Game Theory and Time Utility Functions for a Radio Aware Scheduling Algorithm for WiMAX Networks

Transcription

1 Wireless Networks manuscript No. (will be inserted by the editor) Game Theory and Time Utility Functions for a Radio Aware Scheduling Algorithm for WiMAX Networks Rosario G. Garroppo Stefano Giordano Davide Iacono Luca Tavanti Received: date / Accepted: date Keywords IEEE Mobile WiMAX Radio Aware Scheduler Game Theory Time-Utility Functions Abstract In WiMAX systems the Base Station scheduler plays a key role as it controls the sharing of the radio resources among the users. The goal of the scheduler is multiple: achieve fair usage of the resources, satisfy the QoS requirements of the users, maximize goodput, and minimize power consumption, and at the same time ensuring feasible algorithm complexity and system scalability. Since most of these goals are contrasting, scheduler designers usually focus their attention on optimizing one aspect only. In this scenario, we propose a new scheduling algorithm (called GTS N ) whose goal is to contemporaneously achieve efficiency and fairness, while also taking into account the QoS requirements and the channel state. GTS N exploits the properties of Time Utility Functions (TUFs) and the Game Theory. Simulations prove that the performance of GTS N, when compared to that of several well-known schedulers, is remarkable. GTS N provides the best compromise between the two contrasting objectives (i.e. fairness and efficiency), while QoS requirements are in most cases guaranteed. However, the exponential complexity introduced by the game theory technique makes it rather impractical and not computationally scalable for a large number of users. Thus we developed a suboptimal version, named sub-gts N. We show that this version retains most of the features and performance figures of its brother, but its complexity is linear with the number of users. 1 Introduction The Worldwide Interoperability for Microwave Access (WiMAX) Forum [21] is a consortium of manufacturers and network operators whose aim is to define a set of system profiles and certification profiles for the interoperability of equipments for Wireless Metropolitan Area Networks (WMANs). Specific focus of the WiMAX Forum are radio interfaces based on the IEEE family of standards. IEEE gives the Department of Information Engineering University of Pisa Via G. Caruso 16, Pisa I-56122, Italy {name.surname}@iet.unipi.it

2 2 specifications for fixed and mobile broadband wireless access networks. It uses a pointto-multipoint architecture, in which the Base Station (BS) offers network access to the Subscriber Stations (SSs). In particular, IEEE e [7] has been ratified to deal with terminal mobility, which is a critical issue when considering the wireless nature of this technology. However, while the scope of IEEE is limited to MAC and physical layers only, the WiMAX Forum considers the whole network system, for which it has defined the reference network architecture. There are various features that differentiate WiMAX systems from other wireless access technologies for WMANs. Specifically, WiMAX systems use Orthogonal Frequency Division Multiple Access (OFDMA), a resource allocation model for the frequency-time domain that has the advantage of producing a higher system capacity than that obtained with time domain allocation only, such as OFDM. Furthermore, WiMAX defines systems with diverse channelization (from 1.25 MHz to 20 MHz) to obtain the scalable exploitation of any spectrum width, and two duplexing schemes (Time Division Duplexing, TDD, and Frequency Division Duplexing, FDD) to provide the network operator with highly flexible management of the radio resources. In detail, TDD permits a flexible distribution of channel resources in presence of services generating asymmetric traffic. Still to improve the efficient use of the bandwidth, WiMAX uses AMC (Adaptive Modulation and Coding) schemes on a per-ss basis. Another relevant feature is the definition of multiple Quality of Service (QoS) classes suitable for a combination of data, voice and video services. WiMAX is a connection-oriented technology in which SSs are not allowed to access the wireless media unless they initially register and request bandwidth allocation to the BS (except for certain time slots specifically reserved for contention-based access). The radio resources are managed by the BS, which is responsible for the resolution of the physical/mac resources contention among the different SSs. Specifically, the BS scheduler decides the resource allocation for downlink traffic and grants uplink transmission opportunities to the various SSs as a function of its buffer occupancy and the bandwidth requests from the SSs. Furthermore, in the resource allocations process, the BS scheduler must also take into account the QoS requirements specified by the users (e.g. in terms of delay, delay jitter, throughput). However, QoS support in wireless networks is much more challenging than in wired networks because the characteristics of the wireless links are highly variable and unpredictable both on a time and a location dependent basis. In this framework, the BS scheduler becomes a critical point of the WiMAX system. In order to differentiate the market of WiMAX equipments, the IEEE standard does not define any mandatory scheduling mechanism. Therefore, the design of scheduling algorithms is of special interest to all WiMAX equipment manufacturers and service providers. The key issue for this task is how to allocate the resources among the users 1 not only for satisfying the QoS requirements, but also to maximize goodput, to minimize power consumption, and at the same time to obtain a feasible algorithm complexity and ensuring system scalability. In this scenario, we propose a new scheduling algorithm called Game-theory and TUF -based Scheduler (GTS N ). As the name suggests, GTS N exploits the concepts of Game Theory and the features of Time Utility Functions (TUFs). The scheduling decision is performed in two steps. In the first, every user employs the TUFs for defining the utility of the packets in its buffers, and then sorts them in order of descending 1 Throughout the paper with the term user we mean a Subscriber Station (SS).

