Design and Management of Complex Distributed Systems: Optimization and Game-Theoretic Perspectives. Amir Nahir

Transcription

1 Design and Management of Complex Distributed Systems: Optimization and Game-Theoretic Perspectives Amir Nahir

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3 Design and Management of Complex Distributed Systems: Optimization and Game-Theoretic Perspectives Research Thesis Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Amir Nahir Submitted to the Senate of the Technion Israel Institute of Technology Nisan 5774 Haifa April 2014

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5 The research thesis was done in the Computer Science Department under the supervision of Prof. Ariel Orda and Prof. Danny Raz. Many people have helped and supported me during my PhD studies. Above all, I would like to thank my supervisors, Ariel and Danny, who have guided me during my studies, and have taught me the nature of real research; my parents, Tamar and Dan Nahir, who have instilled in me a never ending desire to learn; last but not least, to the love of my life, my wife, Karin, who has always been there for me pushing me forward, or holding me back, and always listening. The generous financial support of the Technion is gratefully acknowledged.

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7 The results presented in this thesis have been published in the following papers: Amir Nahir, Ariel Orda, and Ari Freund. Topology Design and Control: A Game-theoretic Perspective. In proceedings of the 28 th Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM 09. Pages , Amir Nahir, Ariel Orda, and Danny Raz. Workload Factoring with the Cloud:: A Game-theoretic Perspective. In proceedings of the 31 st Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM 12. Pages , Amir Nahir, Ariel Orda, and Danny Raz. Distributed Oblivious Load Balancing Using Prioritized Job Replication. In Proceedings of the 8 th International Conference on Network and Service Management, CNSM 12, pages 55 63, Amir Nahir, Ariel Orda, and Danny Raz. Schedule First, Manage Later: Network-aware Load Balancing. In proceedings of the 32 st Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM 13 pages , Amir Nahir, Ariel Orda, and Ari Freund. Topology Design of Communication Networks: A Game-theoretic Perspective. In IEEE/ACM Transactions on Networking, 22(2): , Several other manuscripts are currently under submission. 7

8 Contents Abstract 1 1 Introduction 2 2 Topology Design of Communication Networks: A Game- Theoretic Perspective Related Work The Model Link Prices (Solely) Link Prices and Delay The Directed Network Creation Game Relaying Extent - Product Form Relaying Extent - Bottleneck Form Relaying Extent (Product Form) and Delay Relaying Extent (Bottleneck) And Delay Relaying Extent and Link Costs Conclusion Workload Factoring: A Game-Theoretic Perspective Related Work Background & Motivation Workload Factoring in Cloud Computing Model Nash Equilibrium Existence and Uniqueness Existence Uniqueness Price of Anarchy i

9 3.6 Admission Control Allocation within the Shared Resource Preventing Large User Service Degradation Creating System-wide Fairness Conclusion Replication-based Load Balancing Related Work Model & Basic Scheme Motivation Removing the Scheduling Decision from the Job s Critical Path Improving Scheduling Decision Accuracy Workloads The No Delay Case PS/EXP PS/BP Trace/Web The Effect of Delay PS/EXP PS/BP Trace/Web Addressing Delays Conclusion Reversing the Supermarket: a Distributed Approach for Cloud Elasticity Background and Related Work Our Scheme Static System Dynamic System EC2 Implementation Application to Power Conclusion and Future Work Summary 157 ii

10 Abstract in Hebrew א iii

11 List of Figures 2.1 A network topology with price of anarchy Wmax W min The worst price of anarchy computed based on simulation results, W(v i,v j ) = d(v i,v j ) The worst price of anarchy computed based on simulation results, W(v i,v j ) = d(v i,v j ) The worst price of anarchy computed based on simulation results, W(v i,v j ) = log 10 d(v i,v j ) A figure-8 graph well connected graphs including relays with in-degrees eqal to AWS SimpleDB, Entry test results, 10, 000 repeatitions AWS SimpleDB, Entry test results, 100 repeatitions AWS SimpleDB, Query test results, 10, 000 repeatitions AWS SimpleDB, Query test results, 100 repeatitions Network-based storage on AWS vs. local storage AWS Storage access time Job offloading probability as a function of the aggregated load Expected error rate, as a function of the number of jobs in the short queue Correctness of scheduling decisions as a function of the load Correctness of scheduling decisions as a function of the load Example of Bounded-Pareto cumulative distribution function, H = Trace-based load pattern, Wikipedia, Trace-based load pattern, RTP, iv

