Bandwidth packing with queuing delay costs ... - Semantic Scholar

European Journal of Operational Research 112 (1999) 635±645

Theory and Methodology

Bandwidth packing with queuing delay costs: Bounding and heuristic solution procedures Ali Amiri

a,1

, Erik Rolland

b,*,2

, Reza Barkhi

c,3

a College of Business, Weber State University, Ogden, UT 84408-3804, USA Department of Accounting and Management Information Systems, Fisher College of Business, The Ohio State University, Columbus, OH 43210, USA Department of Accounting, Pamplin College of Business, Virginia Polytechnic and State University, Blacksburg, VA 24061-0101, USA b

c

Received 1 November 1996; accepted 1 September 1997

Abstract In this paper we propose a new formulation for the bandwidth packing problem (BWP) in telecommunications networks. This problem is one of selecting calls, from a list of requests, to be routed in the telecommunications network. We consider both revenue losses and costs associated with communications delay as parts of the objective. An ecient Lagrangean relaxation based heuristic procedure for ®nding bounds and problem solutions is demonstrated. Computational results from a large array of instances are reported. We demonstrate that the procedure is ecient in ®nding good solutions while expending a modest amount of computational e€ort. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Bandwidth packing; Path assignment; Call routing; Telecommunications networks; Lagrangean relaxation; Sub-gradient search; Heuristics

1. Introduction One important problem in managing a telecommunication network involves deciding which calls on a list of requests, called a call table, should be routed on the network, and subse-

*

Corresponding author. E-mail: [email protected]. 2 E-mail: [email protected]. 3 E-mail: [email protected]. 1

quently determining a path for each call to be routed. The path for each routed call is to be selected from all possible paths in the network. Typically, the network topology, the capacities of the links, the call table (including revenue and trac requirements/demand of each call), and a unit delay cost are given. Versions of this problem have been studied by Anderson et al. (1993), Laguna and Glover (1993), Cox et al. (1991), Parker and Ryan (1995) and Park et al. (1996), and this problem is typically referred to as the bandwidth packing problem (BWP). The objective of the BWP has in these past research e€orts

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 4 0 1 - 3

636

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

been to maximize the total revenues from calls that are routed. Route (or path) selection is a signi®cant factor in determining response time experienced by network users, and has a major e€ect on the utilization of network resources (e.g., node bu€ers and link capacities). A good routing policy could also allow new users to utilize the network without signi®cant deterioration of the quality of service to existing users and without incurring the costs of establishing new links or upgrading the capacities of existing links. During the management process of the network, tradeo€s have to be made between revenue maximization and response time to users. If the revenue maximization factor alone is considered, network users will experience signi®cant delays, and the quality of service will deteriorate. The model developed in this paper addresses this problem by incorporating both revenue loss and response time (total delay) costs in the objective function. Anderson et al. (1993), Laguna and Glover (1993), Cox et al. (1991), Parker and Ryan (1995) and Park et al. (1996), all proposed a version of the BWP that considers only revenue maximization. They formulated the problem and presented heuristic solution procedures based on tabu search (Anderson et al., 1993; Laguna and Glover, 1993), genetic algorithms (Cox et al., 1991), column generation (Parker and Ryan, 1995), as well as integer programming (Park et al., 1996). Motivated by the important applications for bandwidth packing, as well as by the limitations of the current methods, we present a new formulation that seeks to maximize the di€erence between total revenue of calls to be routed and total delay cost. We develop a procedure which generates feasible solutions as well as bounds for this problem. The remainder of this paper is organized as follows. In Section 2, a mathematical formulation of the path assignment 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 Section 4. Computational results are reported in Section 5. The conclusions are summarized in Section 6.

