________________________________________________________________________

Queueing Delay Guarantees in Bandwidth Packing

________________________________________________________________________

ERIK ROLLAND1, ALI AMIRI2 and REZA BARKHI3

1 Department of Accounting & Management Information Systems, Fisher College of BusinessThe Ohio State University, Columbus, OH 43210E-mail: rolland.1@osu.edu

2 Department of Management, College of Business AdministrationOklahoma State University, Stillwater, OK 74078E-mail: aamiri@okway.okstate.edu

3 Department of Accounting & Information Systems, Pamplin College of BusinessVirginia Polytechnic and State University, Blacksburg, VA 24061-0101E-mail: reza@vt.edu

Draft date: September 29, 1998

This paper is to appear in Computers and Operations Research, 1998.________________________________________________________________________

Abstract.

This paper proposes a new formulation for the bandwidth packing problem, assuringmaximum service delay in telecommunications networks. The bandwidth packing problem isone of selecting calls, from a list of requests, to be routed in the network. We limit themaximum queueing delay, while maximizing revenues generated from the routed calls. Anefficient Lagrangean relaxation based heuristic procedure for finding bounds and solutions tothe problem is demonstrated, and computational results from a variety of problem instances arereported. We show that the procedure is both efficient and effective in finding good solutions.

________________________________________________________________________

Key words: Bandwidth packing, call routing, queueing delay, telecommunications networks,Lagrangean relaxation, subgradient search, heuristics.

1

Statement of Scope and Purpose

The bandwidth packing problem is one of selecting and routing calls in a

telecommunications network. The selection is normally performed as to maximize the

revenues from the calls routed. However, this may cause serious queueing delays in the

network, possibly causing lost profitability and lost customer satisfaction for the network

owners. The scope of this paper is to propose a mathematical formulation that addresses the

bandwidth packing problem – one that maximizes revenues but also at the same time limits the

maximum queueing delays in the network. In addition, we propose a Lagrangian-based

solution procedure that produces both lower bounds and high quality solutions for the

bandwidth packing problem.

1

Ali Amiri is an Assistant Professor of Management Sciences and Information Systems atOklahoma State University. He received the BS degree in Business Administration in 1985from the IHEC, Tunisia, the MBA in 1988 and the Ph.D. in Management Sciences andInformation Systems in 1992 from Ohio State University. His research interests include datacommunication and computer network design and analysis, databases, combinatorial anddiscrete optimization and general OR/MS modeling and analysis. He has published inComputer Communications, Computers & Operations Research, European Journal ofOperational Research, and Naval Research Logistics.

Reza Barkhi is an Assistant Professor of MIS in the Department of Accounting andInformation Systems, Pamplin College of Business, at Virginia Polytechnic Institute & StateUniversity. His current research interests are in the areas of collaborative technologies andproblem solving, and topological design of telecommunication networks. Dr. Barkhi haspublished in journals such as Location Science, European Journal of Operational Research,Group Decision and Negotiation, and Decision Support Systems. He received a BS in CISfrom the College of Engineering, and an MBA, an MA, and a Ph.D. from the College ofBusiness all from The Ohio State University.

Erik Rolland is an Assistant Professor of MIS in the Department of Accounting andManagement Information Systems, Fisher College of Business, at The Ohio State University.His research interests include management and design of telecommunications systems,combinatorial modeling and analysis, and strategic MIS. He has published in journals such asComputers & Operations Research, European Journal of Operational Research,Transportation Science, and Annals of Operations Research. He received a BS in CIS fromthe College of Engineering, and an MA and a Ph.D. from the College of Business all from TheOhio State University.

1

1. Introduction

The reliability and response times of telecommunications networks are major

factors affecting perceived quality of telecommunications services. Users have come to

expect 100% reliability, and virtually immediate response times. Telecommunications

companies must not only satisfy their customers by providing reliable and responsive

systems, but they also have a commitment to stakeholders to maximize profits.

Maximizing profits translates into decisions related to improving the utilization of the

network capacity.

Decisions that affect capacity utilization involve deciding which calls on a list of

requests, called a call table, should be routed on the network. Subsequently, a path for

each call to be routed must also be determined. This path should be selected from all

possible paths in the network. The complete network topology, as well as the call table,

the revenues, and the traffic requirements are given. This problem is typically referred

to as the bandwidth packing problem (BWP). Versions of this problem have been

studied by Amiri, Rolland & Barkhi [1], Anderson et al. [2], Laguna & Glover [13],

Cox et al. [4], Parker & Ryan [15], and Park et al. [14]. The objective of the BWP has

in these past research efforts been to maximize the total revenues from calls that are

routed without consideration to quality of service to users.

