P. Lorenz (Ed.): ICN 2001, LNCS 2094, pp. 148–157, 2001 © Springer-Verlag Berlin Heidelberg 2001
Hong-Hsu Yen and Frank Yeong-Sung Lin
Department of Information Management, National Taiwan University, Taiwan
{d4725001, yslin}@im.ntu.edu.tw
Abstract. In this paper, we consider the backbone network design problem
with a full set of QoS requirements. Unlike previous researches, we consider both the transmission line cost and the switch cost. And the QoS requirements that we considered include the average packet delay, end-to-end packet delay and node disjoint paths. We formulate the problem as a combinatorial optimiza-tion problem where the objective funcoptimiza-tion is to minimize the total network de-ployment cost subject to the aforementioned QoS constraints. Besides the inte-grality constraints, the nonlinear and the nonconvex properties associated with the problem formulation make it difficult. Lagrangean relaxation in conjunction with a number of optimization-based heuristics are proposed to solve this prob-lem. From the computational experiments, the proposed algorithms calculate creditable solutions in minutes of CPU time for moderate problem sizes.
1 Introduction
How to design a usually sophisticated backbone network with the minimum deploy-ment and operation cost subject to various and often stringent QoS requiredeploy-ments is a common challenge faced by network designers and managers. Intensive research has been conducted to address this issue. However, most research tackles this backbone network design problem without considering a full set of QoS requirements.
Gavish model the network topological design problem as a nonlinear combinato-rial optimization problem. The objective is to minimize the network installation cost and the queueing cost imposed on the network users. However, network installation cost and the queueing cost are two different concepts such that it is not appropriate to put them together in the objective function. In addition, end-to-end QoS constraints are not considered in [3].
A bunch of heuristics based on genetic algorithms (GA) and Tabu-Search algo-rithms are proposed to tackle the backbone network design problem [2, 5, 6]. How-ever, in [2, 5, 6], the cost for network access point is not considered in the network installation cost. Furthermore, average and end-to-end delay constraints and reliabil-ity constraint are not jointly considered at the same time.
The delay QoS requirement is crucial to modern application services (e.g. VOD, tele-conferencing). And in particular, backbone network usually requires high avail-ability, that is, redundant links and switching nodes are needed in case of failure. As a
result, unlike more of previous research, the QoS requirement considered in this pa-per can be classified into two parts. The first part is the delay QoS (including the average packet delay and end-to-end delay for each O-D pair) and the second part is the reliability and availability of services. On the other hand, the network construction cost includes both the switch and link installation cost to reflect the cost structure of real network. This problem is a well-known difficult NP-hard problem.
This paper is organized as follows. In Section 2, mathematical formulation of the backbone network design problem is proposed. In Section 3, the dual approach for the backbone network design problem based on the Lagrangean relaxation is pre-sented. In Section 4, the getting primal heuristics are developed to get the primal feasible solutions from the solutions to the dual problem. In Section 5, the computa-tional results are reported. In Section 6, the concluding remarks are presented.
2 Problem Formulation
This QoS based backbone network design problem is modeled as the graph where the users and switches are depicted as nodes and the communication channels are de-picted as arcs. We show the definition of the following notation.
L The set of candidate local loop links and backbone links in the communi-cation network.
W The set of origin-destination (O-D) pairs in the network.
w
λ The traffic requirement for each O-D pair w∈W.
l
C a capacity upper bound in the candidate capacity configurations for link
l
∈
L.Pw a given set of simple directed paths from the origin to the destination of
O-D pair w.
Uk a set of potential incoming links to switch k.
pl
δ the indicator function which is one if link l is on path p and zero otherwise. pk
ε
the indicator function which is one if switch k is on path p and zero other-wise.gl the aggregate flow over link l∈L, which is
∑ ∑
∈Pw ∈ p wW pl w p x λ δ . Dw the maximum allowable end-to-end delay requirement for O-D pair w.
K the maximum allowable average cross network delay requirement. Tw the minimum number of node disjoint paths required for O-D pair w.
O the set of candidate locations for switches.
H the set of link pairs that are with the same end points but in opposite direc-tions.
Al the set of candidate capacity configurations for link l.
Rk the set of admissible switching fabric configuration for switch at location
Ek the set of candidate port configuration for switch at location k.
) ( l
l C
ϕ the cost for installing capacity Cl on link l, including the fixed and variable
cost. l
ξ the fixed link installation cost for link l.
Qk(Jk, Sk) the cost for installing a switch at location k with switching fabric capacity
Jk and number of ports Sk.
