On the point-of-presence optimization problem
in IP networks
Sur le problème de l’optimisation des points de
présence dans les réseaux IP
Steven Chamberland, Marc St-Hilaire, and Samuel Pierre This paper presents a model for the point-of-presence (POP) optimization problem in Internet protocol (IP) networks, where a POP is a node composed of
several interconnected co-located backbone routers. This problem consists of selecting the number of routers and their types, selecting the interface card
types, connecting the access and the backbone links to the ports, selecting the link types between the co-located routers, and routing the traffic within the
POP. A detailed example solved by using a branch-and-bound algorithm is presented along with results for a set of randomly generated problems. The
results show that fair-sized problems can be solved to optimality within a reasonable amount of time.
Cet article présente un modèle pour le problème de l’optimisation des points de présence dans les réseaux IP (Internet protocol). Ce problème consiste
à sélectionner le nombre de routeurs et le type de chacun, sélectionner les types des cartes à insérer dans les routeurs, brancher les liens d’accès et de
transmission aux ports, sélectionner les types de liens entre les routeurs co-localisés et acheminer le trafic dans le point de présence. L’on présente un
exemple détaillé en utilisant un algorithme de séparation et évaluation progressive, suivi de résultats pour des problèmes générés aléatoirement. Les
résultats montrent que des problèmes de taille modérée peuvent être résolus en temps raisonnable.
on average, within % of the proposed lower bound. Finally, in
[5], the authors studied several network topologies (ring, tree and fullmesh) for the inter-router network within a POP.
I. Introduction
Due to the increasing demand for Internet network services, telecommunication carriers are investing continuously in their Internet protocol (IP) infrastructure. As a result, several backbone routers are colocated in the points of presence (POPs) as illustrated in Fig. 1. A POP
is defined as an access point to the IP network owned by a telecommunication carrier. It usually includes routers and may also include
other equipment such as digital/analogue call aggregators, servers and
switches. Considering that IP routers and interface cards are still expensive, POP optimization is an important issue for these carriers in
the quest to remain competitive.
Actually, the POP optimization is typically done by hand or by using
a network simulator to evaluate several scenarios. Since the number
of routers increases continually, the scenario-oriented approaches are
time-consuming, and the solutions found are rarely optimal.
The literature of network engineering and operations research contains many articles relating to network optimization and planning, but
the POP optimization problem has been considered by only a few authors. The difficulty of the problem was pointed out by Chamberland
et al. [1]–[2], who first proposed a global approach for the POP optimization problem. The authors demonstrated that the problem is
hard and proposed a greedy heuristic algorithm for finding solutions
rapidly. For problems generated, the algorithm produced solutions
that were, on average, within % of a proposed lower bound (found
by solving a relaxed version of the problem with a branch-and-bound
algorithm). Extended versions of the problem were proposed in subsequent papers by adding additional performance constraints [3] and
reliability constraints [4] to the original problem. In those papers, tabusearch-based meta-heuristic algorithms were explored to find good solutions within a reasonable amount of time. The solutions found were,
Steven Chamberland, Marc St-Hilaire, and Samuel Pierre are with the
Department of Computer Engineering, École Polytechnique de Montréal,
C.P. 6079, Succ. Centre-Ville, Montréal, Québec H3C 3A7. E-mail: steven
.chamberland@polymtl.ca
Can. J. Elect. Comput. Eng., Vol. 30, No. 3, Summer 2005
In this paper, we tackle the POP optimization problem in IP networks. This problem consists of
selecting the number and types of routers to install in the POP
(where a router type is characterized by its number of slots and
its switch fabric capacity);
selecting the interface card types (where an interface card type
is characterized by its technology and port rate, its number of
ports, and the number of slots necessary for its insertion into the
router);
connecting the access and backbone links to the ports;
selecting the link types between the co-located routers; and
routing the traffic within the POP.
The objective is to minimize the cost of the POP. In fact, this paper extends the work of [6] by including additional constraints for the traffic
routing and the inter-router link dimensioning. Moreover, in this paper
we solve the POP optimization problem exactly, a feat which has never
been done before in the literature.
This paper is organized as follows. Section II presents a mathematical model for the POP optimization problem in IP networks. Section III
presents a detailed example solved by using a branch-and-bound algorithm along with results for a set of randomly generated problems.
Conclusions and further work are presented in Section IV.
