Bandwidth packing with priority classes

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European Journal of Operational Research 192 (2009) 313–325

O.R. Applications

Bandwidth packing with priority classes

Indranil Bose *

The University of Hong Kong, School of Business, Room 730, Meng Wah Complex Pokfulam Road, Hong Kong

Received 9 August 2006; accepted 5 September 2007Available online 14 September 2007

Abstract

The bandwidth packing problem is defined as the selection and routing of messages from a given list of messages with prespecifiedrequirements on demand for bandwidth. The messages have to be routed over a network with given topology so that the generated rev-enue is maximized. Messages to be routed are classified into two priority classes. An integer programming based formulation of thisproblem is proposed and a Lagrangean relaxation based methodology is described for solving this problem. A general purpose heuristicis then developed for generating feasible solutions of good quality. Several numerical experiments are conducted using a number of prob-lem parameters such as number of messages, ratio of messages for lower and higher priority classes, capacity of links, and demand dis-tribution of messages belonging to different classes and high quality solutions to the priority bandwidth packing problem are generatedunder the different situations.� 2007 Published by Elsevier B.V.

Keywords: OR in telecommunications; Bandwidth packing; Integer programming; Lagrangean relaxation; Priority classes

1. Introduction

Networks of today often suffer from congestion problems due to the tremendous increase in traffic in recent times as wellas irrational allocation of bandwidth to support this increased traffic. One of the fundamental problems related to design ofnetworks is determination of which messages to route among a given list of messages and determination of the routes to beused for delivering messages between communicating nodes so that the revenue generated from routing these messages ismaximized. This is known as the bandwidth packing problem. The objective of this research is to use an optimization basedapproach for solving the bandwidth packing problem for messages belonging to multiple service classes. This will involveselection of a target group of messages from a list of messages provided by the users, and determination of the best pathsfor routing these messages. Usually the topology of the network, the capacities of the links, the revenues to be generated byrouting the messages, and the demand requirements of the messages are specified prior to the start of network design. Themessages are listed in the form of a message table and are prioritized based on the demand requirements. Since the networkcapacity is usually insufficient to route all messages, a selected group of messages are routed during a given period of time.This is known as the static bandwidth packing problem (as opposed to dynamic bandwidth packing where the demandrequirements of the messages change over time) and is studied in this paper. My goal in this paper is to find an appropriatemessage selection and routing scheme that provides an efficient resource allocation mechanism and maximizes the revenuegenerated from the usage of the network.

The problem addressed in this research is important since many telecommunication companies have invested heavily inbuilding networks that will be used for carrying multimedia traffic. As more delay sensitive applications (e.g., interactive

0377-2217/$ - see front matter � 2007 Published by Elsevier B.V.

doi:10.1016/j.ejor.2007.09.011

* Tel.: +852 2241 5845; fax: +852 2858 5614.E-mail address: [email protected]

mailto:[email protected]

314 I. Bose / European Journal of Operational Research 192 (2009) 313–325

television, real-time virtual simulations, collaborative computing, online interactive gaming, telemedicine) become com-mon in the future, the routing issues for networks will become very important. As bandwidth becomes scarce for such net-works, there will be an increasing need to develop intelligent message selection and routing schemes that will maximize thegenerated revenue for the company. Technology Futures Inc. has forecasted that the percentage of households in US thatwill demand high definition IP video will increase from 20% in 2010 to 80% in 2020 (Vanston and Hodges, 2004). The ana-lytical model that I propose is a close representation of the present networks that use connection oriented protocols forrouting of messages. It gives significant insights about the various operational issues which are of importance to networkservice providers, network designers, and network users. This paper aims to help network operators maximize the gener-ated revenue by providing design guidelines about various network operating parameters that can help them meet the ser-vice requirements of the different classes of messages.

Various versions of the bandwidth packing problem have been studied in the past. This includes research published byCox et al. (1991), Anderson et al. (1993), Laguna and Glover (1993), Parker and Ryan (1993), Park et al. (1996), Rollandet al. (1999), Amiri et al. (1999), Amiri and Barkhi (2000) and Amiri (2003, 2005). Most of these papers strive to maximizethe revenue earned by routing the messages subject to some service related constraints. A notable exception is the paper byAmiri et al. (1999) where the objective is to maximize revenue as well as minimize the delay cost associated with the use ofthe network. An extension to the basic bandwidth packing problem that considers multi-hour traffic is provided by Amiriand Barkhi (2000) and a further extension that solves the multi-hour bandwidth packing problem with delay guarantees isdescribed in the paper by Amiri (2003). Also, a new version of the bandwidth packing problem which involves schedulingthe messages within a prespecified time window is discussed in the recent paper by Amiri (2005). Various methods are usedin these papers including tabu search (Anderson et al., 1993; Laguna and Glover, 1993), genetic algorithms (Cox et al.,1991), column generation (Parker and Ryan, 1993), and Lagrangean relaxation (Rolland et al., 1999; Amiri et al., 1999;Amiri and Barkhi, 2000; Amiri, 2003; Amiri, 2005). On a different vein, an interesting paper by Dantzer et al. (2000) studiesthe stability properties of the bandwidth packing algorithm. However, the available research on bandwidth packing con-siders only a single message class. This assumption is not realistic as users use the networks for running different applica-tions. Some of these applications are delay sensitive but others are not. So it is more appropriate to model the bandwidthpacking problem in the presence of multiple priority classes. To the best of my knowledge, this is the first paper thataddresses the priority bandwidth packing problem. As opposed to the existing literature this problem is considerably moredifficult because exact analytical expressions for the average delay of the priority classes are difficult to obtain and this isturn complicates the formulation of the problem.

The remainder of this paper is organized as follows. In Section 2, an integer programming based formulation of thebandwidth packing problem with multiple classes is presented. In Section 3, a Lagrangean relaxation of the original prob-lem is obtained by dualizing a subset of the constraints and the relaxed problem is decomposed into two solvable sub-prob-lems. An analytical solution of the sub-problems is proposed. A heuristic to obtain a feasible solution to the problem isdescribed in Section 4. The different numerical results obtained using various choices of problem parameters are detailedin the next section. Section 6 concludes the paper with a summary and discussion of directions for future research.