Route, or path, selection influences response time experienced by network users

and has a major effect on the utilization of network resources (e.g. node buffers and

communications links). A good routing policy would allow new users to use the

network without significant deterioration of the quality of service to existing users.

Lack of a good routing policy may require unnecessary capacity expansions to the

network.

In managing the network, one has to make tradeoffs between revenue

maximization and response time to users. If the only consideration is revenue

maximization, then network users may experience significant delays, and the quality of

2

service will suffer. The model developed in this paper incorporates response time by

including a non-linear constraint that limits the maximum queueing delay in the network

to a management specified upper level.

A version of the path assignment problem that considers only revenue

maximization has been previously addressed in [2], [13], [4], [15], and [14]. A wide

variety of solution procedures have been proposed: tabu search [2], [13], genetic

algorithms [4], column generation [15], as well as integer programming [14].

The authors in [1] addressed the issue of minimizing queueing delay in the

network. They include a cost term associated with this queueing delay in the objective

function of their model. This term is computed by multiplying total link queueing delay

by a unit delay cost. The main justification for using this term was to control the delay

in the network and therefore response time to users. It may be difficult to assign a

weight to this unit delay cost, and further the nature of the solution to the problem may

change adversely with this value. A better way to control response time to users

through queueing delays is to impose an upper limit on the link queueing delay in the

network. Since the upper limit (or bound) for the delay may not be exactly known, the

network designer or manager can start with an estimate of this bound (e.g., the delay

bound to obtain 60% average utilization of link capacity). With an increase in this

bound, total profit increases. By deciding on the level of tradeoffs between total profit

and quality of service to users, the network designer or manager can decide what delay

bound to impose depending on the strategy and priorities of the organization operating

the network.

Motivated by the important applications for path assignment in call routing,

customer satisfaction (i.e., reasonable response times) and the complexity of the

problem, we present a formulation that seeks to maximize total revenues of calls to be

routed while guaranteeing a certain level of quality of service to users. We develop a

procedure that generates feasible solutions as well as bounds for this problem.

3

The remainder of this paper is organized as follows. In section 2, a

mathematical formulation of the BWP problem is presented. A Lagrangean relaxation

of the problem obtained by dualizing a subset of the constraints is presented in section 3.

A heuristic solution procedure is developed in the following section. Computational

results are reported in section 5. The conclusions are summarized in the last section.

2. A Mathematical Problem Formulation

We introduce the following notation necessary for developing a mathematical

model for the BWP:

N the set of nodes in the network

E the set of undirected links (or arcs) in the network

M the set of calls. Each call is represented by a communicating node pair

dm the demand of call m ∈ M (e.g., the demand for network resources: bandwidth)

rm the revenue from call m ∈ M (e.g., monetary units)

O(m) the source node for call m ∈ M

D(m) the destination node for call m ∈ M

Qij the capacity of link (i,j) (bandwidth)

δ the upper limit on the queueing delay (network independent delay surrogate)

The bandwidth packing problem is defined as follows:

Given a graph G=(N, E) and a set of call requests (a call table) M, we seek to

maximize the profits from the routed calls, while assuring that the queuing delay

does not exceed a pre-specified acceptable limit. Further, we cannot exceed the

capacities on the communication links.

A graphical representation of a simple network structure with two calls is

provided in Figure 1. The dashed line shows a call being routed from node 3 to node 5

via intermediate nodes 8 and 7. The thick solid line shows a call being routed from node

4

1 to node 4 via intermediate nodes 7 and 8. The input parameters that need to be

known in order to successfully solve this routing problem include the network topology,

the capacity of the links, and the traffic requirements and revenues for all the calls. In a

typical telecommunications network the topology graphs are often sparse, necessitating

the use of shared resources. This is exemplified in Figure 1, where both calls use one

common resource: link (7,8).

1 2

3

4

5

6

7

9

8

Figure 1. An Example Network Topology

We assume that all nodes in the graph (Figure 1) have infinite buffers to store

messages waiting for transmission on the links. Further, the arrival process of messages

to the network follows a Poisson distribution, whereas the message lengths follow an

exponential distribution. Also, the propagation delays in the links are negligible1. Note

that we only consider a single class (or type) of service for each communicating node

pair.

