Some Synchronization Issues in OSPF Routing

Anne Bouillard

, Claude Jard

and Aurore Junier

ENS/INRIA, Paris, France

LINA, University of Nantes, Nantes, France

INRIA, Rennes, France

Keywords:

OSPF Routing, Synchronization, Simulation, Time Petri Nets.

Abstract:

A routing protocol such as OSPF has a cyclic behavior to regularly update its view of the network topology. Its

behavior is divided into periods. Each period produces a ﬂood of network information messages. We observe a

regular activity in terms of messages exchanges and ﬁlling of receive buffers in routers. This article examines

the consequences of possible overlap of activity between periods, leading to a buffer overﬂow. OSPF allows

“out of sync" ﬂows by considering an initial delay (phase). We study the optimum calculation of these offsets

to reduce the load, while maintaining a short period to ensure a protocol reactive to topology changes. Such

studies are conducted using a simulated Petri net model. A heuristic for determining initial delays is proposed.

A core network in Germany serves as illustration.

1 INTRODUCTION

Routing protocols generally work in a dynamic en-

vironment where they have to constantly monitor

changes. This function is implemented locally in

routers by a programming loop that generates regu-

lar behaviors. Open Shortest Path First (OSPF) pro-

tocol (Moy, 1998) is an interesting example, widely

used in networks. OSPF is a link-state protocol that

performs internal IP routing. This protocol regularly

ﬁlls the network with messages “hello” to monitor the

changes of network topology and messages “link state

advertisements” (LSA) to update the table of shortest

paths in each router.

A lot of work (Francois et al., 2005; Basu and

Riecke, 2001) has been devoted to stability issues.

The stability is required if there is a change in the

network state (e.g., a link goes down), all the nodes

in the network are guaranteed to converge to the new

network topology in ﬁnite time (in the absence of

any other events). The question is difﬁcult when the

change is determined as a result of a bottleneck in

a router (as possible in the OPSF-TE (Katz et al.,

2003)). If the response to a congestion is the exchange

of additional messages, the situation may become

critical. But it has been proved (Basu and Riecke,

2001) that OSPF-TE is rather robust in that matter.

In this article we look at a related problem which

is to focus on the possibilities of congestion of the

input buffers of routers due to LSA trafﬁc. Indeed,

we believe that there are situations where the cycli-

cal behavior of routers may cause harmful timings in

which incoming messages collide in a very short time

in front of routers.

In current implementations, the refresh cycle is

very slow and congestion is unlikely in view of the

routers’ response time. Nevertheless, we address the

question to increase the refresh rate to ensure better

responsiveness to changes. This article shows a pos-

sibility of divergence, and discusses the possibilities

of avoiding harmful synchronization by adjusting the

phase shift of cyclical behavior.

The approach is as follows. We modeled LSAs ex-

changes using TimePetri Nets (in a fairly abstract rep-

resentation). This model was simulated for a topology

of 17 nodes representing the heart of an existing net-

work in Germany (data provided by Alcatel). We then

demonstrated the possibility of accumulation of mes-

sages for well-chosen parameter values. Accumula-

tion is due to a possible overlap of refresh phases in

terms of messages. To validate this model, and thus

the reality of the observed phenomenon, we repro-

duced it on a network emulator available from Alca-

tel. Curves could indeed be replicated. Parameter val-

ues were different, but it was difﬁcult to believe that

the model scaled with respect to the rough abstraction

performed. Once the problem identiﬁed, the question

is then to try to solve it by computing optimum initial

Bouillard A., Jard C. and Junier A..

Some Synchronization Issues in OSPF Routing.

DOI: 10.5220/0004506800050014

In Proceedings of the 4th International Conference on Data Communication Networking, 10th International Conference on e-Business and 4th

International Conference on Optical Communication Systems (DCNET-2013), pages 5-14

ISBN: 978-989-8565-72-3

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

delays. Such a computation can be performed using

linear integer programming on a simpliﬁed graphical

model. We will show using simulation that the com-

puted values are relevant to avoid message accumula-

tion in front of routers.

The rest of the paper is organized as follows: we

ﬁrst present in section 2 the modeling of the LSA

ﬂooding process and its validation. In section 3, simu-

lation shows a possible overload of buffers depending

on the refresh period. Then, in section 4, we study a

possible adjustment of the initial delays, which aims

at minimizing the overload. We show how to compute

these delays. The impact is then demonstrated using

simulation.

2 TPN MODELING OF THE LSA

FLOODING PROCESS

2.1 LSA Flooding Process

The network is represented by a directed graph G =

(V,E), whereV is a ﬁnite set of n vertices (the routers)

and E is a binary relation on V to represent the links.

The i

router is denoted by R

. The set V (R

) denotes

the set of neighbors of R

, of cardinality |V (R

)|. To

help the reader Table 1 gives the list of the main nota-

tions introduced in this paper.

The LSA ﬂooding occurs periodically every T

seconds (30 minutes in the standard). Thus, the LSA

ﬂooding process starts at time kT

, ∀k ∈ N.