II. The mathematical model
A. The assumptions
We make the following assumptions about the organization of
each POP:
2
CAN. J. ELECT. COMPUT. ENG., VOL. 30, NO. 3, SUMMER 2005
Number of access
links
1
2 OC-3
Number of backbone
links
2
1 OC-3
1 OC-12
1
2 OC-12
1 OC-48
3
1 OC-12
2
4
1 OC-12
2 OC-48
2 OC-3
5
1 OC-3
1 OC-12
3
1 OC-48
The POP
to
optimize
6
1 OC-12
4
7
2 OC-12
2 OC-3
Point of presence
Point of presence
8
Interconnected routers in
the POP
1 OC-3
1 OC-12
Client
3 OC-48
Backbone router
Backbone link
1 OC-12
Backbone link
10
1 OC-3
1 OC-12
Inter-router link
1. The POP is composed of co-located interconnected routers.
2. The number of routers installed in the POP cannot exceed the
maximum allowed.
3. The sum of the rate of the ports installed in a router cannot exceed
its switch fabric capacity.
Figure 2: Illustration of the problem.
B. The notation
The following notation is used throughout this paper. The notation is
composed of sets, decision and traffic variables, cost parameters, link
and router parameters, and constants.
1.
Sets
the set of link/port types used in the access network;
We assume that the following information is known:
1. the clients connected to the POP and their access link types
(where an access link type is characterized by its technology and
rate);
2. the backbone links connected to the POP and their types (where
a backbone link type is characterized by its technology and rate);
3. the maximum number of routers that can be installed in the POP;
4. the different types of routers (where a router type is characterized
by its number of slots and its switch fabric capacity);
5. the different types of interface cards (where an interface card type
is characterized by its technology and port rate, its number of
ports, and the number of slots necessary for its insertion into the
router);
6. the cost of purchasing each router type and installing it (including
the floor space, cables, racks, electric installation and labour);
7. the cost of purchasing each interface card type and installing it
(including the patch panel space, the cables and labour);
8. the cost of interconnecting two routers within the POP using a
link and ports of a given type (including the cables and labour);
9. the traffic passing through the POP.
To grasp the significance of the previous assumptions, we provide
in Fig. 2 an example in which the clients are connected to the POP
through OC-3 ( Mb/s) and OC-12 ( Mb/s) access links and
the POPs are interconnected through OC-12 and OC-48 ( Gb/s)
backbone links. OC-192 ( Gb/s) links and ports should be used
to interconnect the routers within the POP. The problem consists in
finding the minimum-cost POP subject to all the previous assumptions
and facts.
Note that optical carrier (OC-) represents an optical signal and
that denotes the number of increments of Mb/s defined in the
synchronous optical network (SONET) technology (for the technical
aspects related to SONET networks, see [6], and for more information
about IP over SONET, see [7]).
Access link
9
Access link
Figure 1: IP network architecture.
Client
5
the set of link/port types used in the backbone network;
the set of link/port types used to interconnect the routers within
the POP;
the set of link and port types (i.e., );
the set of clients connected to the POP;
the set of other POPs (O-POPs) in the network (i.e., the POPs in
the network connected to the POP that we want to optimize);
the set of potential router locations in the POP (i.e., each potential
location corresponds to a rack space used to install a router);
the set of interface cards with ports of type ;
the set of interface card types (i.e., );
the set of router types.
2.
Decision and traffic variables
a - variable such that at location
is of type
the number of cards of type
the number of links of type ;
if and only if the router installed
;
the number of links of type location
;
installed at location
from client from O-POP
;
to location
to
the number of links of type installed between location
and location
for
(we impose the condition
because the communication links are supposed to be full
duplex and symmetric);
the traffic flow (in bits per second) from client to
location
(from location
to client ) originating
from ;
the traffic flow (in bits per second) from O-POP to location
(from location
to O-POP originating from ;
the traffic flow (in bits per second) from location
location originating from .
)
to
CHAMBERLAND / ST-HILAIRE / PIERRE: ON THE POP OPTIMIZATION PROBLEM IN IP NETWORKS
3.
parameters
Cost
the cost of purchasing a card of type
4.
location
;
the cost of purchasing a router of type
and installing it at
location
;
the cost (including the installation cost) of connecting location
to location
using a link of type .
router.
in the
Constants
the traffic demand from to passing through the POP (in bits per second);
the number of links of type from to the POP.