12 4.7 Trace-based load pattern, RTP, Trace-based load pattern, World Cup, Redis-based job processing time distribution Markov chain Expected service rate, as a function of the number of jobs in the system, N = Average queuing overhead, PS/EXP, no delays, low load values Average Queuing Overhead, PS/EXP, no delays, high load values Average Queuing Overhead, PS/EXP, no delays, the Supermarket Model vs. replication, high load values Average queuing overhead, PS/BP, α = 0.5, H = 10,000, no delays, the Supermarket Model vs. replication Average queuing overhead, PS/BP, α = 1.4, H = 10,000, no delays, the Supermarket Model vs. replication, high load values Average queuing overhead, Wiki trace, no delays, the Supermarket Model vs. replication Average queuing overhead, RTP trace, no delays, the Supermarket Model vs. replication Average queuing overhead, RTP trace, no delays, the Supermarket Model vs. replication Average queuing overhead, World Cup trace, no delays, the Supermarket Model vs. replication Average Queuing Overhead, PS/EXP, the Supermarket Model vs. replication, d = 2, high load values Average Queuing Overhead, PS/EXP, the Supermarket Model vs. replication, T d = 0.25, high load values Average queuing overhead with bounded Pareto distributed job lengths and delays, α = 0.5, H = 10,000, d = Average queuing overhead with bounded Pareto distributed job lengths and delays, α = 1.4, H = 10,000, d = 2, low loads Average queuing overhead with bounded Pareto distributed job lengths and delays, α = 1.4, H = 10,000, d = 2, high loads Average queuing overhead, Wiki trace, T d = 0.1, the Supermarket Model vs. replication v

13 4.27 Average queuing overhead, RTP trace, T d = 0.1, the Supermarket Model vs. replication Average queuing overhead, RTP trace, T d = 0.1, replication with d = Average queuing overhead, RTP trace, T d = 0.25, the Supermarket Model vs. replication Average queuing overhead with Location-in-Vector Selection, EXP/PS, d = 2, T d = 0.75, load = 85% Average queuing overhead with Location-in-Vector Selection, EXP/PS, d = 3, T d = 0.5, load = 91% Average queuing overhead with Location-in-Vector Selection, EXP/PS, d = 3, T d = 0.5, load = 97% Average queuing overhead with Location-in-Vector Selection, EXP/BP, α = 1.4, H = 10,000, d = 2, load = 97%, T d = Average queuing overhead with Location-in-Vector Selection, EXP/BP, α = 1.4, H = 10,000, d = 2, load = 99%, T d = Average queuing overhead with Location-in-Vector Selection, Trace/Web, RTP , d = 3, T d = Average queuing overhead with Location-in-Vector Selection, Trace/Web, WorldCup, d = 5, T d = Common elasticity control architecture Our fully distributed elasticity control architecture Average queuing overhead as a function of the load, T P = 8,N = M = Probability of VM holding a certain number of jobs, as a function of the number of jobs. T P = 5,d = Probability of VM holding a certain number of jobs, as a function of the number of jobs. d = 5,λ eff = Probability of VM holding a certain number of jobs, as a function of the number of jobs. T P = 5,λ eff = Probability that an arriving job would trigger the activation of a new VM, T P = 5, high loads Synthetic load pattern, gradually changing Synthetic load pattern, sharply changing vi

14 5.10 Average time in the system under fixed arrival rate for different t wait values Number of active VMs in the system under fixed arrival rate for different t wait values Average time in the system under sharply changing arrival rate for different t wait values Number of active VMs in the system under sharply changing arrival rate for different t wait values th percentile time in the system for synthetic arrival patterns. T P = 5,T A = 8,t wake = 120,t wait = VM states under a fixed job arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of idle VMs under gradually changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of busy VMs under gradually changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of VMs in wakeup under gradually changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of VMs in shutdown under gradually changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of idle VMs under sharply changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of busy VMs under sharply changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of VMs in wakeup under sharply changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = Number of VMs in shutdown under sharply changing arrival rate. T P = 5,T A = 8,t wake = 120,t wait = The effective load in the system for synthetic arrival patterns. T P = 5,T A = 8,t wake = 120,t wait = th percentile time in the system for HTTP traces. T P = 5,T A = 8,t wake = Number of idle VMs when running the Wikipedia trace Number of busy VMs when running the Wikipedia trace Number of idle VMs when running the World Cup 1998 trace Number of busy VMs when running the World Cup 1998 trace145 vii

15 5.30 Number of VMs in shutdown when running the World Cup 1998 trace Number of idle VMs when running the RTP trace Number of busy VMs when running the RTP trace Number of VMs in wakeup when running the RTP trace Number of idle VMs when running the RTP trace Number of busy VMs when running the RTP trace Number of VMs in wakeup when running the RTP trace Job processing time distribution of Redis-based runs, EC2 Implementation EC2 test, fixed load. T P = 5,T A = 8,t wait = EC2 test, sharply changing load. T P = 5,T A = 8,t wait = EC2 test, gradually changing load. T P = 5,T A = 8,t wait = viii

