1.pdf

Minimizing The Payment Function inShortest Path Games

AM.Tech Phase I Report

Submitted in Partial Fulfillment of the Requirementsfor the Degree

of

MASTER OF TECHNOLOGY

by

Archana Shokeen(144101043)

under the guidance of

Dr.S. Sajith

to the

DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI

GUWAHATI - 781039, ASSAM

Contents

List of Figures 3

1 Introduction 31.1 Area of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Shortest Path Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background / Preliminaries 52.1 SP Games and Related Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Algorithm Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 General Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Literature Survey 93.1 Channel-unaware and QoS-unaware strategies . . . . . . . . . . . . . . . . . . . 93.2 Channel-aware and QoS-unaware strategies . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 Maximum Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.3 Proportional Fair Scheduler . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.4 Joint Time and Frequency domain schedulers . . . . . . . . . . . . . . . 11

3.3 Channel-aware and QoS-aware strategies . . . . . . . . . . . . . . . . . . . . . . 113.4 Buffer Aware Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Cross Layer Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Motivation of Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Summary of Work Done 154.1 Graph G can be created from graph G ,by adding new edges such that payment of

edges is∑e∈SP ′(g′)

P e <∑

e/∈SP (g)

P e

ie.where P is payment for ’e’ edges , SP and SP are the shortest path in respectivegraphs ,only in case when no. of edges involved in SP are less than that involvedin SP of graph G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Every chosen alternative from set of SP costs equal to the system designer,though payments to corresponding edges may differ. . . . . . . . . . . . . . . . . 184.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2

4.3 The payment of edges done by system is directly proportional to the differencebetween the current shortest path SP and the alternate SP.ie p (SP-SP) . . . . 194.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Plan of Future Work and Conclusion 215.1 Plan of Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

References 23

3

List of Figures

2.1 Representation Of Coalitional Games . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1 Two layer QoS aware Scheduler . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1 graph showing ’a’ as starting point and ’d’ as sink . . . . . . . . . . . . . . . . . 154.2 graph showing ’a’ as starting point and ’d’ as sink with new edge bd introduced 164.3 graph showing ’a’ as starting point and ’d’ as sink with alternate new edge bp

and pd introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 graph showing ’a’ as starting point and ’d’ as sink with 3 new edgebp,pq and qd

introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 graph showing ’a’ as starting point and ’d’ as sink with two SPi, SPj . . . . . . 184.6 graph showing ’a’ as starting point and ’d’ as sink with new edge bd introduced 194.7 graph showing ’a’ as starting point and ’d’ as sink with new edge bd introduced 20

1

Abstract

A class of coalition games arising from shortest path problems is defined These shortest pathgames are shown to be totally balanced and allow a Payment allocation scheme Possible meth-ods for obtaining core element ie accumulative reward are indicated First by relating to theallocation rules in minimizing the taxation and bankruptcy problems second by constructingan explicit rule that takes opportunity costs into account by considering the costs of the secondbest alternative and that rewards players who are crucial to the construction of the shortestpath. In particular, we study the Payment computation to different variants of shortest pathgames, as well as the relation that can be retrieved from the number of edges involved inShortest path chosen and the Payment they gets.

2

Chapter 1

Introduction

1.1 Area of Research

The operations research literature offers numerous algorithms and heuristics for network op-timization problems In game theoretic approaches to network optimization problems differentagents control the elements of the network Next to finding optimal solutions,this adds the prob-lem of dividing the costs or benefits generated by such solutions over the involved agents.Thuswe have several classes of games arising from problems in operations research In this researchpaper we study a class of cooperative shortest path games in which the transportation of agood from a source to a sink of a network generates an externally given income, in the formof Payment. The nodes in the network are owned by the players. The value of a coalition isdefined by the net Payment this coalition can realize by transporting the good through thenetwork via a shortest path using only nodes that are owned by the players in the coalition orzero if the net Payment happens to be Infinite .where the Payment or reward for transportinga good from source to sink is not associated with a specific player only. The definition of thecooperative shortest path games is given in Section Every efficient allocation in which eachplayer contributes a non negative amount his reward to the costs of the shortest path yieldsa core element of the shortest path game.we here introduces the notion of Payment allocationscheme for a cooperative game such that it specifies for each coalition S ⊂ N of players anallocation of the valuev ⊂ S over its members . The Payment allocation rule is constructed thattakes opportunity costs into account by considering the costs of the second best alternative ofShortest path costs and that rewards players who are crucial to the construction of the shortestpath.

1.2 Shortest Path Games

Shortest path problems and their associated cooperative games are defined In the shortestpath problems considered in this paper there is a finite set of players Each player owns arcs orconnections in a finite network There are costs associated to the use of each arc Each playerreceives a nonnegative payment or reward if he manages to transport a good from the sourceof the network to its sink.

