Technical Report 04-EMIS-12 An equal-opportunity-loss MPLS-based network design model by Richard S. Barr 1 Richard V. Helgason 1 Maya G. Petkova 1 and Saib Jarrar 2 1 {helgason, barr, maya}@engr.smu.edu EMIS Department School of Engineering Southern Methodist University Dallas, TX 75275 2 [email protected]MCI Data Network Engineering Richardson, TX 75081 September 2004 Presented at the CORS/INFORMS Joint International Meeting, May 2004, Banff, Alberta, Canada
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Technical Report 04-EMIS-12
An equal-opportunity-loss MPLS-based network design model
by
Richard S. Barr1
Richard V. Helgason1
Maya G. Petkova1
and
Saib Jarrar2
1{helgason, barr, maya}@engr.smu.edu EMIS Department
School of Engineering Southern Methodist University
The objective function simply maximizes the guaranteed percentage of delivered traffic demand
for all commodities. Constraints (1), (2), (3), and (4) are flow conservation equations, which ensure a
connected path for each routed commodity. Constraints (5) guarantee that a link can be used no more than
once in the path designed to route each commodity. Constraints (6) are traffic performance constraints,
which ensure that the number of hops along any path cannot exceed a predetermined upper hop limit h .
Constraints (7), (8), (9) and (10) are used to ensure a single traffic delivery path for each commodity. (We
are only interested in networks which have bandwidth capacity enough to deliver at least 51% of the
requested demand for each OD pair). Constraint sets (11) and (12) enforce the link capacity resource
constraints. Constraints (13) impose the natural upper bound for the percentage fulfillment of the traffic
demand for each commodity. Constraints (14) impose the uniform lower bound on the percentage delivered
traffic demand for each commodity. Constrains (15) are nonnegativity constraints.
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5. The Basic EOL Model ( Stage II).
The Stage II equal-opportunity-loss model uses the optimal uniform traffic flow lower bound
determined using the Stage I model to design traffic engineered paths by routing as much demand as
possible ( thus optimizing the revenue), so that each commodity’s delivered demand is at or above the
guaranteed lower bound.
5.1.Mathematical Model: Stage II.
Let *D is the optimal solution obtained using the Stage I model. The second stage model is:
Maximize
∑∑ ∑∈<∈ ∈
−+−Nnc
cneeTraffNnc Ee
ecncn HopswYXcwTraffDelivcn ,
20:,
1 )())(( σσµ
Subject to:
Constraint sets (1) to (13), and (15) from stage I and a new constraint set (14*) as follows.
0: , * <∈∀∈∀≥ cncn TraffNnNcDDeliv (14*)
In the II stage formulation, the objective function consists of three terms with the first one being
the primary objective and the dominant term. The first term represents the total revenue generated from the
routed commodities (i.e. the delivered traffic). The total demand delivered (the revenue) is maximized.
The second term represents the total delay incurred by all the delivered traffic. The purpose of this term is
to select the solution with the lowest delay among multiple alternate optimum solutions (yielding the same
revenue) that may exist. The total delay is multiplied by the scaling factor , 01 1 >> ω . Typically, 1ω is
set so that the second term will be small relative to the first (dominant) term. The third term represents the
total number of links used (total number of hops). Its purpose is to minimize the total number of hops in
order to avoid cycling, which could be generated in attempting to maximize the revenue (delivered traffic).
It is also multiplied by a scaling factor 01 2 >> ω , 2ω usually greater than second term’s scaling factor
1ω .The new constraint set (14*) ensures that each commodity’s delivered demand is at or above the
guaranteed lower bound.
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6. Parametric Study.
If the total requested packet traffic for all OD pairs cannot be delivered, a set of congested links
based on the flows from stage II is constructed. A link is considered to be congested if at least 98% of its
capacity has been used. A parametric study was conducted to assess the outcomes from individually
doubling the capacities of the congested links in an attempt to find the link which will give the largest
revenue increase.
7. Optimization Model. To suppliment the parametric study, an ancillary optimization model was built as an enhancement
of the Stage II model. The goal was to determine by optimization the link (links), which contribute to the
largest revenue increase if their capacities are doubled.
Two new parameters for the optimization model are introduced.
eSatur
=otherwise ,0
saturated is link if ,1 eSature
nl the number of links, which capacity is to be doubled, 20 ≤< nl
Further a new decision variable is introduced.
eDcap
=otherwise ,0
doubled be tohas link ofcapacity theif ,1 eDcape
The optimization model for finding the optimal link (links), whose capacities have to be doubled is
described below.
Maximize
∑∑ ∑∈<∈ ∈
−+−Nnc
cneeTraffNnc Ee
ecncn HopswYXcwTraffDelivcn ,
20:,
1 )())(( σσµ
Subject to:
Constraint sets (1) to (10), constraint sets (13), (14*), and (15).
The capacity constraints sets (11) and (12) are replaced by
EeDcapbbX eeee ∈∀+≤ σ (11’)
EeDcapbbY eeee ∈∀+≤ σ (12’)
The objective function is the same as in the second stage model. The new constraint sets (11’) and (12’)
account for the new, increased link capacity.
Two new constraints sets were added to this optimization model
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EeSaturDcap ee ∈∀≤ (16)
∑∈
≤Ee
e nlDcap (17)
Constraint set (16) ensures that only the capacity of the saturated (congested links) can be doubled.