1 The packet travel time is assumed to be negligible.

5

Even though the list of calls is known in advance, the traffic requirement for each call

may typically be bursty. For example, both video and data transmission exhibit variable bit

rates, for which queueing delays can be approximated by using an M/M/1 model. The validity

of this approximation is supported by experimental evidence: it has been shown that the

optimal routing is insensitive to the shape of the delay versus link load curve, and is only

affected by the asymptotic value of the link capacity [9].

Given the above assumptions, the telecommunications network is modeled as a

network of independent M/M/1 queues ([11], [12]). In this network, links are treated as

servers with service rates proportional to the link capacities. The customers are

messages whose waiting areas are the network nodes. Using the notation described

above and the decision variables defined below, the queueing delay in link (i,j) is

∑

∑

∈

∈

−Mm

mij

mij

Mm

mij

m

XdQ

Xd

and the average link queueing delay in the network is given by

||

1

E ∑ ∑∑

∈∈

∈

−Eji

Mm

mij

mij

Mm

mij

m

XdQ

Xd

),(

.

The number of links in a network (|E|) is constant. Therefore the queueing delay

requirement can be represented by constraint (5) in the formulation of the model below.

The decision variables are:

Ym = 1 if call m is routed

0 otherwise

Xmij =

1 if call m is routed through a path that uses link (i,j)

0 otherwise

6

Wmij =

otherwise 0

to of direction thein

),(link uses that patha throughrouted is call if 1

ji

jim

Problem P:

ZP = Max ∑∈Mm

rm Ym (1)

subject to:

∑∈Nj

Wmij - ∑

∈NjW

mji =

∈∈=−

=

otherwise

MmandNimDiifY

mOiifYm

m

0

)(

)(

(2)

Wmij + W

mji ≤ X

mij ∀ (i,j)∈E and m∈M (3)

∑∈Mm

dmXmij ≤ Qij ∀ (i,j)∈E (4)

δ≤−∑ ∑∑

∈∈

∈

Eji

Mm

mij

mij

Mm

mij

m

XdQ

Xd

),(

(5)

Xmij ∈ (0,1) ∀(i,j)∈E and m∈M (6)

Ym ∈ (0,1) ∀ (i,j)∈E (7)

Wmij ∈ (0,1) ∀ (i,j)∈E and m∈M (8)

The objective function (1) represents total revenues of routed calls. Constraint

set (2) contains the flow conservation equations, which define a route for each call

represented by a communicating node pair. Constraints in set (3) links together the Xmij

and Wmij variables. The problem can be correctly formulated with either X

mij or W

mij

variables only. However, both variable sets are useful in the Lagrangean relaxation

developed in the next section. The capacity constraints on the links are considered by

constraint set (4). Constraint (5) enforces the upper limit on the queueing delay in the

7

network. The integrality conditions on Xmij ,Ym and W

mij variables, are enforced in

constraint sets (6)-(8), respectively.

3. Problem Relaxation.

Problem P is a combinatorial optimization problem with a nonlinear constraint

(5). The problem studied in Anderson et al. [2], Laguna & Glover [13], and Park et al.

[14] is a special case of problem P and is known to be NP-complete [7]. Problem P is a

nonlinearly constrained (0,1) integer programming problem, and it is difficult to solve

this problem to optimality with standard mixed-integer programming tools. Hence, we

propose a composite upper and lower bounding procedure based on a Lagrangean

relaxation of the problem.

Consider the Lagrangean relaxation of problem P obtained by dualizing

constraint set (3) using nonnegative multipliers αmij for all (i,j) ∈ E and m ∈ M,

respectively, and relaxing constraint (5) using a nonnegative multiplier ψ:

Problem L:

ZL = Max ∑∈Mm

rm Ym-ψ ∑ ∑∑

∈∈

∈

−Eji

Mm

mij

mij

Mm

mij

m

XdQ

Xd

),(

+ ∑∑∈∈ EjiMm ),(

αmij (X

mij - W

mij - W

mji ) +ψδ (9)

Subject to (2), (4), (6), (7) and (8).