The LSA of a router R

records the content of its

database. Then, R

shares this LSA (denoted LSA

)

with its neighbors to communicate its view of the net-

work at the beginning of each period. The router R

sends LSA

after an initial delay d

. More precisely,

sends LSA

at d

+ kT

, ∀k ∈ N. Suppose that a

router R

receives LSA

and that it starts processing it

at time t. Then, R

ended the processing of LSA

time t + T

, where T

is the time needed by any router

to process an LSA or an acknowledgment(Ack). Dur-

ing this processing, R

updates its database and sends

a new LSA to its other neighbors if some new infor-

mation is learned. Consequently, R

could send a new

LSA at time t + T

, and its neighbors will receive it at

time t + T

+ T

, where T

represents the time to send

a message.

Note that any information received by R

can be

taken into account if some properties are satisﬁed.

The most important one is the age of the LSA. An

LSA that is too old is simply ignored. In all cases, at

time t + T

, R

sends an Ack to R

. The objective is

to inform R

that LSA

has been correctly received. In

parallel, R

waits for an Ack from all of its neighbors

before a given time. If an Ack is not received before

the end of this time, R

sends LSA

again until an Ack

is properly received.

The LSA ﬂooding process ends when every router

has synchronized to the same database.

2.2 The Simulation Model

Time Petri Net (TPN) (Jard and Roux, 2010) is an

efﬁcient tool to model discrete-event systems and to

capture the inherent concurrency of complex systems.

In the classical deﬁnition, transitions are ﬁred over an

interval of time. Here, transitions are ﬁred at a ﬁxed

time. This assumption is justiﬁed by observations of

actual OSPF traces whose data processing time does

not vary that much. In our case, the formal deﬁnition

of TPN is the following:

Deﬁnition 2.1 (Time Petri Net). A Time Petri Net

(TPN) is a tuple (P,T,B,F,M

,ϕ) where

• P is a ﬁnite non-empty set of places;

• T is a ﬁnite non-empty set of transitions;

• B : P×T → N is the backward incidence function;

• F : T × P → N is the forward incidence function;

• M

: P → N is the initial marking function;

• ϕ : T → N is the temporal mapping of transitions.

The remainder of this part is devoted to the con-

struction of the TPN that models message exchanges

of the LSA ﬂooding process. The objective is to

model and observe the dynamic behavior of a given

network.

Router Modeling The TPN that models the behav-

ior of the LSA ﬂooding process in a router R

needs

three timers: d

, T

and T

. Their functions are:

creating LSA

, managing a message received and re-

transmitting a received LSA when needed. Messages

are processed one by one. The following paragraphs

present each functional part of the TPN that models a

router.

• Place Processor. Initially this place contains

one token, representing the processing resource of a

router that is used to process LSAs and Acks. This

place mimics the queuing mechanism of R

and guar-

anties that only one message is processed at once. For

each different kind of messages (LSA

and Ack) the

processing mechanism is the following: an instanta-

neous transition is ﬁred, to reserve the resource of R

Note that it can only be ﬁred if a message is wait-

ing. Then the successor transition with timing T

can

be ﬁred, modeling the processing time of the router,

and Processor becomes marked again, enabling the

processing of a new message.

DCNET2013-InternationalConferenceonDataCommunicationNetworking

ACKs arrivals from a neighbor

LSAs arrivals from a neighbor

Acks sendings to the sender

ACKs arrivals from a neighbor

LSAs arrivals from a neighbor

Acks sendings to the sender

Start

LSAsend

i→k

LSArec

j→i

ACKrec

j→i

ACKsend

i→ j

Retransmission

Destruction

bound

LSArec

k→i

ACKrec

k→i

ACKsend

i→k

Retransmission

Destruction

bound

Processor

LSAsend

i→ j

Figure 2: TPN of a router R

that has two neighbors, R

and R

LSAsend

i→k

LSAsend

i→ j

Processor

Start

Figure 1: Part of TPN that creates the LSA of a router R

• Creation of LSA. Figure 1 represents the part

of the TPN that creates LSA

s at time d

+ kT

, for

k ∈ N in router R

. Initially Start

contains one to-

ken, t

ﬁres at time d

and a token appears in p

at time d

for the ﬁrst time. Afterward, the cycle

, p

generates a token in p

at times d

+ kT

k ∈ N. Those token will be processed using the mech-

anism described above, generating tokens in places

LSAsend

i→ j

, R

∈ V (R

• Reception of an Ack (dotted rectangles on Fig-

ure 2) A token in ACKrec

j→i

represents this event.

It is processed using the mechanism described above

and does not generate any new message.

• Reception of an LSA from a neighbor (dashed

rectangles in Figure 2). A token in place LSArec

j→i

represents this event. It is processed using the mech-

anism described above and generate an Ack, that is

sent to the sender. It can also possibly generate an

LSA message that will be retransmitted to its other

neighbors (transition Retransmission). Otherwise, the

token is destroyed (transition Destruction). In the

ﬂooding mechanism, an LSA

is retransmitted only

if it is received for the ﬁrst time during one ﬂooding

period. That way, the LSA ﬂooding process ensures

that every router converges to the same database be-

fore the end of every period. To model this, we bound

the number of retransmissions per period (for R

, the

number of retransmissions of an LSA received from

is b

, that is modeled by placing b

tokens in each

place bound of R

at the beginning of each period).