16 Abstract The design and management of current distributed systems is a very complex task. This is mainly due to the fact that typical systems are very large and are often not controlled by a single entity. For example, the Internet is composed of independent administrative entities, called Autonomous Systems (ASs), and the overall behavior is determined by a non-trivial combination of the different policies of each AS and the actions of the end-users. When designing such a system, one must consider the fact that, while the system s designer may have some idea of optimal system-wide behavior, it has to consider very different possible policies and end-user actions, which determine the actual system performance. Cloud computing is an emerging computing paradigm in which tasks are assigned to a combination of connections, software and services accessed over a network. This network of servers and devices is collectively known as the cloud. Computing at the scale of the cloud allows users to access supercomputer-level power using a thin client or another access point, like a smartphone or a laptop. Since end-users are given access to supercomputerlevel resources, their effect over the system s overall performance is greater than ever. This raises multiple research questions related to the management and performance of cloud computing systems in light of the end-user s selfishness. In this work we specifically study the topologies of networks constructed by selfish users and the overall system performance when selfish end-users may split work between a shared resource (cloud) and private resources. We also consider task assignment policies that are specifically adequate for large-scale distributed systems, and show that they provide new capabilities in improving system performance. In particular, we develop new resource allocation algorithms that converge to a working point that balances the end-user experience with the operational costs of leasing resources from the cloud provider. 1

17 Chapter 1 Introduction Complex distributed systems surround us and affect many aspects of our lives. Often, such systems are not controlled by a single entity. The Internet, for example, has over eight billion devices connected to it at any point in time, and that number is expected to grow to over forty billion by 2020 [80]. The traffic in the Internet is managed by over thirty thousand autonomous systems (ASes), each of which is driven by its own interests, and employs its own routing policies. The design and management of such large scale systems is a complex task. Some of these systems are not designed by a central entity, but are selfarranged by the different elements building the system. The structure of a peer-to-peer network, for example, is defined by its users and the connections they create between each other. The design of the system plays a crucial role in its usage. For example, in peer-to-peer networks, data can only be routed between directly-connected users. Even when the system is designed by a central entity, such as in the case of a large scale data center, one must consider the fact that, while the system s designer may have some idea on what would make the system-wide behavior be optimal, it has to consider very different possible policies and end-user actions, which determine the actual system performance. The coordination among the different actors (namely, those who build the system, those that manage it, and those that use it) is often impossible or impractical for several reasons. In some cases, as is the case of the ASes described above, each user is driven by its own incentives. The incentives of one user may be in conflict with those of other users. In such scenarios, 2

18 different users may be reluctant to cooperate in order to preserve their own interests. In addition, coordination is often not practical due to the scale of the system. This may arise even in cases where the system is structured and managed by a single entity. For example, coordinating the assignment of search queries to processing servers in a data center is not practical. Finally, coordination among users requires computational resources and may incur undesired overheads. Consider the above mentioned example of search queries. While the assignment of the query to the processing server may have crucial implications to the quality of the service provided to the user, taking into account the complete state of the system for any single query introduces significant overhead. To collect the state of the system, each processing server needs to be interrupted in order to report its current state. This interruption comes at the expense of providing service to users. In addition, once the required state is collected, further computation isrequiredtodeterminewhichserverwillbeassignedwiththequeryathand. This processing action requires additional computation resources. Moreover, during all this time, the query is suspended, awaiting the assignment to the processing server, incurring further delay to its completion. In addition, by the time that the processing server is at last chosen, the state of the system might already have changed i.e., the well-known the stale data problem[53] and therefore the selection of the server may be far from optimal. Our goal is to study the structure and management of networks and systems that arise in non-cooperative setups, and suggest resource management schemes that address the challenges described above. These non-cooperative setups give rise to game-theoretic analysis aiming at quantifying the stability of the emerging structures and the performance associated with them. Game theoretic models have been employed in various networking contexts, such as flow control [2, 44], routing [3, 67] and bandwidth allocation [46]. These studies mainly investigated the structure of the network operating points i.e., the Nash equilibria of the respective games. Such equilibria are inherently inefficient [19] and, in general, exhibit suboptimal network performance. As a result, the question of how much worse the quality of a Nash equilibrium is with respect to a centrally enforced optimum has received considerable attention e.g., [45, 76, 77]. In order to quantify this inefficiency, two conceptual measures have been proposed in 3

