Game Theory and Time Utility Functions for a

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 802.16 · 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 GTSN ) whose goal is to contemporaneously achieve efficiency and fairness, while also taking into account the QoS requirements and the channel state. GTSN exploits the properties of Time Utility Functions (TUFs) and the Game Theory. Simulations prove that the performance of GTSN , when compared to that of several well-known schedulers, is remarkable. GTSN 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-GTSN . 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 802.16 family of standards. IEEE 802.16 gives the Department of Information Engineering University of Pisa Via G. Caruso 16, Pisa I-56122, Italy E-mail: {name.surname}@iet.unipi.it

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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 802.16e [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 802.16 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 users1 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 (GTSN ). As the name suggests, GTSN 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).

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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 GTSN , 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 GTSN scheduler and, after analysing its complexity, introduces its suboptimal version, denoted as sub-GTSN . 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.

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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 flows2 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, wi , 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 WF2 Q (Worst-Case Weighted Fair Queuing) and WF2 Q+. WF2 Q+ works like WF2 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 WF2 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.

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Fig. 1 Example of link sharing hierarchy (left) and corresponding WF2 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 WF2 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 Ri (t) and the average throughput Ti (t) experienced by user i, and then selects the flow of user i∗ to be served next according to: ∗

i = argmaxi∈I



Ri (t) Ti (t)

 .

Note that this formulation does not take into account the QoS requirements.

(1)

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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, Ui (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 Ri (t) to the average rate of the user Ri (t). Then, the user to be served is selected according to: ∗

i = argmaxi∈I

 |Ui (t)|

Ri (t) Ri (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 GTSN 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 10.94 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

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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, nlc . Figure 2 shows an OFDMA frame when operating in TDD mode3 . 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 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 (nsc ) No. pilot subcarriers per symbol No. null/guard subcarriers per symbol Subcarrier spacing Symbol duration Useful symbol time Guard time No. symbols in a frame (Ns ) No. of data subcarriers in a slot (nlc ) Frame duration (Tf )

1.25 128 72 12 44

5 10 512 1024 360 720 60 120 92 184 10.94 kHz 102.9 µs 91.4 µs 11.4 µs 48 48 5 ms

20 2048 1440 240 368

Still assuming to use DL-PUSC, a slot is composed of nlc = 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 NlDL in a downlink subframe is given by: NlDL =

nsc Ns · DLR , nlc DLR + 1

(3)

where nsc represents the number of data subcarriers in a symbol (360 in our example), and Ns 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 RDL of the downlink subframe can be evaluated as: RDL =

NlDL · nlc · b . Tf

(4)

DL The downlink subframe can reach the maximum data rate Rmax when all served users employ the most spectral-efficient scheme (i.e. 64-QAM with coding rate r = 5/6, DL for an overall b = 5). In such a case, Rmax = 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 DL b = 0.67), the downlink subframe can serve the minimum rate Rmin = 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.

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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

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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

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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 Rk (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 fU having the highest overall payoff. In other words, the solution consists in the coalition for which the sum of the players’ throughputs is maximized: fU = argmaxF

K X

Rk (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: fN = argmaxF

K Y

Rk (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:

 fE = argmaxF

 min Rk (f )

k∈K

.

(7)

– Kalai-Smorodinsky: This solution extends the egalitarian one by weighting each payoff with respect to its maximum: Rk,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:

 fK = argmaxF

R (f ) min k k∈K Rk,max

 .

(8)

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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 GTSN 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 GTSN 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)N K) = O(M N K). 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 GTSN is O(M N K + N K ). Clearly, since the complexity increases exponentially with the number of users, GTSN 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 GTSN 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: fN = argmaxF

Y

Rk (f ) = argmaxF

k∈K

X

ln(Rk (f )),

(9)

k∈K

where, as usual, K is the number of users and Rk (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 fk = {pk1 , pk2 , . . . , pkik } indicate the set of served packets for user k in coalition f , and Ik,j = {pk1 . . . , pkj }, j = 1 . . . ik indicate all the possible subsets4 of fk (clearly Ik,ik = fk ). Let then define ∆k,i as the utility increase for user k due to the service of packet pi : ∆k,i = ln(Rk (Ik,i )) − ln(Rk (Ik,i−1 )), 4

We recall that packets must be served in FIFO order.

