90-02, 91A30, 91B16, 93-02, 94-02, 94A40, 94A99, 90B35,

1. Introduction. Fairness is an important property of a resource allocation algorithm: when network resources are insufficient to satisfy demand, they should be divided fairly among network users [1, 2, 3]. The network users and designers are frequently concerned with fairness in the usage of network resources. Because of the subjective nature and the multiplicity of the issues involved, the term fairness cannot be uniquely defined. Devising fair allocation strategies is also an active area of research in economic systems (see e.g., [4, 5]). In its simplest form, the fairness of a resource allocation mechanism means equitability of users in allocating resources according to a prearranged agreement. This survey aims at compiling and discussing the relevant definitions and criteria available in the literature on fairness in wireless cellular packet networks. The fairness objective for a single wired node is immediate and is defined by maxmin fairness [1] and realized by fair queueing [6]. For wireless networks a primary challenge in fair resource allocation is compounded by the fact that when a certain amount of “resource” is allocated to a flow, there is no guarantee that the allocated resources will be used efficiently. This is due to unpredictable factors in wireless medium such as fading, noise, interference and hand-offs, which can dramatically increase bit error rates, leading to increased packet loss rates and degradation in service quality. Therefore, strict assurance of fairness can be relatively inefficient in a wireless environment [7]. With a unique bottleneck (the radio interface) and with equal weights for all users, fairness means equal throughput for all users. But, if a mobile terminal is far from the base station, a higher proportion of the air interface resource (e.g., total transmit power) needs to be allocated to that terminal so that it could have the same throughput as the others. On the other hand, completely ignoring fairness will enables ∗ This work was supported in part by Nortel Network, National Capital Institute of Telecommunications (NCIT) and Mathematics of Information Technology and Complex Systems (MITACS). † K. Navaie is with the Broadband Communications and Wireless Systems Center, Department of System and Computer Engineering, Carleton University, Ottawa, Ontario, Canada, K1S 5B6 (email: [email protected]). ‡ Delfin Y. Montuno is with the School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada, K1S 5B6 . He is also with Nortel Networks, Nepean, Ontario, Canada, K2H 8E9 (email: [email protected]). § Y. Q. Zhao is with the School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada, K1S 5B6 (email: [email protected]).

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us to maximize the total instantaneous throughput by serving users with a good radio efficiency factor. This improves the utilization of radio resources [8]. Fairness can be also approached from the viewpoint of either allocating resources or utilizing resources. This leads to two types of fairness: effort fairness and outcome fairness. In a wireline environment, effort fairness equals outcome fairness, but they can be substantially different in a wireless environment. Outcome fairness is defined based on the amount of received data. It guarantees that flows of equal weights receive the same amount of data. This concept, however, is inadequate in wireless networks to maximize network throughput. In wireless networks, an erroneous flow, with severe channel errors, can exhaust the wireless resources, and as a result, other flows may receive limited resources even if their channel conditions are good. On the other hand, effort fairness guarantees equitable utilization of wireless resources in each flow. Therefore, effort fairness is more attainable in wireless networks, however it may not guarantee individual throughput requirements nor maximize resource utilization. Because of the time varying nature of wireless channels, fairness is usually defined and implemented over a period of time. Based on the length of that period, two kinds of fairness are considered: short-term fairness and long-term fairness. Intuitively, short term fairness means the ability to provide equitable allocation of resources to all active connections over short time-scales; and long term fairness, in contrast, is concerned with the amount of resources assigned over a longer time-scale. Obviously, short-term fairness implies long-term fairness. Short-term fairness has significant impacts on the performance and Quality of Service (QoS) of applications, especially those with stringent real-time requirements. Note that in many non-realtime applications, short-term fairness cannot be guaranteed when the resource allocation (e.g., scheduling) is based on non-fading channels. Asymptotically long-term fairness in wireless networks is not a critical issue, because a mobile user with a “bad” link at some instance will likely, in the statistical sense, have a “good” link at some other instance. A good news is that for some multimedia and data applications there is a service flexibility afforded by the ability of some multimedia and data applications to tolerate and adapt to transient changes in QoS along with channel state variations. This service flexibility has been referred as service elasticity [3]. Service elasticity can be exploited in the design of schemes for resource control to significantly improve the overall performance of wireless systems (see e.g., [9]). In this Chapter, we focus on the elastic traffic where the major concerns of resource allocations are efficiency and fairness. Since wireless links may be different not only in instantaneous qualities but also in average qualities, achieving perfect fairness and maximizing bandwidth efficiency are two conflicting objectives in most cases. When wireless links are fast-changing and possess multiple states, maintaining short-term fairness becomes impractical and may be unnecessary. In addition, when links differ in average quality, guaranteeing bandwidth for flows with inferior link qualities while striking a balance between efficiency and fairness objectives is a difficult task. For elastic traffic, scheduling and queueing are the main resource allocation procedures. The main objective of this Chapter is to address different approaches to fairness concept in cellular network. Therefore the details of scheduling algorithms is beyond the scope of this work. A very comprehensive review on scheduling algorithms for wireless networks can be found in [10]. Since our focus is on the fairness in cellular wireless networks, the fairness issues in the other wireless communication networks