Fl(fl, Cl) the average delay on link l∈L, which is a function of fland Cl.
Bl(fl, Cl) the average number of packets on link l
∈
L, which is a function of fland Cl, and by the Little’s results, which is equal to λw*Fl(fl,Cl).
And the decision variables are depicted as follows.
xp 1 when path p
∈
Pwis used to transmit the packets for O-D pair w∈
W and 0 otherwise.zp 1 when path p
∈
Pwis the node disjoint path for O-D pair w∈W and 0other-wise. y
wl
1 when link l∈L is on the path chosen for O-D pair w∈W and 0 otherwise. fl the estimated aggregate flow on link l∈L.
M
l
1 when a link is installed at location l∈Land 0 otherwise. C
l
the capacity assignment for link l∈L.
Jk the switching fabric capacity assignment for switch at location k.
S
k
the number of ports for switch at location k.
We formulate the QoS based backbone network design problem as a nonlinear and nonconvex combinatorial optimization problem, as shown below.
min
∑
∑
∑
∈ ∈ ∈ − + = H l l l l l k k O k k l L l l IP C Q J S M M Z ) , ( ) , ( ) ( ξ ϕ (IP) subject to:∑
∑
∈ ∈ l L l l l W w w C f B ( , ) 1 λ≤
K (1)∑
∈ ≤ L l wl l l l w D C f F y ( , ) ∀w∈W (2)∑
∈ = w P p p x 1 ∀w∈W (3) xp= 0 or 1 ∀p∈Pw,w∈W (4)∑
∈Pw p pl p x δ≤
ywl ∀w∈W,l∈L (5) wl y = 0 or 1 ∀w∈W,l∈L (6)l W w p P w pl p l x f g w ≤ =
∑ ∑
∈ ∈ δ λ L l∈ ∀ (7) l l C f ≤ ∀l∈L (8) l l A C ∈ ∀l∈L (9) Ml = 0 or 1 ∀l∈L (10) l l l CM C ≤ ∀l∈L (11) k k R J ∈ ∀k∈O (12)∑
∈ ≤ k U l l k S M ∀k∈O (13) k k E S ∈ ∀k∈O (14) k W w p P w pk p J x w ≤∑ ∑
∈ ∈ λ ε ∀k∈O (15)∑
∈ = w P p w p T z ∀w∈W (16)∑
∈ ≤ w P p l pl p M z δ ∀w∈W,l∈L (17) zp= 0 or 1 ∀p∈Pw,w∈W . (18)The objective is to minimize network installation cost. There are three terms in the objective function. The first term is to compute the total link installation cost, includ-ing the fixed cost and the variable cost. The second term is to compute the total switch installation cost. The third term is to compute one fixed cost for each installed opposite links. The necessity of subtracting the third term is to ensure that only one rather than two fixed cost is calculated for two links with the same attached nodes but in opposite direction. Constraint (1) enforce the average cross network delay con-straint. Constraint (2) enforce the end-to-end packet delay for each O-D pair. Con-straints (3) and (4) require that the all the traffic for each O-D pair should be transmit-ted over exactly one path. The decision variable
y
wl in Constraint (6) is an auxiliary decision variable, which is equal to∑
∈Pw
p pl p
x δ . Hence, the equality in Constraint (5) is replaced by inequality due to the ease use of the Lagrangean relaxation. Constraints (7) and (8) are the link capacity constraints. Constraint (9) determines the possible capacity configurations of all links. Constraints (10) and (11) require that the link must be installed first before link capacity assignment. Constraints (12) and (14) determine the possible switching fabric and number of ports of all switches. Con-straint (13) is the switch termination conCon-straint, which means the number of incoming links to the switch should not exceed the number of ports on that switch. Constraint (15) is the switch capacity constraint Constraints (16) and (18) are the path diversity (node disjoint) requirement for each O-D pair. Constraint (17) guarantees that link must be installed first before it could be adopted on the node disjoint path for each O-D pair.
