The Pennsylvania State University The Graduate School College of Engineering STOCHASTIC POST-PROCESSING FOR TOPOLOGY DESIGN AND

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1 The Pennsylvania State University The Graduate School College of Engineering STOCHASTIC POST-PROCESSING FOR TOPOLOGY DESIGN AND CAPACITY ASSIGNMENT ON A MULTI-CLASS NETWORK WITH GUARANTEES ON ROBUSTNESS AND QUALITY OF SERVICE. A Thesis in Industrial Engineering and Operations Research by Emmanuelle J. Wallach c 2012 Emmanuelle J. Wallach Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2012

2 The thesis of Emmanuelle J. Wallach was reviewed and approved by the following: M. Jeya Chandra Professor of Industrial Engineering Thesis Advisor Jack C. Hayya Professor Emeritus of Supply Chain and Information Systems Paul Griffin Professor of Industrial Engineering Department Head, Industrial and Manufacturing Engineering Signatures are on file in the Graduate School. ii

3 Abstract This thesis considers a domain in the Internet and proposes a strategic problem of designing its network. In particular, an optimization problem is formulated and solved for planning the capacities of the links of the multi-class network so that (a) the network is robust, and (b) quality of service (QoS) can be guaranteed for traffic flowing across the network. The complexity of the optimization problem is that the QoS constraint, in terms of the decision variables, can neither be expressed in closed-form nor as a convex function. Therefore standard optimization techniques cannot be used. A two-stage heuristic is developed, which first solves a routing problem, ignoring the QoS constraints and then uses the QoS constraints to determine the capacity of the various arcs. Several examples show that the heuristic performs remarkably well, both in terms of speed and solution quality. iii

4 Table of Contents List of Figures List of Tables Acknowledgments vi vii viii Chapter 1 Introduction Definitions The Internet Quality of Service Specific problem Thesis organization Chapter 2 Literature review Modelling techniques to obtain performance measures Multicommodity flow networks Queueing networks Stochastic knapsack problem Resolution techniques to solve the problems to optimality Knapsack problems The Capacity and Flow Assignment Problem Non linear optimization techniques Chapter 3 Problem statement Notation Optimization Problem Formulation The Quality of Service constraint Chapter 4 The DRASTIC Heuristic Deterministic Route Assignment STochastic Implementation for Capacities Deterministic Route Assignment and STochastic Implementation for Capacities (DRASTIC) iv

5 Chapter 5 Numerical tests An example Distance to optimality Small problems Behavior for larger problems Chapter 6 Conclusions 34 Bibliography 36 v

6 List of Figures 1.1 Schematic view of the Internet A portion of the Internet A domain Schematic representation of a queue A generic loss system Link capacities obtained by solving S vi

7 List of Tables 2.1 Performance measures available for single-server queues depending on the arrival process and service time distribution in the case of service on a first-come firstserve basis Parameters for the commodities in the example problem Routes for each commodity Deterministic approximation of the capacity on each arc Total loss for each commodity before capacity optimization Optimized capacity on each arc Total loss for each commodity after capacity optimization Comparison between the heuristic solution and the optimal solution Computational times Loss probabilities in the deterministic case Loss probabilities in the stochastic case Capacity and cost ratios vii

8 Acknowledgments I received assistance from three advisers to complete this degree. I am particularly indebted to Dr. Natarajan Gautam, who suggested this problem. I would also like to thank Dr. Tom Cavalier, who agreed to take over when Dr. Gautam moved to Texas A&M, and Dr. M. Jeya Chandra, who inherited me when Dr. Cavalier retired. This research was partially supported by NSF Award No. ANI , as well as by Mr. and Mrs. William L. Weiss through the Weiss Fellowship program. I am grateful to Professor Raj Acharya, Professor George Kesidis, Dr. Donna Ghosh and Dr. Venkatesh Sarangan for all the meaningful discussion and valuable criticisms that helped formulating and solving the research issue posed in this thesis. viii

9 Chapter 1 Introduction Real-time Internet applications, such as Internet telephony, video-on-demand, and distributed computing, in addition to requiring much bandwidth, need some assurance pertaining to their timely transmission: what is called Quality of Service (QoS) [17]. The current Internet service is called best effort : all data are treated equally and served on a first-come, first-serve basis. This is fine for messages such as , but real-time applications cannot suffer the least delay. Users are therefore demanding that the Internet support different levels of guaranteed QoS appropriate to each real-time application. Since consumers are willing to pay for such a service, it is of interest for Internet service providers (ISPs) to hear these complaints, as they pave the way for the generation of revenue from Internet traffic. Now imagine that a handful of new computers need to be connected to the Internet. Which ones should be linked to each other? How much capacity should each new link be assigned? The necessary materials are expensive, and the installation, cumbersome, so the resulting design will be used for years to come. How does one create the most efficient network, one that would allow a smooth flow of information? And how does one make the design process efficient, that is, arrive at a satisfactory solution quickly and easily? This problem generates two conflicting concerns: users want the assurance of QoS (a smooth flow of information ), and service providers are being forced by the economy to review their designs and operations in order to make them more efficient. Operations Research provides the necessary resources for designing such a network: it has the tools to adequately model and optimize Internet networks for the service providers, while addressing requirements of the customers. A number of characteristics of such problems make them hard to model and solve. Internet traffic is by nature stochastic, meaning it cannot be predicted with certainty; at best, the distribution of demand is kwown. In general, the distribution itself is hard to obtain and to deal with (e.g., fat tail, self-similarity). Moreover, the model must be detailed enough to provide relevant QoS measures such as delays of thousandths of seconds. All these considerations indicate that telecommunication networks generate complex problems that are hard to model and solve. The contribution of this research is two-fold: from the application standpoint, it means to