The tokens are inserted in these places by weighted

arcs between t

and each place bound.

• Global TPN Figure 2 represents the behavior

for one router. Such a net is built for each router. Fi-

nally, place LSAsend

i→ j

(resp. ACKsend

i→ j

) is con-

nected to place LSArec

i→ j

(resp. ACKrec

i→ j

) by in-

serting a transition LSA

i→ j

(resp. ACK

i→ j

) with ﬁring

time T

between them.

2.3 Model Validation

We performed our experimentations on the 17-node

German telecommunication network represented in

Figure 3. This article focuses on the study of router R

that has the largest number of neighbors (|V (R

)| =

6).

SomeSynchronizationIssuesinOSPFRouting

Figure 3: German telecommunication network.

The arrivals of LSAs and Acks in the actual net-

work are captured by an emulation using the Quagga

Routing Software Suite (Ishiguro, 2012), where each

node is set from an Ubuntu Linux machine that hosts

a running instance of the Quagga Routing Software

Suite. Figure 4 represents the arrival of messages in

by the emulation of the LSA ﬂooding on the Ger-

man topology during 8000s with T

= 1800s.

100

150

200

250

300

350

0 1000 2000 3000 4000 5000 6000 7000 8000

umber of messages arrived

time(s)

Figure 4: Emulation of the arrivals to R

During the emulation, the processors of routers

are parametrized with a 900 MHz CPU, and the mean

size of an LSA (resp. an Ack) is 96 bytes (resp. 63

bytes). The processing time of an LSA (resp. an Ack)

is approximately 0.8 µs (resp. 0.5 µs). The transmis-

sion time of an LSA (resp. an Ack) in 96 ms (resp. 64

ms).

Unfortunately, these parameters can not be used

directly to parametrize the TPN, as the TPN only

represents the behavior of the LSA ﬂooding process.

However,an actual router is much more loaded. Thus,

and T

must be adjusted to include the whole load

of the router.

The simulations presented in this article are pro-

duced by the software Renew (see (Kummer et al.,

2003)) which can simulate Time Petri Nets. Note

that the TPN are automatically generated (the TPN

that models the German Telecommunication network

is not represented here due to its size). Figure 5 rep-

resents the simulation of message arrivals using the

TPN where T

= 1800s, T

= 15s, T

= 30s. To corre-

spond to the sendings emulated in Figure 4 the num-

ber of LSAs retransmitted per neighbour during a pe-

riod is b

= ⌈

(n−1)

4|V (R

⌉.

One can observe that Figure 4 and 5 are quite sim-

ilar: the parameters chosen as above are deﬁned to

represent the actual behavior of an LSA ﬂooding pro-

100

150

200

250

300

350

0 1000 2000 3000 4000 5000 6000 7000 8000

Number of messages arrived

time(s)

Figure 5: Message arrivals to R

with T

= 1800s.

cess. The two curves are both composed of periods

that last 1800s. They show on each period a burst of

message arrivals that lasts approximately 800s, then

message arrivals stop until the next period. We there-

fore conclude that our abstract model correctly cap-

tures the phenomenon of LSA ﬂooding.

From now on we ﬁx the parameters ((b

)

i∈{1,...,n}

, T

and T

) as deﬁned above.

3 STUDY OF PERIOD LENGTH

We study here the effect of the period length T

both message arrivals and queue length. We ﬁrst dis-

cuss the normal case where T

= 1800s. Then, we

present a congested case where T

= 514s. Finally,

we observe a limit case where T

= 1000s.

3.1 Low Trafﬁc Case

Figure 6 represents the simulated queue length of R

during 10

s (approx. 1 day), where T

= 1800s. One

can observe a lot of ﬂuctuations. At the beginning

of each period R

receives and processes messages.

However, the number of messages that are received

is much larger than those which are processed. Con-

sequently, the queue length increases. Afterward, the

sendings stop, and R

keeps processing messages. The

queue length decreases.

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Queue length

time(s)

Figure 6: Buffer length of R

with T

= 1800s.

DCNET2013-InternationalConferenceonDataCommunicationNetworking

3.2 Congested Case

Figure 7 represents the message arrivals in R

during

8000s, and Figure 8 the queue length of R

during

s, where T

= 514s. One can observe that mes-

sages arrive continuously on router R

. Then, R

never idle and never empties its queue. Consequently

the queue length permanently increases.

100

200

300

400

500

600

700

800

900

1000

0 1000 2000 3000 4000 5000 6000 7000 8000

Number of messages arrived

time(s)

Figure 7: Message arrivals to R

with T

= 514s.