Problem L can now be decomposed into two subproblems as follows:

Problem L1:

Max ∑∈Mm

rm Ym - ∑∑∈∈ EjiMm ),(

αmij (W

mij + W

mji ) (10)

Subject to (2), (7) and (8).

Problem L2:

8

Max ∑∑∈∈ EjiMm ),(

αmij X

mij - ψ ∑ ∑

∑∈

∈

∈

−Eji

Mm

mij

mij

Mm

mij

m

XdQ

Xd

),(

(11)

Subject to (4) and (6).

Problem L1 can be further decomposed into |M| subproblems (one for each call) as

follows:

Max rm Ym - ∑∈Eji ),(

αmij (W

mij + W

mji ) (12)

subject to:

∑∈Nj

Wmij - ∑

∈NjW

mji =

∈∀=−

=

otherwise

NimDiifY

mOiifYm

m

0

)(

)(

(13)

Ym ∈ (0,1) ∀ (i,j)∈E and m∈M (14)

Wmij ∈ (0,1) ∀ (i,j)∈E and m∈M (15)

Problem L2 includes a non-linear term in the objective function. In essence, this

optimization problem resembles a non-linear multi-dimensional knapsack problem. It is

difficult to obtain an optimal solution procedure or a heuristic that would solve this

problem well. To be able to solve problem L2, we decompose it into |E| sub-problems

(one for each link) in the following manner:

Max ∑∈Mm

αmij X

mij - ψ ∑ ∑

∑∈

∈

∈

−Eji

Mm

mij

mij

Mm

mij

m

XdQ

Xd

),(

(16)

subject to:

∑∈Mm

dmXmij ≤ Qij (17)

Xmij ∈ (0,1) ∀(i,j)∈E and m∈M (18)

9

Problem L does not satisfy the integrality property, since the relaxation of L does

not necessarily have an integer solution. Hence, the relaxation of problem P can

theoretically give a lower bound which is at least as good as, and possibly better than

the relaxation of P.

Each subproblem of problem L1 can be solved by solving the shortest path

problem from O(m) to D(m) using the nonnegative multipliers α ijm as the cost of the

links (i.e., link distances). If the revenue from the call is greater than the cost of that

shortest path, then the call is routed through that path. if not, the call is not routed and

we set Ym = 0 and Wmij = 0 ∀ (i,j)∈E.

Each subproblem of problem L2 corresponding to a link (i,j) is equivalent to a

single constraint (0,1) knapsack problem with a nonlinear objective function. We relax

the integrality constraints and solve the continuous version of this problem using the

following greedy type procedure.

Procedure Greedy:

Step 1: Reorder the Xmij variables by sorting them in nonincreasing order of

αmij /dm;

Re-indexed the variables in this order, and Let m=0.

Step 2: Let m=m+1 and set

Xmij =

>>

otherwise

XandifX mij

0

0 0 0 0 α

where X0 = min{ 1 , 1

dm [(Qij - S) - (ψdmQij

αmij

)1/2]} and

S = ∑k<m

dkX

kij .

Step 3: If m=|M| stop; if Xmij < 1 then stop and set X

kij =0 for k=m+1,...,|M|.

Otherwise go to step 2.

10

4. The Solution Procedure

Feasible solutions as well as lower bounds for the optimal solution of problem P,

can be obtained by using the relaxation presented above. As for all relaxation

procedures, the success of the approach depends heavily on the ability to generate good

Lagrangean multipliers [8]. Theoretically, let ZL(α,ψ) be the value of the Lagrangean

function with a multiplier vector (α,ψ), then the best bound using this relaxation is

derived by calculating ZL(α∗,ψ∗) = Min )( ψα , {ZL(α,ψ) }. In practice, a good but not

necessarily optimal set of multipliers is often derived using iterative methods such as

subgradient optimization method and various multiplier adjustment methods known as

ascent (descent) methods [14]. We use the subgradient optimization method to search

for “good” multipliers. The subgradient method is a modified version of the gradient

method in which subgradients replace gradients [8]. Since this method is well

understood, we do not provide its implementation details; we summarize, however, the

particulars of our algorithm in appendix A.

We now outline a heuristic procedure to solve problem P. The below stated

procedure (Procedure Examine) attempts to generate a feasible solution to problem P at

every iteration of the subgradient optimization algorithm using information provided by

the solution to problem L1. The best feasible solution is retained when the subgradient

algorithm is terminated. Note that in the solution to problem L1 every call is either

routed through the links in the network or not routed at all. However, there may be

some links with loads higher than their available capacities. This simple heuristic

attempts to route calls through the network without exceeding link capacities. Thus, the

heuristic guarantees to generate a feasible solution at every iteration of the subgradient

optimization procedure. The complete heuristic is stated below.