19 the literature. The first, termed the price of anarchy [69], corresponds to a worst-case analysis and is the ratio between the worst Nash equilibrium and the social optimum. The second, termed the price of stability [7], is the ratio between the best Nash equilibrium and the optimum, and it quantifies the degradation in performance when the solution is required to be stable (i.e., with no agent having an incentive to independently defect out of it once being there). In Chapter 2 we present a rigorous analysis of topology design issues in the context of non-cooperative networks. We study the effects of several topology design considerations on the performance of non cooperative networks. Specifically, we consider the price of establishing a (directed) link between network elements, the delay of the resulting routing paths, and the relayingextent ofnodesalongtheroutingpaths, wherethelatterisameasure of the path s proneness to congestion. We analyze the effect of each consideration, as well as tradeoffs among them, on the resulting network topology and on its implied performance. As mentioned, we conduct our investigation within the realm of noncooperative games; however, this requires us to consider also the more basic framework of the (system) optimization problems. We show that for all considered cases but one, the existence of a Nash equilibrium point is guaranteed. For the remaining case, we show an example that admits no Nash equilibrium, but indicate, through simulations, that practical scenarios do tend to admit a Nash equilibrium. In addition, we demonstrate that the price of anarchy, i.e., the performance penalty incurred by non-cooperative behavior, may be prohibitively large; yet, we also show that such games usually admit at least one Nash equilibrium that is systemwide optimal, i.e., their price of stability is 1. This finding suggests that a major improvement can be achieved by providing a central ( social ) agent with the ability to impose the initial configuration on the system. Cloud computing is a new and emerging paradigm in which jobs are processed by several services deployed on a set of distributed servers and accessed via a common network. This network of servers and devices, collectively known as the cloud, can offer a significant reduction in computing and storage costs due to economy of scale. In fact, computing at the scale of the cloud allows users to reach into the cloud for resources as they need them, and to attain supercomputer-level power using any standard client 4

20 with network connectivity. The Cloud Computing paradigm is based on three fundamental types of entities: the cloud provider, the service provider and the end user. The cloud provider owns and manages the infrastructure, constructed in form of large scale data centers. The service provider deploys its service in the data center, and the end user accesses it to have its jobs processed. Many end users may access a single service, and many services can be deployed in the same data center. Therefore, while the data center is designed and managed by a single entity(the cloud provider), the scale of the cloud prohibits central coordination and brings about many of the challenges described above. For many typical users, the cloud constitutes an additional means of computation, i.e., in additional to other (local) resources. In Chapter 3 we explore a setup in which users can split their work between a shared resource (such as the cloud) and a private resource. Unlike the private resource, which provides guaranteed performance, the performance of the shared resource is highly dependent on the usage pattern of other users, which in turn influences a user s decision if and to what extent to make use of the shared resource. The intrinsic relation between the utility that a user perceives from the shared resource and the usage pattern followed by other users gives rise to a non-cooperative game, which we model and investigate. We show that the considered game admits a Nash equilibrium. Moreover, we show that this equilibrium is unique. In addition, we show that, while in some cases of interest the Nash equilibrium coincides with a social optimum, in other cases the price of anarchy can be arbitrarily large. We demonstrate that, somewhat counter-intuitively, exercising admission control to the shared resource may deteriorate its performance. Furthermore, we demonstrate that certain (heavy) users may scare off other, potentially large, communities of users. Accordingly, we propose a resource allocation scheme that addresses this problem and opens the shared resource to a wide range of user types. With the growth in adoption of the Cloud Computing paradigm, more and more users choose to completely give up keeping any significant private computational resource. In such a setup, users no longer split their job between a private and a shared resource they send the entire job for processing in the cloud. Consequently, the user s performance is completely dependent on the performance of the shared resource (the cloud). Once a 5

21 job is sent to the cloud, the scheduler s assignment of the user s job to the specific processing server (resource) may have crucial implications on the quality of the service provided to the user. Furthermore, existing schedulers often incur a high communication overhead when collecting the data required to make scheduling decisions, hence delaying job requests on their way to the processing servers. We propose a novel scheme that incurs no communication overhead between the users and the servers upon job arrival, thus removing any scheduling overhead from the job s critical path. Our approach, presented in Chapter 4, is based on creating several replicas of each job and sending each replica to a different server. Upon the arrival of a replica to the head of the queue at its server, the latter signals the servers holding replicas of that job, so as to remove them from their queues. We show, through analysis and simulations, that this scheme significantly improves the expected queuing overhead over traditional schemes under various load conditions and different job length distributions. In addition, we show that our scheme remains efficient even when the inter-server signal propagation delay is significant (relative to the job s execution time). We provide a heuristic solution to the performance degradation that occurs in such cases and show, by simulations, that it efficiently mitigates the detrimental effect of propagation delays. A key feature that makes cloud computing attractive to service providers is elasticity, i.e., the ability to dynamically change the amount of allocated resources. This is typically done by adjusting the number of virtual machines (VMs) running a service based on the current demand for that service. For large scale services, centralized management is impractical and distributed methods are required. In such settings, no single component has full information on demand and service quality, thus elasticity becomes a real challenge. In Chapter 5 we address this challenge by proposing a novel elasticity scheme that enables fully distributed management of large cloud services. Our scheme is based on two main components, namely, a task assignment policy and a VM management policy. The task assignment policy strives to pack VMs while maintaining SLA requirements. The VM management policy is based on local activation of new VMs and self-deactivation of VMs that are idle for some duration of time. Through analysis, simulations and an implementation, we demonstrate that our scheme quickly adapts to changes in job arrival rates and minimizes the number of active VMs so as 6

