−H)(Cmax − λ/n)H/(H−1), where Cmin andCmax are the minimum and the maximum link capacity ofthe network, respectively. Thus, d(n)− d(n) is bounded by

nlα−1HH

1−H

((Cmin − λ/n)

HH−1 − (Cmax − λ/n)

HH−1

). �

As the end-to-end delay over n paths can be approximatedby d(n), the optimal number of multiple paths that minimizesthe sum of end-to-end link delays can be obtained by solvinga minimization problem of the function d(n) if d(n) is convex.When it is assumed that the number of paths n is larger thanλ/C, the function d(n) is convex as shown in Lemma 3.

HAN AND CHUNG: SELF-SIMILAR TRAFFIC END-TO-END DELAY MINIMIZATION MULTIPATH ROUTING ALGORITHM 2123

Lemma 3: The function d(n) is convex and satisfies theconvexity condition d′′(n) ≥ 0 for all n ≥ λ/C, where d′′(n)

is the second derivative function of d(n). �Proof: As the derivative function d′(n) can be

obtained as d′(n) =lα−1HH/(1−H)(C − λ/n)1/(H−1)(C +

λ(H − 1)−1n−1), the second derivative becomes d′′(n) =lα−1HH/(1−H)(C − λ/n)(2−H)/(H−1)(λ2H(H − 1)−2n−3).As n ≥ λ/C > 0 and 0.5 ≤ H < 1, (C − λ/n)(2−H)/(H−1) ≥0 and λ2H(H − 1)−2n−3 ≥ 0, which results in d′′(n) ≥ 0. �

As d(n) is convex for n ≥ λ/C, the optimal solution of thefollowing problem can be obtained from the derivative of d(n),d′(n) = (C − λ/n)1/(H−1)(C + λ(H − 1)−1n−1). Therefore,the optimal number of paths can be obtained from the followingstatement.

Minimize d(n) = nlα−1HH

1−H (C − λ/n)H

H−1 (6)

Subject to n ≥ λ/C (7)

As (C − λ/n)1/(H−1) ≥ 0, d′(n) = 0 for n which causes(C + λ(H − 1)−1n−1) = 0. Thus, the optimal solution is ob-tained when n = λ(1−H)−1/C, and the optimal number ofmultiple paths n∗ which minimizes the average link delay isobtained as (8), where �· is the ceiling function.

n∗ =

⌈λ

C(1−H)

⌉(8)

The routing update mechanism to re-calculate the distribu-tion ratio r∗ is defined so that the difference bound betweend(n) and d(n) doesn’t exceed a certain threshold level. Assum-ing the average traffic inflow rate λ has changed into λr, thebounds of ratio difference is obtained from Lemma 2 and therouting stability condition is obtained as

THl<

((Cmin−λr/n)

HH−1 −(Cmax−λr/n)

HH−1

)((Cmin−λ/n)

HH−1 −(Cmax−λ/n)

HH−1

) <THh

(9)

where THl is the low threshold and THh is the high thresh-old. Using the stability condition, SDM prevents performancedegradation due to being too sensitive or too insensitive oftraffic changes in the network.

C. Overall Operation and Pseudo Code

The pseudo code for SDM is used to update the routing con-figuration when the routing stability condition is not satisfied.In the first stage of the algorithm, the optimal number of routingpaths n∗ is obtained. The network information required for thecalculations is obtained in step 1, and the optimal number ofrouting paths is obtained by (8) in step 2. In the second stage,SDM uses Suurballe’s algorithm [9] to construct minimum costnode-disjoint routes. In step 3 the number of paths setup n isinitially set to 0. For each link (u, v) in G, link cost wu,v =1/Cu,v is assigned in step 4. If n < n∗, Suurballe’s algorithmrepeatedly runs through steps 5 through 8. After updating thenumber of paths n in step 6, Suurballe’s algorithm is used to

find a node-disjoint path in step 7. Suurballe’s algorithm firstadopts Dijkstra’s algorithm to find the minimum cost path.After the information of the path P

nis stored in the feasible

path set P f , the algorithm modifies the weights of edges,creating a residual graph Gt. The WHILE loop is terminatedwhen no disjoint path is found on Gt or n ≥ n∗ in step 8.Thus, the number of routing paths n becomes n when thenumber of total disjoint paths n is smaller than n∗, or wouldequal n∗ as stated in step 9. Although the SDM algorithm isapplied to Suurballe’s algorithm, the main idea of SDM holdsfor other disjoint multipath routing protocols. The third stage ofthe SDM algorithm optimizes the traffic distribution for the ndisjoint multipaths. To solve the SDM distribution ratio over nrouting paths, the Shapley value is calculated using (5) and theSDM distribution ratio r∗i for the ith routing path is obtainedas r∗i = ϕi(v) in step 10. Finally, the algorithm distributesthe information of all n disjoint paths P f , and the optimaldistribution ratio r∗ in step 11.