3 3 utility. This list represents the user-preferred transmission order of its packets. The utility is obtained from an association between a TUF and the Traffic Class (TC) to which the packet belongs. In the second step, the scheduler uses a Game Theory formulation to select, from the lists provided by each user, the set of packets to be transmitted in the next WiMAX frame. The set of packets is obtained as a specific game solution targeting the proportional fairness objective (which is the Nash solution, hence the N subscript in the name). The constraints on available radio resources are also taken into account. The proposed scheduling algorithm employs a cross-layer approach that accounts for application requirements by means of TUFs and for the physical constraints by means of the Channel State Information (CSI) of each user. Due to the high computational complexity of GTS N, which is based on the Nash solution of the game, a sub-optimal scheduler is developed and presented. The performance of the two schedulers are compared to that of other well-known scheduling algorithms by means of simulation analysis. The paper, after a summary of related works in Section 2, introduces the main features of the MAC layer of the WiMAX technology in Section 3. Then, in Sections 4 and 5, the paper describes the basic definitions of Game Theory and of the TUFs, which are necessary to understand the basic principles of the proposed scheduler. Section 6 presents the GTS N scheduler and, after analysing its complexity, introduces its suboptimal version, denoted as sub-gts N. The simulation analysis and the performance comparison among the various schedulers is detailed in Section 7. Finally, Section 8 gives some concluding remarks about the proposed schedulers. 2 Related work Over the years the problem of managing and sharing the resources of the communication medium has been widely discussed, for both wired and wireless systems. With the rapid diffusion of wireless technologies, the problems of service guarantee and fairness, which had long been specific to wired networks, have also been extended to the wireless domain. However, in the case of wireless networks, the peculiarities of the radio channel should be taken into account, in order to avoid the problems rising from a communication link that could become degraded or even unavailable during the transmission. For this reason, the algorithms developed for wired networks, which assume an errorfree data channel, hardly adapt to wireless networks. Furthermore, specific features of WiMAX, such as the utilization of OFDMA, do not permit the use of scheduling algorithms developed for other wireless technologies, such as CSMA/CA-based WLANs or CDMA-based cellular systems. In this context, various scheduling algorithms have been developed [3]. Most of them have the common assumption that the channel condition does not change within the frame period. Furthermore, the channel information is assumed to be known at both the transmitter and the receiver. A detailed survey of recently proposed scheduling algorithms can be found in [3]. In this section we focus and report just the main features of the algorithms considered for the comparison analysis. Scheduling algorithms for WiMAX systems can be classified into two main classes: channel unaware and channel aware. The schedulers belonging to the former class do not care about information on the radio channel quality experimented by the SSs. On the contrary, channel aware algorithms use the reports on channel quality provided by the SSs in the scheduling decision.

4 4 Among the channel unaware schedulers, the basic algorithm is Round Robin (RR), a very simple solution for providing fairness among the users. This algorithm selects one by one each user to be served in a circular order. When a user is selected, the packet in the head of its queue (Head Of Line, HOL, packet) is extracted and served. Since the average packet size can vary among the different traffic flows 2 and users, such a policy could allocate a larger portion of the available bandwidth to some flows/users to the detriment of others. To address this problem, a variant of RR was developed, which is called Deficit Round Robin (DRR). This variant defines a quantum of bytes to be served. Every time a user is selected, the quantum is added to a counter that determines the amount of bytes to be served for that user. If the HOL packet size is less than or equal to the counter, the packet is served and its size is scaled from the counter. Otherwise it is not served and the scheduler passes to the next user. Both RR and DRR do not support QoS. To add this feature, variants of them were developed, such as Weighted Round Robin (WRR) and Deficit Weighted Round Robin (DWRR). These algorithms use some weights, w i, whose purpose is to adjust the throughput performance. In other words, the weights define the number of packets to be served (WRR) or the number of quantum to be added to the counter (DWRR) every time a user is selected. Note, however, that none of the Round Robin family of schedulers provide end-to-end delay guarantees. Weighted Fair Queuing (WFQ) has been proposed to bring the concept of virtual time into the networking area. Virtual time has been introduced in the early 90 s for solving the problems associated to the sharing of a processor that has to serve several sessions. The solution, obtained with the ideal model of infinitesimal work units, is the Generalized Processor Sharing (GPS). However, the assumption of infinitesimal packet size can not be directly put into practice, as networking algorithms and devices work on finite-sized packets. Hence WFQ is an approximation of GPS that overcomes the problems due to the infinitesimal packet size assumption. In practice, the virtual time is the time when the packet is served in the corresponding GPS and the packets are served in a increasing virtual finish service time order. With further refinements new versions of WFQ were developed, like WF 2 Q (Worst-Case Weighted Fair Queuing) and WF 2 Q+. WF 2 Q+ works like WF 2 Q, but it establishes hierarchy of traffic; the rate of the physical link to be shared among all sessions is split according to the percentage of traffic assigned to each user and, within its traffic, to each application. An example of WF 2 Q+ is reported in Figure 1. All schedulers cited so far are general algorithms that can be applied to any kind of underlying wired or wireless technology. Some of them have been applied to WiMAX networks, such as variations of RR and WFQ. However, as already mentioned, all these algorithms did not take into account the conditions of the radio channel, i.e. they are radio unaware schedulers. On the other side there are the radio aware schedulers. They can be further classified into four subclasses depending on the primary objective considered in the scheduling design, i.e. provide fairness, guarantee QoS, maximize system throughput or optimize power. The subclass of algorithms maximizing the system throughput are characterized by the allocation of the radio resources to the users with better channel quality; these are usually called pure opportunistic schedulers. This approach permits to achieve the maximum throughput, but is highly unfair since the scheduler could decide to allocate 2 With the term flow we mean a sequence of packets with homogeneous features and QoS requirements. A user may be associated with one or more flows.