Notation: A shortest path problem is a tuple N,V,( Ai)iN , w, (ri ) iN ,

where

N is a Finite set of players;

3

V is a finite set of vertices with two special elements the source So and the sink Sieach player i N owns a set Ai ( V V )of directed arcs in the network;w is a weight assigned on edges; weight assigned to arc (a,b) owned by player i N is w(i,(a,b))

R+ ;each player i N receives a Payment Pi R+for transporting his goods from the source to

the sink.In this work we propose some relevant theorems Considering a graph G (V,E) ,such that V

are vertices and E are edges. If all the edges of the Graph are held by Agents then What couldbe the possible lowest payment that the system designer can pay off to those edges involvedin shortest path chosen. Is there any relation between no. of edges involved and paymentfunction?

The remainder of the report is organized as follows. Chapter 2 describes the backgroundbasics of Shortest Paths games and related areas with the basic terminologies involved. Chapter3 provides the literature survey on the previous relevant work related to the problem .It alsodescribes the motivation and hence the objectives of the current work.Chapter 4 describes thesummary of work done in this thesis along with the examples. Chapter 5 describes the plan offuture work and conclusion.

4

Chapter 2

Background / Preliminaries

2.1 SP Games and Related Games

In this chapter we given an overview of different variants of shortest path games, and wefurthermore present various graph-based games, like flow games, minimum cost spanning treegames, etc., as well as other interesting coalitional games, like market and linear productiongames. For each type of game representation, which we consider in our work, we also introducea formal model. This is essential in the following chapters, when we prove properties andcomplexity results and determine how the various classes relate to each other. Market gamesand linear production games are not directly involved in the research we conduct, but they arenevertheless interesting in a more indirect way. Both types of game representations, with theircorresponding games, characterize an important class of coalitional games, namely the class oftotally balanced games. This is also the case for flow games. As will be seen later, variousvariants of shortest path games are actually totally balanced as well. So, they can be related tothe other coalitional games via the class of totally balanced games. We furthermore present adirect correspondence, which can be used to transform games, given as a flow, market or linearproduction games into one of the other two game representations. The correspondence betweenthese classes of coalitional games may offer useful indications regarding properties and resultsfor totally balanced games. we present some additional graph-based coalitional games, whichhave been analysed in the literature with respect to the computation of solution concepts.We continue by determining characteristics for shortest path games and other graph basedcoalitional games. Afterwards we introduce different variants of shortest path games, which arebased on several preselected characteristics. we refer to Figure . Here we can distinguish twodifferent layers:

1. Upper Layer: In this layer, a coalitional game is given in its original representation,thus in its actual problem domain. So, we can talk about concepts like shortest paths,maximal flow, etc. of a game.

2. Lower Layer: In this layer, coalitional games are represented in characteristic functionform hN, vi. This is an elegant way to work with coalitional games, stripped from rep-resentation details, as a formal concept in cooperative game theory. The definitions andpropositions, like for example balancedness, monotonic, null-player, etc., are all based onthis basic representation of coalitional games.

3.

5

Fig. 2.1 Representation Of Coalitional Games

Our focus in this thesis will be on Payment function and edges getting involved. To given anoverview over games and game representations and how they are related to each other

2.1.1 Algorithm Perspective

A path P (So ! Si) in a directed graph is a sequence of vertices such that from each of itsvertices there is an arc to the next vertex in the sequence. A path may be infinite, but a finitepath always has a first vertex So, called its source, and a last vertex Si, called its sink. Nowwe present a minimal class of shortest path games, called VSPG (standing for value shortestpath game). Let be a shortest path pre-problem, where the arcs of the graph hV,Ai are ownedby a finite set of players N according to a total bijective map o : A ! N, such that o(a) = imeans that player i is the owner of edge a. Hence, every arc is assigned to exactly one player.We have a cost map c that assigns to every arc a 2 A a non-negative real number c(a) (c : A !R+0 ). Given the simple structure of VSPG, a path owned by players of coalition S is simply asequence of vertices (v1, v2, ..., vm) such that v1 = So, vm = Si and for each k 2 1, ...,m1 thearc (vk, vk+1) is owned by player ik 2 S. Let P(S) denote the collection of all paths owned bycoalition S. For any path P we denote by o(P) the set of owners of the arcs in P. Suppose thatthe transportation of a certain good from the source to the sink of produces an income r and acost given by the length of the path that was used. In particular, the costs associated to a pathp = (v1, v2, ..., vm) 2 P(S) are defined as the sum of the costs of its arcs: cost(p) =Pm1k=1c(vk, vk+1). Note that a coalition S N can transport the good only through paths owned byit and a path P is basically owned by a coalition S if o(P) S. Obviously, if a coalition S hasto find a path from source to sink, it will choose among its alternatives in P(S) the path withminimal costs. Define for each S 2 2N ;: c(S) = minp2P(S)cost(p) if P(S) 6 ; 1 otherwise Weoverloaded c at this point, but it will be obvious from the context which function is meant. Notethat the shortest path can be computed in polynomial time using for instance the well-knownalgorithm of Dijkstra [25]. We now put all the information necessary to describe a shortestpath game together, and introduce the notion of a shortest path game environment , which isany such tuple hN, o, c, ri. Remember that a coalitional game with transferable utility can berepresented as a pair (N, v), where N is a finite set of players and v : 2N ; ! R is a functionthat assigns to each coalition S 2 2N ; its value v(S) 2 R. We can associate with the TU-gamehN, v i whose characteristic function vis given by: v(S) = maxr c(S), 0 =r c(S) if S owns a