22 to reduce the operational costs of the service, while adhering to strict SLA requirements. Overall, our results show that taking into consideration key aspects of large scale systems, such as the users inability to coordinate, as early as in the design phase of the system may have detrimental impact of the system s performance. On the other hand, a robust system design, coupled with some of the suggested resource management techniques, may support scaling the system with minimal to no performance degradation. The rest of the thesis is organized as follows. In Chapter 2 we study network topology design issues in the context of non-cooperative networks. Next, we formulate and study the workload factoring problem using game theory in Chapter 3. In Chapter 4 we propose using job replication as a scalable method for job scheduling, and finally, in Chapter 5 we propose a novel method for managing elasticity in a fully distributed fashion. 7

23 Chapter 2 Topology Design of Communication Networks: A Game-Theoretic Perspective In this chapter we study the performance of non-cooperative networks, that is, networks structured by non-cooperating network elements, in light of three major topology design considerations. The metrics we consider are: link establishment cost, path delay, and path proneness to congestion (modeled through the relaying extent of the nodes). We analyze the effect of each of these considerations, as well as tradeoffs amongst them, on the resulting network topology and its implied performance. We conduct our investigation within the realm of noncooperative games; however, this requires us to consider also the more basic framework of the (system) optimization problems. Our findings can be summarized as follows: For all considered cases but one, we establish the existence of a Nash equilibrium. For the remaining case, we provide a generic counter-example. Furthermore, we show, through simulations, that practical scenarios tend to admit at least one Nash equilibrium with a reasonable price of anarchy. The results described in this chapter also appear in [59] and [57]. 8

24 We show that, typically, the considered games may perform poorly from a systemwide perspective, as exhibited by large values of the price of anarchy. Moreover, we show that such poor operating points may be reached through a natural course of the network game. On the other hand, we show that, in many cases, the price of stability is 1, i.e., there is a Nash equlibrium that is systemwide optimal. Therefore, efficient performance can be often achieved by just controlling the initial configuration of the system. In some of the considered cases we show that, similarly to [23], Nash equilibria are obtained in very simple topologies, such as a complete graph (clique) or a ring. Moreover, when the relaying extent is the only design consideration, we show that the system optimum too is obtained in such a simple topology, namely a ring. This finding may suggest that a practically appealing model should address the relaying extent only in conjunction with additional design considerations. We achieve this by adding delay as a second consideration. 2.1 Related Work Topology design of computer networks focuses on finding the network configuration with the best possible performance given some optimization criteria [10]. Classical works on this subject aimed at finding optimal network topologies with respect to criteria such as fault tolerance [82], reliability [79] and delay [70]. Bottleneck models have many practical applications in the context of network design and management. Such models can be used to reflect the remaining battery life in a wireless network [90], to minimize the usage of loaded buffers in traffic engineering [10], or to avoid congested links when routing traffic [84]. In the context of game theory, bottleneck models have also been studied in [9]. Prior game-theoretic work on topology design has focused on the tradeoff between the price of establishing a link and the delay of the implied routing paths. In [23], the authors presented a fundamental model where players 9

25 set up links to construct an undirected graph, trading off link establishment costs with path lengths. They focused on the case of undirected networks, with homogeneous link prices, that is: once a link has been established between nodes v i and v j, either node could use that link to transmit data to its adjacent node; furthermore, the price of establishing a link was assumed identical throughout the network. In [1], the authors followed the model of [23] and improved some of the results. In [15], the authors used a model similar to the one above, but demanded bilateral agreement of both sides for every link establishment, that is, a link is established only if both nodes choose to establish it. In [20], the authors followed up and improved some of the results of [15] and [23]). In [11], the authors extended [23] by limiting the cost of non-connectivity to a finite value. In [8, 7], the authors studied a similar problem, where players aspire to connect to only some of the other players, and may buy non-adjacent links. Finally, the authors of [56] looked at a similar problem from a peer-topeer network perspective, where the links are directional and link delay may be arbitrary. In this study, we consider directional links, and furthermore, investigate several novel design criteria. While we focus our work on network design, our findings are relevant also to topology control in wireless networks. Several studies have addressed such topology control games [58, 42, 21], however in the context of omnidirectional antennas, i.e., connectivity is (solely) derived by the radius covered by the power level set by the node. Our results, on the other hand, are relevant to the case of directional antennas, where a node can choose its neighbors on a per-node basis. 2.2 The Model Our unified network model is based on a set of N network elements. We assume that network elements establish links among themselves during the network s setup phase, and continue using these links when transmitting throughout the remainder of the network s lifetime. We refer to the network elementsasnodes, anddenotethembyv = {v 0,v 1,...,v i,...,v N 1 }. Edges are used to represent links between nodes, and are denoted by a set E. Note that the existence of a link from v i to v j means that v i can send data directly 10