,
(from C. The model
In this section, we define the POP optimization model, denoted P. This
model has the goal of minimizing the objective function subject to
constraints:
minimize
(1)
(10)
(11)
(12)
(13)
(14)
(9)
(8)
Inter-router network topology constraints:
subject to the following:
)
5.
Link and router parameters
the bit rate (in bits per second) of the link/port of type ;
the number of slots in a router of type
;
the switch fabric capacity (in bits per second) of a router of type
;
the number of ports on the card of type
;
the number of slots necessary to insert a card of type
and installing it at
3
Access link capacity constraints:
Client assignment constraints:
O-POP assignment constraints:
(3)
(4)
(5)
(6)
(15)
(16)
(17)
(18)
(19)
(20)
Traffic-flow conservation constraints:
Router-capacity constraints (port level):
Router-capacity constraints (switch fabric level):
Inter-router link capacity constraints:
Router-capacity constraints (slot level):
Backbone link capacity constraints:
Router-type uniqueness constraints:
(2)
(7)
(22)
(23)
(21)
(24)
4
CAN. J. ELECT. COMPUT. ENG., VOL. 30, NO. 3, SUMMER 2005
L EMMA 1: The inequalities
(25)
Integrality and nonnegativity constraints:
IR IN
IN
IN
IN ¾
are valid for P.
(27)
(28)
P ROOF : We prove only (27), since (28) can be similarly proven. Let
be a given location. If
(5)
, then using constraints
or (6) and constraints (7), (10), (11) and (13) we obtain for
and . However, if
all , then using con
straints (2), we obtain for all . Thus, inequalities (27) are valid for DP.
(26)
The following inequalities give lower and upper bounds on the number of routers to install in the POP.
The objective function (1) of P is composed of three terms: the cost
of the links, the cost of the cards and the cost of the routers. Constraints (2) are client assignment constraints that require each client to
be connected to the POP with the required access link types, and constraints (3) are O-POP assignment constraints that require each O-POP
to be connected to the POP with the required backbone link types. The
router-type uniqueness constraints (4) require that at most one router
type be installed at location
, and constraints (5) require that the
total number of slots used by the cards at location
be less than
or equal to the number of slots available for the router type installed
at this location. Constraints (6) require that the sum of the rates of the
ports installed at the router at location
be less than or equal
to its switch fabric capacity, and constraints (7) to (13) dictate that the
number of links of type connected to a router be less than or
equal to the number of ports of this type. Constraints (14) necessitate
that the topology of the inter-router network be the one specified by
the network planner. The full-mesh topology is considered in this paper. This topology has been chosen as the preferred topology since the
network planner typically wants to minimize the number of hops between the clients in order to minimize the end-to-end delay and the
jitter. However, if the number of routers is large (e.g., more than six
routers), another topology such as a two-level topology should be used
in order to minimize the number of inter-router links (where the first
level of routers is used to connect the access links, and the second level
to connect the backbone links). In fact, constraints (14) require at least
one link between two locations if and only if routers have been installed at these two locations. These constraints were first described in
[8], in the context of the topological network design problem with a
two-level structure. Other topologies such as the tree or the ring topology were also considered (see [5] for more details concerning these
topologies).
Since P is
-hard (transformation from the knapsack problem [9]), it is unlikely that large instances of this problem can be solved
to optimality within a reasonable amount of time.
Note that P can also be used for the POP expansion problem. Indeed, the network planner has only to fix a subset of the variables for
representing the already existing configuration of the POP and solving
the model to find the other variables.
!
!
! ! !
!"
where
and
Access link capacity constraints (15) and (16) require that the traffic flow (in bits per second) from client to the POP be less than
or equal to the total access link capacities from this client (in both directions), and backbone link capacity constraints (17) and (18) require
that the traffic flow from to the POP be less than or equal to the
total backbone link capacities installed between and the POP. Interrouter link capacity constraints (19) and (20) require that the traffic
flow (in bits per second) from location
to location
be
less than or equal to the total link capacities installed between and
. Constraints (21) to (25) are traffic-flow conservation constraints,
and constraints (26) are integrality and nonnegativity constraints.
L EMMA 2: The following inequalities are valid for P:
!"
(29)
(30)
(31)
(32)
# where ! is the maximum number of ports of type per slot,
# "
and " is the minimum number of slots required to connect the access
and the backbone links.