6

path in and c(S) ¡ r0 otherwise for every S N. Hence, the environment and the definition of vis our game representation GN. The coalitional its goods from source to sink, it will receive atotal reward r and incur costs c(S), the costs of the shortest path owned by S. If r c(S) ¿ 0,coalition S makes a profit. If r c(S) 0, coalition S can generate profit zero by simply doingnothing. Therefore, coalition Ss profit is maxrc(S), 0. Lets sum it up: A shortest path game oftype VSPG is any such game hN, vi generated by a game representation GN as defined above

2.1.2 General Characteristics

1. Graphs: Directed vs Acyclic Directed

The options are: The underlying graph of a graph-based game is directed or acyclicdirected For some games, like shortest path games, there does not seem to be a strongmotivation at first to distinguish both options. This is the case, because Dijkstras shortestpath algorithm can be applied for both kinds of graphs to determine the shortest path.So,it can be quite interesting in some situations to consider the reduction to acyclic directedgraphs.

2. Source/Sink: Vertex vs Set

The options are: The source and sink of a shortest path or flow game are simple verticesor sets of vertices. The question, if it is meaningful to distinguish these cases is easy toanswer. For example, if players can own several arcs and arcs can be owned by severalplayers in shortest path games, then it does not make sense to distinguish these cases.We can easily see that there is an equivalent game, based on a slightly altered gamerepresentation GN2 . Lets take an arbitrary shortest path game representation GN1 witha corresponding pre-problem = hV, A, s, ti (G = hV,Ai), where the source s is a setof vertices. We can simply introduce a new vertex So to the graph, connect So to allvertices t 2 s (from So to t), assign costs 0 to all these arcs, declare So the new source andassign every player as the owner of the newly added arcs. Hence, the newly added arcsare public. For the target vertex we follow the same procedure, whereas the directionalityof the arcs is reversed. The rest of the diagram stays unchanged. Lets call the alteredgame representation GN2 . Then vGN1 (S) = vGN2 (S) for all S N. A similar reasoningholds for flow games. Of course, this trick does not work anymore if players can own onlyone arc and every arc is only owned by exactly one player. In this case we might not beable to represent the game. Hence, there could be a difference in expressive power. Butdespite this aspect, we could still simulate the function by computing all combinations ofsource/sink pairs So Si (m n if m = —So— and n = —Si—) and choose the best outcome.Hence, the simulation is polynomial and therefore there is no difference in computationalcomplexity. Due to the fact that the impact of this characteristic is rather insignificantfor most games, we mostly concentrate on shortest path and flow games where both, thesource and sink, are a vertex. The only exception of a shortest path game with sets ofvertices is the game introduced by Fragnelli et al.

3. Reward Scheme: Global vs Individual

The options are: A shortest path game can have a global reward scheme or an individualreward scheme. Shortest path games we , have a global reward scheme. But anotherpossibility to reward coalitions of players for transporting goods, is to assign an individualreward in the form of payment to each player. By definition, the reward of a coalition

7

in games with an individual reward scheme is the sum of the individual rewards of thecoalitions players. Of Course, this reward is only granted in the case that the coalitiontransports the good successfully from the source to the sink. As we will see later, thischaracteristic heavily influences the expressive power of shortest path games.

8

Chapter 3

Literature Survey

Nash Equilibrium : It is a game to possess dominant strategies solution so we need a generalizedand a more applicable solution which was given by John

Nash. This approach says that there exist a strategy s is said to be a Nash Equilibrium.Any agent i cannot deviate to another strategy

s’ assuming others stick to the original strategy and in turn increase its payoff.Mechanism Design : A mechanism constitute of few rational agents of which preferences

are private. So mechanismmake sure that a fair game is held and no agent strategically change the outcome of the

game according to his preference. Social Choice : A Social Choice is a joint decision takenby all the agents in a game environment. For example, in an election a social choice can becalculated

by counting the majority of votes casted by voters.Due to randomly changing environment and Channel Quality Information (CQI), Scheduler

play a key role to fulfill the QoS requirement of heterogeneous User Equipment (UE) and alsohas to solve the challenges discussed in the previous chapters. There are many schedulingalgorithms with their pros and cons are available in the literature. On the basis of inputparameters, objective and service target we can classify scheduling algorithm into groups of thefollowing strategies [1].

3.1 Channel-unaware and QoS-unaware strategies

These types of strategies are first introduced in wired networks and assumed that propagationenvironment is time-invariant and error free. These strategies are not applicable to wirelessnetworks directly but they can be use jointly used with channel-aware approach to improve sys-tem performance. Some of the cochannel-unaware strategies available in literature are describedbelow FIFO, Round Robin and Blind Equal Throughput [2].