1000

2000

3000

4000

5000

6000

0 10000 20000 30000 40000 50000 60000 70000 80000

90000

100000

Queue length

time(s)

Figure 8: Buffer size of R

with T

= 514s.

3.3 Limit Case

Figure 9 represents the message arrivals in R

during

8000s, and Figure 10 shows the queue length of router

during 10

s, where T

= 1000s. This time, the

sendings of a period are not merged with the sendings

of the next period. Then, each period is long enough

so that R

can process messages from its queue be-

fore the beginning of the next one. Figure 10 shows

the ﬂuctuations of the queue length that correspond to

this. However the queue length is not empty at the

end of each period. Consequently, the stability of this

router is not ensured.

3.4 Sufﬁcient Condition for Congestion

Suppose being in the worst case where each router

learns some new information from each router and let

us now focus on the quantity of messages received

during a period.

Theorem 3.1. Let n( j) be the number of messages

received by a router R

during a ﬂooding period T

100

150

200

250

300

350

400

450

500

0 1000 2000 3000 4000 5000 6000 7000 8000

Number of messages arrived

time(s)

Figure 9: Message arrivals to R

with T

= 1000s.

0 10000 20000 30000 40000 50000 60000 70000 80000

90000

100000

Queue length

time(s)

Figure 10: Buffer length of R

with T

= 1000s.

Then

n( j) > n(|V (R

)|).

Proof. Let us ﬁrst focus on the case of networks with

a tree topology. In this case, we show that the above

inequality is in fact an equality. Two kinds of mes-

sages can be received: LSAs and Acks. Let us ﬁrst

count the number of messages received by router R

concerning the ﬂooding from router R

. Consider R

as the root of the tree, R

can receive LSA

from its fa-

ther only: R

will receiveone and only once LSA

. Af-

terward R

sends LSA

to its children and will receive

an Ack (as illustrated in Figure 11). As a consequence,

the number of messages received for the ﬂooding of

LSA

is the number of neighbors of R

. Consider the

ﬂooding of LSA

. The router R

sends the LSA to its

neighbors and will receive an Ack from them. Glob-

LSA

sent at step j

ack sent in response to LSA

received

Figure 11: Flooding of LSA

: LSA and ACKs transmissions

in a tree topology.

SomeSynchronizationIssuesinOSPFRouting

ally, R

will then receive exactly n(|V (R

)|) mes-

sages.

For networks with a general topology, one can ob-

serve that the ﬂooding of LSA

deﬁnes a spanning tree

of the graph: (R

) is an edge of the spanning tree

if R

ﬁrst received LSA

from R

. Then for the ﬂood-

ing of LSA

, R

receives at least the messages it would

receivedif the topology were the spanning tree, which

gives the desired inequality.

The number of messages processed by router R

during a ﬂooding period is 1+ n( j): it processes the

received messages plus LSA

. Deﬁne N( j) the num-

ber of messages processed during a ﬂooding period

by R

, we have

N(j) = n(|V (R

)|) + 1.

If a router can not process every message of its

buffer before the end of each period a congestion oc-

curs. Also, given the minimal bound of Theorem 3.1

the congestion is ensured by the following threshold

on T

Lemma 3.2. If T

< T

N(j) then the queue length of

tends to inﬁnity.

Proof. The proof is straightforward from Theo-

rem 3.1.

Example 3.3 (Simulation of TPN by Renew soft-

ware). Consider the tree topology network of Fig-

ure 11. Theorem 3.1 ensures that the number of mes-

sages received by R

(|V (R

)| = 4) is N(j) = 9× 4+

1 = 37. Therefore, if T

is set to 15 s in the TPN, if

< 15 × 37 = 555 s the network is congested. Sim-

ulation of the TPN, representing this topology, with

= 15s, T

= 554s, T

= 30s has been made during

4.10

s to illustrate this result. The evolution of the

queue length of router R

is shown in Figure 12. The

queue length of R

clearly increases during the simu-

lation, showing that the network is congested. Finally,

as the simulation has been made with the largest pe-

riod length that ensures congestion, during each pe-

riod, R

has enough time to process many messages

from his queue. Consequently, one can observe that

the queue length varies a lot.

4 COMPUTING OPTIMUM

INITIAL DELAYS

In Section 2.2, we emulated the ﬂooding phenomenon

of the OSFP protocol using Time Petri nets. The ini-

tial idea was to consider initial delays for each router

0 50000 100000 150000 200000 250000 300000 350000 400000

Queue length

time(s)

Figure 12: Queue length of R

with T

= 554s of tree topol-

ogy.

as parameters. The question is then to infer con-

straints on these parameters that ensure a minimum

size of the input buffers. Even if this kind of question

can be theoretically solved using symbolic model-

checking (Lime et al., 2009), the computation com-

plexity is high. The state of the art of the current ex-

isting tools did not allow us to automatically produce

such symbolic constraints.

In order to compute initial delays, we adopt the

following method. We only take into account the mes-

sage contributing to the ﬂooding mechanism: when

an LSA message concerning router R

is received at

router R

, it is forwarded only if it is received for the

ﬁrst time. Then, we will model neither the LSA mes-

sages that are not the ﬁrst to be received at a node, nor

the Acknowledgments.