2. Problem formulation

In this section I provide a mathematical representation of the bandwidth packing problem. A network with a knowntopology is used for transmission of messages (that belong to two different priority classes) between several origin and des-tination pair of nodes. The messages belonging to the higher priority class are shorter in length, generate more revenue, andare more demand intensive compared to the messages belonging to the lower priority class. The objective of the networkmanager is to select and route the messages through the network in such a way that the revenue generated from the processis maximized. At the same time, the number of messages in the network belonging to the higher and the lower priorityclasses must not exceed some predetermined bounds and the bandwidth consumed by all the messages routed on any linkof the network must be less than the available capacity of the link. I introduce the following notation for developing aninteger programming model for the priority bandwidth packing problem:N set of nodes in the networkE set of undirected links in the networkM set of messagesM1 set of messages of lower priorityM2 set of messages of higher priority1=l1 average length of messages of lower priority1=l2 average length of messages of higher prioritydm1 demand for message m1 2 M1

dm2 demand for message m2 2 M2

rm revenue generated by routing message m 2 M

I. Bose / European Journal of Operational Research 192 (2009) 313–325 315

OðmÞ source node for message m 2 MDðmÞ destination node for message m 2 MQij capacity of link ði; jÞd1 maximum number of messages of lower priority in the networkd2 maximum number of messages of higher priority in the networkT m1

ij average delay experienced on link ði; jÞ by messages of lower priorityT m2

ij average delay experienced on link ði; jÞ by messages of higher priorityLm1

ij number of messages of lower priority on link ði; jÞLm2

ij number of messages of higher priority on link ði; jÞ

The decision variables are

Y m ¼1 if message m is routed

0 otherwise

�

X mij ¼

1 if message m is routed through a path that uses link ði; jÞ0 otherwise

�

W mij ¼

1 if message m is routed through a path that uses link ði; jÞin the direction of i to j

0 otherwise

�

In order to obtain a mathematical formulation of the priority bandwidth packing problem we have to make severalassumptions. These are listed below

• The nodes have infinite buffers to store messages waiting for transmission.• Arrival process of messages entering the network follows a Poisson distribution.• Length of messages follows an exponential distribution.• Propagation delay in the links is negligible.• The average message length for each type of message is used.• The link and message system is studied as a preemptive priority queue.• The higher priority class of messages is more demand intensive compared to the lower priority class of messages.

The above bulleted list is a standard set of assumptions that appear in literature related to topological design of net-works. The interested reader may refer to Altinkemer and Bose (2003), Amiri et al. (1999), Gavish and Hantler (1983)and Rolland et al. (1999) for an explanation on the need for these assumptions. Based on these assumptions, the networkis modeled as a network of M/M/1 queues. In this network, links are treated as servers with service rates proportional tothe link capacities. The messages are customers waiting to be routed at a particular node. The queuing delay experienced inlink ðijÞ by each type of message ðT m1

ij ; Tm2ij Þ is given below (See Appendix A for the derivation)

T m1ij ¼

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij Þ=l1 þ ðP

m12M1dm1 X m1

ij =l1 þP

m22M2dm2 X m2

ij =l2Þ=2

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ;

T m2ij ¼

Qij �P

m22M2dm2 X m2

ij =2

l2QijðQij �P

m22M2dm2 X m2

ij Þ:

If delay is measured in terms of messages, then delay can be defined as the total number of messages to be routed on aparticular link. This is equivalent to measuring delay in absolute terms. Since the goal of the network designer is to limit theupper bound of delay, he/she can use a relaxed formulation of the delay in terms of maximum number of messages on alink for lower and higher priority messages. This is helpful for formulating the problem because it simplifies the analyticalformulation to an extent. Using Little’s Law (Kleinrock, 1976), upper bounds for Lm1

ij and Lm2ij can be obtained (see Appen-

dix A for the derivation)

Lm2ij 6

Pm22M2

dm2 X m2ij

ðQij �P

m22M2dm2 X m2

ij Þ;

Lm1ij 6

ðQij �P

m22M2dm2 X m2

ij ÞP

m12M1dm1 X m1

ij þ aP

m12M1dm1 X m1

ij

Pm22M2

dm2 X m2ij

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ;

where a ¼ l1=l2.The nature of the problem imposes certain restrictions on the characteristics of higher priority messages. They are

shorter in length than lower priority messages, i.e., a 6 1: In addition, the higher priority messages generate more revenue


while they are not as tolerant to queuing delay as the lower priority messages. So the number of messages belonging to thehigher priority class that are allowed in the network is smaller than that for the lower priority class (i.e., d2 6 d1Þ. With thenotations defined, the following model of the priority bandwidth packing problem can be developed.

Problem P

MaxX

m2M1;M2

rmY m ð1Þ

Subject to :Xj2N

W mij �

Xj2N

W mji ¼

Y m if i ¼ OðmÞ;

�Y m if i ¼ DðmÞ; i 2 N and m 2 ðM1;M2Þ;

0 otherwise;

8>><>>: ð2Þ

W mij þ W m

ji 6 X mij 8ði; jÞ 2 E and m 2 ðM1;M2Þ; ð3ÞX

m12M1

dm1 X m1ij þ

Xm22M2

dm2 X m2ij 6 Qij 8ði; jÞ 2 E; ð4Þ

Xði;jÞ2E

Pm22M2

dm2 X m2ij

ðQij �P

m22M2dm2 X m2

ij Þ6 d2; ð5Þ

Xði;jÞ2E

ðQij �P

m22M2dm2 X m2

ij ÞP

m12M1dm1 X m1

ij þ aP

m12M1dm1 X m1

ij

Pm22M2

dm2 X m2ij

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ6 d1; ð6Þ

Y m 2 ð0; 1Þ 8m 2 ðM1;M2Þ; ð7Þ

X mij 2 ð0; 1Þ 8ði; jÞ 2 E and m 2 ðM1;M2Þ; ð8Þ

W mij 2 ð0; 1Þ 8ði; jÞ 2 E and m 2 ðM1;M2Þ: ð9Þ

The objective function (1) represents the total revenue earned by routing messages. Constraint set (2) represents the flowconservation equations, which define a route for each message represented by a communicating node pair. Constraint set(3) links together the X m

ij and W mij variables. The constraint set (3) is redundant but useful for performing Lagrangean relax-

ation (described in the next section). Constraint set (4) guarantees that total flow on a link does not exceed capacity of thelink. Constraints (5) and (6) impose upper bound on number of messages belonging to each priority class for the network.Constraint sets (7)–(9) are the integrality constraints.