10 2 help design a QoS-aware Internet; from an Operations Research standpoint, the intention is to design efficient algorithms to solve, either optimally or approximately, the complex problems generated by the search for a QoS-aware Internet. This complexity is due mainly to the importance of stochastic considerations for QoS. The main objective of this research is thus to apply optimization methods from Operations Research to telecommunication networks. The remainder of Chapter 1 is organized as follows. Section 1.1 motivates the present research through some telecommunication vocabulary and a brief description of the computer network background. Subsequently, Section 1.2 defines the precise problem under consideration. Section 1.3 then details the organization of the remainder of this thesis. 1.1 Definitions The Internet The Internet, or web, is essentially a connected network of computers. These computers fall into one of many categories: there are PCs, but also specialized computers such as routers (which are usually high-degree nodes) or servers (used to store information). On a smaller scale, a single computer can itself be viewed as a network consisting of its various components such as cpu, memory, etc. Depending on the level of accuracy required, such networks may be modelled using the well-developed deterministic network theory, or as queueing networks if it is wished to add stochastic flavor. Using standard network modeling terminology, nodes represent computers or routers and directed arcs represent the physical connections between the nodes. Data constitutes the traffic flow. The Internet has a hierarchical structure. It is divided into domains, also known as autonomous systems, which are independently owned and operated. Figure 1.1 shows a schematic view of the Internet consisting of independent domains. Figure 1.2 shows a portion of the real Internet, circa 1999; domains are represented by nodes in this illustrative depiction. This is due to the hierarchical nature of the Internet: in terms of inter-domain travel, domains can be approximated as nodes, whereas in intra-domain travel the domain s own network must be considered. Figure 1.3 presents a schematic view of a domain. Nodes in a domain may be divided according to their position. Core routers are internal to the domain; they are connected only to nodes belonging to the same domain. Edge routers provide a connection between the domain they belong to and other domains. Note that personal computers are end nodes with degree 1 and therefore are not usually modelled, as they would not bring additional detail. Although the size of a domain (its number of nodes) may be quite variable, in [8] domains are assumed to contain an average of 10 nodes. The domain administrator is responsible for designing and maintaining the domain network, as well as routing all calls within his domain. He has no control over other domains, as domains do not routinely exchange topology information. A call, or request, is loosely defined as a series of related packets emanating from a com-

11 3 Figure 1.1. Schematic view of the Internet. puter and destined to a single recipient. For actual transmission, these packets are treated independently of each other and no differently than any other data traveling on the network [18]. However, considering calls instead of packets is useful in describing the transmission as experienced by the user, so the concept of a call is fundamental to a QoS guarantee. Both call [6] and packet arrival processes have been studied over the Internet in order to help design and plan a network, allocate resources efficiently, and develop efficient algorithms to handle both calls and packets. An admission policy decides which calls to accept and which to drop at the outset. If accepted, a routing protocol decides where to send the call Quality of Service In the problem considered in this research, the objective is to minimize cost while guaranteeing QoS. Note that there are nearly as many definitions of QoS as there are researchers currently working on the subject. The most common and relevant ones are end-to-end delay, or latency (the time it takes for a message to reach its destination from its origin) and loss probability (the probability that the message is dropped en route due to lack of capacity). Delay is more relevant for real-time applications such as video conferencing, while loss probability contrarily affects data transfers most. Other commonly mentioned QoS parameters include delay variation (or jitter), availability (uptime), and data transfer rate (throughput) [17]. Moreover, different classes of applications (say, telephone calls versus video-on-demand) may have different requirements for various QoS measures. These measures are difficult to obtain for various reasons. For additive parameters, such as end-to-end delay, the difficulty is not so much in obtaining the delay for each arc as in keeping track of the total along the whole network. For loss probability, the difficulty

12 4 Figure 1.2. A portion of the Internet. Figure 1.3. A domain. lies in the existence of different classes of traffic, with each having its own characteristics. Other QoS parameters, say, bandwidth, are called bottleneck parameters, in that the QoS value of a path is the one of its weakest link. Those are easier to solve. In addition, all the data is of stochastic nature, which means that QoS measures are themselves stochastic variables. All this makes QoS measures very difficult to obtain. Another topic of interest is robustness or survivability. Nodes in the Internet are notoriously prone to failures. A node failure is of little importance in the case of best effort service, but it