3.2 Channel-aware and QoS-unaware strategies

In 4G wireless system, Channel Quality Information (CQI) feedback is periodically sent by eachUE to the base station in the uplink for each subchannel. Depending upon the CQI value sentby the UE, eNodeB predices the current channel quality and achievable throughput (data rate)based on the best modulation scheme that has to be used for the current channel quality in

9

order to achieve maximum data rate. The common drawback of these startegies is that theycannot be deployed to serve flows with QoS requiirements which was the basic requirement ofLTE specification by 3GPP.

3.2.1 Channel model

Propagation environment for wireless networks is random in nature and due to the dynamicnature of the network we cannot define exact channel model for it, but it can be approximated.It mainly depends upon five components [3]: distance loss, shadowing, multipath-fading, intercell interference, intra cell interference.

User Equipment calculate SNR value from the above five components and map these SNRvalue to Channel Quality information (CQI) value as

CQI =

0 if SNR ≤ −16SNR1.02

+ 16.62 if −16 ≤ SNR ≤ 1430 if 14 ≤ SNR

(3.1)

CQI value periodically sent from UEs to base station using ad hoc control message in theuplink. From CQI value scheduler can estimate the channel quality perceived by each userand hence it can predict the maximum achievable throughput. Scheduling strategies whichuses CQI information in allocation of resources to UE are called channel-aware strategies andfollowing are the basic channel-aware strategies available in literature.

3.2.2 Maximum Throughput

Aim of maximum throughput strategy is to maximize the system throughput in term of spectralefficiency. To achieve its aim, Maximum Throughput(MT) (also called Max CQI )resource allo-cation algorithm allocates RB’s to those users which have maximum achievable instantaneousthroughput or in other words having maximum CQI value in the current transmission timeinterval (TTI). This strategy maximizes throughput of system because it always serves userswhich have the best channel condition due to which it is also called as opportunistic strategies.Main disadvantage of this strategy is that it is unfair to users which have bad channel condition.

3.2.3 Proportional Fair Scheduler

This strategy [4] is able to manage the trade-off between maximizing throughput and fairnessbetween users. Proportional fair (PF) strategies maximize throughput of system by usingMaximum throughput scheme and give throughput fairness to users by Blind Equal throughputscheme.

This is one of the very popular strategies because as it gives fairness among users and at thesame time tries to optimize spectral efficiency and was used in [4], [5], [6]. The past runningthroughput acts as a fairness factor for all users. User which is not served from long timeperiod has very low past running throughput and hence its metric value is high and servedwithin certain amount of time.

10

3.2.4 Joint Time and Frequency domain schedulers

This strategy allocates resources in two layers. In top layer a Time Domain Packet Scheduler(TDPS) selects active users from the set of all users which are connected to the base station.Inlower layer Frequency Domain Packet Scheduler (FDPS) allocate resources to the active users.The final outcomes of resource allocation depends upon both TDPS and FDPS strategies.Main advantage of this strategy is that time complexity for FDPS (RBs allocation) is reducedbecause it checks for allocating resources only to active users identified by TDPS. Both TDPSand FDPS use any type of strategy depending on the requirement of the scheduler. Like RRcan be use for time fairness and PF can be use for throughput fairness by TDPS. In [7], [8],[9],[10] authors uses two level strategies. In [8] authors implements PF scheduler for both TDPSand FDPS maximize throughput and fairness simultaneously in low time complexity. [10]implements QoS-aware strategy for TDPS and maximum throughput strategies for FDPS and[9] implements Time Domain Priority Set Scheduler (TD-PSS) for TDPS and Proportional Fairscheduler for FDPS.

3.3 Channel-aware and QoS-aware strategies

Modern wireless cellular technology concurrently serves diverse real-time applications with dif-ferent QoS requirements. Each user has its own QoS requirement in terms of guaranteed datarates, delivery delay, or different quality level of multimedia assets due to heterogeneous enduser devices.Following are some of the channel-aware and QoS-aware strategies available in theliterature. Some of these startegies guarantee date rate and others bound the packet delay.

A generic QoS-aware solution that guarantee data rate for traffics/flows. It works in bothtime/frequency domains. In [9] authors implemented a strategy as shown in fig 3.1.

Fig. 3.1 Two layer QoS aware Scheduler

In schedulability check step they check following condition for every queue

1. Pending retransmission for transmission queue if yes then selected for scheduling

11

2. Available data in transmission queue is enough for transmit in frame if no then it is notselected for scheduling

In TDPS, Time Domain Priority Set Scheduler (TD-PSS) is used that try to achieve targetbit rate (TBR) for all flow/users and Separate users in two sets. Generally users with flowsbelow their target bitrates form a high priority set called set 1 and rest of the users form lowpriority sets i.e. set2 and high priority group users serve first. For higher priority set authorschedule Users using BET scheme and for low priority set PF scheme for resource allocation isused.

Once TDPS selects the candidate users, in frequency domain proportional fair (PS) schemeis used for all active flows.A similar approach is followed in [11] where the authors set the priority for flow depends uponthe QoS class identifier (QCI). On the basis of guarantee bit rate (GBR) sort them and eachflow is divided in to guarantee bit rate (GBR) and non-guarantee bit rate (non-GBR) class.Allocate resources to GBR class first and after that if RBs are unallocated then allocate it tonon-GBR flow.