3. Problem relaxation

Problem P is a combinatorial optimization problem with non-linear constraints. It is an NP-complete problem since theproblem described by Rolland et al. (1999), which is a special case of Problem P, is also NP-Complete (Garey and Johnson,1979). So I propose a heuristic based on Lagrangean relaxation by dualizing constraint set (3) using non-negative multi-pliers am

ij for all ði; jÞ 2 E and m 2 ðM1;M2Þ, then further dualizing constraints (5) and (6) using non-negative multipliersw1 and w2. The relaxed problem (Problem L) is shown below.

Problem L

ZL ¼MaxX

m2M1;M2

rmY m � w1

Xði;jÞ2E

ðQij �P

m22M2dm2 X m2

ij ÞP

m12M1dm1 X m1

ij þ aP

m12M1dm1 X m1

ij

Pm22M2

dm2 X m2ij

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ

� w2

Xði;jÞ2E

Pm22M2

dm2 X m2ij

ðQij �P

m22M2dm2 X m2

ij ÞþX

m2M1;M2

Xði;jÞ2E

amijðX m

ij � W mij � W m

jiÞ þ w1d1 þ w2d2

Subject to constraints (2), (4), (7), (8) and (9).Problem L can be decomposed into sub-problems L1 and L2 as follows:Problem L1

MaxX

m2M1;M2

rmY m �X

m2M1;M2

Xði;jÞ2E

amijðW m

ij þ W mjiÞ

Subject to constraints (2), (7) and (9).


This sub-problem can be further decomposed into jMj sub-problems (one for each message) as follows:

Max rmY m �Xði;jÞ2E

amijðW m

ij þ W mjiÞ ð10Þ

Subject to :Xj2N

W mij �

Xj2N

W mji ¼

Y m if i ¼ OðmÞ;�Y m if i ¼ DðmÞ; i 2 N and m 2 ðM1;M2Þ;0 otherwise;

8<: ð11Þ

Y m 2 ð0; 1Þ 8m 2 ðM1;M2Þ; ð12ÞW m

ij 2 ð0; 1Þ 8ði; jÞ 2 E and m 2 ðM1;M2Þ: ð13Þ

Each sub-problem can be solved by solving the shortest path problem from OðmÞ to DðmÞ using the non-negative multi-pliers am

ij as the cost of the link. If the revenue from the message is greater than the cost of the shortest path, then the mes-sage is routed through that path, otherwise, the message is not routed and I set Y m ¼ 0 and W m

ij ¼ 0 8ði; jÞ 2 E. Whensolving the shortest path problem, the messages belonging to the higher priority class are routed first and then messagesbelonging to the lower priority class are routed.

Problem L2

MaxX

m2M1;M2

Xði;jÞ2E

amijX

mij � w2

Xði;jÞ2E

Pm22M2

dm2 X m2ij

ðQij �P

m22M2dm2 X m2

ij Þ

� w1

Xði;jÞ2E

ðQij �P

m22M2dm2 X m2

ij ÞP

m12M1dm1 X m1

ij þ aP

m12M1dm1 X m1

ij

Pm22M2

dm2 X m2ij

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þþ w1d1 þ w2d2

Subject to constraints (4) and (8).This sub-problem can be decomposed into jEj sub-problems as follows:

MaxX

m2M1;M2

amijX

mij � w2

Pm22M2

dm2 X m2ij

ðQij �P

m22M2dm2 X m2

ij Þ

� w1

ðQij �P

m22M2dm2 X m2

ij ÞP

m12M1dm1 X m1

ij þ aP

m12M1dm1 X m1

ij

Pm22M2

dm2 X m2ij

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þþ w1d1 þ w2d2 ð14Þ

Subject to :X

m12M1

dm1 X m1ij þ

Xm22M2

dm2 X m2ij 6 Qij 8ði; jÞ 2 E; ð15Þ

X mij 2 ð0; 1Þ 8ði; jÞ 2 E and m 2 ðM1;M2Þ: ð16Þ

This problem is equivalent to a single constraint knapsack problem with a non-linear objective function. The integralityconstraints are relaxed and the continuous version of this problem is solved using the following greedy procedure, whichis similar to the one used by Rolland et al. (1999).

Procedure Greedy

Step 1: Reorder the X mij variables by sorting them according to priority class first (higher to lower) and then within each

priority class in non-decreasing order of amij=dm and re-index the variables in this order, and let m ¼ 0.

Step 2: Let m ¼ mþ 1 and set

X mij ¼

X 0 if X 0 > 0;

0 otherwise:

�

X 0 is determined as follows: If m 2 M1;X 0 ¼ min 1; 1dm ðQij � S1 � S2Þ �

w1dm½ðQij�S2ÞþaS2�am

ij

� �1=2� ��

. If m 2 M2, thenX 0 is the solution of the following equation:

amij �

w2dmQij

ðQij � S2 � dmxÞ2� w1S1d2

ðQij � S1 � S2 � d2xÞ2�

aS1dmw1½ðQij � S2 � dmxÞðQij þ S2 þ dmxÞ � S1Qij�ðQij � S1 � S2 � dmxÞ2ðQij � S2 � dmxÞ2

¼ 0:

In particular, when S1 ¼ 0;X 0 ¼ min 1; 1dm ½ðQij � S2Þ � ð

w2dmQij

amijÞ1=2�

n owhere

S1 ¼X

k<m;m2M1

dkX kij and S2 ¼

Xk<m;m2M2

dkX kij:

Step 3: If m = jMj stop; If X mij < 1 then stop and set X k

ij ¼ 0 for k ¼ mþ 1;mþ 2; . . . ;M :A theoretical justification of the greedy procedure is provided in Appendix B.