5 5 Fig. 1 Example of link sharing hierarchy (left) and corresponding WF 2 Q+ scheduler (right) for best effort (BE) and real time (RT) traffic. no resources to the users experiencing high error rates. Furthermore, these algorithms often do not care about QoS requirements. Because of these drawbacks, this subclass will no longer be considered in this work. For the same reason, scheduling schemes based on power constraints are out of the scope of our work and will not be considered. Since WiMAX users usually pay for their QoS assurance, schedulers must consider users QoS requirements (such as the minimum reserved rate), and may need to introduce some compensation mechanisms. A compensation technique accounts for the missed opportunities of transmission experienced by each flow and tries to compensate them later in time. In this context, a typical approach in the design of wireless schedulers is to run an error-free scheduling policy and then use a compensation scheme. For example, the authors of [1] present an algorithm with compensation, denoted as Channel Aware Compensated Scheduler (CACS), which employs the WF 2 Q+ scheduler. CACS first classifies the incoming packets by their class of service (CoS) and puts them in different queues. Then, to introduce some opportunism, every time a packet is selected the channel towards its destination is checked. If the RSSI is below the sensibility of the receiver, the link is marked with a bad label. When the channel is marked as bad, the scheduler proceeds to check another packet in the same queue to be sent in place of the selected one. For each flow there is a counter of debit/credit. The flow gains one point every time its HOL packet is not transmitted and replaced by the HOL packet of another flow. Conversely, the flow loses a point when its HOL packets replaces the a packet of another flow. The packet that is chosen as a replacement is the one having the counter with the highest value. In this way the authors try to provide at the same time a certain amount of fairness between users together with some opportunistic features to improve the overall exploitation of the channel. Proportional Fairness Scheduling (PFS) is one of the most common approaches used in the subclasses of channel aware schedulers aimed at providing fairness. PFS takes into account the current achievable rate R i (t) and the average throughput T i (t) experienced by user i, and then selects the flow of user i to be served next according to: { } i Ri (t) = argmax i I. (1) T i (t) Note that this formulation does not take into account the QoS requirements.

6 6 On the contrary, Modified Largest Weighted Delay First (M-LWDF) can provide QoS guarantees, and it is provable that the obtained throughput is optimal for LWDF [2]. M-LWDF serves the packet experiencing the biggest delay. In addition, the delay experienced by the packet is weighted according to the type of application and to the quality of the link that the receiver is measuring. In [19] the authors present a version of M-LWDF in which a TUF is used to quantify the urgency of serving a given packet, U i (t). For each user, a list of packets is created in decreasing order of urgency. The packet having the greatest urgency represents its user. Every urgency is then weighted by an efficiency factor, which depends on the signal quality experienced by the destination user. This factor is the ratio of the currently achievable rate R i (t) to the average rate of the user R i (t). Then, the user to be served is selected according to: i = argmax i I { U i (t) R i(t) R i (t) }. (2) Once a user is served, its representing packet, which is in the head of its ordered list, is inserted into the MAC frame, and then the procedure is repeated for the next packet. The algorithms presented above are the most common baseline schedulers. However, in most cases, they either optimally utilize radio resources or provide fairness among users. Rarely do they achieve both goals contemporaneously. To this end, the proposed GTS N scheduler tries to make the best of the radio channel and at the same time guarantee a certain fairness among users, while also taking into account the QoS requirements. 3 Overview of mobile WiMAX PHY and MAC layers Mobile WiMAX has several features that allow for a very flexible deployment. In this Section, we just recall some main features of Mobile WiMAX that will be used in the presentation of the proposed scheduler. Details on Mobile WiMAX and key issues on the scheduling design for this technology can be found in [3]. The mobile WiMAX physical layer (PHY) is based on Orthogonal Frequency Division Multiplexing (OFDM), a modulation scheme that offers good resistance to multipath and allows to operate in non line of sight conditions. The multiple access is handled in both uplink and downlink by Orthogonal Frequency Division Multiple Access (OFDMA). Both uplink and downlink resources are allocated on a per-user basis (where a user is in fact a SS), and the standard allows for resources to be allocated in time, frequency, and space. Specifically, OFDM subcarriers are the resources that are shared and dynamically allocated to the users. When using OFDMA, a scheduler in the BS assigns different subsets of subcarriers to different users. Scheduling algorithms can allocate resources based on demand, QoS requirements, and channel conditions. The subcarriers of mobile WiMAX are always spaced by khz. Having a fixed subcarrier spacing is of paramount importance for the robustness against the Doppler effect caused by node mobility. The standard defines four system profiles as a function of the used channel bandwidth. As a result, in order to maintain the fixed subcarrier spacing, the number of subcarriers is determined by the profile bandwidth. For example, the slimmest 1.25 MHz profile employs a 128 subcarriers modulation, whereas the widest 20 MHz profile uses 2048 subcarriers per OFDM symbol. The subcarriers can be divided into data, pilot and guard (or null) subcarriers. Data subcarriers are then