Procedure Examine:

11

(1): Order the calls by sorting them in non-increasing order of the optimal values of

the objective functions of their corresponding sub-problems of problem L1.

Start with the first call.

(2): If the call can be routed using the path determined in the solution of its

corresponding sub-problem of problem L1 without exceeding the available link

capacities and the queueing delay upper limit, then route the call and update the

available link capacities.

(3): If all calls have been examined then Stop.

Otherwise proceed with the next call and go to step 2.

5. Computational Results

We coded the above solution procedures in Pascal, and performed a number of

computational experiments using a 7610 VAX computer.

5.1 Test Problem Generation.

To evaluate the effectiveness of the proposed procedures, we randomly generated

various networks and call tables containing information about calls to be routed on those

networks. We generated the networks as follows:

First, the generator locates the specified number of nodes on a 100x100 grid. Each

node has a degree equal to 2, 3 or 4 with probability of 0.6, 0.3 and 0.1, respectively. We

repeat the following procedure for each node i∈Ν: Determine node i’s closest neighbor (in

terms of Euclidean distance) with unsatisfied degree requirement, label this node j. Add arc (i,j)

and repeat this until node i’s degree requirement is satisfied or all the nodes with unsatisfied

degree requirements have been considered. If the degree requirements are not met for node i,

then connect node i to its closest neighbors to which it is not already connected until the degree

requirement of node i is satisfied. At the end check if the network is connected; if not, add links

necessary to make it connected.

12

The parameters that define the network (node degrees and their associated probability of

occurrence) are chosen in order to generate realistic telecommunications networks, where there

are some redundant communication links. A typical graph resulting from the network

generation using the parameters stated above is depicted in Figure 2a. Figure 2b shows the

same nodes, but each node has a degree equal to n-1. The latter case (2b) is not realistic in

telecommunications networks, since full duplication of communications links is prohibitively

expensive.

Figure 2a. A telecommunications network. Figure 2b. A fully connected network

We generated five sets of networks with 20, 25, 30, 35 and 40 nodes respectively. The

average numbers of links for each of the networks are shown in Table 1.

Table 1. Average number of links in the networks

Number of nodes Avg. number of linksin the network in the network

20 28.825 36.030 43.435 52.140 59.8

13

Each link in the network is randomly assigned a capacity equal to 48, 96, 192, or 500

with equal probabilities. These capacity choices approximately correspond to real line choices:

a T3 line, OC2, OC4, and OC9, respectively (approximations in Mbps). This provides realistic

variations in the capacity limits to test the robustness of our algorithm.

The test problem generator also produced the call tables. The call table contains

information about the origin and destination nodes for all communicating node pairs, as

well as the revenues and communication demand for those calls. For each call, the

generator randomly determines an origin and a destination node. The traffic

requirement, or demand (dm), as well as the revenue (rm) for each call are generated

randomly from two uniform distributions between 20 and 40 and between 10 and 50,

respectively2.

As depicted in Table 2, we generated 5 sets of networks with parameters as described

above, to capture a wide range of problem structures. For each network set (20, 25, 30, 35 and

40 nodes), we generated five problem groups, where only the number of calls differed. For

each problem, we used random call tables to create 10 problem instances (each line in Table 2

reports averages from these 10 problem instances). The number of calls (P) varies between 40

and 80 percent of all possible calls. For example, in a 40 node network, using a percentage of

60 (P=60), there are 936 calls (or 40x39x0.6)3. Let DM, a delay multiplier, be the upper limit

on the average link utilization4. This delay multiplier was fixed at 60% in order to enable us to

study the effects of variations in number of calls.

To study the implications of queuing delay, we generated a new set of problems (Table

3). The five problem groups are each based on one corresponding problem instance from Table

2 Various other limits on the traffic requirements were used in separate experiments to investigate thesensitivity of our proposed algorithm to these parameters. We found that the algorithm was not sensitive to anyspecific parameter limits in the range [5, 100].3 For reasons of comparison, we mention that Anderson et al. [2] reported results of computational experimentsconducted using networks ranging in size from 14 nodes and 35 calls to networks with 192 nodes and 20 calls.Park et al. [14] report results for problems with up to 30 nodes and 75 arcs, and with a maximum of 75 calls.4 The upper limit on the total queueing delay, δ, in the network is equal to |E|*DM/(1-DM).

