A New Multi-path Routing Methodology
Based on Logit Type Assignment
Yudai Honma
and Masaki Aida

Hideyuki Shimonishi
and Atsushi Iwata

Faculty of System Design
Tokyo Metropolitan University
Hino, Tokyo 191-0065
Email: (yudai, maida)@sd.tmu.ac.jp

NEC System Platforms Research Laboratories
Kawasaki, Kanagawa 211-8666
Email: h-shimonishi@cd.jp.nec.com
Email: a-iwata@ah.jp.nec.com

Abstract—We present a new multi-path routing methodology
called MLB-routing based on multinomial logit model, which iswell known as the random utility theory. The key concept of thestudy is to incorporate multiple paths from same origin to destination, and distribute packets followed by the multinomial logit
type probability. Since MLB-routing is pure multi-path routing,it reduce the severe convergence to same links and increases the
bandwidth utilization in the network. Compared to the existing
multi-path routing schemes that select pre-determined alternate
paths, the proposed method can dynamically distribute packets
to every possible paths and thus is more efficient than them.
Furthermore, it should be mentioned that this methodology could
be implemented as both link-state protocol and distance-vector
protocol. Therefore, it has enough affinity for present Internet
mechanism. Using simulations, we have also shown that thismethodology produces more efficient use of network and causes
significant improvements in end-to-end delays and jitter times.

I. I NTRODUCTION
Lots of routing mechanisms in the Internet network havebeen based on single-shortest path routing in various metrics.
This strategy seems rational because all packets on the networks should be delivered as soon as possible. However, it iscautious that some of same links are heavily used as a shortest
paths in different node pair. Hence, routing protocols based on
single-shortest path method cause the severe convergence of
packets to same links, and results to traffic congestion.
Traditionally, the main methodology to reduce traffic congestion is end-to-end control. It is well known that TCP
mechanism [10], which is the most commonly used in Internet,
has implemented end-to-end congestion controls. It basicallymeans to control the “quantity” of packets on the networks.
On the other hand, there are several idea to solve the traffic
congestion by controlling the “routing” methodology. For example, PB-routing protocol proposed by Basu et al. considers
the traffic potential fields on the network, and tries to avoid thecongested links by making a detour to an alternate route [3].
However, even if they consider link traffic (i.e. adopting thetraffic-aware routing), the fact of “single-shortest path” routing
is not changed at all. That is, same links would be often used,and some alternative links become new bottlenecks.
Meanwhile, the techniques of multi-path routing have been
focused as another methodology to ease traffic congestion. The
basic idea of multi-path routing is to split the packets withthe same origin and destination along multiple alternate paths.
Hence, multi-path routing encourages the effective utilizationof network resources. First, the number of packets per eachpath is decreased, so the emergence of bottleneck will be
prevented. In addition, multi-path routing means to use more
links than single routing, and it is favorable in terms of theload-balancing of traffics. Consequently, multi-path routing
contributes the various things, such as the improvement of
the quality of service (QoS), minimizing delay, the increaseof reliability of delivery, and so on.
It is well known that some protocols such as OSPF and BGP
implement equal-cost multi-path (ECMP) [7], [11]. However,there is less likelihood that multiple paths with equal distanceexist between each source-destination pair, which means thefull connectivity of the network is still not used for loadbalancing. Therefore, other multi-path routing methodologies,
which also use “near-optimal paths”, have been discussed in a
number of researches (see e.g. [2], [4], [8] and the references
therein). Although, to calculate the multi-path routing, it isnecessary to enumerate the multiple candidate paths to split.
Hence, lots of multi-path routing protocols adopted the ideaof the link-state protocols, and only a few researches are
implemented as distance-vector protocols [14], [15].
In this paper, we present a new multi-path routing methodology which is based on multinomial logit model (MNL).
We call this methodology “(multinomial) logit based routing”,
or “MLB-routing”. Since MLB-routing is pure multi-pathrouting, it is supposed to reduce the severe convergence tosame links and increases the bandwidth utilization in the
network. It also should be mentioned that this methodology
could be implemented as both link-state protocol and distancevector protocol. Therefore, it has enough affinity for present
Internet mechanism.
The key idea of MLB-routing is to determine the route ofpackets based on MNL. MNL is a classical model of random
utility theory, which is used to assess the individual’s choosing
behavior [12]. Since the behavior of “choosing something” isbrought on by various psychological factors, it is difficult toestimate exactly what to be selected. Hence, in random utility
theory, we estimate the “probability” to be chose toward toeach alternatives, and it is the most interesting point of thetheory.
MLB-routing has many advantages over traditional multipath routing system. There are enormous candidate paths
between same origin-destination pair. In lots of existing multipath routing protocol, nevertheless, some of paths are only
considered because of limitations of their algorithms. Meanwhile, it is noteworthy that MLB-routing aims to consider “allof paths”. Thus, it contributes to more effective utilization of
network resources. It seems impractical to consider all possible
paths in large scale networks. However, this strategy paradoxically decreases the computational complexity, and enables us
to implement MLB-routing as distance-vector protocol. This
discussion results from the equivalency between MNL andMarkov model in network flow [1].
Furthermore, owing to the property of MNL, in MLBrouting, the choice probability for each path decrease exponentially as the length of path becomes longer than shortest
one. It means that high utility paths are frequently selected
and less utility ones are not, which is a favorable structure wehope. Hence, we can avoid another problem that impracticalpaths are often used, because the probability for impractical
path becomes quite low value. We believe that the generalframework of MLB-routing could be adapted for various
situation through the careful arrangement of algorithm.
The rest of the paper is organized as follows. In the next
section, we describe the MLB-routing model in greater detailand clarify the structure of system analytically. This is followed by Sections 3 where we describe our implementation of
the routing and evaluate its performance. Finally, we conclude
and mention about future works in Section 4.
II. T HE L OGIT BASED ROUTING M ETHODOLOGY
In this section, We present the basic idea of MLB-routing
methodology. We first explain the brief summary of multinomial logit model (MNL) which is well known in random
utility theory. We then describe how to apply the MNL to therouting system, and formulate MLB-routing system with analgorism to calculate the routing effectively.