P ROOF : Using equations (7) to (13), we obtain
# (33)
# With the full-mesh topology, we have
, and using (5), we obtain
for all and
# # (34)
# # If we sum
inequality:
"
on the two sides of (34), we obtain the following
!
(35)
CHAMBERLAND / ST-HILAIRE / PIERRE: ON THE POP OPTIMIZATION PROBLEM IN IP NETWORKS
5
where ! and " are respectively defined by (31) and (32). From (35)
we obtain
"
!
f(x)
(36)
and then
!"
!
or
(37)
x-
!
!"
(38)
!
!
!
!
! !
!"
is an integer in all
Two link types (i.e., OC-3 and OC-12) are used in the access network, two in the backbone network (i.e., OC-12 and OC-48), and one
in the inter-router network (i.e., OC-192). The costs of the router types
are given in Table 1, and the costs of the interface card types in Table 2.
These costs include the purchase and the installation costs (including
the floor space, the patch panels, the cables and connectors, the racks,
the electric installation and labour). The cost of interconnecting two
co-located routers is (including the cables and labour).
!"
L EMMA 3: The inequality
!
is obtained for
The proposition follows because
feasible solutions of P.
In this section, we first present a detailed design example, followed by
results for a set of randomly generated problems. The CPLEX Mixed
Integer Optimizer (see [10] for more information about CPLEX) is
used to solve the model. Note that the algorithm used by the CPLEX is
the branch-and-bound algorithm. For the tests, the branch-and-bound
node limit was set to , the upper limit on the tree size was
Mb, and the time limit was hours. For the computing platform,
we used a Sun Ultra 5 workstation with Mb of RAM.
and the maximum value of III. Numerical results
!"
!
(40)
is valid for P.
P ROOF : In Lemma 2, we showed that the righthand side
of (37) is obtained when
is equal to
and
! #
! # .
in (37), we obtain (40).
Then, if we replace T HEOREM 1: If P is feasible, then (40) is respected and
!
!
Figure 3: Equation .
x
P ROOF : Because (40) and (41) are obtained from constraints of P,
therefore if P is feasible, then these inequalities are respected for all
feasible solutions.
The solutions of (39) are
x+
(39)
!"
where IR. Then is used to define the feasibility region of P
as the function of the number of routers installed. In Fig. 3, we depict
equation and indicate the interval of such that .
x max
-AB
Consider the following equation:
;;;
!"
A. An illustrative example
For the example, the number of clients in is , the number of OPOPs in is , and the maximum number of routers that can be
installed in the POP is (i.e., ). The number and types of
access links are presented in Table 3, and the number and types of
backbone links are presented in Table 4.
The traffic demand is generated randomly using a uniform distribution law in the interval from to Mb/s between each pair of
clients, from to Mb/s between each pair of O-POPs, and from to Mb/s between each client and O-POP.
, therefore using Lemma 2, we know that
Since ! and "
the number of routers installed in the POP will be between and .
Moreover, since (40) and (41) are satisfied, we cannot draw a conclusion about the feasibility of the example using Theorem 1.
(41)
The solution obtained with CPLEX took s, and its value was
$585 500, obtained after exploration of branch-and-bound nodes.
6
CAN. J. ELECT. COMPUT. ENG., VOL. 30, NO. 3, SUMMER 2005
Table 3
Router type B
Number and types of access links for each client
Router type B
OC-192
Client
Location 1
Location 3
Type of card
Number
Type of card
A
B
C
D
E
F
0
0
1
3
6
1
A
B
C
D
E
F
Number
8
0
2
0
3
1
Figure 4: Solution of the example.
Table 1
Characteristics of the router types
Type
Number of slots
Switch fabric capacity (Gb/s)
Cost ($)
A
7
80
17 000
B
14
160
28 000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Technology
A
B
C
D
E
F
OC-3
OC-3
OC-12
OC-12
OC-48
OC-192
Number of
ports
4
16
4
16
4
1
Number of
slots required
1
2
1
2
1
1
Cost ($)
3000
20 000
10 000
35 000
30 000
50 000
The solution, illustrated in Fig. 4, has two routers of type B (one at
location
and the other at location
), and an OC-192 link
interconnecting them.
B. Other results
In this subsection, we present the results of a systematic set of experiments designed to assess the performance of the branch-and-bound
algorithm.