14

2, where P=60. Thus, for all these problems, we kept the number of calls constant. We varied

the surrogate measure for link utilization, DM, from 40% to 70%, in steps of 5% for each

problem group. In order to achieve a reasonable level of confidence about the performance of

the solution procedure versus the problem structure, we generated 10 instances by randomly

creating different call tables for each instance (that is, each line in Table 3 reports the average

results from 10 problem instances).

5.2 Analysis and Discussion.

Tables 2 and 3 show the average performance measures for different networks.

The results of the experiments are described by providing the number of nodes in the

network (|N|), the percentage of the number of routed calls (P), the delay multiplier

(DM), the gap between the “best” feasible solution value and the upper bound expressed

as a percentage of the upper bound, the total revenue, the average and maximum link

utilization, and the CPU times in seconds. The numbers describing the queueing delay

multiplier (DM) are unit-less. This surrogate measure can be applied to specific

telecommunications networks to calculate delays in actual time-units.

Table 2 shows the results for different percentages of the total number of

available calls. As the number of calls increases, the average link utilization and total

queueing delay do not change significantly because of the requirement of minimum level

of response time to users. However, when this number increases, total revenue of

routed calls increases as a result of better selection of calls to be routed from among a

larger set of available calls. Also, there is higher probability that more calls will not be

routed. The average gap between the feasible solution value and upper bound varies

between 1.46% and 5.80% with a mean of 2.82%. The CPU time (in seconds) varies

between 33 for networks with 20 nodes and 846 for networks with 40 nodes.

The effects of changes in delay multiplier (DM) are reported in Table 3. As the

delay multiplier increases, a tradeoff between total revenue and response time to users is

15

made. When the delay multiplier increases, total revenue of routed calls increases at the

expense of quality of service (response time) to users. The deterioration in response

time is reflected in the increase in the total queueing delay. For example, for the 40

node networks, the total queueing delay on average increases from 39.2 when the delay

multiplier is 40 to 138.6 when this multiplier is 70. The deterioration in response time is

also indicated by the increase in average and maximum link utilization. For example, for

the 40 node networks, the average and maximum link utilization increase from 34.2% to

54.4% and 68.4% to 91.0%, respectively. The average gaps between feasible solution

values and upper bounds vary between 0.90% and 5.68% with a mean of 2.79%. The

CPU time (in seconds) varies from 46 for networks with 20 nodes to 642 for networks

with 40 nodes.

In summary, it is observed that as the delay multiplier increases, the number of

calls routed increases, thus revenue increases. However, this increase in revenue comes

at the expense of increased delays. This pattern is expected beyond the tested range of

the delay multiplier (40-80%). Further, our experiments have shown that as the average

node degree goes up, keeping everything else constant, the delay will diminish because

of additional routing capacity. The number of nodes in the graphs seems not to directly

have any impact on the solution quality, but it indirectly impacts CPU requirements.

The main drivers of CPU requirements are the number of arcs and the number of calls.

The number of arcs increases as the number of nodes increases. Due to the

decomposition of the problem, the CPU requirements also increase as the number of

calls increases. The total delay in the system is related to the number of calls since it is

the sum of the products of the number of calls and the delay per call.

6. Conclusion

We have provided a formulation and an efficient bounding procedure for the call

routing problem with a minimum service quality threshold. Incorporating this threshold

16

is important in assuring a maximum acceptable delay for the incoming calls in

telecommunications networks. Our contributions are as follows:

1. We propose a new formulation for the call routing problem that considers all

possible paths for routing the calls, and selects the best subsets.

2. We maximize the revenue generated from selecting the calls, while we assure a

minimum service quality with respect to maximum acceptable delay.

3. We propose efficient bounding and solution procedures for this problem, based on a

two-level decomposition of the problem.

In our computational experiments, we demonstrated that, on average, our heuristic

produced consistently good solutions with average optimality bounds of approximately

2.8% in less than five minutes of CPU time.