SDM Algorithm

Optimize the number of routing paths1. SCAN the network status and obtain the average link

capacity C and traffic characteristic parameters H and λ2. COMPUTE the optimal number of routing paths n∗

using (8)Construct disjoint multipath routes3. SET the number of paths found to 0 (i.e., n ← 0)4. SET wu,v = 1/Cu,v for each link (u, v) ∈ G5. WHILE disjoint path can be found and n < n∗

6. SET n ← n+ 17. DO Suurballe’s algorithm to find a disjoint path P

n,

which is added to P f (i.e., P f ← P f ∪ {Pn})

8. END WHILE9. SET the number of routing paths n ← min (n∗, n)Optimize the traffic distribution ratio10. COMPUTE the optimal distribution ratio r∗ using (5)

for each path in P f

11. RETURN P f and r∗

IV. PERFORMANCE ANALYSIS

To evaluate and validate the performance of SDM on mul-tihop networks, the average end-to-end delay and data lossrate are compared through MATLAB simulation based on anoptical United States backbone network shown in Fig. 1, whichconsists of 17 optical switches connected by 27 links [10]. EachOC-3 equivalent path has a 155.52 Mbps link rate. The targettraffic flows are generated at node 1 and are destined to node 8.

The performance is also examined over a random network,formed by 40 nodes that are produced by a uniform randomdistribution over a circular area. The transmission radius isassumed to be one third of the radius of the entire region, andthe link capacity is uniformly random within the range from 4to 8 Gbps. The multiplexed traffic inflow rate λ varies from 1 to

2124 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 12, DECEMBER 2014

Fig. 1. United States backbone network.

Fig. 2. End-to-end delay and loss rate performance comparison.

4 Gbps, and the Hurst parameter H is 0.8. The average end-to-end delay and the loss rate by applying random early detection(RED) is shown in Fig. 2.

As SDM distributes the inflow traffic into multiple routeswith a proper distribution rate, it outperforms MDM and ADMin terms of end-to-end delay as shown in Fig. 2(a). Averagingthe simulation results with λ ranging from 1 to 4 Gbps, theaverage end-to-end delay of SDM is only 51.40% (32.47%)

compared to when using MDM and 66.06% (45.80%) com-pared to when using ADM on the backbone (and the random)network. Fig. 2(b) shows the loss rate over 10,000 s. As SDMuses multiple paths to distribute packets adequately over thenetwork using fBm estimations, less packets are stacked inqueues, which results in a lower loss rate by switches execut-ing RED. Under various inflow rates, SDM achieves a lowerloss rate than the other algorithms and shows the same trendof performance gain on the backbone and random network.Averaging the simulation results with λ ranging from 1 to4 Gbps, the loss rate of SDM is reduced by 74.44% (77.08%)and 67.78% (68.21%) compared to when using MDM andADM, respectively, on the backbone network (and on therandom network).

V. CONCLUSION

This paper proposes a multipath routing algorithm SDMthat can reduce the average end-to-end delay and data lossrate of self-similar traffic. SDM considers the characteristicsof self-similar traffic and conducts multipath routing basedon a N -person game path selection and distribution scheme.None of the existing algorithms on multipath routing take self-similar traffic characteristics into consideration, and therefore,show deteriorated performances over actual high-speed back-bone networks. SDM constructs disjoint paths and minimizesinterference among routing paths, which makes it easier forcongestion control. Possible extensions of SDM can considerthe variation on traffic characteristics according to TCP con-gestion control and use of non-disjoint paths. However in somecases, non-disjoint paths may yield a better performance, butmay result in instability and would require more control.

REFERENCES

[1] S. M. Mostafavi, E. Hamadani, R. Kuehn, and R. Tafazolli, “Delay min-imisation in multipath routing using intelligent traffic distribution poli-cies,” IET Commun., vol. 5, no. 10, pp. 1405–1412, Jul. 2011.

[2] Z. Sahinoglu and S. Tekinay, “On multimedia networks: Self-similartraffic and network performance,” IEEE Commun. Mag., vol. 37, no. 1,pp. 48–52, Jan. 1999.

[3] J.-M. Chung, J.-H. Seol, T. Yeoum, S. Choi, and H. Lim, “Statisticaldelay control scheme for DiffServ networks with self-similar traffic,” IETElectron. Lett., vol. 44, no. 9, pp. 606–607, Apr. 2008.

[4] Z. Quan and J.-M. Chung, “Statistical admission control for real-timeservices under earliest deadline first scheduling,” Comput. Netw., vol. 48,no. 2, pp. 137–154, Jun. 2005.

[5] C. Raiciu, M. Handley, and D. Wischik, “Coupled Congestion Con-trol for Multipath Transport Protocols,” Fremont, CA, USA, RFC 6356,Oct. 2011.

[6] A. Rizk and M. Fidler, “Non-asymptotic end-to-end performance boundsfor networks with long range dependent fBm cross traffic,” Comput.Netw., vol. 56, no. 1, pp. 127–141, Jan. 2012.

[7] A. Ghosh, S. Ha, E. Crabbe, and J. Rexford, “Scalable multi-class traf-fic management in data center backbone networks,” IEEE J. Sel. AreasCommun., vol. 31, no. 12, pp. 2673–2684, Dec. 2013.

[8] L. Shapley, “A value for n-person games,” in Annals of MathematicsStudies, vol. 2. Princeton, NJ, USA: Princeton Univ. Press, 1953,pp. 307–317.

[9] J. W. Suurballe, “Disjoint paths in a network,” Networks, vol. 4, no. 2,pp. 125–145, 1974.

[10] M. C. Wendel, G. Scheets, and J.-M. Chung, “Mesh optical QoS andmulti-layer restoration,” in Proc. NFOEC, CA, CA, USA, 2001, vol. 3,pp. 916–924.