(10)

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where Ik,i−1 represents the set of served packets for user k up to the previous step. The combination of (9) and (10) yields: fN = argmaxF

X

ln(Rk (f )) = argmaxF

ik XX

∆k,i .

(11)

k∈K i=0

k∈K

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 fN 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). sub We thus identify the coalition fN as a function of N K maximizations (being N the number of packets per user, as assumed in Section 6.1): sub fN ← argmaxk,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|pki is HOL}5 . ∗ – Allocate packet pki∗ 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:

( sub fN

←

argmaxk,i

∆k,i pk

!

) ∀i, k

,

(13)

nl i pk

where nl i is the number of slots required for transmission of packet pki . The complexity of this algorithm (which refers to the inter-user scheduling phase) is simply O(N K). Thus its computation time grows linearly with the number of users, whereas the optimal GTSN 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(M N K + N K) ≈ O(M N K).

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

7.1 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 GTSN algorithm, is reported in Figure 5.

Fig. 5 Architecture of the simulator with the GTSN scheduler.

Our main focus was on the downlink subframe. Note however that the GTSN 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 Tf = 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 hb = 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 · log10



4πd0 λ



+ 10 · γ · log10



d d0



+ χf + χh + s,

(14)

15

Fig. 6 Architecture of the simulated system.

where λ is the wavelength, γ = a − b · hb + c/hb is the path loss exponent, hb is the BS height, d is the distance from the BS, and d0 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 = 0.0075 and c = 12.6. χf and χh are corrective factors applied, respectively, for frequencies higher than 2 GHz and for receiver heights (hR ) between 2 and 10 meters. They are evaluated according to (in this case f represents the frequency): f , 2 · 103   hR . χh = −10.8 · log10 2

χf = 6 · log10





(15a) (15b)

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 RSSIk (Received Signal Strength Indicator of user k). RSSIk is evaluated according to: RSSIk = EIRP − Lk ,

(16)

16

where Lk 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 802.16e [7], the receiver sensitivity threshold, RSSIthr , for the 5 MHz profile is equal to -90 dBm. Hence, the CNIR for user k can be evaluated as: CNIRk = RSSIk − RSSIthr ,

(17)

The IEEE 802.16e 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 QPSK 1/3 QPSK 1/2 QPSK 2/3 QPSK 3/4 16-QAM 1/2 16-QAM 2/3 16-QAM 3/4 64-QAM 1/2 64-QAM 2/3 64-QAM 3/4 64-QAM 5/6

CNIR 0.5 ÷ 6 6 ÷ 7.5 7.5 ÷ 9 9 ÷ 12 12 ÷ 14.5 14.5 ÷ 15 15 ÷ 18 18 ÷ 20 20 ÷ 21 21 ÷ 23 ≥ 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 λi , expressed in terms of packets per second (pps): λi =

DL ρ · Rmax . M · E[Si ]

(18)

DL In the formula, ρ is the utilization factor (set to 0.95, see above), Rmax is the maximum downlink data rate (as computed by (4)), M is the number of traffic classes (M = 4 in our case), and E[Si ] is the mean packet size of traffic belonging to TC i. The traffic of

17

every TC is equally split among all users. Thus the intensity parameter λi,j per user j and TC i is simply λi,j = λi /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 WF2 Q+; – Radio aware schedulers: CACS, M-LWDF and PFS. We recall that DWRR, WF2 Q+ and CACS use some weights wi 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 WF2 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 ith TC is equal to: RDL , ri = wi · P M i=1 wi

(19)

where wi is the weight associated to TC i and defined a priori as reported above, M is the number of TCs, and RDL is obtained from (4). In case of ideal fairness of the system, the rate per user and per TC is ri /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 Ri (t) and the average throughput Ti (t) a sliding window is used: Ri (t) , W T (t) Ti (t) = Ti (t0 ) · (1 − 1/W ) + i , W

Ri (t) = Ri (t0 ) · (1 − 1/W ) +

(20)

where t and t0 stand for the present and past computation times, Ri (t) and Ti (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

7.3 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. 7.3.1 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 =


2 P Rk , Pk 2

K·

k

(Rk )