Another method for satisfy guarantee Data-Rate is to give different priority for each usersas done in [12] and serve highest priority user first. Priority for individual user is calculated asthe ratio of HeadOfLine(HOL) delay to the expiration time of the packet

Delivery of packets within certain dead line is one of the main requirements for real timeapplication like VoIP, streaming video. Aim of these set of strategies is to deliver packets withintheir bounded delay.

Modified Largest waited delay first (M-LWDF)[13] strategies is presented in literature andmetric is calculation taking into account of the HeadOfLine Delay and the expiration time ofthe packets and also takes the proportional fair metric into its metric calculation to providefairness and spectral efficiency. M-LWDF shaping the behavior of PF and assuring balanceamong spectral efficiency, fairness, and QoS provisioning. The major problem with M-LWDFscheduler is it cannot put a bound on the the packet delays of QoS flows and also doen’t handleoverload situations.In [14] the authors present a QoS aware scheduler which works in two levels where the first levelTime Domain scheduler classifies flows into GBR and non GBR flows and classifies all GBRflows into two QCI levels and all non GBR flows into 3 QCI levels and then prioritize eachflow by the ratio of the QCI Class Weight to the accumulated bearer data rate. The secondlevel scheduler is a FDPS which maximizes spectral efficiency by allocating RB’s based on thepriority calculated by first level TDPS scheuler and selecting the best AMC scheme by takinginto account CQI information of the user.The major drawback of this approach is that a set offlows with different QoS requirements are put into the same QCI class and are treated as sameeven though each of them have a different QoS requirements.In [15] the authors present a downlink scheduling algorithm called Modified Earliest DeadlineFirst Proportional Fair(M-EDF-PF) for handling real time traffic in LTE systems. The algo-rithm schedules users whose packets has the least deadline among all the users and also theymultiply their metric of Earliest Deadline with the proportional fair metric to increase fairnessand spectral efficiency. Even though the algorithm has low packet loss rate , the algorithmdoesn’t impose strict delay bounds for QoS flows and doesn’t handle overloads efficiently.In [16] the authors present a downlink scheduling algorithm called Delay-Prioritized Schedul-ing(DPS) for handling Real Time Traffic in LTE systems. The DPS algorithm uses the CQIinformation and the instantaneous packet delay of Real time flows and works by giving the

12

highest priority to the user with minimum remaining time for the packet to expire and allo-cates RB’s to that user and iterates till all RB’s are allocated.The major drawback with thisscheme is that if all flows have same QoS requirements then the algorithm will not be able toallocate resources in a fair manner as some flows might starve for resources.In [17] the authors presented a new QoS aware downlink scheduling algorithm for LTE networkswhich is split into three phases. In the first phase the UE’s are allocated RB’s based on the CQIvalue reported in Frequency Domain and then in second phase the concept of virtual queuesare introduced to predict the packet delay of the incoming packets based on the current size ofthe queue, current packet loss rate and on the inter arrival packet time and rate of transmissionof the flow. Now with the predicted number of packets for each flow , in the third phase a cutin process is called to allocate more RB’s to the flows which will have larger packets arrivingin the near future. The RB’s are taken from such flows which creates the least decrease in thespectral efficiency of the system and allocated to the flows which are predicted to have morepackets coming in the future.In [18] the authors propose a two level packet scheduler algorithm(FLS Algorithm) where thefirst level computes how much data is to be sent by each real time flows in the next LTEframe(10ms) to satisfy the QoS requirements using a discrete control theory and then schedulesthe users in the second level with the PF scheduler. The authors also present results whichclaims that the algorithm provides the best Quality Of Experience(QOE) among the variousstate of the art scheduling algorithms. FLS algorithm bounds the packet delay for all QoS flowsand satisfies QoS requirements at LTE Frame level boundaries and also optimizes on spectralefficiency and provides a considerable degree of fairness to non real time flows. The majordrawback of FLS algorithm is that the algorithm doesn’t use the Channel Quality Informationin determining the amount of data that can be sent by the scheduler in LTE frame as it justfinds out how much data is to be sent by each Real Time Flow to satisfy its QoS requirementsin the next LTE frame. This makes the scheduler unsuitable for handling overload conditions.

3.4 Buffer Aware Strategies

In [19] presented an channel adapted and buffer aware packet scheduling algorithm called CABAwhich considers the UE buffer status to avoid buffer overflow while allocating radio reseourcesto the flows. CABA algorithm minimizes the packet losses due to buffer overflow problems atthe receiver queue and also provides fairness among users by proportional fair metric . Themajor drawback with this algorithm is that it doesn’t enforce strict QoS requirements on theQoS flows and also it contains many weighing parameters which need to be fine tuned to providebetter performance. algorithm is that the

3.5 Cross Layer Strategies

In [20] the authors presents a cross layer downlink scheduling algorithm called Weighted RoundRobin (WRR) which gives importance to the perceptual qualtiy of the video flow sent. Theweights assigned to the Real Time flows depends on SNR calue,QoS delay requiirements andprevious average data rate achieved by the user. WRR algorithm has the objective function tominimize the video distortion under the given packet delay constraints.In [21] the authors proposed a Multi Service QoS guaranteed cross layer resource allocationalgorithm in LTE systems which uses channel state information of physical layer and queue