4. Solution procedure

The Lagrangean relaxation described above produces an upper bound of the optimal solution of problem P. The feasiblesolution gives the lower bound of the maximization problem. The difference between the two bounds gives the gap in thesolution and is a measure of how close the relaxation can approximate the optimal solution. The gap is calculated as apercentage gap using the formula:

Gap ¼ Upper bound� Lower bound

Lower bound� 100:

Like all relaxation procedures, the success in obtaining a tight lower bound depends heavily on the ability to generategood Lagrange multipliers. Theoretically, the best bounding Lagrange multipliers must satisfy the conditionZLða�;w�1;w

�2Þ ¼ minða;w1;w2ÞfZLða;w1;w2Þg. In practice, subgradient optimization method can be used to obtain good values

of the multipliers. Since this is a standard method, the procedure is not provided in detail. The interested reader may referto Bazaraa and Goode (1979) for a description of the subgradient optimization technique.

A feasible solution can be calculated in each iteration of the subgradient optimization by using information provided bythe solution to problem L1. In obtaining the solution to problem L1, I decide whether to route a message or not, but I donot take into account the link capacity constraint and constraint on maximum number of allowed messages for each pri-ority class. The simple heuristic described below routes the messages through the network without exceeding the link capac-ities or violating constraints related to maximum number of messages for each priority class. Thus it guarantees to generatea feasible solution to problem P.

Procedure for generation of feasible solution

1. Start with the higher priority class.2. Order the messages by sorting them in the non-increasing order of the optimal values of the objective function in prob-

lem L1.3. If the message can be routed using the path determined in the solution of problem L1, without exceeding the link capacities

and upper bound on number of messages in the network, then route the message and update available link capacities.4. If all messages have been examined then move to step 5 and proceed with next priority class. If no more messages

belonging to lower priority class is present then stop. Otherwise proceed with the next message and go to step (2).5. Repeat using the lower priority class.

5. Numerical experimentation

In order to test the effectiveness of the solution procedure I conduct several numerical experiments. The values of theproblem parameters that are used in the experiments are listed in Table 1. The experiments are conducted using networkswhere number of nodes is 10, 15, 20, and 25, respectively. The networks are generated in such a fashion that each node hasa degree equal to 2, 3, or 4 with probability of 0.6, 0.3, and 0.1, respectively. For a prespecified number of nodes, a singlenetwork topology is constructed and tested subsequently. The method of generation of the network topology is similar tothat adopted by Rolland et al. (1999). The network is assumed to be made of OC4 links and has capacity of 192 Mbps.I perform some experiments using different values of link capacities as well. After the generation of the networks, the mes-sage tables are generated. The total number of messages in the message table is dependant on the number of nodes of thenetwork and is equal to k � N � ðN � 1Þ, where k is a fraction whose value is controlled by the network designer. For thebase case, k is assumed to be 0.6. Hence, for the 10 node case, the total number of messages generated is equal to 54. Someexperiments are performed when the value of k is changed from 0.4 to 0.8 in steps of 0.1. The message table lists two typesof messages and indicates the origin and the destination node for each message. The ratio of the number of messagesbelonging to lower priority to higher priority is taken to be 80:20 in the base case. When the total number of messagesis 54 for a 10 node case, the number of higher priority messages is approximately 11 and the number of lower priority mes-sages is approximately 43. Several experiments are performed in which the ratio of messages is changed to 90:10, 70:30,60:40, and 50:50 as well. The messages belonging to the higher priority class are more demanding and the demand require-ment for these messages is assumed to be uniformly distributed between 30 Mbps and 40 Mbps. On the other hand, formessages belonging to the lower priority class the demand requirement is uniformly distributed between 5 Mbps and10 Mbps. New applications like high definition (HD) television are likely to be bandwidth intensive and it is known that‘‘three simultaneous HD streams alone require 24 megabytes, without even considering the implications of upcoming appli-cations, such as video telephony and personal broadcast. This could drive the bandwidth requirements for the digital hometo 50 Mbps’’ (Marcus, 2005). The revenue generated from the routing of the two classes of messages are also assumed to be

Table 1Choice of problem parameters for numerical experiments

Parameters Values

Number of nodes 10, 15, 20, 25Capacity of links 96 Mbps, 192 Mbps, 500 MbpsMessage generator fraction 0.4, 0.5, 0.6, 0.7, 0.8Ratio of low priority to high priority traffic 60:40, 70:30, 80:20, 90:10Demand for high priority traffic 20–30 Mbps, 30–40 Mbps,

40–50 Mbps, 50–60 MbpsDemand for low priority traffic 5–10 Mbps, 10–15 MbpsRevenue generated by high priority traffic 26–50 dollar unitsRevenue generated by low priority traffic 10–25 dollar unitsMaximum number of high priority messages 400Maximum number of low priority messages 800


uniformly distributed between [10,25] dollar units and [26, 50] dollar units for the lower and the higher priority class,respectively. Some experiments are conducted using different demand profiles for the lower and the higher priority class.The upper bound on the number of lower and higher priority messages in the network is taken to be 800 and 400, respec-tively. From the experiments the feasible solution, the percentage gap between the Lagrangean solution and the feasiblesolution, the maximum and average utilization of the links, and the number of lower and higher priority messages thatare routed over the network, are obtained.