(21)

where Rk represents the average throughput of the kth 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 Popt = (ηopt , UFIopt ) = (1, 1) in it, and then measure the Euclidean distance of every scheduler from Popt . Hence, the new metric ED is:

q ED =

(ηopt − ηS )2 + (UFIopt − UFIS )2 ,

(22)

where ηS and UFIS are the efficiency and fairness of a generic scheduler S. 7.3.2 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

ranging from 100 (best) to 0 (worst). The R-factor is related to MOS through the following expression: MOS = 1 + 0.035R + 7 · 10−6 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 − Id − Ie ,

(24)

where Id is the delay contribution and Ie is the packet loss contribution. Id and Ie can be evaluated according to: Id = 0.024d + 0.11(d − 177.3) · H(d − 177.3),

(25a)

Ie = γ1 + γ2 · ln(1 + γ3 · e),

(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 γi 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 = dcodec + ddejitter + dnetwork ,

(26a)

e = enetwork + (1 − enetwork )edejitter ,

(26b)

where dcodec is the algorithmic and packetization delay associated with the codec and the IP packet processing, ddejitter is the delay associated with the dejittering buffer required to smooth out the delay variation in the arriving packet stream, dnetwork is the one way transit delay across the transport network, enetwork is the loss probability in the transport network, and edejitter 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 γi ’s take the following values: γ1 = 0, γ2 = 30, γ3 = 15 [9]. We assume that the dejitter buffer size is set to ddejitter = 50 ms. Assuming the rest of the network does not introduce jitter6 , 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 edejitter = 0. The codec and IP processor delay is dcodec = 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 dnetwork = 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.

20

experimented in the BS buffer and measured during the simulations is added to this value for the computation of the R-factor. MOS and the E-model are applicable to voice only. For the other TCs, and specifically for video, we assessed the perceived QoS directly in terms of percentage of expired packets. 7.3.3 Throughput Throughput quantifies the amount of information transported in a given period. Specifically, we measured the bits of MAC payload per second. Note that GTSN has no explicit mechanism to guarantee a minimum bandwidth to the various applications. Yet the ordering of packets according to their utility can indeed provide implicit chances to transmit even for low priority traffic such as data and BE, as it will be shown in Section 7.4.

7.4 Simulation Results The simulation study has been carried out considering three scenarios characterized by a different number of users: 5, 20 and 50. The study compares the performance of the DWRR, WF2 Q+, PFS, M-LWDF, CACS, and the proposed GTSN and sub-GTSN . All simulation results have been obtained from several runs, each one generating 10000 frames (or equivalently a time period of 50 s). 7.4.1 Scenario with five SSs The results concerning efficiency and fairness of the compared schedulers in this scenario are reported in Table 3. GTSN provided the best efficiency (0.98), achieving nearly the ideal value. It also showed an acceptable value of UFI (0.91), even though all the others provided a higher one. These results are due to the good ability of GTSN to exploit the good channel quality of the users closer to the BS, while trying to maintain an acceptable throughput for the distant users. As for sub-GTSN , the drastic complexity reduction comes at the expenses of a more than acceptable worsening of the performance. Indeed, Table 3 highlights that sub-GTSN has an average performance loss of about 2%.

Table 3 Efficiency and fairness. Efficiency (η) UFI ED

DWRR 0.78 0.92 0.236

WF2 Q+ 0.89 1 0.112

PFS 0.87 0.96 0.136

M-LWDF 0.84 0.99 0.160

CACS 0.87 0.99 0.130

GTSN 0.98 0.91 0.092

sub-GTSN 0.96 0.89 0.117

The results also confirm what we anticipated in Section 7.3.1, i.e. that efficient schedulers are the least fair, whereas fair schedulers are the least efficient. For example, WF2 Q+, M-LWDF and CACS get the highest UFI, but also a very low efficiency (DWRR only is worst). Conversely, GTSN is the most efficient, but pays in terms of fairness.

21

The last row of Table 3 reports the values of our combined efficiency-fairness metric, ED . It is apparent that the simplest scheduler, DWRR, is the farthest from the ideal point, whereas GTSN is the closest. This means that GTSN offers the best compromise between fairness and efficiency. The others are somewhere in between, with WF2 Q+ quite close to GTSN . Table 4 shows the user fairness indexes specific to the various TCs. Data confirm the previous statements, i.e. the fair behaviour of CACS, WF2 Q+, and PFS. Differently from above, but not surprisingly, M-LWDF does not behave fairly with the applications, heavily favouring high priority traffic (in fact, this is its goal). As for GTSN , it is somewhere in between, while sub-GTSN shows slightly worse UFI values. This proves that GTSN strives to strike a balance among the different performance objectives. It yields the best fairness to the high priority traffic (voice above all, and video), but also an acceptable value to data and BE. A similar but qualitatively worst trend is achieved by M-LWDF.