In view of these facts, let us assume the utility Uj when an
individual choose the alternative j as follows:
Uj = Vj + εj ,(2)where Vj is a function estimated by a researcher, and εj is arandom variable. Vj depends on the observed characteristics ofthe alternative j, and generally Uj = Vj . Hence we introduce
the term εj .
The definition stated above indicates that each utility Uj isa random variable, so it is impossible to identify one and only
alternative the individual would choose. This fact implies that
we should analyze the choice behavior as a stochastic event.
Now, choosing the alternative j means that Uj is the highest
value. Therefore, we can formulate the probability Pj that theindividual choose the alternative j as follows:
Pj = Pr [Uj > Uj  (for ∀j  ∈ N, j  = j)] .

(3)

Specific form of (3) is identified when we define the distribution of random variable εj . the procedure explained above isthe formulation of random utility model.
In the MNL, it is assumed that the random factors of utilities
ε1 , ε2 , · · · , εJ are independent random variables from theidentical Gumbel distribution, where the Gumbel distribution
with parameters (α, γ) isc.d.f. F () = exp [− exp [−γ ( − α)]] ,(4)
p.d.f. f () = γ exp [−γ ( − α)] exp [− exp [−γ ( − α)]] .
(5)
In this case, we can obtain the probability Pj as follows:
exp [γVj ].
Pj = 
exp [γVj ]

(6)

j∈N

This is the probability derived by the MNL. As understoodby (6), when γ = ∞, only the best alternative is used, and if
γ = 0, then all alternatives are used equally likely.
B. MLB-Routing Assignment

A. Multinomial Logit Model
Let us suppose a situation that an individual is going
to choose an alternative j from an alternative set N ∈{1, 2, · · · , J}. Furthermore, we assume that the individual
recognizes the following utility
def.

Uj = [utility when he chooses the alternative j] ,(1)

and the individual is supposed to choose the alternative tomaximize his utility. Our first objective is to analyze whichalternative the individual would choose.
If we “precisely” know his utility Uj , then we can figureout which alternative the individual would choose. However,
it is impossible to figure out the individual’s utility without
the least difference, because utility Uj would be affected tovarious elements such as his character, preference, and so on.
Therefore, it is undeniable that some errors are mixed in “theutility estimated by researchers”.