3 Lagrangean Relaxation
We dualize Constraints (1), (2), (5), (7), (8), (11), (13), (15) and (17) of Problem (IP) to get the following Lagrangean relaxation problem (LR).
min ZD = ( ) ( k, k) O k k l L l l C
∑
Q J S∑
∈ ∈ + ϕ∑
∈ − H l l l l lM M ) , ( ξ +a∑
−∑
∈ ∈ l L l l l W w w C f B ( , ) 1 [ λ K] +∑
[ ∈W w w b∑
( , ) ] ∈ − L l w l l l wlF f C D y +∑ ∑
[ ∈W ∈ w wl L l c∑
− ∈Pw p pl p x δ ywl]+ [ l l] L l l f C h −∑
∈ + + −∑ ∑
∑
∈ ∈ ∈ ] [ l W w p P p pl w L l l f x d w λ δ [ l l l] L l l M C C e −∑
∈ + [ k] W w p P p pk w O k k J x n w −∑ ∑
∑
∈ ∈ ∈ ε λ + +∑
[∑
] ∈ ∈ − k U l l k O k k S M m +∑
∑
[∑
] ∈ ∈ ∈ − w P p p pl l L l wl W w M z q δ (LR) subject to:∑
∈ = w P p p x 1 ∀w∈W (19) xp= 0 or 1 ∀p∈Pw,w∈W (20) wly
= 0 or 1 ∀w∈W,l∈L (21) l l A C ∈ ∀l∈L (22) Ml = 0 or 1 ∀l∈L (23) k k R J ∈ ∀k∈O (24) k k E S ∈ ∀k∈O (25)∑
∈ = w P p w p T z ∀w∈W (26) zp= 0 or 1 ∀p∈Pw,w∈W . (27)We can decompose (LR) into five independent subproblems. Subproblem 1: for xp: min
∑ ∑ ∑
∑
∈ ∈ ∈ ∈ + + O k k w p pk W w p P l L wl l w p pl x n x d c w ] ) ( [ λ δ λ ε (SUB1)subject to: (19) and (20). Subproblem 2 : for Cl, ywl and fl:
min ( l) L l l C
∑
∈ ϕ +a∑
∑
∈ ∈ l L l l l W w w C f B ( , ) 1 λ +∑
∈ l∑
∈L − l l l wl W w w y F f C b ( , )∑ ∑
− ∈W ∈ w wl wl L l y c l L l lf d∑
∈ + l L l lC e∑
∈ +∑
− ∈L l l lf h l L l lC h∑
∈ (SUB2) subject to: (21) and (22).Subproblem 3: for Ml: min l l L l l M C e
∑
∈ − +∑
∑
∈ ∈ lUk l k O k M m∑
∈ − H l l l l lM M ) , ( ξ∑
∑
∈ ∈ − L l wl l W w M q (SUB3) subject to: (23).min ][ ( k, k) k k k k O k k J S n J m S Q − −
∑
∈ (SUB4) subject to: (24) and (25).Subproblem 5: for zp: min
∑
∑ ∑
∈ ∈ ∈ l L p P pl p wl W w w z q δ (SUB5)subject to: (26) and (27).
In order to deal with the nodal weight of (SUB1), the node spiltting technique [4] is used. As a result, (SUB1) could be further decomposed into W independent shortest path problem with nonnegative arc weights. It can be easily solved by the Dijkstra’s algorithm. (SUB2) could also be decomposed into L independent sub-problems. For each link l∈L,
Subproblem 2.1: for Cl, ywl and fl:
min )ϕl(Cl +a ( , ) 1 l l l W w w C f B
∑
∈ λ +∑
∈W w w b ywlFl(fl,Cl) wl W w c∑
∈ −y
wl −dlfl+elCl+ − l lf h hlCl (SUB2.1)subject to: ywl= 0 or 1 ∀w∈W and Cl∈Al.
(SUB2.1) is a complicated problem due to the coupling of three decision variables, Cl, ywl and fl . Since the possible capacity configurations of links are finite, such as
64kbps, 128kbps, 256kbps, 512kbps, T1 and T3 for example. We can exhaustive search all different possible link configuration by finding the best ywl and fl. In [1],
Lin proposed an efficient algorithm to solve ywl and fl at a given link capacity under
M/M/1 queuing model. Therefore, the algorithm to solve (SUB2.1) under M/M/1 queuing model is proposed as bellow. In addition, the formulation could be extended to any non M/M/1 model with monotonically increasing and convexity performance metrics.
Step 1. For each possible link capacity configuration, applying the algorithm devel-oped in [1] to solve (SUB2.1) as to find the optimal ywl and fl .
Step 2. Finding the minimum objective value of (SUB2.1) from the objective value associated with each possible link capacity configuration. Then ywl and fl can
be determined from the optimal link capacity.