Table 4 User Fairness Index (UFI) per Traffic Class. Voice Video Data BE

DWRR 0.983 0.855 0.806 0.795

WF2 Q+ 0.999 0.994 0.958 0.935

PFS 0.993 0.986 0.991 0.991

M-LWDF 0.999 0.978 0.582 0.551

CACS 0.999 0.980 0.987 0.991

GTSN 0.999 0.921 0.703 0.650

sub-GTSN 0.999 0.883 0.674 0.621

The second performance parameter we evaluated is MOS, computed for each user according to (23), and reported in Table 5 as a function of the distance from the BS (user 1 is the closest, user 5 is the farthest).

Table 5 MOS for the different users. User User User User User

1 2 3 4 5

DWRR 4.17 4.17 4.17 4.17 1.53

WF2 Q+ 4.15 4.15 4.15 4.15 4.15

PFS 1.94 1.92 1.58 1.29 1.07

M-LWDF 4.13 4.13 4.13 4.13 4.12

CACS 1.53 1.51 1.52 1.50 1.46

GTSN 4.15 4.15 4.14 4.13 4.04

sub-GTSN 4.16 4.15 4.14 4.13 3.92

We can observe that the schedulers with the highest UFI are the ones that yield a very similar MOS across the users (WF2 Q+, M-LWDF, CACS). Conversely, DWRR and PFS heavily penalize the farthest user. Finally, PFS, GTSN , and sub-GTSN provide a gradual decrease in the MOS. However, the absolute values achieved by PFS are well below the acceptable service quality, whereas GTSN offers a very good service level to all users, with the maximum difference smaller than 3% (6% for sub-GTSN ). In terms of absolute MOS values, WF2 Q+, M-LWDF and GTSN provide a good QoS to the voice sessions, whereas the quality offered by PFS and CACS is absolutely insufficient. Note, however, that the good performance of WF2 Q+ and M-LWDF is achieved at the expenses of efficiency, as we have seen in Table 3. The performance of sub-GTSN is quite similar to that achieved by GTSN , except for the MOS of user 5, for which we can observe a appreciably worse value; however, the measured MOS still remains in the range of an acceptable quality of service.

22

As for the video traffic, the expiration ratio is shown in Table 6, which reports the average values and the 98% confidence interval (over 10 runs) as well. At least for the first three users, the result provided by GTSN is very good, and is overcome by DWRR only. However, similarly to voice, DWRR puts all the impairment on the farthest user, for which more than three packets out of four expires. Again, the fairest schedulers, WF2 Q+ and M-LWDF, are the ones that protect best the far users, but to achieve this goal WF2 Q+ penalizes the first users (it has one of the worst results), while M-LWDF pays in terms of efficiency (as shown in Table 3). Finally, the average expiration ratio shows that M-LWDF has the best result, followed by GTSN , sub-GTSN , and DWRR. The other schedulers are all quite far.

Table 6 Expiration Ratio of Video traffic. User 1 User 2 User 3 User 4 User 5 Average