As described above, MNL is not only simple but alsoquite persuasive model, so it is used in various fields. One
typical example is a routing behavior in transportation networkmodel. The general objective of transportation network model
is to describe transportation flows in road networks, andtherefore it is an essential issue to predict driver’s routing
behavior [9]. MNL is also utilized to describe such routing
behavior, and the number of researches have been discussed
for effective calculation. We incorporate their researches andpropose MLB-routing in the following.
In order to describe the MLB-routing protocol, we first
define a system model assumed in this study. We assume a
network of nodes connected by directed links as a directed
graph G = (N, E). The set of nodes in the networks is labeledby the N , and the set of edges E. Furthermore, eij index adirected edge from vertex i to vertex j with cost metric cij .
For the rest of this paper, we shall use the terms nodes (links)

and vertices (edges) interchangeably. Each node v can act asa traffic source and /or sink. Finally, every node v has a setof Z (v) neighbors denoted by nbr (v).
Now we explain the concept of MLB-routing protocol. let
us consider any packet p at a origin node o going to destination
node d. There are not only the shortest path but also several
way of paths to reach the destination d, and we try to use“all” of paths (theoretically). However, it doesn’t mean to use
all route equally; naturally the shortest path should be mostoften used, and as the longer the cost of path becomes, theless the path used. To incorporate this idea, let us define theprobability that packet p use the rth path Prod as follows:
exp −γCrodod
Pr = 
(7).
exp −γCrodr∈Φod

The above expression comes from the definition of MNL,
where Crod denotes the costs for the rth path between nodeso and d, and Φod is the alternative set of paths from nodeo to d. In this study, the each pass cost is calculated by thesummation of link cost, namely
def.
Crod = cov1 +Λ−1

cvl vl+1 + cvΛ d

(8)

l=1

where rth pass is o → v1 → v2 → · · · → vΛ → d.
Now let us discuss how to define the alternative set Φod . The
natural definition of the alternative set is “simple path”; thepath which does not pass through same link more than once.
In this study, however, we do not introduce any restriction, andconsider all of path including “cyclic path”. This strategy has
a possibility that the definition generates very long detours
which should not be assigned. Furthermore, it seems to bedifficult to calculate the probability in a large scale network.
On the contrary, this definition is made from the viewpointof the computational feasibility. As shown in later sections,this definition enables us to calculate the stochastic assignment
in large scale networks. Moreover, the definition is useful insome iterative algorithms for solving the assignment whichmeans to expand to distance-vector protocol. From next subsection, we discuss the structure of MLB-routing system, andeffective algorisms for calculation.
C. The Structure of MLB-routing Model
Let us now discuss the structure of the MLB-routing system.
First, we explain how to allocate packets to various paths.
Obviously we can allocate packets to any arbitrary paths bysource routing. However, to apply MLB-routing to the present
Internet network, it had also better to setup MLB-routing as
hop-by-hop routing system.
Fortunately, it is proved that the network flow generated by(7) is equivalent to the following Markov assignment flow [1]:Wjd,
p (j|i) = exp [−γcij ]
Wid



def.
Wid =
exp −γCrid ,
r∈Φid

(9)
(10)

where p (j|i) is a transition probability that a packet in node
i select the next hop j. This formulation satisfy the conditionj∈nbr(i) exp [−γcij ] Wjd
p (j|i) =
=1
(11)
Wid
j∈nbr(i)which is required to be regarded as the conditional probabilities. Furthermore, the equivalency between two assignments
is easily confirmed by
Prod = p (v1 |o) ×Λ−1p (vl+1 |vl ) × p (d|vΛ )l=1

Wv 1 dWv 2 d× exp [−γcv1 v2 ]
× ···
Wod
Wv 1 d Wv Λ d
Wdd
× exp −γcvΛ−1 vΛ
× exp [−γcvΛ d ]WvΛ−1 dWv Λ d