2. Problem formulation In order to formulate the BWP in a telecommunications network, we introduce the following notation: N E M

the set of nodes in the network the set of undirected links in the network the set of calls. Each call is represented by a communicating node pair the demand of call m 2 M dm rm the revenue from call m 2 M O(m) the source node for call m 2 M D(m) the destination node for call m 2 M capacity of link (i,j) Qij C unit delay cost The BWP can now be de®ned as follows: Given a graph G ˆ (N, E) and a set of call requests (a call table) M ˆ (O, D, r, d), we seek to maximize the pro®ts from the routed calls, while minimizing the queuing delay costs, and while not exceeding the capacities on the communication links. We assume that the network topology, the capacity of the links, and the trac requirements and revenues for all the calls are known. We also make some assumptions which are typically used in modeling the queueing phenomena in telecommunications networks. Speci®cally, we assume that nodes have in®nite bu€ers to store messages waiting for transmission on the links, that the arrival process of messages to the network follows a Poisson distribution, and that message lengths follow an exponential distribution. We further assume that the propagation delay in the links is negligible, and that there is only a single class of service for each communicating node pair. Even though the list of calls (requests) is assumed to be known in advance, the trac requirement for each call is usually bursty, as is the case with video and data transmission. That is, calls exhibit variable bit rates. As an approximation, we will use the M/M/1 model for link queueing delays. The validity of this assumption is supported by experimental evidence (in the design of packet-switched networks): the optimal routing is insensitive to the shape of the delay versus link

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

load curve, and is only a€ected by its asymptotic value, i.e. the link capacity (which is generally the same for all models) (Gerla, 1973). Under these assumptions, the telecommunications network is modeled as a network of independent M/M/1 queues (Kleinrock, 1964, 1976), in which links are treated as servers with service rates proportional to the link capacities. The customers are messages whose waiting areas are the network nodes. The queueing delay in link (i, j) is 1/ (lQij ÿ Aij ), where 1/l is the average message length, Qij is the capacity of link (i, j), and Aij is the arrival rate of messages to link (i, j). The average end-to-end delay in the network can be estimated as the weighted sum of the expected delays of the links in the network. We de®ne the decision variables as follows: Ym ˆ



1

if call m is routed;

0 otherwise; 8 > < 1 if call m is routed through a path that uses link …i; j†; Xijm ˆ > : 0 otherwise;

Wijm

8 1 if call m is routed through a path > > > < that uses link …i; j† in the ˆ > direction of i to j; > > : 0 otherwise:

Now, the average end-to-end delay becomes: P m 1 X m2M xij P ; T …i;j†2E Qij ÿ m2M xm ij where T ˆ

1X m A : m m2M

Note that T is constant, and therefore can be removed from consideration. Now, since C is de®ned as the unit queueing delay cost, the total queueing delay cost is represented by the term: P m m X m2M d Xij P : C Q ÿ m2M d m Xijm …i;j†2E ij

637

The organization managing the network incurs an opportunity cost associated with this queuing delay. This opportunity cost could represent the lost revenues from customers who ®nd the delay unacceptable, and hence switch to competitors. Alternatively, seen from a customer viewpoint, the delay cost could be viewed as the cost of lost productivity or the cost of delayed decision-making. Thus, the delay costs can be viewed as being similar to the backorder costs in a manufacturing/ inventory system; that is, the delay cost represents an expression of the potential loss of customers, customer loyalty or customer productivity. In this paper, the objective is to maximize the revenues of routed calls less this backorder, or queueing delay, cost. Given our objective of minimizing the delay costs while maximizing revenues, we can introduce arti®cial delay costs to achieve a desired level of tradeo€ between total revenues and average response time to customers. Our proposed model explicitly addresses this tradeo€, and some implications of this tradeo€ are further discussed in Section 5. The problem can now be properly formulated as follows: Problem P: X X rm Y m ÿ C ZP ˆ max m2M

…i;j†2E

P m2M

Qij ÿ

P

d m Xijm

m2M

d m Xijm …1†

s.t. X X Wijm ÿ Wjim ˆ j2N

j2N

8 m Y > > > > > > > < ÿY m > > > > > > > :

if i ˆ O…m† if i ˆ D…m† 8…i; j† 2 E and m 2 M;