(SUB3) can be decomposed into H independent subproblems. For each pair of bi-directional links (l,l)∈H,
Subproblem 3.1: for Mland Ml:
min −elClMl −elClMl+G1Ml+G2Ml −ξlMlMl wl l W w M q
∑
∈ − wl l W w M q∑
∈ − (SUB3.1) subject to: Ml= 0 or 1 and Ml= 0 or 1.1. If the link l is the incoming link to any potential switch, say k1, then assign G1 to
1
k
m , else assign G1 to zero.
2. If the link l is the incoming link to any potential switch, say k2, then assign G2 to
2
k
m , else assign G2 to zero.
In (SUB3.1), two opposite direction links are considered at the same time. As a re-sult, the algorithm to optimally solve (SUB3.1) is proposed as follows.
Step 1. Let N1 = 0, N2 =−elCl +G1 wl W w q
∑
∈ − , N3 =−elCl+G2 wl W w q∑
∈ − , N4 = l lC e − +G1−elCl+G2 −ξl wl W w q∑
∈ − wl W w q∑
∈ − .Step 2. Identify the Ni with the minimum value, where i = 1, 2, 3, 4.
Step 3. If i = 1, then assign Ml= 0 and Ml= 0, else if i = 2, then assign Ml= 1 and l
M = 0, else if i = 3, then assign Ml= 0 and Ml= 1, else if i = 4, then assign l
M = 1 and Ml= 1.
(SUB4) can be further decomposed into O independent subproblems. For each independent subproblem, due to the number of possible switch configurations (in-cluding number of ports and switching fabric) is finite and manageable within com-putational time, we can exhaustively search all possible combination of switch con-figurations as to find the optimal Jk and Sk.
(SUB5) can be further decomposed into W independent node disjoint shortest path problem with nonnegative arc weights. Suurballe propose an efficient algorithm to optimally solve link disjoint path problem [7]. Hence, (SUB5) could be optimally solved by the Suurballe’s algorithms in conjunction with the node splitting technique. According to the algorithms developed above to solve each subproblem, we could successfully solve the Lagrangean relaxation problem optimally. By using the weak Lagrangean duality theorem (for any given set of non-negative multipliers, the opti-mal objective function value of the corresponding Lagrangean relaxation problem is a lower bound on the optimal objective function value of the primal problem), ZD is a
lower bound on ZIP. We could construct the dual problem to calculate the tightest
lower bound and solve the dual problem by using the subgradient method.
4 Getting Primal Feasible Solutions
To obtain the primal solutions to the (IP), solutions to the (LR) are considered. We develop sophisticated getting primal heuristics to getting the primal feasible solutions. This getting primal heuristic start with the routing assignment obtained from the (SUB2.1). From the routing assignment in (SUB2.1), the aggregate flow on each link can be calculated. In order to satisfy the end-to-end delay requirement for each O-D pair, the tightest end-to-end delay for all O-D pairs is located by searching the mini-mum end delay requirement among all O-D pairs. From the tightest
end-to-end delay, the tightest link delay can be calculated by dividing the tightest end-to-end-to-end-to-end delay to the maximum hop number in any routing path. The maximum hop number for any O-D pair is equal to the number of potential switches plus one, since the source node must home to the switch first, and then route to the other switches, and finally route to the destination node. From the tightest link delay, we can determine the minimum link capacity in order to satisfy the tightest link delay requirement. From the above statement, we could satisfy the delay requirements.
In order to satisfy the node disjoint requirement for each O-D pair, the node dis-joint path assignment from (SUB5) is used. If the associated link on any node disdis-joint path did not install at the above procedure, the minimum nonzero capacity is installed on that link. After the link capacity is determined, the number of links incoming to each potential switch can be determined. Also from the aggregate flow on each link, the total aggregate flow incoming to each potential switch can also be determined. As a result, the minimum cost switch configuration in order to satisfy the number of ports and switch fabric constraints can be determined as well.
5 Computational Experiments
The network planning algorithms developed in Section 3 and 4 are coded in C++ and performed at PC with INTELTM PIII-800 CPU. The input parameters include the
loca-tions for the users and the potential switches, admissible configuraloca-tions and cost structures of potential switches and links, traffic requirements and survivabil-ity/connectivity requirements. And the output parameters include the switch and link configuration assignment, routing assignment, node disjoint paths assignment, aver-age end-to-end delay and individual end-to-end delay for each O-D pair.
The maximum number of iterations for the proposed dual Lagrangean algorithm developed above are 1000, and the improvement counter is 30. The step size for the dual Lagrangean algorithm is initialized to be 2 and be halved of its value when the objective value of the dual algorithm does not improve for 30 iterations.