Λ−1

Wdd
= exp −γ cov1 +cvl vl+1 + cvΛ d
Wod
l=1


exp −γCrod
= ∞
.
(12)



od
exp −γCr= exp [−γcov1 ]k=1

The existence of equivalent Markov assignment implies that
we can apply the simple rule to all nodes (routers). Moreclearly say, if each node choose the next node to pass throughthe packet following (9), the network flow consequently equal
to the assignment which discussed in II-B.
Since the above method distributes packets to every possible
next nodes in a packet-by-packet manner, there would be apossibility of out-of-order packet delivery. In this paper, we
will not further argue this problem, but we believe the problem
can be solved by, for example, re-sequencing at the end hostor using special random functions that select same next nodefor packets of the same flow.
Now, please note that p (j|i) does not depend on the nodeof origin. It is confirmed that there are no subscript notation
related to origin in (9). Thus, if each router follows the above
Markov assignment, every packets which start from arbitrary
node spontaneously hold by (7). Generally, existing routing
table also does not depend on the origin, so it is quite favorable
feature of the MLB-routing. The example of the routing table
assumed in the MLB-routing is shown in Table I. Since it ismulti-path routing, there are multiple candidate nodes for next
hop even if packets go to same destination.
D. How to Calculate Wid
To determine the value of p (j|i), we have to calculate the
Wid which is included in p (j|i). Since the summation in Widmeans to consider all of path from node i to d, it is hard toderive them natively.
Consider a matrix A with N rows and N columns, whose[i, j]element is given as follows:aij =exp [−γcij ] (if node i to j is connected).
0(otherwise)(13)

E. As a Distance-vector Routing Protocol
From the above discussion, we represented the methodology to implement MLB-routing as a link-state protocol.
Meanwhile, such as RIP [6], distance-vector routing protocol
which is based on Bellman-Ford algorithm is still a major
method. Hence, it is worth enough to implement MLB-routing
as distance-vector protocol. We explain the methodology toachieve this idea in the following.
The idea of distance-vector routing protocol is that exchanging information (such as routing tables) between neighborhoodrouters. Thus, we present how to calculate Wid (and it meansp (j|i)) by iterative exchange of information. It is easy tounderstand that

TABLE IA N EXAMPLE OF ROUTING TABLE
destinationnext routerrouterj1..
.
routerjN
routerj1..
.
routerjN
routerj1.
.
.
routerjN

network A

network B

network C

probabilitypA (j1 |i)..
.
pA (jN |i)pB (j1 |i)..
.
pB (jN |i)pC (j1 |i).
.
.
pC (jN |i)weight
WiA

WiB

WiC

W = A I + A + A2 + A3 + · · ·= A (I + W )The element of A2 yields
[2]

aij =

N


is satisfied from (17). Furthermore, it is equivalent toWid =exp [−γcij ] (Wjd + δjd ) ,aik akj

k=1



=


exp [−γ (cik + ckj )]
exp −γCrij ,

(14)

r∈Φij
2

where Φij
L is the set of paths which connect nodes i and j in
L steps. Similarly, the typical element of AL is given by



[L]
(15)
exp −γCrij .
aij =
r∈ΦL

From the definition, we haveijΦij = Φij
1 ⊕ Φ2 ⊕ · · · .

(16)Consequently, we can obtain the value of Wid as follows:
W = A + A2 + A3 + · · ·(17)where the element of matrix W is



Wij =
exp −γCrij

(18)

r∈Φij

Furthermore, if the matrix A satisfied the Haskins-Simon
condition, the following formula is obtained:[I − A]−1(22)

j∈nbr(i){k|eik ,ekj ∈E}

=

(21)

= I + A + A2 + A3 + · · · .

(19)Therefore, we derive the value of Wij as follows:
W = [I − A]

−1

− I.