DWRR 0±0 0±0 0±0 0.194 ± 0.02 0.757 ± 0.04 0.198

WF2 Q+ 0.251 ± 0.03 0.237 ± 0.05 0.293 ± 0.05 0.246 ± 0.07 0.162 ± 0.03 0.238

PFS 0.345 ± 0.03 0.367 ± 0.02 0.432 ± 0.02 0.53 ± 0.025 0.534 ± 0.02 0.441

M-LWDF 0.025 ± 0.008 0.02 ± 0.006 0.037 ± 0.011 0.108 ± 0.026 0.116 ± 0.022 0.06

CACS 0.418 ± 0.03 0.432 ± 0.028 0.453 ± 0.033 0.44 ± 0.039 0.452 ± 0.033 0.439

GTSN 0±0 0 ± 0.003 0.03 ± 0.016 0.297 ± 0.038 0.418 ± 0.055 0.149

sub-GTSN 0±0 0.0008 ± 0.0025 0.044 ± 0.011 0.382 ± 0.045 0.556 ± 0.051 0.197

Throughput achieved by the various schedulers for each specific application is reported, together with the aggregated overall throughput, in Table 7. The first thing to note is that GTSN provides the highest overall throughput, thus proving its ability to exploit the channel features to dispatch a high number of packets. The main beneficiaries of such a throughput gain are the data and BE traffic classes. Yet, this is not at the expenses of voice and video. A confirmation is in the comparison with M-LWDF. We have seen that M-LWDF is a QoS-oriented scheduler, which privileges voice and video over data traffic. Hence it can be considered as a benchmark in terms of the maximum attainable throughput by high priority traffic. Since the results achieved by GTSN for voice and video are very close to those of M-LWDF, we can safely state that GTSN , which is not a strict priority scheduler, is able to provide the highest throughput to voice (and the second highest to video), and at the same time allocate data and BE more bandwidth than just the remaining. This is due to the utilization of the TUFs and to the ordering of the packets per decreasing utility values, which may sometimes result in low priority traffic to pre-empt high priority traffic so that its packets do not expire. It is worth noting that the sub-optimal scheduler provides almost the same performance of GTSN , with an aggregated throughput loss smaller than 1%.

Table 7 Throughput (in kbps) for the various Traffic Classes. Voice Video Data BE Total

DWRR 2553 ± 27 2179 ± 46 557 ± 26 505 ± 20 5794 ± 74

WF2 Q+ 2713 ± 12 1916 ± 85 598 ± 69 446 ± 10 5673 ± 76

PFS 1786 ± 28 1441 ± 18 1600 ± 26 1667 ± 34 6494 ± 82

M-LWDF 2725 ± 29 2397 ± 64 296 ± 52 258 ± 36 5677 ± 63

CACS 1829 ± 61 1446 ± 122 1149 ± 85 1667 ± 85 5894 ± 667

GTSN 2733 ± 56 2200 ± 26 1381 ± 50 1252 ± 45 7566 ± 39

sub-GTSN 2729 ± 33 2118 ± 65 1370 ± 40 1276 ± 64 7493 ± 50

As for the other schedulers, a good throughput for voice and video is also guaranteed by DWRR and WF2 Q+, but at great expenses of the other two applications. Finally, PFS and CACS did not guarantee any specific TC, but simply, as in their principle of

23

operation, guaranteed quite the same throughput for all applications. Unfortunately, as shown in Tables 5 and 6, this strategy turned out to be a looser, since it resulted in a very poor perceived quality by all users. Then, examining the figures for the schedulers employing the TC-weights wi (i.e. DWRR, WF2 Q+, and CACS), it is easy to check that the throughput achieved by the various TCs does not match the expected ratios (i.e. 1/2 for voice, 1/4 for video and 1/8 for data and BE, see Section7.2). Specifically, DWRR and WF2 Q+ give video much more bandwidth than its quota (even though WF2 Q+ almost succeeds in respecting the voice weight), whereas CACS privileges data and BE at the expenses of voice. Indeed, the non-ideal operation conditions produced a severe shift from the theoretical performance of these schedulers. As a final remark, the results obtained in this simple scenario show that the suboptimal scheduler (sub-GTSN ) provides a quite similar performance to that of GTSN , with the advantage of a minor complexity. The analysis of the results shows just one appreciable difference: sub-GTSN turns out to be more opportunistic than its optimum counterpart, as reflected by the lower MOS of the farthest user and the higher MOS of the closest user (see Table 5). The gap, however, is very small. Hence, in the remaining of the simulation analysis only sub-GTSN will be considered. This choice is obviously due to the more acceptable time complexity of sub-GTSN . 7.4.2 Scenarios with 20 and 50 SSs As a consequence of the lower complexity of sub-GTSN , it was feasible to run simulations with a more considerable number of users. Specifically, we tested the scheduler and compared it with the other reference solutions in two scenarios with 20 and 50 users. The users were equally distributed at the same distances as before (i.e. 300, 600, 900, 1200, and 1500 meters from the BS). The first result, shown in Tables 8 and 9, refers to efficiency and fairness. For both 20 and 50 users, sub-GTSN provides the best efficiency, whereas it suffers in terms of UFI. However, the gain in the former compensates the poor result in the latter, as proven by the result for ED , which is always the best. Table 8 Efficiency and Fairness for 20 users. Efficiency UFI ED