Constraint set (17) imposes number of links whose capacity is to be doubled concurrently.
8. Computational Experiments.
8.1. The Test Network.
The EOL model was tested on a realistic network, which has the typical topology of a nationwide
data communications network The example network is shown in Fig.1 and its description is given below.
? 20 nodes, 31 links
? Average node degree ~3
? Link capacities:
• 2488 Mbps (15 links) - OC48 transmission line
• 622 Mbps ( 16 links) - OC12 transmission line
? Trunks connecting the nodes are bi-directional and full duplex
Figure 1. Network topology of realistic test network
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17
20
11
9
13
8
19
4
515
1
14
2
3
6
7
10
12
18OC-48 (2488 Mbps)
OC-12 (622 Mbps)
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8.2. Computing Environment.
Tests were performed on a Compaq AlphaServer DS20E with dual 667 MHz processors and 4096
MB RAM. Machine is configured as a Network Queuing System executing batch jobs. Each job has access
to approximately 20 MB RAM. Models were implemented using AMPL 8.0. Integer programming
solutions were generated using the CPLEX Linear Optimizer 8.0. Default settings for CPLEX were used
except that the MIP time limit was set to 1500 seconds.
8.3. Data sets .
A traffic generator was used to generate multiple sets of commodities and traffic demands. OD
pairs were selected randomly and uniformly from the set of nodes (no duplicates allowed). The demands
associated with the OD pairs were selected randomly using a uniform distribution over the range specified
by the min and max demands. Table 1 summarizes the characteristics of the data sets used and the
solutions obtained from by the EOL stage I model.
The experimental results for each individual data set are given in the Appendix. The tables show
three performance metrics for each of the presented solution strategies: percentage of revenue missed,
bandwidth utilization, and bandwidth efficiency (see [4]). The percentage of revenue missed is defined as
the ratio of the total demand not delivered to the total demand. Bandwidth utilization is defined as the ratio
of total flow on all arcs to the total bandwidth of all arcs. Bandwidth efficiency is defined as the ratio of
total demand delivered to the total bandwidth of all arcs. These metrics are helpful to make general
observations about the behavior of the different solution strategies and can be further used for hypothesis
testing.
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SET # OD
Pairs Demand Range
Mean Demand
Guaranteed % Delivered
Demand
DS1 80 320 240 74.67
DS2 80 320 240 69.57
DS3 160 320 240 < 51.00
DS4 320 320 240 < 51.00
DS5 80 160 120 100.00
DS6 160 160 120 82.49
DS7 160 160 120 90.67
DS8 320 160 120 51.41
DS9 80 80 60 100.00
DS10 160 80 60 100.00
DS11 320 80 60 95.55
DS12 320 80 60 89.20
DS13 80 160 480 < 51.00
DS14 80 80 480 < 51.00
TABLE 1. DATA SETS
9. Summary and Conclusions.
A two stage equal-opportunity-loss model for solving a funadmental TE problem for MPLS -based
networks is formulated. The first stage of the model finds the guaranteed equal level (percentage) of traffic
that can be delivered for all commodities by determining the maximal concurrent traffic flow lower bound.
The concept of EOL fairness in traffic delivery was introduced. The model treats all demand pairs fairly
and guarantees and guarantees that there is a bandwidth allocation for each commodity which will allow
this lower bound percentage of given demand to be delivered for all commodities. The second stage
designs the paths by routing as much demand as possible so that each commodity’s delivered demand is at
or above the guaranteed lower bound. A parametric study was conducted to assess the outcomes from
individually doubling the capacities of the congested links in attempt to find the link (links) which will give
the largest revenue increase. An optimization model was built as an enhancement of the Stage II model to
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determine which link (links) will produce the largest revenue increase if their capacities are doubled. The
optimization model result can be compared to the results from the parametric study and used for making
cost-effective decisions. Network managers can benefit from utilizing the parametric study along with the
optimization model for capacity planning, revenue management, optimal resource allocation.
REFERENCES
[1] AWDUCHE, D. O. MPLS and traffic engineering in IP networks. IEEE Communications Magazine 37:12 (1999), 42-47.
[2] GUICHARD, I., AND PEPELNJAK, I. MPLS and VPN Architectures, Cisco Press, Indianapolis, IN, 2001.
[3] GUICHARD, I., PEPELNJAK, I., AND APCAR, J. MPLS and VPN Architectures, Volume II, Cisco Press, Indianapolis, IN, 2003.
[4] JARRAR, S. Formulation and evaluation of optimization models for MPLS traffic engineering with QoS requirements. D.Eng Praxis, Southern Methodist University, Dallas, TX, 2004.
[5] KENNINGTON, J. L. EMIS 8392 Class Notes: Prospects for Operations Research in the Design and Analysis of Telecommunications Networks, (Summer 2002).
[6] MATULA , D. W., AND SHAHROKHI, F. The maximum concurrent flow problem. JACM 37 (1990), 318-334.
[7] POSTEL, J. DoD standard transmission control protocol. RFC 761, Internet Engineering Task Force, http://www.ietf.org, 1980.
[8] POSTEL, J. Internet protocol. RFC 791, Internet Engineering Task Force, http://www.ietf.org, 1981.
[9] ROSEN, E., VISWANASAN, A., AND CALLON, R., Multiprotocol Label Switching Architecture, RFC 3031, Internet Engineering Task Force, http://www.ietf.org, 2001.