26 to v j, but it does not imply that v j can send data directly to v i, therefore the network s topology is represented by a directed graph, denoted as G. We use η in (v i ) to denote v i s in-degree, and Adj G (v i ) to denote v i s adjacent nodes, i.e., Adj G (v i ) = {v j < v i,v j > E}. A non-cooperative game [68] is comprised of three components, namely players, strategies and costs. In the non-cooperative games we analyze, the network elements (nodes) are the players. The strategies, s i, of each player v i, are the sets of links that it may choose to establish, i.e., each player v i chooses its direct neighbors, Adj G (v i ). A specific choice of the players strategies, s = i=n 1 i=0 {s i }, defines the strategy profile, which induces the network s topology G. Given the network s topology G, we denote by l G (v i,v j ) the routing path from v i to v j. Routing paths are chosen so as to (self-) optimize some design considerations, which are captured by a cost function. More specifically, in the games we investigate, the goal of each node (player) is to achieve full connectivity with all other nodes, and, under this constraint, optimize the design considerations. We term this class of games as connectivity games. As mentioned, we study the impact of three design considerations. The first is the price of establishing a link. To model it, we define a weight function W : V V R, i.e., W(v i,v j ) defines the price (to v i ) of establishing a link from node v i to node v j. The second design consideration is the delay of the resulting routing path, which is captured through the number of hops along the routing path. The third design consideration is the relaying extent, which is captured by the in-degree of the nodes along the path. Specifically, in-degrees quantify the relaying extent in two possible ways, namely the product of the in-degrees of nodes along the path, or the bottleneck of these values. When nodal in-degree reflects the amount of traffic expected to go through a node, a high in-degree may indicate a high probability of packet loss (due to congested buffers). In such a case, the product value of nodal in-degrees along a path represents the probability of a succesful transmission of a packet through a congested relay. Whereas the bottleneck value may be chosen when nodes share the bandwidth of the relay. In both cases, a lower value of the relaying extent implies a better routing path. 11

27 Each of the three design considerations, namely establishment price, delay and relaying extent, translates into a corresponding nodal cost function, which captures the performance of the node. Specifically, with each player v i V, we associate a (non-negative) cost value c(v i ), which accounts for various factors, depending on the design consideration, e.g., the node s power consumption, its distance from some other node, path interference, etc. The precise definition of the node s cost value is detailed in the following respective sections. Each player (node) strives to minimize its cost c(v i ). Player v i s best-response move is a strategy which, given the strategies of all other players, yields the lowest value to c(v i ). A network topology G is said to be at Nash equilibrium if each player considers its chosen strategy to be the best under the given choices of other players. The selfish behavior of the players typically leads to network-wide inefficiency. We quantify this inefficiency through the ratio between the cost of the worst possible Nash equilibrium topology and the cost of an optimal solution. In keeping with common terminology [45, 69], this ratio is called the price of anarchy and it quantifies the penalty incurred by lack of cooperation (or coordination) among the players in a noncooperative game. We also consider the price of stability [7], which is the ratio between the cost of the best Nash equilibrium and the cost of an optimal solution and quantifies the inefficiency of the noncooperative game in cases where the initial operating point of the system can be chosen by a social agent. Note that, in order to quantify the price of anarchy and price of stability, the related system optimization problems need to be defined. In general, the goal of system optimization is to minimize the total cost, namely C(G) = v i V c(v i). An exception is whenever the nodes costs depend on a bottleneck function, as when a bottleneck relaying extent is considered; there, total cost is defined as the networkwide bottleneck, i.e., C(G) = max vi V c(v i ). 2.3 Link Prices (Solely) The first game we consider entails each player v i with the total price of the links it chooses to establish, i.e., c(v i ) = v j Adj G (v i ) W(v i,v j ). In case player v i fails to connect to one (or more) of the other nodes, its cost is infinite (i.e., c(v i ) = ). We term the respective game as the connectivity game with link prices. We show that this game has a Nash equilibrium 12

28 point. In addition, we show that the game s price of stability [7] is 1. On the other hand, we show that this game has a high price of anarchy, and that an iterative best-response move by each player may lead to that state. Theorem 1 The connectivity game with link prices has a Nash equilibrium. Proof. By proving that this game is an exact potential game [55]. To thatend,wedefinethefollowing(potential)functionφ: Φ(G) = v i V c(v i). One needs to prove then, that when a player v i improves its cost, the potential decreases by the exact same amount. Let G be a valid topology (i.e., all players have routing paths to all other players) induced by a strategy profile s. Assume player v i changes its strategy from s i to s i to improve its cost. It is easy to see that when starting from a valid strategy profile, player v i must remove at least one link to improve its cost. Let G be the topology induced by the strategy profile after player v i has changed its strategy. For any player v j v i, since G is a valid topology, it holds that v j has a routing path to v i. Note that when player v i changes its strategy (i.e., adds or removes links), it cannot affect v j s connectivity to itself. It follows that v j is connected to v i in G. Since v i improved its cost, it is clear that v i has routing paths to all other nodes in G, and so v j has routing paths to all nodes in G. We conclude that for any player v j v i,c G (v j ) = c G (v j ). Hence, Φ(G ) Φ(G) = c G (v j ) c G (v j ) = c G (v i )+ v j V v j v i c G (v j ) c G (v i ) c G (v i ) c G (v i ). v j V v j v i c G (v j ) = In view of [55], the theorem follows. Next, we prove an important property on the correlation between the Nash equilibrium strategy profiles of the connectivity game with link prices and the local optima of the corresponding (system) optimization problem. We will later rely on this lemma to prove our results on the price of stability and price of anarchy of the game. 13