Two sets of computational experiments are performed. The computational time for these two sets of computational experiments are all within fifteen minutes under the network size of 30 user/switch nodes. Hence, the proposed algorithms are efficient in time complexity.
In these computational experiments, the cost of the link assignment is divided into two parts, fixed cost and variable cost. The fixed cost is calculated from the Euclidean distance between two end points that the link connected, and the variable cost is based on the link capacity configuration. There are fifteen discrete potential link ca-pacity configurations, from 0 to 500, for the computational experiments. And the cost associated with these potential capacity configurations is a concave function to reflect the economy-of-scale effect. On the other hand, the switch installation cost is based upon the switching fabric and the number of ports on the switch. There are nineteen discrete potential switch configurations in the computational experiments. And the cost associated with these potential switch configurations is also a concave function.
In the first set of computational experiment, we want to test the solution quality when the input delay requirements are loose as compared to the output of the delay requirement. And node disjoint path requirement is not considered. In the second set of computational experiment, we want to test the solution quality when the input delay requirements are tight as compared to the output of the delay requirement. And the 2-connected node-disjoint-path requirement for each O-D pair is considered.
Table 1 depicts the computational results for the various network sizes and traffic demand without node disjoint requirement and loose delay requirements. The first column is the network size. The location, x-axis and y-axis, of user nodes and poten-tial switch nodes are randomly distributed between the 0 and 500. The second column is the traffic demand of the user nodes, are randomly distributed between 30 to 400. The third column reports the lower bound of the primal problem. The forth column reports the upper bound of the primal problem. The fifth column reports the error gap between the lower bound and upper bound. The seventh column reports the average network delay requirement, and the sixth column reports the average network delay calculated by the proposed algorithms. As could be seen from the fifth column in Table 1, the more number of user/switch nodes the looser the error gap. And we have a tighter error gap for heavy traffic in the same network topology.
Table 1. Solution quality obtained by various network sizes and traffic demand without node disjoint requirement
# of
Us-ers / switches
Traffic
Demand boundLower boundUpper Gap(%) Error network delayAverage K
9 30~200 943.4 1639.9 73.8 0.168 10 9 60~400 1852.4 2778.0 49.9 0.068 10 12 30~200 1639.8 2664.8 62.5 0.079 10 12 60~400 3048.0 4353.1 42.8 0.069 10 15 30~200 1895.1 2918.9 54.0 0.294 10 15 60~400 3695.3 5291.5 43.2 0.131 10 30 30~200 2496.9 6042.3 141.9 0.397 10 30 60~400 4442.8 7936.1 78.6 0.117 10
Table 2. Solution quality obtained by various network sizes and traffic demand with two node disjoint requirement
# of Users
/ switches Demand Traffic boundLower boundUpper Gap(%) Error network delayAverage K
9 30~200 941.5 1980.3 110.3 0.195 1 9 60~400 1852.6 2909.0 57.0 0.054 1 12 30~200 1636.2 3220.0 90.7 0.087 1 12 60~400 3048.0 4353.1 42.8 0.069 1 15 30~200 1887.3 3369.6 78.5 0.203 1 15 60~400 3694.9 5667.0 53.4 0.130 1 30 30~200 2472.0 7098.8 187.2 0.452 1 30 60~400 4388.6 8628.5 96.6 0.130 1
Table 2 depicts the computational results for the various network sizes and traffic demand with two node disjoint requirement and loose delay requirements. As com-pared to Table 1, the average cross network delay and the end-to-end delay require-ments are more stringent, delay = 1 instead of 10. On the other hand, there are two node disjoint requirement for each user nodes. As could be seen from the third col-umn of Table 2, we have a looser lower bound in Table 2. However, we still have a reasonable good upper bound in Table 2.
6 Concluding Remarks
In this paper, the mathematical formulation and algorithms for the backbone network design which considers the system and user specified QoS requirements are proposed. The objective of this backbone network design problem is to minimize the total in-stallation cost of link and switch inin-stallation cost. The system QoS requirement is the average cross network delay requirement. The user specified QoS requirements in-clude the end-to-end delay requirement and node disjoint path requirements. Besides integrality constraints, the non-convexity of the delay performance metric makes this problem difficult. By using the Lagrangean relaxation method and the subgradient method to construct the dual problem and calculate the tightest lower bound, we pro-vide getting primal feasible solution heuristic to obtain the primal feasible solution based on the solutions to the dual problem. Based on the solution quality and the computational time, we propose effective and efficient algorithms for this problem.
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