DWRR 0.82 0.9 0.206

WF2 Q+ 0.892 1 0.108

PFS 0.87 0.998 0.13

M-LWDF 0.836 0.98 0.165

CACS 0.87 0.99 0.13

sub-GTSN 0.97 0.9 0.104

M-LWDF 0.83 0.986 0.17

CACS 0.87 0.99 0.13

sub-GTSN 0.94 0.91 0.108

Table 9 Efficiency and Fairness for 50 users. Efficiency UFI ED

DWRR 0.82 0.88 0.216

WF2 Q+ 0.88 0.97 0.124

PFS 0.88 1 0.12

Comparing the two scenarios with the five users one (see Table 3 in Section 7.4), we can note that some schedulers offer a slightly worst performance when the number of

24

users increase (M-LWDF and WF2 Q+), whereas others (i.e. DWRR and PFS) have a small improvement. CACS yields more or less the same figures. Therefore, all schedulers seem to scale pretty well, with the best results achieved, on average, for the 20 users scenario. As for sub-GTSN , it improves its performance when passing from 5 (refer to Table 3) to 20 users, and it has a slight decrease when dealing with 50 users. Hence its behaviour is more than satisfactory. As for MOS, the results are reported in Tables 10 and 11. The results have been grouped as a function of the distance from the BS and the figures represent the average values measured for the users of each group. In both scenarios the best performance is achieved by M-LWDF and sub-GTSN . More in detail, sub-GTSN provides a smooth decrease for the MOS of the first four groups of users, whereas the last one is offered only a barely sufficient service (which becomes less than acceptable for the 50 user scenario).

Table 10 MOS for 20 users. Users @ 300 m Users @ 600 m Users @ 900 m Users @ 1200 m Users @ 1500 m

DWRR 3.10 2.91 2.72 2.40 1.93

WF2 Q+ 3.79 3.79 3.84 3.87 4.03

PFS 1.61 1.61 1.58 1.57 1.51

M-LWDF 4.14 4.14 4.14 4.14 4.13

CACS 1.50 1.49 1.50 1.48 1.48

sub-GTSN 4.17 4.17 4.15 4.15 3.39

WF2 Q+ 1.80 1.82 1.82 1.85 2.98

PFS 1.83 1.83 1.81 1.80 1.77

M-LWDF 4.14 4.14 4.14 4.14 4.13

CACS 1.48 1.48 1.48 1.48 1.48

sub-GTSN 4.17 4.17 4.16 4.15 2.60

Table 11 MOS for 50 users. Users @ 300 m Users @ 600 m Users @ 900 m Users @ 1200 m Users @ 1500 m

DWRR 2.52 2.46 2.37 2.27 2.07

A more useful insight can be obtained comparing these tables with the one referred to the 5 users scenario (Table 5). DWRR and WF2 Q+ now present much worse results, thus they are not able to properly manage the increase in the number of users. PFS and CACS achieve more or less the same performance, which, however, is well below the acceptable level. M-LWDF maintains its very good MOS values across all scenarios. Finally, sub-GTSN maintains its good performance in the scenario with 20 users; on the contrary, in the other scenario we can observe that the MOS measured for the farthest users decay to an unacceptable value. In this case, in order to maintain a certain level of efficiency, sub-GTSN loses the possibility to guarantee a satisfactory voice service to the farthest users. Looking at Tables 12 and 13, reporting the measured throughput, we can observe the high price paid by the M-LWDF in terms of throughput for guaranteeing good quality to voice services also to the farthest SSs in the 50 users scenario. The proposed sub-GTSN records an overall throughput that is about 41% higher than the one measured with M-LWDF. In summary, sub-GTSN offers the highest overall throughput, whereas M-LWDF, the sole scheduler that provided a good MOS for all users and all

25

scenarios, pays this choice by heavily penalizing the low priority TCs, as data and BE experienced a throughput that is less than one tenth of the offered load (2.8 Mbps).