13

state of data link layer into the metric calculation. The users/flows are classified into threesets based on their instantaneous severity of the packet delays and then flows are allocatedRB’s starting from the most urgent set and then once all users on this set is scheuled then theusers from the second urgent set2 are allocated RB’s and the same process is repeaated forthird set of urgent users too.The priority level of each user inside an urgent set is defined to getthe relative urgencie of the flows inside their set. The above scheduling scheme has the majorproblem of not being spectrally efficient and also can lead to starvation of most of the flows ifthere is a very high QoS requirement flow in the system as always this flow will be present inthe most urgent set and will utilize most of the radio resources.

In summary, the major limitations of the above related work is the absence of a resourceallocation mechanism for 4G LTE Systems which is low overhead and is able to satisfy theQoS demands of Multimedia services and can handle overload situations and provide fairnessto NRT flows.

3.6 Motivation of Current Work

We have seen from the previous sections of this chapter that ther is no resouce allocationmechanism available currently in 4G LTE Systems which can solve the major challenges that wehave stated in the previous chapters. Even though they are lot of proposed resource allocationalgorithms available in the literature as stated in the previous work, many of those algorithmsare unsuitable and cannot be deployed in a real world situaions because of various reasons likethey have high complexity or they are not able to handle overload situaions or they are notable to satisfy the QoS demands of RT flows and provide the best system throughput.

From the various shortfalls of the algorithms discussed in the literature we propose themotivation and the objectives of the current work is to create a Resource Allocation Strategyin the downlink of 4G Wireless Networks with the following key objectives :

1. Low Complexity which ensures scalability of the algorithm

2. Fullfil QoS Requirements of Multimedia flows

3. Maximize Spectral Efficiency of the system

4. Handle Trasnsient Overloads by graceful degradation

5. Provide Fairness among Non Real Time Flows

14

Chapter 4

Summary of Work Done

To show the variation in payment function with the edges involved in the Shortest path con-sidered , we came across through certain observation described in following Theorems.

4.1 Graph G can be created from graph G ,by adding new edgessuch that payment of edges is∑

e∈SP ′(g′)

P e <∑

e/∈SP (g)

P e

ie.where P is payment for ’e’ edges , SP and SP are the shortestpath in respective graphs ,only in case when no. of edgesinvolved in SP are less than that involved in SP of graph G.

4.1.1 Example

Lets consider the sample graph shown as below , where a is the starting point and d is desti-nation.

Fig. 4.1 graph showing ’a’ as starting point and ’d’ as sink

Thus total Payment of edges is p=60.

15

Edges removed Alt SP Alt. Path sp-(sp-w) P’ab 30 ad 30-(16-1) 15bc 30 ad 30-(16-10) 24ce 24 abcd 24-(16-2) 10ed 24 abcd 24-(16-3) 11

Table 4.1 Payment calculation

Now if we add some weighted edge bd in graph g , so that the new SP contains less no. ofedges than in SP, we can see the fall in payment of edges.

Fig. 4.2 graph showing ’a’ as starting point and ’d’ as sink with new edge bdintroduced

Now the SP is abd ie 3;

Edges removed Alt SP Alt. Path sp-(sp-w) P’ab 30 ad 30-(3-1) 28bd 16 abced 16-(3-2) 15


total Payment of edges is p=43.Thus payment of edges decreasedAs our main focus is to retrieve relation between no. of edges involved in some SP and

payment .Lets consider a case where the edge bd is split by a node p , increasing the no. ofedges involved in SP,while the value of SP remain same .ie. 3

the SP is abpd ie 3; To calculate p :Total Payment of edges is p=56. Thus payment of edges increased.Now again extending this test case in such a manner where no. of edges involved in SP is

equal to the no. involved in previous SP. And having the weight summation as previous valuethe SP is abpqd ie 3;To calculatep :total Payment of edges is p=67.99.

16

Fig. 4.3 graph showing ’a’ as starting point and ’d’ as sink with alternate newedge bp and pd introduced

Edges removed Alt SP Alt. Path sp-(sp-w) P’ab 30 ad 30-(3-1) 28bp 16 ad 16-(3-1) 14pd 16 abced 16-(3-1) 14


Fig. 4.4 graph showing ’a’ as starting point and ’d’ as sink with 3 new edgebp,pqand qd introduced

Thus payment of edges increased from the base case.(unfavourable) Thus from illustrationof above test cases, we can conclude that the payment of edges can be decreased only in thecase when no. of edges involved in the new SP is strictly less than that involved in SP.

17

Edges removed Alt SP Alt. Path sp-(sp-w) P’1 ab 30 30-(3-1) 282 bp 16 16-(3-0.33) 13.333 pq 16 16-(3-0.33) 13.334 qd 16 164-(3-0.33) 13.33

Table 4.4 Payment Calculation

4.2 Every chosen alternative from set of SP costs equal to thesystem designer, though payments to corresponding edges maydiffer.