(20)

As indicated above, if we know the link weighted matrix
A, then transition probability p (j|i) is clearly determined.
It means that all routers (nodes) will work appropriately for
MLB-routing. Since the link weighted matrix A is derived
by assuming a link-state routing protocol (such as OSPF [7]),we obtained the methodology to implement MLB-routing as
link-state routing protocol.

where δjd is Kronecker delta. The recurrence formula (22)implies that each node can update their own weight Wid byexchanging the weight information for neighborhood nodes.
In other word, Wid could be calculated by Bellman-Fordalgorithm. As a result, we obtain another method to calculate
MLB-routing as a distance-vector protocol.
F. Traffic-aware Case
We now show how MLB-routing could be used to constructtraffic-aware routing algorithms. This expansion enables us toincorporate the traffic congestion in the model. In order todo this, we have to design the link costs to include a traffic
component. In this paper, we use the outgoing queue sizes toa network link ij as a measure of traffic at each link.
Let Qij denote the queue length on the outgoing link eij
and Bij be the bandwidth associated with eij . Furthermore,
we assume that dij is the standard delay in the link eij . Then,
we redefine the cost of link eij as follows:
cij = dij +

Qij
.
Bij

(23)

The above expression (23) represents the expectation time topath through the link eij under the condition of current queue
length Qij . Hence, using (23) when we calculate the MLBrouting, packets are supposed to avoid the traffic congestion
spontaneously (because the cost of congested link will take agreat value).
G. Properties of MLB-routing
Finally, we discuss some properties of MLB-routing
methodology. At first, we summarize the stability of MLBrouting algorithm. Earlier, we mentioned that the matrix A
should be satisfied the Haskins-Simon condition for equation(20). It means that Wid and p (j|i) are both stable. Furthermore, Haskins-Simon condition is equivalent toρ (A) = max {|λ|} < 1,(24)

where ρ (A) is the spectral radius of the matrix A and λ is the
eigenvalue of A. The value of max {|λ|} clearly decrease as γ
become small. Therefore, setting the value of γ appropriately,
we can construct the stable routing system.
The advantages of MLB-routing depend on the situation.
Clearly, if the sending rates are low compared to the link
capacities, then end-to-end delays do not happen even insingle-shortest path algorithm. In such situation, there are no
significant improve by using MLB-routing. Some packets areallocated to non-optimal route in MLB-routing, it results tothe increase of end-to-end delays.

time], which means the 400 packet generation at each unit
time. The simulation is continued till all the data packets intransit reached their destinations.
To compare the performance of the MLB-routing algorithm
to the single-shortest path algorithm (SPP), we set the maximum queue size at each network node to infinity. In otherwords, no packet is lost at any node. This assumption enables
us to make meaningful comparisons between the delay values
for viable source-destination pairs when the two different
routing schemes are used.

III. P ERFORMANCE E VALUATION

Using the condition stated above, we ran simulation for both
the MLB-routing algorithm and single-shortest path algorithm(SPP). In this paper, we compare two methodologies in termsof (i) end-to-end delay, and (ii) jitter times, respectively.
Now, let us briefly discuss the value of γ for MLB-routing.
In the above example network, γ > 0.875 is required to satisfy
ρ (A) < 1. Hence, we calculated MLB-routing in three casesthat γ = 1.0, 2.0, and 8.0. The simulation result is shown in
Figure 1 and Figure 2.
First, we compare end-to-end delay in each algorithm.
Figure 1 indicates the scatter plot of mean transit times
for each origin-destination pair. Each subfigure compares theresult of MLB-routing and that of SPP, where the diagonalline indicates the break-even point. From the figures, we canconfirm that a large majority of plots lie above the diagonalline for lots of simulation cases. It means that MLB-routing
outperforms SPP with respect to transit times.
The efficiency of MLB-routing scheme depends on thesituation. As previously mentioned, MLB-routing is effective
when traffic is congested (B = 40). On the other hand, when
the link bandwidths is high enough (B = 60), end-to-end
delays do not occur in SPP and there are no improve by
MLB-routing. The value of γ is also an important factor for
MLB-routing. If γ is too high, MLB-routing becomes equalcost multi path, and cannot achieve its full potential. In thecase of {γ, B} = {8.0, 40}, some plots are scattered under
the diagonal line, and it implies that some other links becomebottlenecks. On the contrary, if γ is too low, MLB-routing
allocates some packets to impractical path. Actually, MLBrouting is inferior to SPP in the case of {γ, B} = {1.0, 60}, it
result from the MLB’s ineffective detouring. We are supposed
to determine the value of γ with respect to the congestion of
network (the value of B). In this simulation, γ = 2.0 seems
to enhance the potential of MLB-routing. Especially, the caseof {γ, B} = {2.0, 40} is best-performed, and 39.1% decreaseof mean transit times are observed.
We now compare both protocol in terms of jitter times(standard deviation of transit times). Just as Figure 1, thescatter plot of jitter times for each origin-destination pairis shown in Figure 2. In the case of γ = 1.0 and 8.0,
though significant improvements by our methodology are notobserved, the scatter plot for γ = 2.0 (Figure 2-(b)) lies above
the diagonal line. That is, MLB-routing has a possibility to
improve not only transit times but also jitter times by setting