7 7 grouped into basic resource sets, called slots. A slot is the minimum amount of timefrequency resources that can be allocated to a certain user. A slot consists of a set of subcarriers over one, two, or three symbols, depending on the subchannel allocation algorithm in use. In general, the slot size can be expressed as the number of data subcarriers it is made of, n l c. Figure 2 shows an OFDMA frame when operating in TDD mode 3. The frame is divided into two subframes: a downlink subframe followed by a small guard interval and the uplink subframe. The downlink-to-uplink subframe ratio (DLR) indicates the size of the downlink subframe with respect to the uplink subframe. Fig. 2 The TDD mobile WiMAX frame. As shown in Figure 2, the downlink subframe begins with a preamble that is used for physical layer procedures (e.g. time and frequency synchronization, channel estimation). This is followed by a frame control header (FCH), which provides frame configuration information, such as the MAP message length, the modulation and coding scheme, and the usable subcarriers. The allocations of data regions are specified in the uplink and downlink MAP messages (DL-MAP and UL-MAP) that are broadcast following the FCH in the downlink subframe. The figure also shows the bursts (i.e. groups of slots) assigned to the various users. Mobile WiMAX is quite flexible in terms of how users and packets are multiplexed into a single frame. A downlink frame may contain multiple bursts of varying size and type carrying data for several users. The frame size is also variable on a frame-by-frame basis from 2 to 20 ms, even though the sole mandatory size is 5 ms. The physical parameters of the standard mobile WiMAX profiles are summarized in Table 1. In particular, the Table refers to the Downlink Partial Usage of Subcarriers (DL-PUSC) subchannel allocation algorithm (the mandatory one), which has been assumed to be in use in the simulation analysis. An evaluation of the available physical resources can be easily provided. For example, let us consider the 5 MHz profile and a downlink/uplink ratio DLR = 3 : 1. 3 Though WiMAX supports both Time and Frequency Division Duplex (TDD and FDD, respectively), most implementations favours TDD because of its advantages, such as more flexible sharing of bandwidth between uplink and downlink.

8 8 Table 1 Parameters of mobile WiMAX profiles using the DL-PUSC algorithm. Profile [MHz] No. subcarriers per symbol (FFT size) No. data subcarriers per symbol (n s c) No. pilot subcarriers per symbol No. null/guard subcarriers per symbol Subcarrier spacing khz Symbol duration µs Useful symbol time 91.4 µs Guard time 11.4 µs No. symbols in a frame (N s) 48 No. of data subcarriers in a slot (n l c) 48 Frame duration (T f ) 5 ms Still assuming to use DL-PUSC, a slot is composed of n l c = 48 data subcarriers (independently of the profile). Hence, according to the parameters in [7] and the values in Table 1, the number of data slots Nl DL in a downlink subframe is given by: N DL l = ns c N s DLR n l c DLR + 1, (3) where n s c represents the number of data subcarriers in a symbol (360 in our example), and N s is the number of symbols in a frame, which is always 48 for a 5 ms frame. Let now b be the number of bits per data subcarrier and per symbol. Assuming that all users are employing the same modulation scheme and that all data slots are fully utilized, the data rate R DL of the downlink subframe can be evaluated as: R DL = N l DL n l c b. (4) T f The downlink subframe can reach the maximum data rate Rmax DL when all served users employ the most spectral-efficient scheme (i.e. 64-QAM with coding rate r = 5/6, for an overall b = 5). In such a case, Rmax DL = 11.8 Mbps. On the other hand, when all users utilize the least efficient scheme (i.e. QPSK with coding rate r = 1/3 and b = 0.67), the downlink subframe can serve the minimum rate Rmin DL = 1.73 Mbps. As a result, the mobile WiMAX MAC layer can accommodate a wide range of data rates, and all parameters (i.e. available slots, constellation in use) must be taken into account for an optimal exploitation of the radio resources. 4 Game Theory Game Theory deals with situations whose final result depends on the choices of several decision-makers (the players) [15]. Their target may be common (but not necessarily identical), different or opposite. A strategy is a function that assigns a move to a player for each possible situation of the game in which he/she is the decision-maker. Random elements are also allowed. Players are assumed to be utility maximizers, where utility is a subjective assessment of the advantage the player can obtain from the current situation (hence different aspects, such as economic, monetary, social, should also be taken into account). The payoff is a value assigned to each player for every possible termination of the game.

9 9 A payoff function converts the concepts of preference and utility into actual values, as defined by Von Neumann and Morgenstern [13]. According to Harsanyi classification [12], there are two classes of games: cooperative and non cooperative. In non cooperative games, binding agreements are not allowed, and it is preferable that players cannot communicate, since communications may influence their choices. Conversely, in cooperative games there is the possibility that some players associate and create a coalition, with the aim of improving their profits. Cooperative games can be further divided into transferable utility and non-transferable utility games. In the former case, players in the winner coalition can share the winnings, whereas in the latter they will receive a predefined payoff. The solution of a game corresponds to a suggestion to the players, possibly all of them, about the strategy to choose and, for cooperative games, how to divide the total payoff of a coalition among its members. In other terms, a solution suggests a choice that satisfies global fairness criteria, but that also respects the preferences of each player. 5 Time-Utility Functions Time-Utility Functions (TUFs) have been introduced by Jensen [4] to allow the semantics of soft time constraints to be precisely specified. A TUF, which is a generalization of the deadline constraint, determines the utility U to the system resulting from the completion of an activity (e.g. serving a queued packet) as a function of its waiting time in the system. Examples of TUFs are reported in Figure 3. Fig. 3 Examples of Time Utility Functions. In the figures, A and D represent, respectively, the maximum utility and the deadline. In all cases, if the activity is started after the deadline, the associated utility is null. In detail, and with reference to the above example of serving a packet, (a) is