4.2.1 Example

Lets consider a graph G, having a set of Sp SPi, SPj such that both have same value =24.Now lets check the effect on payment of edges . the SPi = SPj = 24. The p is calculated as

Fig. 4.5 graph showing ’a’ as starting point and ’d’ as sink with two SPi, SPj

follows : For SPi the path is abcd.

Edges removed Alt SP Alt. Path sp-(sp-w) P’1 ab 30 30-(24-1) 72 bc 30 30-(24-10) 163 cd 16 24-(24-13) 13


Total Payment of edges is p=36.For SPi the path is abcd.Thus total Payment of edges is p=36.Thus from above example we can clearly see that although individual payment may differ

for edges in case of multiple shortest paths but the overall summation remains same.

18

Edges removed Alt SP Alt. Path sp-(sp-w) P’ab 30 ad 30-(16-1) 15bc 30 ad 30-(16-10) 24ce 24 abcd 24-(16-2) 10ed 24 abcd 24-(16-3) 11


4.3 The payment of edges done by system is directly proportionalto the difference between the current shortest path SP and thealternate SP.ie p (SP-SP)

4.3.1 Example

Lets consider the sample graph in both cases where diffrence between SPi and SPj is minimumand maximum.

Case1 : SP-SP is maximum Now the SP is abd ie 2.




Thus total Payment of edges is p=44. Thus payment of edges decreased from the base case(p=60), with gain of 16 units for designer.

Now case2 : SP-SP is minimumNow the SP is abd ie 15.Thus total Payment of edges is p=31.

19




Thus payment of edges decreased from the base case (p=60), with gain of 29 units Andfrom previous case 13 units. Thus payment is proportional to the difference between Sps isexplained by above example.

20

Chapter 5

Plan of Future Work and Conclusion

5.1 Plan of Future Work

At this point, we would like to give an overview of possible future directions of research andideas that could be pursued to expand the work of this thesis. To obtain a more precise idea ofthe influence of characteristics on the computational complexity of solution concepts applied toshortest path games, we think that it would be particularly interesting to analyse various sta-bility concepts for different non-simple variants of shortest path games. To design constructiveproofs for analysing all the possible Payment outcomes in various cases. We also think thatit would be interesting to prove several open problems for those graph-based games, which wepresented in this thesis, and to look for further graph-based coalitional games to extend thesample space of graph-based games. So, having a more complete set of results we might beable to isolate influential properties and characteristics of graph-based games. This may evenallow researchers, interested in transferring problems to graph-based coalitional games, to useit as heuristic to specify coalitional games in such a way that the application of interestingsolution concepts is computationally tractable. For many real-world motivated graph-basedgames, which have been analysed in the literature, where the existence of an efficient algorithmis imperative, this situation is of course unsatisfactory. But we have also noticed during ourresearch that particular simplifications of graph-based games, most notably simplifications ofthe underlying graph, can lead to computationally favourable results. For example, the reduc-tion of graphs to trees seemed to be a promising reduction. This is also interesting in a morepractical context, because there are many real-world problems with treestructures in computerscience (Internet and networking), which could be analysed from a gametheoretic perspective.Hence, the reduction to trees and also acyclic directed graphs for graph-based games is notonly a theoretical consideration to obtain polynomial complexity results, but also a promisingway to determine classes of coalitional games, where interesting solution concepts applied togames, which are motivated from real-word problems, are tractable

5.2 Conclusion

At first we discussed which characteristics are interesting in the context of graph-based coali-tional games mainly for shortest path games, with respect to Payment functions. Then weintroduced several variants of shortest path games, which vary over those characteristics

of shortest path games, compared them and related them to other types of coalitional

21

possible in same graphs . It was interesting to see what effect no. of edges and the ownershiprelation had on the Payment or reward received by them. To breif shortly, we came across fewresults that

If a new edge is added to some graph in a manner such that value of Shortest path is lessercan contribute towards lesser payment of edges only in case when the no. of edges involved inpresent chosen SP is lesser than that in the alternate SP.

We came to know that if there are more than one Shortest paths in graph, then no matterwhat path we choose the Payment will amount same , irrespective of no. of edges involved,although individual payment of edges may vary.

The payment of edges done by system designer is directly proportional to the differencebetween the current shortest path SP and the alternate SP.ie p (SP-SP) ie, if we want tominimize the Payment then the new edge we are adding should be close to the accumulativesum of the edges whic it is substituting. A general idea via example is shown in the thesisdescribing the influence of various characteristics of graph-based SP games.

22

References

[1] F. Capozzi, G. Piro, L. Grieco, G. Boggia, and P. Camarda, “Downlink packet scheduling inlte cellular networks: Key design issues and a survey,” Communications Surveys Tutorials,IEEE, vol. 15, no. 2, pp. 678–700, 2013.

[2] P. Kela, J. Puttonen, N. Kolehmainen, T. Ristaniemi, T. Henttonen, and M. Moisio, “Dy-namic packet scheduling performance in utra long term evolution downlink,” in WirelessPervasive Computing, 2008. ISWPC 2008. 3rd International Symposium on, 2008, pp.308–313.