4.1 Constraints Modeling

Our goal is to perform the ﬂoodings as closed as pos-

sible while interacting as little as possible. We say

that two ﬂoodings do not interact if, for each router,

the ﬁrst LSA received from those two ﬂoodings in that

router are not queued at the same time.

More formally, we consider a graph G = (V, E),

where V = {R

,... ,R

} is the set of routers and

E ⊆ V ×V is the set of links between the routers. If

) ∈ E, then τ

i, j

denotes the transmission time

between R

and R

, and τ

i, j

= ∞ if (R

) /∈ E. The

sojourn time of a message in R

, between its recep-

tion and its forwarding, belongs to the interval [δ

,∆

This time also holds for the source of messages.

Let us ﬁrst compute the intervals of time I

i, j

when the ﬁrst LSA originating from R

is received

in R

if the ﬂooding starts at time 0. If i = j, then

i,i

= [0,0], and otherwise, we have I

i, j

= [α

i, j

,β

i, j

[

where α

i, j

= min

k∈{1,...,n}

i,k

+ δ

+ τ

k, j

and β

i, j

min

k∈{1,...,n}

i,k

+ ∆

+ τ

k, j

The quantities α

i,k

+ δ

and β

i,k

+ ∆

respectively

represent the minimal and the maximal departure

times from R

DCNET2013-InternationalConferenceonDataCommunicationNetworking

For the computation of both α

i, j

and β

i, j

, we recog-

nize the computation of a shortest path in a graph with

respective edge lengths (δ

+ τ

i, j

) and (∆

+ τ

i, j

). Let

α = (α

i, j

) and β = (β

i, j

) the matrices of the shortest-

paths. They can, for example, be computed using the

Floyd-Warshall algorithm. Now, the messages origi-

nating from R

are present in R

during an interval of

time included in [α

i, j

,β

i, j

+ ∆

[= [α

i, j

,γ

i, j

[. We de-

note by D

i, j

this interval and D the matrix of these

intervals.

Example 4.1 (Sojourn times in the routers).

[δ

, ∆

[= [1, 2[

[δ

, ∆

[= [2, 3[

[δ

, ∆

[= [1, 3[

[δ

, ∆

[= [1, 2[

Figure 13: Example of a toy topology.

Figure 13 represents a toy topology with 4 vertices.

Matrix D is then:

D =







[0,2[ [2,6[ [5,9[ [8,14[

[2,6[ [0,3[ [3,7[ [6,12[

[5,9[ [3,7[ [0,2[ [3,7[

[9,14[ [7, 12[ [4,7[ [0,3[







Now, if the ﬂooding from server R

starts at time

, its ﬁrst LSA received by R

is present in that server

at most in the interval d

+ D

i, j

= [d

+ α

i, j

+ γ

i, j

Then, in order to have no interference between

the ﬂoodings in router R

, the family of intervals

+ D

i, j

)

i∈{1,...,n}

must be two-by-two disjoint, and

to have no interference at all, the following condition

must hold:

∀i, j,k ∈ {1, . ..,n}, i 6= k ⇒ d

+ D

i, j

∩ d

+ D

k, j

that is,

∀i, j, k ∈ {1,. ..,n}, i 6= k ⇒



+ γ

i, j

≤ d

+ α

k, j

+ γ

k, j

≤ d

+ α

i, j

For each triple (i, j,k), the two constraints above are

exclusive: as γ

i, j

> α

i, j

, if one holds, necessarily, the

other one does not hold.

Now, if we don’t consider the ﬁrst ﬂooding from

each router only, we have to study the interferences

between the ﬁrst and second ﬂooding from each

router (if there is no interference between those two

sets of ﬂooding, then there will be no interference at

all).

If the ﬂooding period is T, then the constraints

must then be transform in

∀i, j,k ∈ {1, ...,n},

+ γ

i, j

≤ d

+ α

k, j

+ γ

k, j

≤ d

+ α

i, j

and

+ γ

k, j

≤ d

+ T + α

i, j

and

+ γ

i, j

≤ d

+ T + α

k, j

(1)

The two cases are illustrated on Figure 14. Note

that, depending on which of the two ﬁrst constraint

is satisﬁed, one of the two last inequalities is trivially

satisﬁed.

′

+ D

k, j

′

+ T + D

k, j

+ D

k, j

+ T + D

k, j

+ D

i, j

+ T + D

i, j

′

+ T + D

i, j

′

+ D

i, j

Figure 14: Different possibilities for the constraints. In the

ﬁrst case, d

i, j

is before d

k, j

and in the second case,

′

+ D

k, j

is before d

′

+ D

i, j

, but in both cases, d

+ D

k, j

before d

+ T + D

i, j

and d

+ D

i, j

is before d

+ T +D

k, j

The problem we want to solve is then to ﬁnd

)

i∈{1,...,n}

such that all the constraints are satisﬁed

and T is minimized.