10 10 suitable for very high priority traffic, whose packets have to be sent as soon as possible (and which do not care about other transmissions); (b) is typical of delay sensitive applications, such as VoIP: the packet utility rises and reaches the maximum very quickly with respect to the service delay, in order to encourage its service; (c) can be associated to streaming applications, which can tolerate delay better than VoIP; (d), instead, is typical for BE traffic, where no stringent requirements are guaranteed. 6 Game-theory and TUF -based Scheduler We modelled the packet scheduling problem for WiMAX systems as a cooperative game with non-transferable utility. The players are identified with the users (i.e. the Subscriber Stations) and the payoff is the throughput each user can obtain. The set of all possible combinations of strategies is determined by the constraint on the available physical resources (i.e. the slots, see Section 3) and represents the game feasible set F. The architecture of the proposed scheduler is shown in Figure 4. A classifier sorts the incoming traffic per user and per traffic class (TC). Each user has several TCs and each TC is associated to a buffer, which is served in a FIFO manner. In order to meet the application requirements, a TUF is assigned to each TC. The TUF is specified by the applications requesting network access, and, at any given time, only one TUF can be associated to a TC. An example of TC-TUF association is given at the end of Section 7.1. Fig. 4 Architecture and working principle of the Game Theory scheduler. The scheduling algorithm is divided in two steps: firstly, every user/player builds its own strategy (intra-user scheduling); then, the scheduler chooses a game solution among those in the feasible set (inter-user scheduling). During the intra-user scheduling phase, every user sorts his packets (belonging to all TCs) in decreasing order of utility. As explained in Section 5, the utility of each packet is determined, as a function of its waiting time and deadline, by the TUF associated to the TC the packet belongs. Note, however, that the service order established in the FIFO queues must be preserved: hence a packet cannot be put in the list before

11 11 packets preceding it in their queue (even if it has a higher utility). In other terms, we can imagine that every position in the list is filled taking into account only the packets at the head of the buffers (HOL packets). Among those, the packet with the highest utility is chosen.the resulting list defines the user preferred service order of the packets, which have to be served from the first to the last element. In terms of game theory, the list represents the user strategy. Once the lists/strategies have been set up, the scheduling algorithm has to choose which combination of them to serve (inter-user scheduling), under the constraint of the available amount of slots. In this decision, the different signal quality among the users must be considered as well, since the packets will consume a different number of slots depending on the employed modulation. Every strategy combination represents a coalition. The payoff R k (f) for player/user k for a coalition f is the throughput it obtains. The scheduling algorithm can choose the coalition to be served according to different game solutions, which provide more or less fairness among the K users. In a previous work [17], we considered four solutions: Utilitarian. The scheduling algorithm chooses the coalition f U having the highest overall payoff. In other words, the solution consists in the coalition for which the sum of the players throughputs is maximized: K f U = argmax F R k (f). (5) k=1 Clearly, this solution does not target any fairness, since serving packets belonging to users with bad channel quality is regarded as a resource wastage, because they consume more slots to obtain the same throughput as users enjoying a good channel. Nash. The scheduler chooses the coalition that has the highest Nash value, which is evaluated as: K f N = argmax F R k (f). (6) k=1 The Nash solution brings more fairness among the players than the utilitarian one. A coalition in which a player has a null payoff (i.e. no served packets) has a null Nash value. Therefore the coalitions taken into account are those which have some packet served for each player. Egalitarian: The scheduler follows the well known max-min throughput concept, trying to maximize the minimum payoff: } f E = argmax F {min k K R k(f). (7) Kalai-Smorodinsky: This solution extends the egalitarian one by weighting each payoff with respect to its maximum: R k,max (also known as the ideal payoff or utopia point, since it can be reached only when user k is the sole player of the game). Thus the scheduler searches a weighted max-min solution: f K = argmax F {min k K R k (f) R k,max }. (8)

12 12 We have shown in [17] that the best choice to optimally exploit the WiMAX radio resources and guarantee a certain amount of fairness among the users is the Nash solution. Hence, in the remaining of the paper, the game theory scheduler is assumed to implement the Nash solution. 6.1 Complexity and scalability We evaluated the worst case computational time complexity of the GTS N scheduler. For this task we assumed that the number of queued packets per TC is the same for all classes. In this context, we define N as the number of packets per user, M as the number of TCs, and K as the number of users. Hence each TC buffer holds N/M packets. The first phase of GTS N involves building the user strategies. Each user repeatedly picks the packet with the highest utility from the M HOL packets. Thus each choice implies M 1 comparisons, and this task is repeated by every user for the N packets in the buffer. Hence the complexity of this phase is O((M 1)NK) = O(MNK). The complexity of the second phase, which consists in finding the best coalition among the F possible ones, depends on the size of F. In the worst case, we must examine all the possible coalitions. Since each strategy on its own yields N possibilities (i.e. serve the first packet; serve the first and second packet; serve the first, second, and third packet; and so on), combining K strategies results in N K coalitions. In summary, the global complexity of GTS N is O(MNK + N K ). Clearly, since the complexity increases exponentially with the number of users, GTS N may present scalability problems (in terms of execution time, not performance). 6.2 Sub-Optimal Game-theory and TUF -based Scheduler The complexity analysis of the GTS N has pointed out that the exhaustive search of the Nash solution among all possible coalitions is not scalable, since the complexity of the algorithm increases exponentially with the number of users in the system. Therefore, finding a sub-optimal, but scalable and computationally feasible solution for the interuser phase is an appealing opportunity. To this aim, (6) can be rewritten exploiting the property of the natural logarithm to be a monotonically increasing function: f N = argmax F R k (f) = argmax F ln(r k (f)), (9) k K k K where, as usual, K is the number of users and R k (f) is the payoff (i.e. the throughput) for user k in coalition f. A further step consists in expanding the summation into a series of elementary operations. Let f k = {p k 1, pk 2,..., pk i k } indicate the set of served packets for user k in coalition f, and I k,j = {p k 1..., pk j }, j = 1... i k indicate all the possible subsets 4 of f k (clearly I k,ik = f k ). Let then define k,i as the utility increase for user k due to the service of packet p i : k,i = ln(r k (I k,i )) ln(r k (I k,i 1 )), (10) 4 We recall that packets must be served in FIFO order.