In the first set of results depicted in Table 2, the experiments are conducted for networks where number of nodes is 10,15, 20, and 25. The links in the networks have capacity of 192 Mbps. The ratio of lower priority to higher priority messageis 80:20. The demand for higher priority message is U[30,40] Mbps and the demand for lower priority message is U[5,10]Mbps. For this experiment, k is increased in steps of 0.1 from 0.4 to 0.8. As k is increased the number of messages in eachnetwork increases and this increases the experimental complexity. For example, for the 10 node case, when k = 0.4, thetotal number of messages is 36. Among these messages 7 belong to the higher priority class and 29 belong to the lowerpriority class. This leads to an optimization problem with 900 variables (not counting the W m

ij variables). On the otherhand, for the 20 node case, when k = 0.8, the total number of messages is 304. Of these, 61 messages belong to the higherpriority class and 243 messages belong to the lower priority class. This leads to an optimization problem with 39824

Table 2Impact of number of messages

Fraction(k)

Feasiblesolution

Percentagegap

Maximumutilization

Averageutilization

Routedmessages forpriority 1

Total messagesfor priority 1



Number ofmissedmessages

10 node

0.4 781.96 1.54 69.27 31.11 29 29 7 7 00.5 903.93 1.34 100 53.13 36 36 9 9 00.6 1164.87 1.05 100 45.73 43 43 11 11 00.7 1307.89 2.53 98.95 63.19 48 50 13 13 20.8 1317.76 8.18 100 56.18 52 58 13 14 7

15 node

0.4 1765.84 0.70 100 52.96 67 67 17 17 00.5 2326.75 0.55 99.47 64.40 84 84 21 21 00.6 2712.57 1.73 100 74.10 99 101 25 25 20.7 3078.17 4.71 100 75.27 108 118 29 29 100.8 3306.41 11.01 100 80.71 112 134 33 34 23

20 node

0.4 3219.76 0.40 85.94 41.66 122 122 30 30 00.5 4105.7 0.32 98.44 48.57 152 152 38 38 00.6 4853.35 4.81 98.96 58.64 170 182 46 46 120.7 5501.46 5.28 100 65.52 194 213 53 53 190.8 5875.72 9.84 100 71.68 206 243 61 61 37

25 node

0.4 5144.63 0.26 86.97 35.50 192 192 48 48 00.5 6370.8 0.43 96.88 38.16 240 240 60 60 00.6 7663.6 1.74 100 44.36 280 288 72 72 80.7 8598.55 3.94 100 53.17 316 336 84 84 200.8 9779.72 5.99 100 61.08 346 384 96 96 38

Table 3Impact of traffic ratio

Trafficratio

Feasiblesolution

Percentagegap

Maximumutilization

Averageutilization

Routed messagesfor priority 1


Routed messagesfor priority 2



10 node

90:10 1070.93 1.13 86.46 38.47 49 49 5 5 080:20 1164.87 1.05 100 45.73 43 43 11 11 070:30 1377.88 1.82 100 61.56 37 38 16 16 160:40 1329.4 4.89 97.92 73.75 30 32 21 22 350:50 1401.66 7.51 100 72.60 23 27 26 27 5

15 node

90:10 2447.80 2.43 98.96 56.71 109 113 13 13 480:20 2712.57 1.73 100 74.10 99 101 25 25 270:30 2791.65 9.79 100 79.13 71 88 38 38 1760:40 2732.05 16.73 100 81.31 62 76 42 50 2250:50 2862.03 21.55 100 84.10 43 63 53 63 30

20 node

90:10 4387.67 1.96 98.44 47.36 199 205 23 23 680:20 4853.35 4.81 98.96 58.64 170 182 46 46 1270:30 5050.04 7.58 99.48 64.89 138 160 68 68 2260:40 5001.76 16.84 100 71.97 98 137 85 91 4550:50 5418.40 19.81 100 76.28 85 114 97 114 46

25 node

90:10 7251.48 0.19 94.79 41.78 324 324 36 36 080:20 7663.60 1.74 100 44.36 280 288 72 72 870:30 8187.08 3.45 98.96 56.37 234 252 108 108 1860:40 8626.35 5.29 99.48 65.10 188 216 143 144 2950:50 9038.43 9.12 100 71.04 157 180 165 180 38


variables leading to a 4324.9% increase in the number of variables. From Table 2, the following can be observed. The fea-sible solution increases with the increase in k across all networks. The percentage gap remains reasonably low for all casesexcept when k = 0.8. In those cases, the average utilization of the links reaches a high value and there is not enough band-width to route all messages. It is also observed that for all cases where the percentage gap is greater than 5%, the maximumutilization reaches 100%, which implies that for such a choice of k, at least one link in the network develops a bottleneck.Since the messages belonging to the higher priority class have preemptive priority over messages belonging to the lower

Table 4Impact of capacity of links

Capacity Feasiblesolution

Percentagegap

Maximumutilization

Averageutilization






10 node

96 972.55 21.11 100 67.08 37 43 8 11 9192 1164.87 1.05 100 45.73 43 43 11 11 0500 1164.96 1.04 41.20 17.67 43 43 11 11 0

15 node

96 2147.13 28.94 98.96 69.73 80 101 16 25 30192 2712.57 1.73 100 74.10 99 101 25 25 2500 2736.83 0.49 48 19.29 101 101 25 25 0

20 node

96 3423.32 47.76 98.96 59.07 107 182 36 46 85192 4853.35 4.81 98.96 58.64 170 182 46 46 12500 5046.42 0.251 50 16.13 182 182 46 46 0

25 node

96 5181.45 50.20 100 61.23 172 288 49 72 139192 7663.60 1.74 100 44.36 280 288 72 72 8500 7770.06 0.16 90.4 17.98 288 288 72 72 0


priority class, the algorithm preferably routes higher priority messages. Hence, it is noted that for high values of k, greaternumber of lower priority messages is dropped compared to higher priority messages, though higher priority messages aremore demand intensive.

In the second set of experiment, the effect of different choices of ratios of lower to higher priority traffic on the generatedrevenue and routed messages is studied. The results are shown in Table 3. All choices of parameters are same as in the firstset of experiment except the value of k is fixed at 0.6 and the traffic ratio is varied. The ratio of lower to higher prioritytraffic used are 90:10, 80:20, 70:30, 60:40, 50:50. For most practical purposes, it can be assumed that in most networksthe proportion of higher priority traffic will be smaller than lower priority traffic. Since, higher priority traffic is moredemand intensive, therefore, it is likely that if the proportion of higher priority traffic increases there will be more droppedmessages. This is observed in the last column of Table 3. Also, it is expected that the generated revenue should increase asthe proportion of higher priority traffic increases since routing it generates more revenue. From Table 3, it is observed thatthis is not always the case because though there are a greater number of higher priority messages available for routing theyare not routed due to limited capacity. Hence, the feasible solution does not always increase with increase in proportion ofhigher priority traffic. This implies that to maximize revenue it is not necessarily true that the designer should try to route asmany higher priority messages as is possible. The percentage gap remains low in general except for four cases where itexceeds 10%. These cases are marked by the development of a bottleneck link (i.e., maximum utilization is 100%), averageutilization of links more than 70%, and large number of dropped messages. This implies that the algorithm performs betterwhen there is enough available capacity but doesn’t fare as well when capacity is scarce.