0

otherwise

Wijm ‡ Wjim 6 Xijm ; X m2M

…2†

d m Xijm 6 Qij ;

8…i; j† 2 E and m 2 M;

…3†

8…i; j† 2 E;

…4†

638

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

Xijm 2 f0; 1g;

8…i; j† 2 E and m 2 M;

…5†

Y m 2 f0; 1g;

8m 2 M;

…6†

Wijm 2 f0; 1g;

8…i; j† 2 E and m 2 M:

…7†

In this formulation, the two terms in the objective function represent total revenues of routed calls and the total queuing delay cost, respectively. The format of the queuing delay component is in accordance with our queuing modeling assumptions. The queuing delay cost (C) implicitly measures the importance of network utilization (higher C means lower network utilization). We address the issue of the choice of C further in our computational results. As the queuing delay costs are to be minimized, we assume that the denominator is always greater than zero. Constraint set (2) contains the ¯ow conservation equations which de®ne a route (path) for each call represented by a communicating node pair. Constraint set (3) links together the Xijm and Wijm variables. Even though the problem can be correctly formulated with either Xijm or Wijm variables only, both of them are useful in the Lagrangean relaxation developed in the next section. Constraint set (4) represents the capacity constraints on the links. Constraint sets (5)±(7) enforce the integrality conditions on Xijm ,Ym and Wijm variables, respectively. 3. A Lagrangean relaxation of the problem Problem P is a combinatorial optimization problem with a nonlinear objective function. The problem studied in Anderson et al. (1993), Laguna and Glover (1993) and Park et al. (1996) is a special case of problem P and is known to be NPcomplete (Garey and Johnson, 1979). Since problem P is a nonlinear mixed integer programming problem and since a typical problem contains thousands of variables and constraints, it is dicult to solve this problem to optimality using standard mixed-integer±linear programming tools. We propose, instead, 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 non-negative multipliers am ij for all …i; j† 2 E and m 2 M, respectively. Problem L: ZL ˆ max

X

m

r Y ÿC

m2M

‡

…i;j†2E

X X m2M …i;j†2E

P

X

m

d m Xijm m m m2M d Xij

m2M

Qij ÿ

P

m m m am ij …Xij ÿ Wij ÿ Wji †

…8†

s.t. Eqs. (2), (4)±(7). Problem L can be decomposed into two subproblems as follows: Problem L1: X X X m m rm Y m ÿ am max ij …Wij ‡ Wji † m2M

…9†

m2M …i;j†2E

s.t. Eqs. (2), (6) and (7). Problem L2: max

X X m2M …i;j†2E

m am ij Xij

ÿC

X …i;j†2E

P m2M

Qij ÿ

P

d m Xijm

m2M

d m Xijm …10†

s.t. Eqs. (4) and (5). Problem L1 can be further decomposed into |M| sub-problems (one for each call) as follows. max

rm Y m ÿ

X …i;j†2E

s.t. X X Wijm ÿ Wjim j2N

j2N

m m am ij …Wij ‡ Wji †

8 m Y > > > < ÿY m ˆ > > > : 0

Y m 2 f0; 1g; Wijm 2 f0; 1g;

…11†

if i ˆ O…m† if i ˆ D…m† 8i 2 N ;

…12†

otherwise …13†

8…i; j† 2 E:

…14†

Similarly, problem L2 can be further decomposed into |E| sub-problems (one for each link) as follows.

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

max

X m2M

m am ij Xij

X

s:t:

m2M

Xijm

ÿC

X …i;j†2E

d m Xijm 6 Qij ;

2 f0; 1g

P

d m Xijm …15† m m m2M d Xij

m2M

Qij ÿ

P

4. A heuristic solution procedure 8 m 2 M:

…17†

Procedure Proc1: Step 1: Reorder the Xijm variables by sorting m them in non-increasing order of am ij =d ; re-index the variables in this order, and let m ˆ 0. Step 2: Let m ˆ m + 1 and set  X0 if am ij > 0 and X0 > 0; Xijm ˆ 0 otherwise 8