13 13 where I k,i 1 represents the set of served packets for user k up to the previous step. The combination of (9) and (10) yields: i k f N = argmax F ln(r k (f)) = argmax F k,i. (11) k K k K i=0 As proven in [5], this approach still leads to attain proportional fairness, which, when utility is the logarithm of throughput (as in our case), is equivalent to the Nash solution. Note that (11) is just a development of (6), hence finding the coalition f N that maximizes (11) still requires an exhaustive search over all coalitions. The sub-optimal approximation consists in separately finding the maximum of each k,i. In other terms, we evaluate the utility brought by serving a single packet, as given by (10), and select the packet that maximizes such an increase. We then repeat this task for all terms in the summation (or until there are no more available resources). We thus identify the coalition fn sub as a function of NK maximizations (being N the number of packets per user, as assumed in Section 6.1): f sub N { argmax k,i ( k,i ) i, k }. (12) Therefore, starting from the beginning of the allocation process, when no packet has been chosen yet, we can recursively execute the following procedure: Find a user-packet pair (k, i ) such that it has the highest marginal utility: k,i k,i k [1, K], i {i p k i is HOL}5. Allocate packet p k i into the download frame (if there is room, otherwise stop the algorithm), and go to the previous step. From the definition of k,i follows that the described method is effective only when all packets have the same size. If that is not the case, the user with the largest packet will always be favoured, as this provides a bigger utility increase than that of small packets. To tackle this shortcoming, we normalized the increase to the number of radio resources (i.e. slots) the packet requires for its transmission. Hence, the final form of the sub-optimal algorithm becomes: ( ) } {argmaxk,i f sub N k,i n pk i l i, k, (13) where n pk i l is the number of slots required for transmission of packet p k i. The complexity of this algorithm (which refers to the inter-user scheduling phase) is simply O(NK). Thus its computation time grows linearly with the number of users, whereas the optimal GTS N had an exponential dependence on K. When accounting for both intra- and inter-user phases, the overall complexity of the sub-optimal scheduler becomes O(MNK + NK) O(MNK). 7 Performance comparison This section presents the outcome of a performance comparison study between the proposed schedulers and other well-established schedulers that can be found in literature. 5 Note that, according to our scheduler, each user has already ordered its packets into a FIFO queue (see Section 6), hence the dependence on i could actually be removed.

14 Simulation Setup All schedulers have been implemented in the C++ programming language. Also the simulator was a custom implementation (still in C++) for this work. A sketch of the simulator architecture, with specific reference to the GTS N algorithm, is reported in Figure 5. Fig. 5 Architecture of the simulator with the GTS N scheduler. Our main focus was on the downlink subframe. Note however that the GTS N scheduler can be applied to the uplink as well, provided that the users are able to inform the BS of their strategies. Also note that, knowing the downlink performance, it is possible to evaluate the uplink performance via the downlink/uplink ratio DLR. In the simulation analysis, we modelled a system using the 5 MHz profile at a working frequency of 3.5 GHz, with frame duration T f = 5 ms and a downlink/uplink ratio DLR = 3 : 1. Furthermore, the mandatory DL-PUSC subchannel allocation algorithm has been employed, and one tenth of the symbols has been assigned to the preamble and signalling parts. The BS transmits with 50 dbm of EIRP (Equivalent Isotropic Radiated Power) and is placed at height h b = 30 meters. We considered a scenario with several user terminals (SSs) placed at ten meters height and at various distances from the BS. Specifically, the SSs are placed at 300, 600, 900, 1200 and 1500 meters from the BS. An example illustration of the simulated system is shown in Figure 6. To make the simulation environment more realistic and to investigate the ability of the schedulers to account for the users CNIRs, and consequently for the modulation schemes, we employed the Hata-Okumura path loss model [14] [22], which is the most widely used for the prediction of the received signal strength in macro-cellular environments. The Hata-Okumura model provides for various terrain types which determines the attenuation grade. They vary from a hilly area with a large concentration of trees to a flat terrain with sparse trees. In this work we employed the first type, a highly attenuating terrain, to enhance the differences among the various CNIRs. The actual value of path loss L is obtained from: L = 20 log 10 ( 4πd0 λ ) + 10 γ log 10 ( d d0 ) + χ f + χ h + s, (14)