13 5 has the potential to deteriorate QoS measures for sensitive traffic. As a result networks designed with QoS in mind should incorporate spare capacity disseminated through the network. This spare capacity allows the reassignment of calls whose route has failed on other routes without having an adverse impact on overall QoS measures [7, 14]. The ultimate goal is to develop efficient QoS-aware inter-domain routing algorithms. This thesis addresses the first step, namely, intra-domain routing that satisfies both QoS and robustness requirements. 1.2 Specific problem Come back to our original problem of integrating a new computer network into the Internet. There are two ways to achieve this goal. One is to solve, via an available algorithm, a less-detailed approximation of the problem. In our case, this means choosing a definition of smooth flow (the simplest one being that there is enough capacity for all anticipated traffic on average) that allows the problem to be approximated by a readily-solved model. The other way consists of searching for the best combination of a fast and accurate heuristic algorithm to solve a more-detailed model of the problem. Here, this means choosing a heuristic algorithm to solve what is deemed to be the most accurate definition of smooth flow, that is, including some Quality of Service (QoS) considerations such as time in transit or probability of being interrupted. In the first approach, deviation from the original problem is mainly due to the simplifying assumptions in the modelling choice; in the second one, while the deviation due to modelling has been reduced, it is now much more difficult to obtain a high-quality solution. There is a large body of literature available dealing with the first approach, as will be reviewed in Chapter 2. This thesis is mainly concerned with the second approach. This seems better suited to deal with the inherent stochasticity of the network design problem, allowing the diverse existing approximations to be measured against a common standard, as well as yielding new solutions obtained in a more straightforward way. Specifically, consider the following strategic problem that a domain in the Internet is interested in solving, namely, what link capacities to use to connect the nodes in the domain. The flow on this domain would consist of calls whose origin or destination nodes are within the domain, and of calls whose origin and destination nodes are located within other domains. Such inter-domain calls travel from one domain to a neighboring one. In this thesis it is assumed that the demand is known; therefore no distinction is made between core nodes and edge nodes (calls said to originate (respectively terminate) at an edge node actually originate from (respectively terminate at) a connected neighboring domain). The domain is a connected network whose topology and demand matrix is known, that is, the following information is given: the set of nodes to be connected by the network, the set of all possible arcs connecting the nodes, and the predicted demand between nodes. The decision variables considered here are the capacities on each arc (the capacity on an arc may be 0, in which case the arc is not included in the final network). The objective is to minimize the installation cost of all arcs with their required capacities. The constraints guarantee quality

14 6 of service (QoS) for all traffic. It is required that the network should support multiple classes of applications. Demand is described as a set of applications, distinguished by their origin and destination nodes, the bandwidth they require, and the interarrival time and call duration distributions. What needs to be determined is the final set of arcs constituting the network and the capacity - in terms of bandwidth - of every arc. Specifically, the objective is to design a robust network at the lowest possible cost, while maintaining the required level of QoS. It is assumed that robustness would be achieved by having at least one alternate route in case of failure of a node or an arc. The cost of the network is simply that of the installation of all the required links; however, no utilization cost is considered. The QoS that the domain would like to assure is that the fraction of calls that are either denied entry into the system or rejected during the middle of a call is guaranteed to be below a threshold. 1.3 Thesis organization The remainder of this thesis is organized as follows. There will first be a review of the available literature on the subject in Chapter 2. Next, the problem will be stated analytically in Chapter 3. Then, a mathematical model will be proposed. The complexity of the formulation will then be discussed. Chapter 4 introduces the DRASTIC algorithm, a heuristic approach to solve the problem, as an exact solution is mathematically intractable for large problems. Chapter 5 contains some numerical results. Finally, Chapter 6 concludes this work with suggestions for future work.

15 Chapter 2 Literature review This chapter presents the Operations Research concepts that will be used in this thesis. These concepts can be classified into two categories: modelling techniques and resolution techniques. Modelling techniques are used to define a suitable model from the original system, for example, modelling the Internet as a network or a sequence of servers as a series of queues. These modelling techniques allow performance measures, such as QoS measures, to be obtained analytically. The stylized process is subsequently embedded in a mathematical program in order to be optimized. The resulting program is then solved using resolution techniques from Operations Research. 2.1 Modelling techniques to obtain performance measures The most frequently used modelling techniques in this thesis are multicommodity flow networks, queueing models, and the stochastic knapsack. Now follows a brief overview of these topics Multicommodity flow networks Multicommodity flow networks [1, 2] allow multiple commodities to flow in a single capacitated network. In traditional networks, each node may generate either supply or demand for a single commodity flowing through the network. The traffic is usually fungible, meaning that the commodities offered from any source node are equivalent and it does not matter from whence a sink node gets its supply. In multicommodity flows, each commodity is uniquely defined by its origin and destination points. Other characteristics, such as bandwidth requirements for Internet networks, may vary across commodities as well. All of these commodities share a single network. The multicommodity minimal cost flow problem determines the flow of each commodity on each capacitated arc in the network such that total travel cost is minimized. It is usually solved using a column generation approach. The multicommodity capacity and flow assignment problem, much harder to solve, finds the optimal capacity for each arc in the network in addition to the flow of each commodity on each arc. The difficulty of the problem is compounded when both stochastic data and a QoS constraint are added, as will be examined in Chapter 3.