Twenty-five problems were randomly generated as follows. For
each client, the number of access links is randomly selected among
OC-3,
OC-3, OC-12 and
OC-12. For each O-POP, the
number of OC-12 and OC-48 links is randomly selected in the interval
from to .
The optimal solution (i.e., OPT) is evaluated for each test problem.
The solutions are presented in Table 5. The first column identifies the
problem, the second contains the number of clients , and the third
gives the number of O-POPs . The next column presents the optimal
value or the best solution found by the branch-and-bound algorithm if
the node limit (NL) or time limit (TL) is reached. The fifth column
contains the number of nodes explored in the branch-and-bound tree,
and the last column presents the CPU execution time.
From Table 5, we note that the CPU execution time of the branchand-bound algorithm increases dramatically with increasing and
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of links
OC-3
OC-12
1
0
2
0
2
0
1
0
0
2
0
2
0
2
1
0
0
2
1
0
0
1
1
0
2
0
1
0
2
0
Table 4
O-POP
Type
Client
Number and types of backbone links for each O-POP
Table 2
Characteristics of the types of interface cards
Number of links
OC-3
OC-12
1
0
1
0
2
0
0
2
2
0
1
0
0
2
2
0
2
0
0
2
0
1
1
0
1
0
2
0
2
0
1
2
3
4
5
6
7
8
9
10
Number of links
OC-12
OC-48
1
2
3
4
3
0
1
0
3
1
0
3
2
2
1
2
2
0
2
0
O-POP
11
12
13
14
15
16
17
18
19
20
Number of links
OC-12
OC-48
4
3
1
1
1
0
1
2
3
4
2
0
2
3
2
1
3
4
4
4
. The results show that fair-sized problems can be solved to optimality within branch-and-bound nodes and hours of CPU
time. Only three out of problems were not solved within these limits. However, P cannot be solved for large problems with this approach.
IV. Conclusions
In this paper, we have studied the problem of designing POPs in IP
networks and present an optimization model for selecting the number
of routers and their types, selecting the interface card types, connecting
the access and backbone links to the ports, and selecting the link types
between the co-located routers.
The model can be adapted to different objectives and situations by
changing the cost function or by adding appropriate constraints. For
instance, the model can be used for the POP expansion problem. Indeed, one has only to fix a subset of the variables for representing the
already existing configuration of the POP and solving the model to find
the other variables.
The model presented has some limitations. For instance, it is difficult to predict the traffic passing through the POP (i.e., the values),
and the cost of adding another client is not taken into account. More-
CHAMBERLAND / ST-HILAIRE / PIERRE: ON THE POP OPTIMIZATION PROBLEM IN IP NETWORKS
7
References
Table 5
Numerical results
Problem
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
P16
P17
P18
P19
P20
P21
P22
P23
P24
P25
10
20
30
40
50
10
20
30
40
50
10
20
30
40
50
10
20
30
40
50
10
20
30
40
50
5
5
5
5
5
10
10
10
10
10
15
15
15
15
15
20
20
20
20
20
25
25
25
25
25
OPT ($)
157 000
221 000
212 000
375 500
368 500
338 000
304 000
427 500
389 500
472 500
328 000
564 500
519 500
537 500
604 500
595 500
656 500
585 500
NL(922 500)
TL(911 500)
710 500
579 500
711 500
TL(1 014 500)
620 500
Nodes
37
38
99
10 721
1475
101
123
5633
21 647
31 273
68
23 863
8979
21 023
36 357
66 805
12 626
4735
100 000
46 163
1139
18 465
22 778
77 301
3492
CPU (s)
4
7
20
782
219
8
28
625
3273
5007
19
2070
2491
6450
12 591
4430
2106
1179
30 509
43 200
355
2800
12 979
43 200
9985
over, performance and reliability constraints need to be included in the
model. We are currently examining these extensions.
A detailed example solved by using a branch-and-bound algorithm
has been presented along with results for a set of randomly generated
problems with up to clients and other POPs. The results show
that fair-sized problems can be solved to optimality within branch-and-bound nodes and hours of CPU time. Only three out
of problems were not solved within these limits. However, P cannot be solved for large problems (with hundreds of clients) with this
approach.
It would be interesting to explore other exact algorithms, such as the
constraints programming techniques, in order to find exact solutions
more rapidly and to tackle large problems.
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