15 15 Fig. 6 Architecture of the simulated system. where λ is the wavelength, γ = a b h b + c/h b is the path loss exponent, h b is the BS height, d is the distance from the BS, and d 0 is the reference distance, set to 100 m. The constants a, b and c depend on the considered terrain category. In our case the values are a = 4.6, b = and c = χ f and χ h are corrective factors applied, respectively, for frequencies higher than 2 GHz and for receiver heights (h R ) between 2 and 10 meters. They are evaluated according to (in this case f represents the frequency): ( ) f χ f = 6 log , (15a) ( ) hr χ h = 10.8 log 10. (15b) 2 Finally, the s term in (14) determines the shadowing effect; s is a lognormal variable, whose standard deviation is comprised between 8.2 and 10.6 db (we set it to 9 db). We modelled the radio channel characteristics so that the values of s between two consecutive frame transmissions are independent. Note that, though our simulation model assumes static users only, it is easy to extend it to mobile terminals. Mobility can be accounted for by modifying (14) so that d is a function of time. Yet, introducing this new variable would make the interpretation of the results more difficult, since the performance would jointly depend on space and time, and distinguishing the two contributions is not immediate. For this reason, the simulations presented in this work are based on fixed terminals, so that the performance of the various schedulers as a function of the propagation conditions is apparent. Simulations with mobile users are left for future investigations. As for the CNIR, it has been derived starting from RSSI k (Received Signal Strength Indicator of user k). RSSI k is evaluated according to: RSSI k = EIRP L k, (16)

16 16 where L k is the path loss of user k yielded by (14). We then assumed that the noise and interference power is equal to the receiver sensitivity. According to IEEE e [7], the receiver sensitivity threshold, RSSI thr, for the 5 MHz profile is equal to -90 dbm. Hence, the CNIR for user k can be evaluated as: CNIR k = RSSI k RSSI thr, (17) The IEEE e standard suggests the modulation and coding scheme to be associated to different normalised CNIRs in order to obtain a low enough bit error rate to have a very high probability of correct packet reception. We employed the suggested association, reported for convenience in Table 2, and consequently assumed a null packet error rate. Table 2 Modulation and Coding Scheme for different CNIR values. Constellation CNIR QPSK 1/ QPSK 1/ QPSK 2/ QPSK 3/ QAM 1/ QAM 2/ QAM 3/ QAM 1/ QAM 2/ QAM 3/ QAM 5/6 23 As for the traffic model, we assumed that each WiMAX user is a SS providing network access to several hosts, and therefore it aggregates several heterogeneous flows (as shown in Figure 6). Specifically, each user is able to generate and receive four types of packet traffic: voice, video, data and BE. The four TCs are characterized by the following packet sizes: Voice: 200 bytes; Video: 1500 bytes with probability 0.9 and 512 with probability 0.1; Data (e.g. FTP): 64 bytes with probability 0.33, 512 bytes with probability 0.33, 1500 bytes with probability 0.34; Best Effort (BE): uniform random value between 64 and 1500 bytes. The total offered load was set to 95% of the maximum rate that can be sustained by the WiMAX frame. Considering that not all users can enjoy the most efficient modulation, the network is overloaded. The packets of all traffic flows have been generated according to a Poisson process. All TCs offer the same load, and each of them is characterized by an intensity parameter λ i, expressed in terms of packets per second (pps): λ i = ρ RDL max M E[S i ]. (18) In the formula, ρ is the utilization factor (set to 0.95, see above), R DL max is the maximum downlink data rate (as computed by (4)), M is the number of traffic classes (M = 4 in our case), and E[S i ] is the mean packet size of traffic belonging to TC i. The traffic of

17 17 every TC is equally split among all users. Thus the intensity parameter λ i,j per user j and TC i is simply λ i,j = λ i /K, where K is the number of users. Finally, the Time-Utility Functions assigned to the different Traffic Classes are the ones already shown in Figure 3. Specifically, voice uses TUF (b), video uses TUF (c), data and BE use TUF (d). We then assumed D = 50 ms. This means that TUF (b) changes its trend at 7.5 and 15 ms (before the typical inter-arrival time of voice packets, which is 20 ms, as specified in Section 7.3.2). As for (c) and (d), the trend changes at 20 ms and 30 ms, respectively. It is worth mentioning that for all compared scheduling disciplines, we assumed that a packet having a delay higher than 50 ms is dropped. 7.2 Parameters of the compared schedulers The schedulers considered in the comparison are the following (divided according to the classification reported in Section 2): Radio unaware schedulers: DWRR and WF 2 Q+; Radio aware schedulers: CACS, M-LWDF and PFS. We recall that DWRR, WF 2 Q+ and CACS use some weights w i to adjust the throughput of the various traffic flows. Similarly to [1], the weights for the four considered TCs were set to 1/2 for voice, 1/4 for video, and 1/8 for data and BE. The quantum used by DWRR has been set to 753 bytes (equal to the average packet size). Two levels of hierarchy were used for WF 2 Q+, for both the pure version and its use in the compensated form (i.e. CACS): one for the sharing of bandwidth among users (the weights have the same value to ensure fairness among users), and the other for the TCs within each user (we used the weights indicated above). In all cases, the rate for the i th TC is equal to: r i = w i R DL M i=1 w, (19) i where w i is the weight associated to TC i and defined a priori as reported above, M is the number of TCs, and R DL is obtained from (4). In case of ideal fairness of the system, the rate per user and per TC is r i /K. As already reported in Section 2, M-LWDF and PFS are radio aware schedulers that aim at providing, respectively, QoS and fairness guarantees. In these two algorithms we exploited the policies defined in (2) for M-LWDF and in (1) for PFS. To update the average rate R i (t) and the average throughput T i (t) a sliding window is used: R i (t) = R i (t ) (1 1/W ) + R i(t) W, T i (t) = T i (t ) (1 1/W ) + T i(t) W, (20) where t and t stand for the present and past computation times, R i (t) and T i (t) are the instantaneous rate and throughput, and W is the sliding window size in terms of number of frames, which has been set to 10.