Table 5Impact of demand distribution of lower and higher priority classes

Demandforpriority 1

Demandforpriority 2

Feasiblesolution

Percentagegap

Maximumutilization

Averageutilization


Totalmessages forpriority 1


Totalmessages forpriority 2


10 node

5,10 20,30 1140.88 1.06 73.44 36.91 43 43 11 11 05,10 30,40 1164.87 1.05 100 45.73 43 43 11 11 05,10 40,50 1137.85 4.07 99.48 57.47 43 43 10 11 15,10 50,60 1190.82 1.04 96.88 67.60 43 43 11 11 0

10,15 20,30 1134.84 1.07 93.75 61.25 43 43 11 11 010,15 30,40 1103.84 1.10 88.02 63.89 43 43 11 11 010,15 40,50 1129.88 1.07 93.23 59.10 43 43 11 11 010,15 50,60 1052.27 7.59 100 76.18 41 43 10 11 3

15 node

5,10 20,30 2523.66 4.74 98.96 54.06 92 101 25 25 95,10 30,40 2712.57 1.73 100 74.10 99 101 25 25 25,10 40,50 2382.28 5.73 100 74.48 83 101 23 25 205,10 50,60 2397.85 14.92 100 73.79 88 101 20 25 18

10,15 20,30 2534.19 6.92 99.48 74.60 88 101 25 25 1310,15 30,40 2494.37 11.73 100 75.75 83 101 25 25 1810,15 40,50 2262.35 18.12 99.48 84.96 75 101 25 25 2610,15 50,60 2180.41 24.60 100 78.33 74 101 21 25 30

20 node

5,10 20,30 4893.34 0.68 98.44 49.84 180 182 46 46 25,10 30,40 4853.35 4.81 98.96 58.64 170 182 46 46 125,10 40,50 4607.61 6.02 100 60.60 166 182 45 46 175,10 50,60 4317.21 14.95 100 66.72 153 182 43 46 32

10,15 20,30 4608.3 8.35 100 69.46 163 182 45 46 2010,15 30,40 4586.94 6.14 100 79.72 162 182 46 46 2010,15 40,50 4237.99 15.16 100 72.13 143 182 44 46 4110,15 50,60 4128.18 20.17 100 75.99 137 182 43 46 48

25 node

5,10 20,30 7604.07 0.19 97.92 39.68 288 288 72 72 05,10 30,40 7663.6 1.74 100 44.36 280 288 72 72 85,10 40,50 7623.53 4.29 100 53.04 274 288 70 72 165,10 50,60 7391.59 7.11 100 59.92 260 288 69 72 31

10,15 20,30 7722.59 2.74 99.48 51.60 274 288 72 72 1410,15 30,40 7129.99 7.25 100 57.53 256 288 72 72 3210,15 40,50 6805.3 13.79 100 57.78 225 288 71 72 6410,15 50,60 6532.34 17.63 100 65 214 288 70 72 76


The impact of capacity of links on the generated solution is reported in Table 4. Prior experiments assumed that all linksin the network have a capacity equal to 192 Mbps, which corresponds to an OC4 link. Next, experiments are conductedwith two more link capacities – 96 Mbps which represents an OC2 link and 500 Mbps which represents an OC9 link. Table4 shows the results for the choice of problem parameters, (which are kept exactly same as the first two experiments and withk = 0.6 and ratio of lower to higher priority traffic fixed at 80:20) and indicates that the use of OC2 links is not suitable.The solution technique fails to generate high quality solutions for a link capacity of 96 Mbps and this leads to a large num-ber of dropped messages. Between 192 Mbps and 500 Mbps it seems that using OC9 links for the network is overkill. Whenthe link capacity is 500 Mbps, the average utilization remains less than 20%, which implies that a large amount of band-width is wasted. Of course, there are no dropped messages when all links are upgraded from OC2 to OC4.

The last experiment studies the effect of demand distributions of lower and higher priority class on generated revenueand routing of messages. The results are shown in Table 5. The experiment uses two different demand distributions for thelower priority class, namely U[5,10] and U[10,15] (all in Mbps). On the other hand, for the higher priority class four dif-ferent choices of demand distributions are used U[20,30], U[30,40], U[40,50], and U[50,60] (all in Mbps). For this choiceof demand distributions, most of the messages can be routed in case of the 10 node problem whereas for the 25 node prob-lem many lower priority messages and some higher priority messages are dropped. There is no apparent relationshipbetween generated revenue and demand distribution of the messages. However, the percentage gap tends to be higher whenthe value of the mean difference between the demand distribution for the lower and higher priority class increases. Asbefore, all cases related to a high percentage gap are marked by the development of a bottleneck link, high average utili-zation of the links, and many dropped messages. It is also observed that when the demand distribution of the lower priorityclass is kept fixed and the demand distribution of the higher priority class is increased it leads to an increase in the numberof dropped messages and the dropped messages mostly belong to the lower priority class. This implies that when the higherpriority class becomes more demand intensive, then the performance of the lower priority class suffers a lot since the algo-rithm preferentially routes messages belonging to the higher priority class.