16 Queueing networks Modelling a computer as a queueing system has been extensively studied in the literature [11, 4]. A schematic representation of a queue is presented in Figure 2.1. A queueing model allows the following performance measures to be computed: time spent waiting in queue, number of items in queue, total time spent in the system (waiting in queue and being served), number of items in the system (in queue or being served), utilization of the servers (percentage of time the servers are busy), and characteristics of the exit process. Depending upon the complexity of the arrival process and the service time distribution, the aforementioned performance measures may be available as distributions (sometimes as their Laplace-Stieltjes transforms), averages, or average approximations (see Table 2.1). The departure process will be Poisson - a sought-after feature because it results in easier computations - only if the arrival process is Poisson and the service times are either exponentially distributed (with an infinite queue) or there is an infinite number of servers. Figure 2.1. Schematic representation of a queue. Table 2.1. Performance measures available for single-server queues depending on the arrival process and service time distribution in the case of service on a first-come first-serve basis. Arrival process Service time distribution Performance measure availability Poisson Exponential distributions Poisson General (non-exponential) Laplace-Stieltjes transforms of the distributions non-poisson Exponential distributions non-poisson General (non-exponential) approximations of and bounds on the averages Stochastic knapsack problem Another problem of interest is the stochastic knapsack problem [21]. It is used to model a loss system, that is, a finite collection of resources shared by calls that may belong to various classes. An arriving call is either admitted into the system, based upon some call admission control policy taking both the call s class and the state of the system into account, or it is blocked and lost. In the complete sharing admission control policy, an arriving call is admitted unless it arrives

17 9 to find insufficient resources available, in which case the call and its potential revenue are lost. If a call is admitted, it remains in the system for the duration of its holding time. This is the fundamental difference between a loss system and a queueing system. In the latter a call may remain in queue until sufficient resources are freed, then be in service for the duration of its holding time. The current Internet is particularly well-suited for modelling as a loss system, as blocked and dropped calls are indeed lost from the network. It is left to the application to try again later, and every trial is treated as a new arrival by the network routers. When admitted, a call holds the network resources for the duration of the call. Upon completion, a call releases the resources and departs from the system. Figure 2.2 represents a generic loss system. Figure 2.2. A generic loss system. The classical deterministic knapsack problem [22, 5] involves a knapsack of capacity C resource units and K classes of objects. A class-k object has size b k and a reward r k is accrued for each class-k object included in the knapsack. The problem is to decide how many objects of each class to include in the knapsack in order to maximize the total reward subject to the constraint that the selection of objects must fit in the knapsack (i.e., K k=1 b k y k C with y k the number of class-k objects included in the knapsack). In the stochastic version of this problem, class-k objects again have heterogeneous resource requirements (most commonly bandwidth requirements), but they now arrive and depart at random times. Assume that the arrival processes for all K classes are Poisson and independent from one another, with rate λ k for class-k objects. Further assume that holding times are exponentially distributed and independent of both each other and arrival processes, with mean 1/µ k for class-k objects. Let X(t) = (X 1 (t), X 2 (t),..., X K (t)) denote the state of the system at time t, that is, the number of calls of class 1, 2,..., k currently in the system, and {X(t)} the associated stationary stochastic process. This process is an irreducible and aperiodic Markov process over the finite state space S = {(n 1, n 2,..., n K ) : K k=1 b k n k C and n k non negative integer k = 1,..., K}. For each n S, let π(n) denote the probability that the knapsack is in state n in equilibrium. Let ρ k = λ k /µ k be the offered load for class-k objects. Then the equilibrium distribution for the stochastic knapsack, defined below, can be obtained from the truncation of the uncapacitated system for which C = ([10]): π(n) = 1 G K k=1 ρ n k k n k!, n S,

18 10 where G = n S K k=1 ρ n k k n k!. Out of the aforementioned modelling techniques, queueing systems and the stochastic knapsack are solely used to get performance measures. These performance measures are then included in optimization programs specific to the problem and objectives. Note that these techniques provide accurate performance measures, which translates into complicated equations, implying that the corresponding optimization programs will likely be non-linear and need to be solved using techniques drawn from non-linear optimization. Multicommodity flows and the deterministic knapsack problem, by contrast, do not just provide performance measures but define a problem. Algorithms are widely available to solve these last two problems. The next section explains how to solve the deterministic knapsack problem, the multicommodity capacity and flow assignment problem, and gives an overview of nonlinear optimization techniques. 2.2 Resolution techniques to solve the problems to optimality Other well-known techniques such as Markov decision processes and various non linear optimization methods are now presented with a focus on solution techniques Knapsack problems The deterministic knapsack problem, as stated in Section 2.1, can be solved optimally using dynamic programming or branch and bound [5]. However, there also exist fast greedy heuristics that are very efficient. A common greedy heuristic algorithm fills the knapsack starting with the object that generates the most revenue per unit size, i.e., the one for which r k /b k is largest among all classes k. Order classes in non-increasing order of revenue per unit size ratio. Fit as many objects of the highest ranking class as possible in the knapsack. Reduce the remaining capacity accordingly and repeat with the second highest ranking class, etc. In practice this gives good approximate solutions [12]. The bin packing problem is related to the deterministic knapsack problem. The bin packing problem tries to fit a collection of variable-size objects into bins of various sizes (in the knapsack problem all objects are one-dimensional and there is only one possible bin), using the smallest number of bins. The stochastic knapsack problem can also be solved easily provided the classes behave well. An example is provided in Section 3.3. It is located there in order to use the specific notation introduced for this thesis in Section 3.1.