Theorem 4.2. Given (α

i, j

)

i, j∈{1,...,n}

, (γ

i, j

)

i, j∈{1,...,n}

and T, the problem of ﬁnding (d

)

i∈{1,...,n}

satisfying

the constraints of Equation (1) is NP-complete.

Proof. The problem is trivially in NP as for any as-

signment of (d

) and period T, it is possible to check

in polynomial time if the constraints are satisﬁed

(there are O(n

) constraints).

Now, to show that the problem is NP-hard, we re-

duce the salesman problem with triangular inequality

to that problem.

Suppose a complete weighted graph, with posi-

tive weights of the edges w(u, v), satisfying the tri-

angular inequality: for all vertices u, v, x, w(u,x) +

w(x,v) ≤ w(u,v). Set γ

i, j

= max

k∈{1,...,n}

w(k,i) and

i, j

= γ

k, j

− w(i,k).

This assignment of the variables is made in such a

way that if for some j, d

− d

≥ γ

k, j

− α

i, j

, then this

holds for all j, as γ

k, j

− α

i, j

= w

i,k

Now, let (d

) and T be a solution of our problem.

There is a Hamiltonian cycle of weight W ≤ T in the

graph: suppose, without loss of generality that d

≤

≤ · · · ≤ d

Then, w(1,2) + w(2,3) + ··· + w(n,1) ≤

− d

) + (d

− d

) + ··· + (d

− d

+ T) = T.

Conversely, suppose that there is a Hamiltonian

cycle of weightW, correspondingwithout loss of gen-

erality to the cycle 1, 2, ...,n. Set d

= 0 and d

SomeSynchronizationIssuesinOSPFRouting

i−1

+ w(i − 1,i). We have for all i, j d

every con-

straint is satisﬁed and T = W is a possible period: if

k > i, d

−d

= w(i,i+1)+···+w(k− 1,k) ≥ w(i,k).

Moreover, (d

+W)−d

= w(k,k+1)+···+w(n,i)+

··· + w(i− 1, i) ≥ w(k, i).

Hence, we have a Hamiltonian path of length at

most T if and only if we can ﬁnd a solution to our

problem with period at most T: the problem is NP-

hard.

4.2 Exact Solution with Linear

Programming

This problem can be solved with a linear program us-

ing both integer and non-integer variables. The trick

is to encode the constraints

+ γ

i,k

≤ d

+ α

k, j

+ γ

k, j

≤ d

+ α

i, j

into a linear program, and this is why we introduce

integer variables.

First, this set of constraints can be rewritten in

− d

≥ b

i,k, j

or d

− d

≥ b

k,i, j

with b

i,k, j

= γ

i, j

− α

k, j

. Set B = max

i, j,k

i,k, j

Lemma 4.3. There is a solution of this problem where

for all i ∈ {1,...,n}, d

∈ [0,nB].

Proof. The assignment d

= (i− 1)B is a solution of

the problem. Indeed, ∀i < k, ∀ j ∈ {1,...,n}, d

−d

(k−i)B ≥ B ≥ b

i,k, j

. Moreover, ∀i,k, j, d

−d

= (n−

k+ i)B ≥ B ≥ b

i,k, j

Lemma 4.4. The following sets of constraints are

equivalent.

(i) d

∈ [0,nB] and (d

− d

≥ b

i,k, j

or d

− d

≥

k,i, j

)

(ii) d

∈ [0,nB], q ∈ {0,1} and d

− d

+ (1 −

q)nB ≥ b

i,k, j

and d

− d

+ qnB ≥ b

k,i, j

Proof. Suppose that the constraints (i) are satisﬁed.

Either d

− d

≥ b

i, j,k

and the constraints in (ii) with

q = 1 are satisﬁed (we have the two constraints d

−

≥ b

i, j,k

and d

− d

+ nB ≥ nB ≥ b

k,i, j

); or d

− d

k,i, j

and similarly, the constraints in (ii) with q = 0

are satisﬁed.

Suppose now that the constraints (ii) are satisﬁed.

If q = 1, then, trivially, d

− d

≥ b

i, j,k

and if q = 0,

then d

− d

≥ b

k,i, j

Consequently, the linear program is

Minimize T under the constraints

∀i, j,k ∈ {1, ...,n}, i 6= k,

0 ≤ d

≤ nB











i,k, j

∈ {0,1}

− d

+ (1− q

i, j,k

)nB ≥ b

i,k, j

− d

+ q

i, j,k

nB ≥ b

k,i, j

− d

≤ T − max

j∈N

k,i, j

Example 4.5. The toy example above gives T = 28,

with d

= 0, d

= 21, d

= 14 and d

= 5.

Computing this exact solution is possible but has

two drawbacks. First, as the problem is NP-complete,

computing the initial delays in larger networks may

be untractable. Second, this solution does not ex-

hibit monotony properties. For example, if the linear

program lead to a period T and the target period is

′

> T, it might be better to stretch the values d

− d

to (d

− d

′

/T. It is unfortunately not ensured with

the solution found. In the next paragraph, we show

how to compute a solution complying with this addi-

tional constraint.