In this section, we evaluate the performance of the MLBrouting algorithm using simulations. We first describe theimplementation of algorithm, and then explain the network
topologies, followed by the experimental results.
A. Implementation
First, we describe our implementation of the MLB-routing
algorithm. As discussed above, MLB-routing methodology isable to be implemented as both link-state routing protocol andthe distance-vector routing protocol.
To implement the MLB-routing as link-state protocol, we
need to determine both the route computation algorithm andthe information to disseminate. The route computation algorithm described in Section II-D requires not only the link
metric information but also the queue length information.
Therefore, in addition to other link-state information, thequeue length information should be added to the information
dissemination process. Similarly, routers running the BellmanFord algorithm compute routes based on information obtained
from neighborhood nodes. To calculate the route based on
Section II-E, besides the routing table which is assumed in
RIP, node weight Wid should be also exchanged (see Table I).
In this study, we implement the MLB-routing protocol
as follows. First, we calculate the initial value of Wid byimplementing the MLB-routing as link-state protocol. Then,
update the each node of weight Wid iteratively by calculating
the MLB-routing based on the Bellman-Ford algorism.
B. Experimental Methodology
We ran simulation under the following methodology. To
begin with, in order to evaluate the MLB-routing algorithm,
we generated the network topologies using the Waxman
model [13]. In this model, nodes are randomly placed on a2-dimensional plane, and links are added incrementally. The
values of the Waxman-parameters were set to α = 0.5 andβ = 0.15 respectively. We distributed 100 nodes in plane, andcreated the networks. The number of links is 392.
The link cost (delay) is set in proportion to the distance of
two nodes, and the link bandwidths B is set to uniform value
in all links; we calculated in three cases (B = 40, 50, and60 [packet/unit time]). In addition, top left 20 nodes is set as
origin nodes, and bottom right 20 nodes as destination nodes.
During 200 [unit time], every node-pair is sent 1 [packet/unit

C. Performance Analysis

80

60

60

60

40

B  40B  50B  60

20

20

40
MLB

60

SPP

80

SPP

SPP

80

40

B  40B  50B  60208020(a) γ = 1.0

40
MLB

60

40

80

20

(b) γ = 2.0
Fig. 1.

40

30

30

30

20

10

20

30

40

80

SPP

40

SPP

40

SPP

50

B  40B  50B  6060

Scatter Plot of Mean Transit Times

50

20

40
MLB

(c) γ = 8.0

50

10B  40B  50B  6020

20
B  40B  50B  60

10

50

10

20

MLB

30

40

B  40B  50B  60

10

50

10

20

MLB

(a) γ = 1.0(b) γ = 2.0
Fig. 2.

30

40

50

MLB

(c) γ = 8.0

Scatter Plot of Standard Deviation

parameters appropriately. Therefore, we can conclude that
there is an appropriate value of γ which ensures maximum
performance of MLB-routing.
IV. C ONCLUSION AND F UTURE W ORKS
In this paper, we have applied the idea of random utility
theory to Internet routing to develop a multi-path routing
methodology called MLB-routing. The key concept of thestudy is to incorporate the multiple paths from same origin
to destination, and distribute the packets followed by themultinomial logit type probability. We have described that
MLB-routing is implemented as not only link-state protocol
but also distance-vector protocol. Furthermore, we have shownthat our methodology produces more efficient use of networkand causes significant improvements in end-to-end delays andjitter times. the performance of MLB-routing clearly depends
on the value of parameter γ, so it would be the subject of
future work how to determine the appropriate value.
ACKNOWLEDGMENT
This research was partially supported by the Grant-in-Aid
for Scientific Research (B) No. 21300027 (2009-2011) fromthe Japan Society for the Promotion of Science.

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