________________________________________________________________________
Queueing Delay Guarantees in Bandwidth Packing ________________________________________________________________________
ERIK ROLLAND1, ALI AMIRI2 and REZA BARKHI3 1 Department of Accounting & Management Information Systems, Fisher College of Business The Ohio State University, Columbus, OH 43210 E-mail:
[email protected] 2 Department of Management, College of Business Administration Oklahoma State University, Stillwater, OK 74078 E-mail:
[email protected] 3 Department of Accounting & Information Systems, Pamplin College of Business Virginia Polytechnic and State University, Blacksburg, VA 24061-0101 E-mail:
[email protected]
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, assuring maximum service delay in telecommunications networks. The bandwidth packing problem is one of selecting calls, from a list of requests, to be routed in the network. We limit the maximum queueing delay, while maximizing revenues generated from the routed calls. An efficient Lagrangean relaxation based heuristic procedure for finding bounds and solutions to the problem is demonstrated, and computational results from a variety of problem instances are reported. 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.
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 at Oklahoma State University. He received the BS degree in Business Administration in 1985 from the IHEC, Tunisia, the MBA in 1988 and the Ph.D. in Management Sciences and Information Systems in 1992 from Ohio State University. His research interests include data communication and computer network design and analysis, databases, combinatorial and discrete optimization and general OR/MS modeling and analysis. He has published in Computer Communications, Computers & Operations Research, European Journal of Operational Research, and Naval Research Logistics.
Reza Barkhi is an Assistant Professor of MIS in the Department of Accounting and Information Systems, Pamplin College of Business, at Virginia Polytechnic Institute & State University. His current research interests are in the areas of collaborative technologies and problem solving, and topological design of telecommunication networks. Dr. Barkhi has published in journals such as Location Science, European Journal of Operational Research, Group Decision and Negotiation, and Decision Support Systems. He received a BS in CIS from the College of Engineering, and an MBA, an MA, and a Ph.D. from the College of Business all from The Ohio State University.
Erik Rolland is an Assistant Professor of MIS in the Department of Accounting and Management 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 as Computers & Operations Research, European Journal of Operational Research, Transportation Science, and Annals of Operations Research. He received a BS in CIS from the College of Engineering, and an MA and a Ph.D. from the College of Business all from The Ohio 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
1
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.
2
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
3
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).
4 9
5 8
7
6 3 2
1
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.
4
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 ∑ d m X ijm m∈M
Qij −
∑d
m∈M
m
X ijm
and the average link queueing delay in the network is given by d m X ijm ∑ 1 m∈M . ∑ | E | ( i , j )∈E Qij − ∑ d m X ijm m∈M
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
m Xij
1 if call m is routed through a path that uses link (i,j) = 0 otherwise
5
m Wij
1 if call m is routed through a path that uses link (i, j ) = in the direction of i to j 0 otherwise
Problem P: ∑ rm Ym
ZP = Max
(1)
m∈M
subject to: Y m if i = O ( m) m m m ∑ Wij - ∑ Wji = − Y if i = D (m ) j∈N j∈N 0 otherwise m m m Wij + Wji ≤ Xij m ∑ dmXij ≤ Qij m∈M
∑d X − ∑d X m
∑
( i , j )∈E
m∈M
Qij
m ij
m
m∈M
m ij
i ∈ N and m ∈ M
(2)
∀ (i,j)∈E and m∈M
(3)
∀ (i,j)∈E
(4)
≤δ
(5)
m Xij ∈ (0,1)
∀(i,j)∈E and m∈M
(6)
Ym ∈ (0,1) m Wij ∈ (0,1)
∀ (i,j)∈E
(7)
∀ (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 m represented by a communicating node pair. Constraints in set (3) links together the Xij m m m and Wij variables. The problem can be correctly formulated with either Xij or Wij 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
6
m m network. The integrality conditions on Xij ,Ym and Wij 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 m constraint set (3) using nonnegative multipliers αij for all (i,j) ∈ E and m ∈ M, respectively, and relaxing constraint (5) using a nonnegative multiplier ψ:
Problem L:
∑d X − ∑d X m
ZL = Max ∑ rm Ym-ψ m∈M
∑
( i , j )∈E
m∈M
Qij
m ij
m
m∈M
m ij
+ ∑
m∈M
m m m m ∑ αij (Xij - Wij - Wji ) +ψδ (9) ( i , j )∈E
Subject to (2), (4), (6), (7) and (8). Problem L can now be decomposed into two subproblems as follows:
Problem L1: Max
∑
rm Ym -
m∈M
∑
m∈M
m m m ∑ αij (Wij + Wji ) ( i , j )∈E
Subject to (2), (7) and (8).
Problem L2:
7
(10)
∑d X − ∑d X m
m m Max ∑ ∑ αij Xij m∈M ( i , j )∈E
ψ
∑
m∈M
( i , j )∈E Qij
m ij
m
m∈M
(11)
m ij
Subject to (4) and (6).
Problem L1 can be further decomposed into |M| subproblems (one for each call) as follows: Max rm Ym -
m m m ∑ αij (Wij + Wji ) ( i , j )∈E
(12)
subject to: Y m if i = O (m ) m m m ∑ Wij - ∑ Wji = − Y if i = D( m) ∀i ∈ N j∈N j∈N 0 otherwise Ym ∈ (0,1) m Wij ∈ (0,1)
(13)
∀ (i,j)∈E and m∈M
(14)
∀ (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:
∑d X − ∑d X m
m m Max ∑ αij Xij m∈M
-
ψ
∑
( i , j )∈E Qij
m ij
m∈M
m
m ij
(16)
m∈M
subject to: m ∑ dmXij ≤ Qij m∈M m Xij ∈ (0,1)
(17) ∀(i,j)∈E and m∈M
8
(18)
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
m ij
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 m we set Ym = 0 and Wij = 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: m Step 1: Reorder the Xij m αij /dm;
variables by sorting them in nonincreasing order of
Re-indexed the variables in this order, and Let m=0. Step 2: Let m=m+1 and set X if α ijm > 0 and X 0 > 0 m Xij = 0 0 otherwise where X0 = min{ 1 ,
S=
1 dm
[(Qij
- S) - (
ψdmQij m αij
)1/2]} and
k ∑ dkXij .
k