4.3 Heuristic using a Greedy Algorithm

To simplify the problem we only use strongest con-

straints: with c

i,k

= max

k∈N

i,k, j

i,k

≤ d

−d

≤ T −c

k,i

or c

k,i

≤ d

−d

≤ T −c

i,k

(2)

Lemma 4.6. If (d

)

i∈{1,...,n}

is a solution to the con-

straints of Eq. (2) with a period T, then for T

′

> T,

(

′

) is a solution for the same constraints with pe-

riod T

′

Proof. If c

i,k

≤ d

− d

≤ T − c

k,i

, then as

′

≥ 1,

′

− d

) ≥ d

− d

≥ c

i,k

. Second,

′

− d

) =

′

(T − c

k,i

) = T

′

−

′

k,i

≤ T

′

− c

i,k

Solving these constraints is still a NP-complete

problem. In fact the proof of Theorem 4.2 is valid

in this case.

Now, in order to assign the values, we can use the

greedy algorithm presented in Algorithm 1. At each

step, the algorithm assigns one initial delay, that is

chosen to be the smallest as possible, given the ini-

tial delays already assigned, while satisfying the con-

straints set by them.

Lemma 4.7. At each step of the algorithm, the con-

straints (2) such that i,k ∈ D are satisﬁed.

Proof. We show the result by induction. When D =

or |D| = 1, then this is obviously true as no constraints

are involved. Suppose this is true for D and let s the

next element that is added to D in the algorithm. From

DCNET2013-InternationalConferenceonDataCommunicationNetworking

Algorithm 1: Initial delays computation.

Data: c

i, j

Result: d

,... ,d

, T.

1 begin

2 D ←

0 ;

3 S ← {1,...,n};

4 foreach i ∈ S do d

← 0;

5 while S 6=

0 do

6 s ← Argmin

i∈S

;

7 S ← S\ {s};

8 foreach i ∈ S do d

← max(d

+ c

s,i

);

9 foreach i ∈ D do

T ← max(T,d

− d

+ c

s,i

);

10 D ← D ∪ {s};

line 8, we know that d

≥ max

i∈D

+ c

i,s

. Then, for

all i ∈ D, d

− d

≥ c

i,s

. Now, from line 9, for all

i ∈ D, T ≥ d

− d

+ c

s,i

, so d

− d

≤ T − c

s,i

. So, the

constraints involving s are satisﬁed. Now, if the con-

straints between i and j, i, j ∈ D are satisﬁed at one

step of the algorithm, they will remain satisﬁed dur-

ing the following steps, as T can only increase.

Example 4.8 (Application of Algorithm 1). With our

toy example, we have

C = (c

i, j

) =







0 8 11 14

6 0 9 12

9 7 0 7

14 12 9 0







If 1 is chosen ﬁrst (d

= 0 ∀i ∈ {1,2,3,4}), the val-

ues are updates to d

= 0, d

= max(0,d

+ c

1,2

) = 8,

= 11 and d

= 14; T = 0. Then, 2 is chosen and

we get d

= max(d

+ c

2,3

) = 17 and d

= 20;

T = max(T, d

− d

+ c

2,1

) = 14. Finally, we have

= 0, d

= 8, d

= 17, d

= 24 and T = 38.

Note that this problem could also have been solved

using a linear program (with integer variables), by re-

placing the variables q

i,k. j

in the linear program of the

previous paragraph by q

i,k

: forgetting the parameter j,

exactly leads to the same constraints of Equation (2).

In this case, we ﬁnd T = 36, with d

= 0, d

= 30,

= 11 and d

= 18. Our heuristic is near this opti-

mal.

In the next lemma, we assume that our target pe-

riod is T

′

< T, that is, we are not able to ﬁnd a solution

so that there is at most one message in the queues of

the routers. We assume here that the sojourn time of

a message does not depend on the queue length.

Lemma 4.9. Let (d

) be a solution for the initial de-

lays with period T. The same assignment with period

′

< T ensures that in each router, there are never

more than ⌈

′

⌉ messages simultaneously.

Proof. Set ⌈

′

⌉ = q. We number the messages: m

is the j-th message originating from router i. For ℓ ∈

{0,...,q − 1}, in each server, simultaneously, there

cannot be several messages among (m

kq+ℓ

)

k∈N,i∈N

because qT

′

≥ T. As a consequence, there cannot be

more than q messages in a router.

4.4 Simulation Results with Initial

Delays

In this section, we present simulations of the TPN

modeling the German telecommunication network

with initial delays deﬁned by Algorithm 1 in the sta-

ble case (T

= 1800s).

We ﬁrst need to deﬁne the transmission and so-

journ times used by the algorithm:

• the transmission time has already been deﬁned to

= T

= 30s, for all the links of the network;

• for each router R

, the sojourn time is at least equal

to the processing time δ

= T

= 15s, the time to

process the message where the queue is empty.