17

Appendix A

Given an initial multiplier vector (α0, ψ0) (set to the zero vector in this study), a

sequence of multipliers is generated by updating the vector at the iteration k using the

formula

(αk+1, ψk+1) = (αk, ψk) + tk (Wk - Xk ),

where (αk+1, ψk+1) and (αk, ψk) are the multiplier vectors at iterations k+1 and k

respectively, (Wk,Xk) is part of the optimal solution to the Lagrangean Problem L with

multiplier vector (αk, ψk) and tk is a positive scalar step size.

It is well known that lim sup ZL(αk, ψk) converges to ZL(α*,ψ*) if tk → 0 and

k=0

Inf

∑ tk → ∞ [16]. Since in general these conditions are very difficult to satisfy, the

subgradient optimization method is always used as a heuristic. In this study, we used

the following step size that has been found to be satisfactory in practice (Bazaara and

Goode [3]):

tk = λk( ZL(αk, ψk) - Zf) / ||Wk - Xk ||2,

where Zf is the value of the best feasible solution found so far and λk is a scalar

satisfying 0 ≤ λk ≤ 2. This scalar is set to 2 at the beginning of the algorithm and is

halved whenever the bound does not improve in 20 consecutive iterations. The

algorithm is terminated after a specified number of iterations (set equal to 500 in this

study) unless an optimal solution is reached before that point. The algorithm is also

terminated if the gap between the best upper bound and the best feasible solution found

is less than 0.01% of the best upper bound, or the best upper bound does not improve in

100 consecutive iterations by at least 0.01%.

18

References

[1] Amiri, A., E. Rolland and R. Barkhi, "Bandwidth Packing with Queuing Delay Costs:Bounding and Heuristic Solution Procedures", European Journal of OperationalResearch, forthcoming, 1997.

[2] Anderson, C.A., K. Fraughnaugh, M. Parker and J. Ryan, “Path assignment for callrouting: An application of tabu search”, Annals of Operations Research 41 (1993) 301-312.

[3] Bazaraa, M. S. and J.J. Goode, “A Survey of Various Tactics for GeneratingLagrangean Multipliers in the Context of Lagrangean Duality”, European Journal ofOperational Research 3 (1979) 322-328.

[4] Cox, L.A., L. Davis and Y. Qui, “Dynamic anticipatory routing in circuit-switchedtelecommunications networks”, in Handbook of Genetic Algorithms, L. Davis (ed.),Van Nostrand/Reinhold, New York, 1991.

[5] Fisher, M.L., “Lagrangean Relaxation Methods for Solving Integer Programming”,Management Science 27 (1981) 1-18.

[6] Ford, L.R. Jr. and D.R. Fulkerson, Flows in Networks, Princeton University Press,Princeton, N.J., 1962.

[7] Garey, M.R. and D.S. Johnson, Computers and Intractability: A Guide to the Theory ofNP-Completeness (1979) 215.

[8] Gavish, B., “On Obtaining the 'Best' Multipliers for a Lagrangean Relaxation for IntegerProgramming”, Computers and Operations Research 5 (1978) 55-71.

[9] Gerla, M., “The design of store-and-forward (S/F) networks for computercommunications”, Ph.D. Dissertation, Computer Science Dept., Univ. of California, LosAngeles (1973).

[10] Held, M., P. Wolfe and H.P. Crowder, “Validation of subgradient optimization”,Mathematical Programming 5 (1974) 62-88.

[11] Kleinrock, L., Communications nets: stochastic message flow and delay, New York,Dover, 1964.

[12] Kleinrock, L., Queueing systems, Volumes 1 & 2, Wiley-Interscience, NewYork, 1975,and 1976.

[13] Laguna, M. and F. Glover, “Bandwidth Packing: A Tabu Search Approach”,Management Science 39 (1993) 492-500.

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[14] Park, K, S. Kang and S. Park, “An Integer Programming Approach to the BandwidthPacking Problem”, Management Science 42 (1996) 1277-1291.

[15] Parker, M. and J. Ryan, “A Column Generation Algorithm for Bandwidth Packing”,Telecommunications Systems 2 (1995) 185-196.

[16] Poljack, B. T., “A General Method of Solving Extremum Problems”, Soviet Math.Doklady 8 (1967) 593-597.