29 Lemma 1 Let G be a topology. G defines a local optimum of the connectivity problem with link prices if and only if G is at Nash equilibrium for the connectivity game with link prices. Proof. Recall that optimum is measured with respect to the total costs of all players, i.e., the optimization objective function is C(G) = v i V c(v i). In one direction, let G be a local optimum for the optimization problem. It is clear that G is a valid topology. Assume, by negation, that G is not at Nash equilibrium. By definition, there exists at least one player that can unilateraly improve its cost. Let v i be such a player. Let G be the topology reached after v i played its best response to G. As shown in the proof of Theorem 1, for every player v j v i,c G (v j ) = c G (v j ). In addition, since v i played its best response move, c G (v i ) < c G (v i ). Hence, C(G ) < C(G) in contradiction to G s local optimality. The other direction holds by definition. Theorem 2 The price of stability for the connectivity game with link prices is 1. Proof. By proving that there exists an optimum point that is at Nash equilibrium. As the global optimum is also a local one, Theorem 2 follows directly from Lemma 1. 1 In order to analyze the game s price of anarchy, several definitions are required. A player s minimal connectivity price is the minimal price a player must payinordertoestablishanylink. Formally,Wmin i = min v j V,v j v i {W(v i,v j )}. A player s maximal connectivity price is the price a player must pay in order to establish the most costly link. Formally, Wmax i = max vj V{W(v i,v j )} (note that this is not the highest value c(v i ) can take since it refers to a single link only). A game s minimal connectivity price is the minimal price required so that some player may establish a link to some other player. Formally, W min = min vi V{Wmin i }. A game s maximal connectivity price is the price of the most costly link (i.e., it s the highest value W(, ) takes). Formally, W max = max vi V{Wmax}. i 1 Actually, the first direction of Lemma 1 proves a claim stronger than required for establishing the theorem. 14

30 Theorem 3 The price of anarchy for the connectivity game with link prices is Θ( Wmax W min ). Proof. By establishing the corresponding lower and upper bounds, in the following lemmas. Lemma 2 The price of anarchy for the connectivity game with link prices is O( Wmax W min ). Proof. By showing a lower bound on the cost of the optimal topology and an upper bound on the cost of any Nash equilibrium. First, we note that each player must establish at least one out-going link in order to have any kind of connectivity with the rest of the network. It follows that the cost of each player is at least W min. Hence, the cost of the optimal topology is at least N W min. Next, we consider the problem of the worst Nash equilibrium. It follows from Lemma 1 that this Nash equilibrium is also the worst local optimum for the optimization problem. The following lemma proves a general property of directed graphs. Using this property, we will establish the desired upper bound. Lemma 3 Let G = (V,E) be a directed clique of N nodes. Let Ĝ = (V,Ê) be a subgraph of G, such that Ĝ is strongly connected, and, in addition, for any edge e i Ê, (V,Ê \ {e i}) is not strongly connected. Then, Ê 2 (N 1). Proof. Assume, by negation, that there exists a strongly connected graph Ĝ = (V,Ê), such that for any edge e i Ê, (V,Ê \ {e i}) is not strongly connected, and, in addition, Ê > 2 (N 1). Let us now run Tarjan s algorithm for strongly connected components [16]. It holds, from the correctness of Tarjan s algorithm and our assumption, that the algorithm will return a single strongly connected component, which includes all of V. However, Tarjan s algorithm is comprised of two runs of the Depth-First- Search algorithm [16]. Since in each run the DFS algorithm traverses N 1 edges, Tarjan s algorithm traverses at most 2 (N 1) edges. It follows that some of the edges in Ê can be removed without damaging Ĝ s connectivity, in contradiction to our assumption. This completes the proof of Lemma 3. 15