The maximum sojourn time is extracted from the

simulation of the TPN of Section 2 (with no ini-

tial delays). During the simulation, the maximum

queue length is Q

in router R

. Then we take

∆

= Q

Note that doing this enables to take into account

all the messages from the LSA ﬂooding mecha-

nism, and not only the ﬁrst LSA message in each

router.

The maximal queue length of each router is

extracted from a simulation of the TPN dur-

ing approximately 3.5 days (3.10

s). Here is

the list of each maximal queue length: Q =

(7,8,13,2,2, 17,8,37,4,5,13, 2, 2,3,13,6,2). Then,

Algorithm 1 returns the following initial delays:

d = (0, 105,1200,810, 75,255,420,1335,1035,

1080, 1155, 1530,630,330,780,330,1680).

Furthermore, Algorithm 1 computes T

rMax

16695s.

Figure 15 represents the result of the TPN simula-

tion with initial delays listed above when T

= 1800s.

The maximum queue length for router R

is now

Max

= 25, which gives a signiﬁcant improvement:

it was Max

= 37 without the computation of initial

delays. Moreover, the queue length is most of the time

below 10.

5 CONCLUSIONS

This article presents a method usable for the OSPF

SomeSynchronizationIssuesinOSPFRouting

0 10000 20000 30000 40000 50000 60000 70000 80000

90000

100000

Queue length

time(s)

Figure 15: Buffer length of R

with T

= 1800s and initial

delays.

protocol and cyclic protocols that use delay parame-

ters. This method aims at increasing the reactivity of

the network to topology changes, and at minimizing

the queue length of routers. Algorithm 1 provides an

efﬁcient way to spread messages over the whole pe-

riod. Furthermore, it showsto be a good tool to reduce

queue lengths.

REFERENCES

Basu, A. and Riecke, J. (2001). Stability issues in ospf rout-

ing. In Proceedings of the 2001 conference on Ap-

plications, technologies, architectures, and protocols

for computer communications, SIGCOMM ’01, pages

225–236, New York, NY, USA. ACM.

Francois, P., Filsﬁls, C., Evans, J., and Bonaventure, O.

(2005). Achieving sub-second igp convergence in

large ip networks. SIGCOMM Comput. Commun.

Rev., 35(3):35–44.

Ishiguro, K. (2012). Quagga, a routing software package

for tcp/ip networks, http://www.nongnu.org/quagga/.

Jard, C. and Roux, O. H. (2010). Communicating Embed-

ded Systems, Sofware and Design, Formal Methods.

ISTE and Wiley.

Katz, D., Kompella, K., and Yeung, D. (2003). Trafﬁc Engi-

neering (TE) Extensions to OSPF Version 2. Updated

by RFC 4203.

Kummer, O., Wienberg, F., Duvigneau, M., Kohler, M.,

Moldt, D., and Rolke, H. (2003). Renew the reference

net workshop. In mi.

Lime, D., Roux, O. H., Seidner, C., and Traonouez, L.-

M. (2009). Romeo: A parametric model-checker for

petri nets with stopwatches. In Kowalewski, S. and

Philippou, A., editors, TACAS, volume 5505 of Lec-

ture Notes in Computer Science, pages 54–57, York,

United Kingdom. Springer.

Moy, J. (1998). RFC 2328 OSPF v2. Technical report.

APPENDIX

Table 1: List of main notations.

Notation Full name

G = (V,E) directed graph representing the network

n number of routers

router in the network

V (R

) set of neighbors of R

|V (R

)| cardinality of V (R

)

initial delay of R

number of retransmission of an LSA

received from a neighbor of R

LSA

link state advertisement message of R

Ack acknowledgment message

(or T) period length of the LSA ﬂooding process

processing time of messages

time to send a message

(P,T,B,F,M

,ϕ) a Time Petri Net (TPN)

Start

initial place of TPN to create LSA

LSAsend

i→ j

place to send LSA

to R

ACKsend

i→ j

place to send an Ack from R

to R

LSArec

j→i

place to receive LSA

in R

ACKrec

j→i

place to receive an Ack from R

in R

Processor place to guaranty that one message

is processed at a time

bound place to bound the number of

retransmission from a neighbor

Retransmission place to retransmit a received LSA

Destruction place to destroy a received LSA

n( j) (resp. N( j)) number of messages received (resp.

processed) by R

during T

i, j

transmission time between R

and R

[δ

,∆

[ sojourn time of a message in R

i, j

= [α

i, j

,β

i, j

[ time of ﬁrst LSA

received in R

α = (α

i, j

) matrix of values α

i, j

β = (β

i, j

) matrix of values β

i, j

D = (D

i, j

) D

i, j

= [α

i, j

,γ

i, j

[ with γ

i, j

= β

i, j

+ ∆

Q = (Q

) maximal queue length of R

i,k, j

and B b

i,k, j

= γ

i, j

− α

k, j

and B = max

i,k, j

C = (c

i,k

) c

i,k

= max

k∈N

i,k, j

DCNET2013-InternationalConferenceonDataCommunicationNetworking