19 The Capacity and Flow Assignment Problem In network design, the Capacity and Flow Assignment Problem (CFA) refers to the problem of simultaneously designing the topology at the lowest cost, determining each arc s capacity to accept all demand between two pair of nodes and computing optimal routes satisfying the QoS constraints. However, the resulting problem is NP-hard. Moreover, most literature on the CFA problem do not make any realistic QoS considerations, as they are often based on deterministic models. A heuristic to solve the CFA problem without any QoS constraint is presented in [23] in the case of concave link cost functions. This procedure is justified by the economies of scale usually observed in practice. An algorithm based on Lagrangian relaxation is presented in [13], where the objective is to minimize the sum of the delays for all commodities. While this is an interesting approach with a nonlinear objective, it does not provide any guarantee on QoS and is therefore not suitable here. One of the rare exact algorithms available [16] solves the CFA problem to optimality by using generalized the Bender s decomposition with simple delay constraints for QoS, assuming that all calls belong to a single class. In this instance, QoS is measured by the average delay of each commodity on each arc. The delay itself is computed by considering the flow of each commodity on an arc as an independent M/M/1 queue, which in fact is unrealistic. The convexity of the QoS constraint ensures that the multi-commodity subproblem obtained by the generalized Bender s decomposition is solvable to optimality. It is also used to generate cuts that accelerate the solution process. Unfortunately, the results in [16] cannot be used in this thesis due to inherent difficulties that will be explained in Chapter 4. Therefore, a new solution procedure specific to the problem studied must be devised. In order to address the issue of node failure in routing, the spare capacity allocation (SCA) problem attempts to decide how much capacity should be reserved on networks for a given traffic flow and network topology, to be used should one node fail [14]. It allocates a back-up path to each commodity and specifies the spare bandwidth that needs to be reserved to guarantee QoS in case of a single node failure. [20] requires at least two arc-disjoint paths for each commodity in order to ensure network survivability in case of arc failure. The problem is then solved approximately via an iterated process with construction and local search phases Non linear optimization techniques The nonlinear optimization techniques of interest in this thesis pertain to the constrained optimization problems with non-convex objectives and non-convex feasible sets. An additional difficulty is that constraints are not available as a closed form function of the design variables. As most efficient algorithms deal with unconstrained optimization [3], one must resort to penalty and barrier functions. In order to solve optimization problems with non-closed form constraints, step searches must be used. Some do not use derivative information, such as the Hooke-Jeeves method [9], or the Nelder-Mead simplex method [19], which are robust and numerically simple. Methods using first- or second-order derivative information, among them quasi-newton methods, are also available [3].

20 Chapter 3 Problem statement 3.1 Notation The parameters used in the problem formulation are defined below. For practical purposes, all individual applications will be referred to as commodities. A commodity is defined by its origin and destination nodes, its arrival rate at the origin node, the bandwidth it requests, and its call duration rate. It is distinct from a class in that it considers origin and destination data. Each commodity is governed by its own network flow constraints, but all share the same network. The terminology is based on the multi-commodity flow problem [1]. The notation used in this thesis is as follows: N the set of nodes in the network, A the set of arcs between two nodes of N that may be included in the network (note that A may be only a subset of the set of all possible arcs connecting two nodes in N; also, if (i, j) A, then (j, i) A), K the set of commodities travelling through the network, κ the total number of commodities, i.e., the cardinality of K, O k the origin node for commodity k K, D k the destination node for commodity k K, b k the capacity or bandwidth requested by commodity k K, λ k the arrival rate of commodity k K at O k, µ k reciprocal of the average call duration of commodity k K,

21 13 c (i,j) the capacity on arc (i, j) A, Γ (i,j) (c (i,j), c (j,i) ) the cost of installing capacity c (i,j) on arc (i, j) A and capacity c (j,i) on arc (j, i) A. Note that Γ (i,j) (c (i,j), c (j,i) ) = χ (i,j) + γ(c (i,j), c (j,i) ). χ (i,j) the fixed cost of setting up arcs (i, j) and (j, i) A, γ(c (i,j), c (j,i) ) the variable cost of installing capacity c (i,j) on arc (i, j) A and capacity c (j,i) on arc (j, i) A. Assume it to be increasing in both c (i,j) and c (j,i). ɛ k the maximal threshold of loss probability for commodity k, M an arbitrary large number useful for the mathematical formulation, f (i,j),k the function giving the loss probability for commodity k K on arc (i, j) A. It will be defined in Section 3.3. In steady-state, there is a non-zero probability that when a new call from commodity k arrives on an arc, the capacity remaining on this arc is insufficient. The call cannot go through and is then blocked, or lost. The loss probability is the probability that an arriving call of commodity k is blocked, i.e., the probability that the capacity remaining on the arc is too small to accommodate a new call from commodity k in addition to all calls (from any commodity) already present. This model does not assume that same class calls (the ones requesting the same amount of bandwidth) have the same arrival or service rates. The variables used in the mathematical formulation are as follows: δ (i,j) the binary variable associated with arc (i, j) A. It is 1 if either arc (i, j) or (j, i) or both have strictly positive capacity, 0 otherwise (if capacity on both (i, j) and (j, i) is zero), r (i,j),k the net arrival rate of commodity k K on arc (i, j) A, l (i,j),k the rate at which commodity k K calls are lost on arc (i, j) A, R (i,j),k the ideal (no-loss) arrival rate of commodity k K on arc (i, j) A, with R (i,j),k = r (i,j),k + l (i,j),k for all commodities k K and arcs (i, j) A. 3.2 Optimization Problem Formulation The mathematical programming formulation of the network planning problem is described as follows:

22 14 OPTIMIZATION PROBLEM FORMULATION min (i,j) A, i<j Γ (i,j) (c (i,j), c (j,i) ), (3.1) s.t. j,(j,i) A r (j,i),k j,(i,j) A ( r(i,j),k + l (i,j),k ) = { λk if i = O k for all k K and i N \ {D k }, 0 otherwise, (3.2) j,(j,d k ) A r (j,dk ),k + (i,j) A l (i,j),k = λ k for all k K, (3.3) r (i,j),k < (1 ɛ k )λ k for all (i, j) A and k K, (3.4) j,(j,i) A r (j,i),k < (1 ɛ k )λ k for all i N such that i D k and k K, (3.5) j,(i,j) A r (i,j),k < (1 ɛ k )λ k for all i N such that i O k and for all k K, (3.6) f (i,j),k (r (i,j),1, r (i,j),2,..., l (i,j),1, l (i,j),2,..., c (i,j) ) = for all (i, j) A and for all k K, l (i,j),k r (i,j),k +l (i,j),k, (3.7) (i,j) A l (i,j),k ɛ k λ k for all k K, (3.8) c (i,j) M(δ (i,j) + δ (j,i) )for all (i, j) A, (3.9) δ (i,j) = 0 for all (i, j) A such that i j, (3.10) r (i,j),k 0 for all (i, j) A and k K, (3.11) l (i,j),k 0 for all (i, j) A and k K, (3.12) c (i,j) 0 and integer for all (i, j) A, (3.13) δ (i,j) = 0 or 1 for all (i, j) A. (3.14)

23 15 The objective (3.1) is to minimize the total cost of installing the links. The operational cost is not included. It is assumed that there is a fixed cost to set up a link, independent of its direction, then a variable cost depending on the capacities installed in both directions. The first set of constraints (3.2 and 3.3) ensures flow conservation: for any commodity, the incoming flow on any node in the network must equal the outgoing flow minus what is generated or consumed at the node. However, some messages may be lost if capacity is insufficient. The rate of lost calls of each commodity on any arc must be recorded for the flow conservation constraints to remain valid. Notice the special structure of the constraint for the destination node of a commodity: the incoming flow must equal the generation rate of messages at the origin node minus what has been lost en route (sum of the rates of loss on all arcs). The second set of constraints (3.4 to 3.6) stipulates that there be at least two distinct routes for each commodity. Failure of one node or arc in the network must not disrupt service: there must be at least one alternate route - not necessarily disjoint from the first one - to be used should an arc or node on the main route fails. This is done to improve the robustness of the network. Had an implicit arc-path formulation been used, the constraint would have required at least two paths to be active for each commodity. However, the difficulty of inserting loss into such a formulation makes it unworkable. In the chosen model, the equivalent constraint states that the rate of a commodity on any arc or through any node must be strictly less than the minimum rate arriving at its destination. The rate of a commodity going through any node in the network is computed as the sum of all rates of the given commodity emanating from or arriving to the node (cf. next set of constraints). The minimum actual rate for a commodity k K through the network is (1 ɛ k )λ k since ɛ k is the maximum loss probability for commodity k over the whole network. The first constraint states that the rate of commodity k K on any arc (i, j) A must be strictly less than the actual arriving rate of commodity k to the network. Consequently, there must be a non-zero rate for commodity k on another arc. The second and third constraints ensure that the rate of commodity k K going through a node i N is similarly limited. A specific percentage required to travel on a secondary route will be implemented in Chapter 4. The third set of constraints (3.7 to 3.10) ensures a satisfactory QoS and deals with capacity determination. The quantity f (i,j),k gives the loss rate of commodity k K on arc (i, j) A as a function of the intended rates of all commodities and of the arc s capacity c (i,j) in constraint 3.7. The intended or potential arrival rate of commodity k K on arc (i, j) A is simply r (i,j),k + l (i,j),k = R (i,j),k. The loss rate l (i,j),k is the product of the intended rate times the loss probability for commodity k on arc (i, j). The total loss rate (i,j) A l (i,j),k for a commodity k K is then constrained to be less than the QoS threshold ɛ k λ k. The function f (i,j),k is very complex: it is non-linear, non-convex and in some the general case cannot be expressed in closed-form. The next subsection (3.3) is entirely devoted to details on how to obtain equation (3.7). Considering the total loss rate over the whole network for each commodity also makes the set of QoS constraints intractable. Additional constraints (3.11 to 3.14) restrict the capacity to zero when no connection is installed on the arc, and set the binary variable recording an arc s utilization to zero for half the