6. Conclusion

In this paper an integer programming based formulation for the bandwidth packing problem with two priority classes ispresented. Since the problem is NP-complete, a Lagrangean relaxation based method and a heuristic for generating a highquality upper bound for this problem is used. A method for obtaining feasible solutions for the priority bandwidth packingproblem is also described. The paper details an analytical solution for this problem and numerical solutions using networksof different sizes. Several experiments are conducted to study the impact of number of messages, ratio of lower to higherpriority traffic, capacity of links, and demand distributions of lower and higher priority classes on the solution and thealgorithm is able to generate high quality solutions under different situations. All experiments take less than 5 secondsto execute and this implies that the proposed method is quite efficient in solving the static bandwidth packing problem.In future, this research can be extended in various ways. First, it is possible to formulate and solve a multi-period versionof the problem with two priority classes with estimated message tables available for future time periods. Another importantextension is to consider a stochastic version of this problem where the demand is not static in terms of messages but wherethe messages and their demand distributions vary stochastically within a given time period. A limitation of the approachadopted in this paper is that a regular M/M/1 queue is used to model the queuing behavior of all messages belonging to thetwo classes. However, if the system reaches saturation in terms of messages, new messages are not allowed to enter thenetwork. This implies that loss systems rather than regular queues should be used to represent the phenomenon that occursat the links on the edge of the network. I have avoided this in this paper to keep the queuing analysis tractable. Combiningloss systems with regular queues is a more exact representation of the real situation and should be pursued in future. Theheuristic provided in this paper works in a greedy fashion and cannot generate high quality solution when bandwidth isscarce. It is possible to develop non-greedy heuristics that work well and generate low percentage gaps even in cases whenthe average link utilization remains high and there are a large number of dropped messages.

Appendix A

Calculation of average queuing delay for each type of message on link ði; jÞ

Several additional notations are introduced in order to derive the expression for queuing delay of messages belonging todifferent priority classes.

i 2 1; 2; . . . ; J priority class i

ki average arrival rate of messages with priority i

xi ¼ 1liQij

average service time of messages with priority i on link ði; jÞ


di ¼ kili

demand for message with priority i

qi ¼ kixi ¼ di

Qijutilization of server for message with priority i

Consider a head-of-the-line queuing system that follows a preemptive queuing discipline. The preemption is of resumetype (i.e., the customer resumes to do what’s left for him/her to do after being preempted). The total stay time in system isstudied from the perspective of a newly arrived message which is denoted as the ‘tagged’ customer for convenience. Thenaccording to Kleinrock (1976), total time in system for the ‘tagged’ customer is composed of three parts:

1. Average service time xi.2. Delay due to the service required by those customers of equal or higher priority whom our tagged customer finds in the

system, equal to the mean waiting time in an M/G/1 queue.3. Delay due to customers of higher priority who enter the system during the stay of our tagged customer.

T j ¼ xj þPJ

i¼jkxi2=2

1� rjþXJ

i¼jþ1

kiT jxi ¼xjð1� rjÞ þ

PJi¼jkixi

2=2

ð1� rjÞð1� rjþ1Þwhere rj ¼

XJ

i¼j

qi:

When there are only two priority classes of messages, i.e., J ¼ 2, the total time in system of each priority class can be for-mulated as follows:

T 2 ¼x2ð1� r2Þ þ k2x2

2=2

ð1� r2Þ¼ ð1� d2=QÞ=l2Qþ d2=2l2Q2

ð1� d2=QÞ

¼ Q� d2=2

l2QðQ� d2Þ¼ 1� d2=2Q

l2ðQ� d2Þ6

1

l2ðQ� d2Þ;

T 1 ¼x1ð1� r1Þ þ ðq1x1 þ q2x2Þ=2

ð1� r1Þð1� r2Þ¼ ð1� d1=Q� d2=QÞ=l1Qþ ðd1=l1 þ d2=l2Þ=2Q2

ð1� d1=Q� d2=QÞð1� d2=QÞ

¼ ðQ� d1 � d2Þ=l1 þ ðd1=l1 þ d2=l2Þ=2

ðQ� d1 � d2ÞðQ� d2Þ¼ ðQ� d1 � d2Þ þ ðd1 þ ad2Þ=2

l1ðQ� d1 � d2ÞðQ� d2Þ

¼ ðQ� d2Þ þ ðad2 � d1Þ=2

l1ðQ� d1 � d2ÞðQ� d2Þ6

ðQ� d2Þ þ ad2

l1ðQ� d1 � d2ÞðQ� d2Þ:

Using the notation used earlier in the paper, expressions for average queuing delay on link ði; jÞ of messages belonging tothe two priority classes can be derived.

Average queuing delay of messages of higher priority on link ði; jÞ

T m2ij ¼

Qij �P

m22M2dm2 X m2

ij =2

l2QijðQij �P

m22M2dm2 X m2

ij Þ¼

1�P

m22M2dm2 X m2

ij =2Qij

l2ðQij �P

m22M2dm2 X m2

ij Þ6

1

l2ðQij �P

m22M2dm2 X m2

ij Þ:

Hence, average number of messages of higher priority on link ði; jÞ using Little’s Law

Lm2ij 6

Pm22M2

dm2 X m2ij

ðQij �P

m22M2dm2 X m2

ij Þ:

Average queuing delay of messages of lower priority on link ði; jÞ

T m1ij ¼

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij Þ=l1 þ ðP

m12M1dm1 X m1

ij =l1 þP

m22M2dm2 X m2

ij =l2Þ=2

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ

¼ðQij �

Pm12M1

dm1 X m1ij �

Pm22M2

dm2 X m2ij Þ þ ð

Pm12M1

dm1 X m1ij þ

Pm22M2

dm2 X m2ij aÞ=2

l1ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ

¼ðQij �

Pm22M2

dm2 X m2ij Þ þ ð

Pm22M2

dm2 X m2ij a�

Pm12M1

dm1 X m1ij Þ=2

l1ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ

6

ðQij �P

m22M2dm2 X m2

ij Þ þ aP

m22M2dm2 X m2

ij

l1ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ:

Hence, average number of messages of lower priority on link ði; jÞ using Little’s law

Lm1ij 6

ðQij �P

m22M2dm2 X m2

ij ÞP

m12M1dm1 X m1

ij þ aP

m12M1dm1 X m1

ij

Pm22M2

dm2 X m2ij

ðQij �P

m12M1dm1 X m1

ij �P

m22M2dm2 X m2

ij ÞðQij �P

m22M2dm2 X m2

ij Þ:


Appendix B

Justification of greedy procedure.