Table 12 Throughput (in kbps) for the various Traffic Classes – 20 users. Voice Video Data BE Total

DWRR 2348 ± 47 2049 ± 68 608 ± 18 581 ± 23 5585 ± 106

WF2 Q+ 2691 ± 24 986 ± 32 853 ± 58 1023 ± 23 5553 ± 61

PFS 1877 ± 28 1118 ± 29 1462 ± 28 1493 ± 12 5950 ± 49

M-LWDF 2742 ± 23 2436 ± 78 229 ± 34 249 ± 33 5656 ± 69

CACS 1810 ± 74 1433 ± 78 1159 ± 56 1456 ± 90 5859 ± 44

sub-GTSN 2711 ± 36 1662 ± 60 1679 ± 36 1912 ± 37 7964 ± 46

Table 13 Throughput (in kbps) for the various Traffic Classes – 50 users. Voice Video Data BE Total

DWRR 2269 ± 51 2011 ± 49 611 ± 16 593 ± 24 5484 ± 95

WF2 Q+ 2118 ± 40 930 ± 46 1079 ± 29 1304 ± 47 5430 ± 65

PFS 2058 ± 22 1018 ± 18 1379 ± 20 1445 ± 30 5900 ± 47

M-LWDF 2742 ± 32 2456 ± 33 211 ± 31 237 ± 28 5645 ± 46

CACS 1799 ± 79 1408 ± 71 1182 ± 94 1468 ± 144 5857 ± 77

sub-GTSN 2668 ± 19 1555 ± 54 1723 ± 57 2040 ± 27 7987 ± 60

Further noteworthy aspects can be highlighted. The first is the difference, in terms of overall throughput, between sub-GTSN and all other schedulers. For both scenarios, while sub-GTSN reaches almost 8 Mbps, the closest competitor stops at less than 6 Mbps. Hence sub-GTSN offers roughly one third more throughput than all other schedulers (doing even better than GTSN ). The second point is on the comparison with the 5 users scenario (see Table 7 in Section 7.4). While most schedulers decrease their sustained rate as the number of users increases, sub-GTSN is able to keep this value almost constant. Finally, referring to the numbers of the specific TCs, we can observe that subGTSN favours voice over all other services. Video in particular is penalized by our suboptimal scheduler. This occurs because of two factors. The first is that voice packets are generally favoured by the ordering of packets according to their utility. The second is that there are now 20 (or 50) groups of voice sessions, i.e. many more voice packets than in the 5 users scenario. Therefore there is a high probability that there are high utility voice packets in the buffers that will be chosen among all HOL packets to be transmitted. Nonetheless, the 1662 and 1555 kbps achieved by sub-GTSN for video, in the 20 and 50 users scenarios respectively, are still better than WF2 Q+, PFS and CACS, with only DWRR and M-LWDF offering more bandwidth.

8 Conclusion The paper presented GTSN , a scheduling algorithm for Mobile WiMAX systems based on the use of Time-Utility Functions (TUFs) and on the concepts of Game Theory. TUFs have been employed to modulate the priority of packets belonging to different traffic classes, in order to minimize the packet expiration probability. Game Theory

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(with specific choice of the Nash equilibrium) has been used to select the best combination of packets to put in the downlink WiMAX frame. The goal is to contemporaneously achieve efficiency and fairness, under the constraint that the various users (i.e. the SSs) experience different channel conditions. The performance of GTSN was compared to several other schedulers via a series of simulations. The outcome of the test is that GTSN provides the best compromise between the two contrasting objectives (i.e. fairness and efficiency) and achieves the highest throughput among all assessed schedulers. It is also able to offer a good level of service to voice and video sessions (overcome by M-LWDF only), while protecting data and BE traffic from starvation. On the downside, the complexity introduced by the game theory techniques makes it rather impractical and not computationally scalable for a large number of users. We have therefore developed sub-GTSN , a sub-optimal version of GTSN that, instead of solving the game in its complete form, iteratively searches for the packets that offer the highest incremental payoff (defined as the natural logarithm of the throughput). Sub-GTSN retains most features of its brother, but its complexity is linear with the number of users. We compared sub-GTSN with its full-fledged counterpart and showed that it has a very bounded loss in terms of efficiency and fairness. We also compared sub-GTSN with the same reference schedulers as before, in scenarios with 20 and 50 SSs. In both cases sub-GTSN presents very satisfactory figures (among them, a notably higher throughput), even though it has a bit more difficulties than GTSN in dealing with the farthest users. As a final remark, thanks to the GTSN scheduler, we were also able to show that TUFs are beneficial both to high priority traffic (e.g. voice and video), which experimented a low delay and expiration rate, and to low priority traffic (such as data), which was preserved a fair amount of bandwidth even in a saturated network.

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