24 16 arcs (those arcs (i, j) A such that i > j: there is no need for them since the binary variable records utilization regardless of the direction). The last set of constraints takes care of variable restrictions. Rates must be non-negative. Capacity on an arc is non-negative and integer, since it represents the number of physical connections to be installed. Finally, the utilization variables are binary. The problem, as formulated above, is then a mixed-integer nonlinear program. It cannot be easily solved because of the non-convexity of the loss probability constraint (3.7) and the difficulty to obtain a closed-form expression. The presence of a non-convex constraint whose values can be obtained only numerically and the integrality of some variables render the problem intractable for existing exact algorithms. In particular, even for the single-class case (b k = b for all k K), it is impossible to use an exact algorithm adapted from the one proposed in [16] for the case when all calls belong to a single class, because of the algorithm s reliance on the convexity of the constraints. 3.3 The Quality of Service constraint The function f (i,j),k gives the fraction of lost traffic, or loss probability, for commodity k K on directed arc (i, j) A as a function of the intended rates R (i,j),1, R (i,j),2,..., R (i,j),κ of all commodities in K and of the link capacity c (i,j). It is obtained by solving the stochastic knapsack problem. The shape of the function depends on the complexity of the problem: in the simplest cases (i.e., single-class and Poisson arrival process), the Erlang loss formula can be used, but even then f (i,j),k is a nonlinear, non-convex function. In general cases, it may not even be available in closed form. Assume that arrivals follow a Poisson process. Start the analysis by letting all calls be of the same class (the multi-class case will be examined later on). The loss probability can then be determined using the Erlang loss formula if it is assumed that the call arrival rates for each commodity are distributed according to a Poisson process and that they are independent from one another. All calls belong to the same class if they all require the same amount of bandwidth (b k = b for all k K) and have the same service rate (µ k = µ for all k K). Note that the various commodities still differ by their arrival rates and their origin and destination nodes. The capacity c (i,j) must therefore be an integer multiple of b, i.e. c (i,j) = α (i,j) b, with α (i,j) a nonnegative integer. Denote the total gross arrival rate (i.e., including the loss probability on the arc) on arc (i, j) A by R (i,j) = k K (r (i,j),k + l (i,j),k ). The fraction of traffic lost on arc (i, j) A is then identical for all commodities k K and can be expressed as follows: ( R(i,j) f (i,j),k (r (i,j),1, r (i,j),2,..., l (i,j),1, l (i,j),2,..., α (i,j) b) = for all (i, j) A and k K. ) α(i,j) µ /α(i,j)! ( ) α (i,j) R(i,j) y, y=0 µ /y! (3.15)

25 17 In the general case when there are multiple classes of applications using the network, for a closed-form solution to be attainable the arrival processes should be Poisson and independent from one another, and the call holding times to be exponentially distributed, as explained in [21]. Define the state of the network X (i,j) (t) = (n 1 (t), n 2 (t),..., n κ (t)), where n k (t) is the number of calls from commodity k currently in progress on arc (i, j) A. Let Z C be the set of feasible states (a state is defined by the number of ongoing calls of each commodity) for arc (i, j) A of capacity C (C being a nonnegative integer), and Zk C be the subset of feasible states in which an arriving call of commodity k K is accepted. Then, Z C = {(n 1, n 2,..., n κ ) : n i is a positive integer for all i K and i K b i n i C}, (3.16) Z C k = {(n 1, n 2,..., n κ ) Z C : i K b i n i C b k }. (3.17) The process {X(t)} t 0, where X(t) = (n 1 (t), n 2 (t),..., n κ (t)), with arc capacity C a strictly positive integer, can be obtained by truncating the same process with unlimited link capacity (C = ), i.e., by truncating the state space from N κ (if the capacity on the arc is infinite) to Z C. Corollary 1.10 in [10] gives the equilibrium distribution of the truncated process given the equilibrium distribution of the original (untruncated) process. Let p (n 1, n 2,..., n κ ) be the equilibrium probability that there are n k ongoing calls of class k K if the arc capacity is infinite, and p C (n 1, n 2,..., n κ ) the equilibrium probability that there are n k ongoing calls of class k K if the capacity on the arc is limited to the positive integer C. If {X(t)} t 0 is a reversible Markov process, then p C (n 1, n 2,..., n κ ) = p (n 1, n 2,..., n κ ) (m 1,m 2,...,m κ) Z C p (m 1, m 2,..., m κ ). (3.18) Thus, once p (n 1, n 2,..., n κ ) is known, p C (n 1, n 2,..., n κ ) can be obtained. Now, the explanation on how to obtain p (n 1, n 2,..., n κ ). Note that if there is no limit on the arc capacity (C = ), then all incoming calls on arc (i, j) A are accepted. In this case, each commodity can be considered independently. If arrivals follow an independent Poisson process for each commodity, and if call holding times are independent and exponentially distributed, then the stochastic process {n k (t)} t 0 is a reversible continuous time Markov chain for each k K, more precisely an M/M/ queue with arrival rate λ k = r (i,j),k + l (i,j),k and service rate µ k. Let p k (n) be the steady-state probability to be in state n for commodity k, that is, the probability that n calls of commodity k are currently in progress. Then, p k (n) = 1 ( ) λ n k e λ k /µ k. (3.19) n! µ k Since all commodities are now independent from each other, the joint distribution can be obtained by multiplication. Let p (n 1, n 2,..., n κ ) be the steady-state probability that there are