To simplify the expression, general notations a;Q; x; d; d1; d2 are used instead of amij ;Qij;X

mij ; d

m; dm1 ; dm2 respectively.Also, let A ¼

Pk<mak

ijXkij, S1 ¼

Pk<m;m2M1

dkX kij; S2 ¼

Pk<m;m2M2

dkX kij.

(i) Greedy algorithm on base of amij=dm for solving Problem L2.

Let F ða; d1; d2Þ ¼ a� w2d2

Q�d2� w1½ðQ�d2Þd1þad1d2�ðQ�d2�d1ÞðQ�d2Þ

. It is observed from the first part of the function that F ða; d1; d2Þ is an

increasing function of a. For the second part of the function w2d2

Q�d2is increasing in d2. The third part of the function

is given by

w1½ðQ� d2Þd1 þ ad1d2�ðQ� d2 � d1ÞðQ� d2Þ

¼ w1d1

Q� d2 � d1

þ w1ad1d2

ðQ� d2 � d1ÞðQ� d2Þ:

This is an increasing function of d1 and d2. So F ða; d1; d2Þ is strictly increasing in a and strictly decreasing in d1 and d2.So a=d can be used as the sorting criterion for solving the knapsack problem.

(ii) Solving X 0 for Problem L2.

Case 1. If m 2 M1

F ðxÞ ¼ Aþ ax� w2S2

Q� S2

� w1½ðQ� S2ÞðS1 þ d1xÞ þ aS2ðS1 þ d1xÞ�ðQ� S2 � S1 � d1xÞðQ� S2Þ

;

oFox¼ a� w1

d1

ðQ� S2 � S1 � d1xÞ þaS2d1

ðQ� S2 � S1 � d1xÞðQ� S2Þþ d1½ðQ� S2ÞðS1 þ d1xÞ þ aS2ðS1 þ d1xÞ�

ðQ� S2 � S1 � d1xÞ2ðQ� S2Þ

" #

¼ a� w1

d1½ðQ� S2Þ þ aS2�ðQ� S2 � S1 � d1xÞ þ d1ðS1 þ d1xÞ½ðQ� S2Þ þ aS2�ðQ� S2 � S1 � d1xÞ2ðQ� S2Þ

¼ a� w1

d1½ðQ� S2Þ þ aS2�ðQ� S2 � S1 � d1xÞ2

;

oFox¼ 0 ) x ¼ 1

d1

ðQ� S1 � S2Þ �w1d1½ðQ� S2Þ þ aS2�

a

� �1=2" #

;

o2Fox2¼ �w1ðd1Þ2½Qþ ð1� aÞS2�

ðQ� S2 � S1 � d1xÞ3< 0:

So F ðxÞ must have a unique maximum value.

Case 2. If m 2 M2

F ðxÞ ¼ Aþ ax� w2ðS2 þ d2xÞðQ� S2 � d2xÞ �

w1ðQ� S2 � d2xÞS1 þ aS1ðS2 þ d2xÞðQ� S1 � S2 � d2xÞðQ� S2 � d2xÞ ;

oFox¼ a� w2

d2

ðQ� S2 � d2xÞ þd2ðS2 þ d2xÞðQ� S1 � d2xÞ2

" #

� w1

S1d2

ðQ� S1 � S2 � d2xÞ2þ aS1d2

ðQ� S1 � S2 � d2xÞðQ� S2 � dx2Þ

"

þ aS1d2ðS2 þ d2xÞ½ðQ� S2 � da2xÞ þ ðQ� S1 � S2 � d2xÞ�ðQ� S1a� S2 � d2xÞ2ðQ� S2 � d2xÞ2

#

¼ a� w2d2Q

ðQ� S2 � d2xÞ2� w1S1d2

ðQ� S1 � S2 � d2xÞ2

� aS1d2w1½ðQ� S2 � d2xÞðQþ S2 þ d2xÞ � S1Q�ðQ� S1 � S2 � d2xÞ2ðQ� S2 � d2xÞ2

:

oFox¼ 0 ) x satisfies the following condition :

a� w2d2Q

ðQ� S2 � d2xÞ2� w1S1d2

ðQ� S1 � S2 � d2xÞ2� aS1d2w1½ðQ� S2 � d2xÞðQþ S2 þ d2xÞ � S1Q�

ðQ� S1 � S2 � d2xÞ2ðQ� S2 � d2xÞ2¼ 0:

This equation can be solved when the values for a; d2;Q; S1; S2;w1;w2 are available. However, there are several special casesin which there is a closed form solution.


(1) If S1 ¼ 0, then the above equation can be simplified as follows:

a� w2d2Q

ðQ� S2 � d2xÞ2¼ 0 ) x ¼ 1

dðQ� S2Þ �

w2d2Qa

� 1=2" #

:

(2) If S1 is small and a is small, the above equation can be simplified as follows:

a� w2d2Q

ðQ� S1 � S2 � d2xÞ2� w1S1d2

ðQ� S1 � S2 � d2xÞ2¼ 0 ) x ¼ 1

dðQ� S1 � S2Þ �

w2d2Qþ w1S1d2

a

� 1=2" #

:

Now the concavity of F can be justified. Let

Q1 ¼ ðQ� S1 � S2Þ; Q2 ¼ Q� S2;

o2F

ox2¼ � aw2Qðd2Þ2

ðQ2Þ3� aw1S1ðd2Þ2

ðQ1Þ3� aw1S1ðd2Þ2

ðQ1Þ2Q2

� aw1aS1ðd2Þ2ðS2 þ d2xÞðQ1Þ

3Q2

� 2aw1S1ðd2Þ2ðS2 þ d2xÞðQ1Þ

2ðQ2Þ2

� aw1S1ðd2Þ2

Q1ðQ2Þ2� aw1aS1ðd2Þ2ðS2 þ d2xÞ

Q1ðQ2Þ3

< 0:

So F must have a unique maximum value with respect to x.

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Bandwidth packing with priority classes