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Table 2. Effect of Changes in Number of Calls

Percent Total Total % Link Utlization

|N| P DM Gap Revenue Delay Average Maximum CPU*

20 40 60 3.96 1049 42.0 47.0 87.6 33

20 50 60 3.94 1178 42.0 47.0 84.8 41

20 60 60 3.68 1207 41.8 46.0 87.8 47

20 70 60 3.44 1456 43.8 51.6 83.0 57

20 80 60 2.88 1636 43.6 51.6 81.6 66

25 40 60 4.06 1344 52.4 47.2 84.0 73

25 50 60 5.80 1324 53.2 45.4 86.2 91

25 60 60 3.96 1714 52.2 51.0 82.8 110

25 70 60 3.50 1714 51.8 49.2 84.8 125

25 80 60 3.44 2055 53.2 52.0 83.0 147

30 40 60 2.76 2786 63.0 47.4 87.0 148

30 50 60 1.88 2970 63.2 46.0 86.8 180

30 60 60 2.18 3067 64.0 45.2 86.8 216

30 70 60 1.46 3568 62.8 48.2 84.8 247

30 80 60 1.66 3837 62.6 50.0 83.0 283

35 40 60 1.78 3612 77.8 47.4 86.2 263

35 50 60 1.94 3759 73.8 47.8 85.6 316

35 60 60 1.54 4013 74.2 49.4 84.0 380

35 70 60 1.82 4349 75.4 49.8 83.4 451

35 80 60 1.72 4752 75.0 51.2 81.4 520

40 40 60 2.98 3869 88.6 46.8 87.0 425

40 50 60 2.38 4322 88.8 48.0 85.0 531

40 60 60 2.84 4667 89.0 47.8 84.4 634

40 70 60 2.22 5047 88.8 48.6 85.6 737

40 80 60 2.68 5365 89.0 49.0 85.6 846

Average: 2.82 278.62

* All CPU times are measured in seconds on a VAX 7610 computer

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Table 3. The Effects of Changes in the Delay Multiplier

Percent Total Total % Link Utlization

|N| P DM Gap Revenue Delay Average Maximum CPU*

20 60 40 2.94 1014 18.6 33.2 66.8 46

20 60 45 2.40 1080 22.4 37.4 72.8 46

20 60 50 2.74 1129 28.0 40.6 77.8 47

20 60 55 3.20 1171 34.2 44.4 81.2 47

20 60 60 3.68 1207 41.8 46.0 87.8 47

20 60 65 4.50 1240 52.0 49.4 88.8 48

20 60 70 5.48 1263 63.0 51.8 91.6 48

25 60 40 3.00 1393 23.2 34.0 65.8 106

25 60 45 2.96 1485 28.8 38.8 69.8 107

25 60 50 2.86 1573 35.0 43.2 75.6 108

25 60 55 3.44 1646 43.0 46.8 79.0 109

25 60 60 3.96 1714 52.2 51.0 82.8 110

25 60 65 4.98 1773 64.4 53.2 87.8 110

25 60 70 5.68 1826 82.6 56.6 92.4 110

30 60 40 1.12 2636 28.4 33.4 71.0 209

30 60 45 1.18 2770 34.8 36.8 73.8 211

30 60 50 1.30 2887 42.4 39.8 81.2 214

30 60 55 1.66 2982 52.0 42.8 83.6 216

30 60 60 2.18 3067 64.0 45.2 86.8 216

30 60 65 3.02 3138 79.0 47.4 90.4 219

30 60 70 5.02 3179 97.8 49.2 93.2 221

35 60 40 0.90 3349 33.0 35.2 66.8 368

35 60 45 1.12 3538 40.2 39.0 70.2 371

35 60 50 1.36 3710 49.0 42.6 74.8 375

35 60 55 1.48 3867 60.6 46.0 79.6 378

35 60 60 1.54 4013 74.2 49.4 84.0 380

35 60 65 2.26 4129 91.6 52.0 86.8 382

35 60 70 3.34 4211 115.2 54.4 91.8 383

40 60 40 1.20 3889 39.2 34.2 68.4 615

40 60 45 1.40 4117 48.2 38.0 73.2 623

40 60 50 1.56 4330 58.8 41.0 78.0 630

40 60 55 2.06 4507 72.4 44.6 81.0 638

40 60 60 2.84 4667 89.0 47.8 84.4 634

40 60 65 3.80 4799 109.8 51.2 89.2 642

40 60 70 5.48 4926 138.6 54.4 91.0 642

Average: 2.79 275.81

* All CPU times are measured in seconds on a VAX 7610 computer