26 18 n k ongoing calls for commodity k K; then κ p (n 1, n 2,..., n κ ) = p k (n k ). (3.20) The equilibrium probability that there are n k ongoing calls for commodity k K on arc (i, j) A with finite capacity C is then k=1 p C (n 1, n 2,..., n κ ) = = ( κ k=1 1 λ k n k! κ (m 1,m 2,...,mκ) Z C k=1 κ k=1 1 n k! µ k ) nk e λ k /µ k 1 m k! ( λ k µ k ) nk (m 1,m 2,...,mκ) Z C κ k=1 ( λ k µ k ) mk e λ k /µ k ( ) λ mk. 1 k m k! µ k, (3.21) The loss probability L k for commodity k K on arc (i, j) A with capacity C can then be expressed as follows: L k = 1 (n 1,n 2,...,n κ) Z C k p C (n 1, n 2,..., n κ ). (3.22) According to the original notation, the function f (i,j),k giving the loss probability for a call from commodity k K on arc (i, j) A with capacity c (i,j) as a function of the net arrival rates r (i,j),y and loss rates l (i,j),y of commodities y K on arc (i, j) A is given by the following equation: f (i,j),k (r (i,j),1, r (i,j),2,..., l (i,j),1, l (i,j),2,..., c (i,j) ) = 1 for all (i, j) A and k K. n Z c (i,j) k n Z c (i,j) y K y K ( ) r(i,j),y +l ny (i,j),y µy /ny! ( ) r(i,j),y +l ny (i,j),y /ny! µy (3.23) However, although there exists a formula for the loss probability in the case when interarrival and call holding times are exponentially distributed, a brute-force summation of the terms in the numerator and the denominator is impractical because the discrete state spaces Z C and Z C k are prohibitively large even for moderate values of C and κ. To address this intractability, [21] provides an efficient recursive algorithm to compute these quantities. The drawback is that the loss probability has then to be obtained numerically. Moreover, this function is highly non-convex, even when the Erlang formula can be used. Let S(c) = {(n 1, n 2,..., n κ ) : n y is a positive integer for all y K and y K b y n y = c}, q(c) = (n p 1,...,n κ) S(c) C(n 1, n 2,..., n κ ), T k (c) = (n n 1,...,n κ) S(c) k p C (n 1, n 2,..., n κ ). Note that S(c) is the set of states for which exactly c resource units are occupied, and q(c)

27 19 is the probability of this event occurring in equilibrium. Let q(c) = 0 and T k (c) = 0 for c < 0. Corollary 2.1 in [21] states that the occupancy probabilities q(c), c = 1, 2,..., C, satisfy the following recursive equations: cq(c) = κ k=1 ( ) λ b k k q(c b k ), c = 0,..., C. (3.24) µ k The recursive algorithm is then presented in Algorithm 2.1 in [21]. It computes the blocking probability B k of commodity k K on arc (i, j) A if arc (i, j) has capacity C. 1. Set g k (0) 1 and g k (c) 0 for c < 0. Set q k (0) 1 and q k (c) 0 for c < For all c = 1,..., C, in turn, 2.1. Set g k (c) 1 ( ) κ c k=1 b λ k k µ k q k (c b k ) Set G k (c) = c t=0 g k(t) Set q k (c) g k (c)/g k (c). 3. Set B k C c=c b k +1 q k(c). The loss probability for all commodities k K on arc (i, j) A can therefore be obtained by the algorithm in [21]. Now the capacity and flow assignment problem described in Section 3.2 needs to be solved while incorporating the QoS constraint.

28 Chapter 4 The DRASTIC Heuristic The capacity and flow assignment problem formulated in Section 3.2 optimizes the capacity of each arc (i, j) A and the net rate r (i,j),k and loss rate l (i,j),k of commodity k K on arc (i, j) A for all k K and (i, j) A with respect to building cost. Due to the complexity of the problem, especially the non-convex QoS constraint discussed in Section 3.3, heuristic algorithms are used to find a solution. Notice that once the rate of each commodity has been determined on each arc, the best solution with respect to cost is necessarily to assign each arc the smallest capacity satisfying the QoS constraint set (3.7 to 3.10). This is due to the fact that cost increases with capacity. Moreover, this results in a simple one-variable search per arc, for which there exist efficient exact algorithms, even when the loss probability constraint (3.7) is in its most complex form. The basic idea for a heuristic algorithm is then to first decide on routes for each commodity, using a simplified QoS constraint, then to use the complex constraint to determine capacities. This method will always yield a feasible solution, although with no guarantee of optimality. Sending many commodities through the same arc permits some kind of risk sharing and the resulting capacity for all commodities is always less than the sum of the capacities that would be needed if commodities were assigned each a capacity that they did not share with others. Choosing routes beforehand with limited attention to QoS (that is, determining the capacity assignment based on a much simplified approximation of the QoS constraint) may therefore result in suboptimal routing because it does not take this possible reduction in overall capacity into account. However, the existence of a fixed cost for each arc does encourage commodities to share arcs, and might lead to better solutions than a purely variable cost would yield. 4.1 Deterministic Route Assignment For the route determination problem, all commodities are considered, but the QoS constraint is simplified greatly and replaced by a linear function. Considering all commodities allows the solution procedure to take advantage of the structure of the cost function, in particular the fixed setup cost. Let S 1 be the subproblem concerned with route determination. Replacing the QoS