31 Figure 2.1: A network topology with price of anarchy Wmax W min Corollary 1 The cost of the worst Nash equilibrium for the connectivity game with basic link prices is 2 (N 1) W max. Therefore, the cost of any Nash equilibrium is upper bounded by 2 (N 1) W max, while the optimal configuration is lower bounded by N W min, yielding a bound on the price of anarchy of O( Wmax W min ). This completes the proof of Lemma 2. Lemma 4 The price of anarchy for the connectivity game with link prices is Ω( Wmax W min ). Proof. By proving that any directed ring topology is at Nash equilibrium and providing an example. Lemma 5 Every directed ring topology is at Nash equilibrium. Proof. Assume, by negation, that there exists a directed ring topology G, such that G is not at Nash equilibrium. Since G is not at Nash equilibrium, it holds that there exists at least one player that can unilaterally reduce it cost. Let v i be such a player, and let v j Adj G (v i ). Since G is a directed ring, it holds that v i is the only node directly connected to v j. And so, it holds that v i cannot remove its link with v j, since any topology resulting from such a move will not be strongly connected. Hence, v i s best response move can only include addition of links. Since W(, ) is a non-negative function, this contradicts the best response definition. This completes the proof of Lemma 5. 16

32 Consider, for example, the network setup (partly) depicted in Figure 2.1. In this network, the price of the link connecting v i to v i 1 is W min, while all other link prices are W max. Incaseeachnodev i choosestoestablishasinglelinktov i 1,theresulting topology G min is a directed ring topology, thus, following Lemma 5, it is at Nash equilibrium. In addition, we note that the cost of this topology is C(G min ) = N W min (which is optimal). Next, we consider the case in which each node v i chooses to establish a single link to v i+1, the resulting topology G max is a directed ring topology, thus, following Lemma 5, it is at Nash equilibrium. The cost of this topology is C(G max ) = N W max, yielding a lower bound on the price of anarchy of Ω( Wmax W min ). This completes the proof of Lemma 4. Theorem 3 follows directly from Lemmas 2 and 4. A natural order game is a game that begins with all players having no outgoing links (i.e., v i V,Adj G (v i ) = ), and it advances as each player, in its turn, plays its best response; the game ends when in equilibrium. We show that, even in such a simple setting, the connectivity game with link prices still yields a price of anarchy of Θ( Wmax W min ). Theorem 4 The price of anarchy for the natural order connectivity game with link prices is Θ( Wmax W min ). Proof. By showing both upper and lower bounds. Note that the upper bound of O( Wmax W min ) proven in Lemma 2 still holds. We prove the lower bound by using the example defined in Lemma 4. First, we show that if players perform best response moves in ascending order of index (i.e., player v 0 plays first, followed by v 1, v 2, and so on), then the optimal Nash equilibrium is achieved. When player v 0 first plays, all other players have no outgoing links established, and so v 0 is forced to directly connect with all other nodes at a cost of W min +(N 2) W max. When player v 1 enters the game, it can either transmit through v 0, or transmit directly to all players. Since transmission through v 0 yields v 1 a cost of W min, it will strictly prefer this startegy. When player v 2 enters the game, its best response is to transmit through v 1 at a cost of W min. In a similar fashion, each player v i,3 i N 1 will establish a single link to v i 1 at a cost of W min. Once all players have played once, only v 0 can improve its cost by removing all links except for a single link to v N 1 (which has the price of W min ). It can be easily seen that, 17

33 after v 0 s second move, the resulting topology is the same as G min defined in Lemma 4, and thus at Nash equilibrium. The total cost of this topology is N W min. Next, we present an entry order that yields a Nash equilibrium with a cost of N W max. In this case players enter in decending order of index (i.e., v N 1 plays first, followed by v N 2, v N 3, and so on). When player v N 1 first plays, all other players have no outgoing links established, and so v N 1 is forced to directly connect with all other nodes at a cost of W min +(N 2) W max. When player v N 2 enters the game, its best response move is to transmit through v N 1 at a cost of W max. In a similar fashion, each player v i,0 i N 3 will establish a single link to v i+1 at a cost of W max (note that in this case, other options with the same cost exist. making other choices will result in a Nash equilibrium with an equally bad cost). Once all players have played once, only v N 1 can improve its cost by removing all links but the one connecting it to v 0. It can be easily seen, that after v N 1 s second move, the resulting topology is the same as G max defined in Lemma 4, and thus at Nash equilibrium. The total cost of this topology is N W max, and the lower bound on the price of anarchy follows. This completes the proof of Theorem Link Prices and Delay In this section we discuss a game in which the cost function of each player v i entails it both with the total price of the links it chooses to establish, as well as its distance from its destinations. This is perhaps the most elementary model that exhibits the link-delay tradeoff: on the one hand, each node aspires to reduce its expenses on link establishment; on the other hand, transmitting to many other nodes implies less hops between the node and its destinations, thus reducing the hops component of the cost. 2 Formally, the cost of each node v i is defined as c(v i ) = α v j Adj G (v i ) W(v i,v j )+ d G (v i,v j ), v j V 2 Note that there is no interest in dealing with delay as the only design consideration, as it boils down to a trivial case where a clique, i.e., complete graph, is the optimal solution as well as the unique Nash equilibrium. 18