[3] A. Haider and R. Harris, “A novel proportional fair scheduling algorithm for hsdpa in umtsnetworks,” in Wireless Broadband and Ultra Wideband Communications, 2007. AusWire-less 2007. The 2nd International Conference on, 2007, pp. 43–43.

[4] V. Vukadinovic and G. Karlsson, “Video streaming performance under proportional fairscheduling,” Selected Areas in Communications, IEEE Journal on, vol. 28, no. 3, pp.399–408, 2010.

[5] P. Svedman, S. Wilson, and B. Ottersten, “A qos-aware proportional fair scheduler for op-portunistic ofdm,” in Vehicular Technology Conference, 2004. VTC2004-Fall. 2004 IEEE60th, vol. 1, 2004, pp. 558–562 Vol. 1.

[6] M. Khedr, I. El-Rube, Y. Hanafy, and H. Abou-zeid, “Subcarrier opportunistic propor-tional fair scheduling for ofdma systems,” in Internet, 2008. ICI 2008. 4th IEEE/IFIPInternational Conference on, 2008, pp. 1–5.

[7] A. Pokhariyal, K. Pedersen, G. Monghal, I. Z. Kovacs, C. Rosa, T. Kolding, and P. Mo-gensen, “Harq aware frequency domain packet scheduler with different degrees of fairnessfor the utran long term evolution,” in Vehicular Technology Conference, 2007. VTC2007-Spring. IEEE 65th, 2007, pp. 2761–2765.

[8] K. C. Beh, S. Armour, and A. Doufexi, “Joint time-frequency domain proportional fairscheduler with harq for 3gpp lte systems,” in Vehicular Technology Conference, 2008. VTC2008-Fall. IEEE 68th, 2008, pp. 1–5.

[9] G. Monghal, K. Pedersen, I. Kovacs, and P. Mogensen, “Qos oriented time and frequencydomain packet schedulers for the utran long term evolution,” in Vehicular TechnologyConference, 2008. VTC Spring 2008. IEEE, 2008, pp. 2532–2536.

[10] G. Piro, L. Grieco, G. Boggia, and P. Camarda, “A two-level scheduling algorithm forqos support in the downlink of lte cellular networks,” in Wireless Conference (EW), 2010European, 2010, pp. 246–253.

23

[11] Y. Zaki, T. Weerawardane, C. Gorg, and A. Timm-Giel, “Multi-qos-aware fair schedulingfor lte,” in Vehicular Technology Conference (VTC Spring), 2011 IEEE 73rd, 2011, pp.1–5.

[12] D. Skoutas and A. Rouskas, “Scheduling with qos provisioning in mobile broadband wire-less systems,” in Wireless Conference (EW), 2010 European, 2010, pp. 422–428.

[13] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, P. Whiting, and R. Vijayakumar,“Providing quality of service over a shared wireless link,” Communications Magazine,IEEE, vol. 39, no. 2, pp. 150–154, 2001.

[14] Y. Zaki, T. Weerawardane, C. Gorg, and A. Timm-Giel, “Multi-qos-aware fair schedulingfor lte,” in Vehicular Technology Conference (VTC Spring), 2011 IEEE 73rd, 2011, pp.1–5.

[15] B. Liu, H. Tian, and L. Xu, “An efficient downlink packet scheduling algorithm for realtime traffics in lte systems,” in Consumer Communications and Networking Conference(CCNC), 2013 IEEE, 2013, pp. 364–369.

[16] K. Sandrasegaran, H. Ramli, and R. Basukala, “Delay-prioritized scheduling 40;dps 41;for real time traffic in 3gpp lte system,” in Wireless Communications and NetworkingConference (WCNC), 2010 IEEE, 2010, pp. 1–6.

[17] W. K. Lai and C.-L. Tang, “Qos-aware downlink packet scheduling for {LTE} networks,”Computer Networks, vol. 57, no. 7, pp. 1689 – 1698, 2013. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S1389128613000352

[18] G. Piro, L. Grieco, G. Boggia, R. Fortuna, and P. Camarda, “Two-level downlink schedul-ing for real-time multimedia services in lte networks,” Multimedia, IEEE Transactions on,vol. 13, no. 5, pp. 1052–1065, 2011.

[19] Y. Lin and G. Yue, “Channel-adapted and buffer-aware packet scheduling in lte wirelesscommunication system,” in Wireless Communications, Networking and Mobile Computing,2008. WiCOM ’08. 4th International Conference on, 2008, pp. 1–4.

[20] H. Luo, S. Ci, D. Wu, J. Wu, and H. Tang, “Quality-driven cross-layer optimized videodelivery over lte,” Communications Magazine, IEEE, vol. 48, no. 2, pp. 102–109, 2010.

[21] Y. Qian, C. Ren, S. Tang, and M. Chen, “Multi-service qos guaranteed based downlinkcross-layer resource block allocation algorithm in lte systems,” in Wireless CommunicationsSignal Processing, 2009. WCSP 2009. International Conference on, 2009, pp. 1–4.

24

Documents

1.pdf