18 Performance parameters The algorithms are compared on the basis of four criteria: efficiency, fairness, perceived quality of service (QoS) and throughput. They are described in detail in the following subsections Efficiency and Fairness Efficiency (η) is defined as the ratio between the slots utilized by the scheduler and the slots available in the WiMAX frames. Obviously, schedulers that waste less resources (η 1) are preferable. To measure the fairness of the schedulers, we computed the User Fairness Index (UFI), defined as [18]: UFI = ( k R ) 2 k K k (R k) 2, (21) where R k represents the average throughput of the k th user. Efficiency and fairness are two well known contrasting objectives. While the former tries to optimally exploit the radio resources, favouring users with good channel quality, the latter strives to provide the same throughput to all users. The ideal (or utopian) scheduler would achieve both η = 1 and UFI=1, but in most cases schedulers are either highly efficient or highly fair, hence the difficulty of getting a unique comparison metric. To the best of our knowledge, no specific metric exists that compares schedulers on the basis of both efficiency and fairness in a joint manner. Therefore we defined a new parameter. Assuming to build a Cartesian plane where the axis are efficiency and fairness, we can place the ideal performance point P opt = (η opt, UFI opt ) = (1, 1) in it, and then measure the Euclidean distance of every scheduler from P opt. Hence, the new metric E D is: E D = (η opt η S ) 2 + (UFI opt UFI S ) 2, (22) where η S and UFI S are the efficiency and fairness of a generic scheduler S Perceived QoS of voice sessions The presence of a deadline constraint implies that packets are dropped if their transmission delay grows beyond a certain value. Clearly, the smaller is the deadline, the higher is the dropping probability. In our test all packets have the same deadline (50 ms), but the effect of dropped packets is different on the various types of traffic. The quality of VoIP flows is affected by both the expiration ratio and the transmission delay. The QoS perceived by the listener can be expressed through the Mean Opinion Score (MOS) [11]. MOS values are comprised into a [1,5] interval, with 5 indicating excellent quality and 1 bad service level. MOS 3.60 usually indicates a satisfactory service level. We evaluated the MOS of each scheduler by means of the E-Model [8]. A basic result of the E-Model is the scalar rating factor R, which is a measure of voice quality

19 19 ranging from 100 (best) to 0 (worst). The R-factor is related to MOS through the following expression: MOS = R R(R 60)(100 R). (23) The R-factor takes into account several parameters, such as packet loss, delay, quantizing distortion, impairments due to the equipment, background noise, echo. Some of them cover intangible quantities or are function of several other parameters. ITU-T G.107 recommends a set of default values for these parameters for planning purposes. A simplified expression for computing the R-factor is the following [16]: R = 94.2 I d I e, (24) where I d is the delay contribution and I e is the packet loss contribution. I d and I e can be evaluated according to: I d = 0.024d (d 177.3) H(d 177.3), I e = γ 1 + γ 2 ln(1 + γ 3 e), (25a) (25b) where d is the one way delay (in ms), H( ) is the Heavyside (or step) function: H(x) = 0 if x < 0 and H(x) = 1 otherwise, e is the total loss probability, and the factors γ i depend on the codec, since every codec is affected by packet losses in a different way. As for the evaluation of the one way delay d and the total loss probability e, they can be computed from: d = d codec + d dejitter + d network, e = e network + (1 e network )e dejitter, (26a) (26b) where d codec is the algorithmic and packetization delay associated with the codec and the IP packet processing, d dejitter is the delay associated with the dejittering buffer required to smooth out the delay variation in the arriving packet stream, d network is the one way transit delay across the transport network, e network is the loss probability in the transport network, and e dejitter is the loss probability due to underflow or overflow in the dejittering buffer of the decoder. In this work we assume that VoIP devices employ the G.711 codec [10], with packetization at IP layer of 20 ms of speech. This produces 200 byte packets with a bit-rate of 80 kbps per flow. For G.711 and random packet losses, the γ i s take the following values: γ 1 = 0, γ 2 = 30, γ 3 = 15 [9]. We assume that the dejitter buffer size is set to d dejitter = 50 ms. Assuming the rest of the network does not introduce jitter 6, the value assigned to the dejitter buffer size is equal to the maximum allowed jitter. This can be easily deduced considering the packet deadline constraint imposed by the TUF and the relation suggested in [6] to calculate the jitter. Therefore, there are no losses in the dejittering buffer and e dejitter = 0. The codec and IP processor delay is d codec = 20 ms. Moreover we assume that for every VoIP session one end is placed in a host and the other somewhere in the worldwide global network behind the BS (the cloud identified by IP network in Figure 6). Then, we set the one way transit delay across the global IP transport network to d network = 150 ms [20]. The delay 6 This assumption permits to directly associate the performance results to the analysed scheduler, without any bias due to the backhaul network.