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such as wireless local area networks, multi-hop wireless networks, and hybrid networks are not examined here. In the literature there are very many publications that deal with the fairness through some adjustments of the concepts and algorithms we present in this manuscript. In this Chapter, we only cite those articles which, to our knowledge, have a novel approach to the fairness problem in wireless cellular packet networks. 1.1. Organization. The outline of this Chapter is as follows: In Section 2 we briefly review the characteristics of wireless cellular medium. In this section we present the key characteristics of a wireless cellular share medium and then review different channel effects in different time scales. Then in Section 3 we investigate the fairness concepts defined for wireless cellular communications. We specifically concentrate on the max-min and proportional fairness concepts while providing the mathematical definitions. We also present the effort-outcome discussion for the fairness in wireless communication systems in Section 3. Since our focus is on packet services in the wireless medium, in Section 4 we review the wireless fair scheduling methods. In Section 5 opportunistic scheduling methods and proportional fairness scheduling are described. The concluding remarks of this Chapter are presented in Section 6. 2. Cellular Wireless Medium. Wireless channels operate through electromagnetic radiation from the transmitter to the receiver. In wireless communications the transmission channel could vary over very short time-scales in the order of microseconds. These add to attenuation effects that could also vary randomly with time due to user mobility. Furthermore, since the medium is shared by several users, the equitable sharing of this limited varying radio resources is an important challenge. The key characteristics of the wireless cellular medium can be summarized as follows: 1. Dynamic variations of wireless channel capacity. 2. Location-dependent and bursty channel error that may be uncorrelated over longer time scales. 3. Contention among multiple users for access to the wireless medium. 4. Lack of network-wide channel state information in mobile users in terms of which other mobile users are contending for the same channel or have the higher priority to access the medium, etc.. 5. Transmit power and processing power constraints in base-stations and mobile terminal. 6. Asymmetry in the capacity demand for the uplink and the downlink. These characteristics strongly affect the performance of resource allocation algorithms in cellular wireless networks. 2.1. Wireless Channel Effects. In a typical outdoor wireless propagation environment, a mobile wireless user is communicating with a wireless access-point (i.e. base-station). The signal transmitted from the mobile user may reach the access point directly (line-of-sight) or through multiple reflections on local scatterers such as buildings, mountains, etc.. As a result, the received signal is affected by multiple random attenuations and delays. Moreover, the mobility of either the mobile users or the scatterers may cause these random fluctuations to vary over time. Channel time variations results is random variations of the received signal strength over time. Finally, in a shared wireless environment the transmitted signal may incur interference due to concurrent transmissions. The attenuation incurred in wireless propagation can be decomposed into three main elements: a signal attenuation due to the distance between communicating nodes

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(path-loss), attenuation effects due to absorption in local structures such as buildings (shadowing), and rapid signal fluctuations due to constructive and destructive interference of multiple reflected radio wave paths (fading). Variations due to path-loss and shadowing occur over relatively large distances that occurs as mobile user moves through a distance of the order of the cell size. Pathloss and shadowing are typically frequency dependent. Variation due to multi-path fading occurs over very short distances, on the order of the signal wavelength and is frequency dependent. The channel characteristics depend on the combination of all three propagation effects. Moreover, it may include the effect of a transmitter/receiver filters. Note that, each of above effects causes channel variations in different timescales. In some cases discussed in this Chapter it is assumed that the path-loss remains constant during the time-scales of interest. To model the wireless channel state in conventional resource allocation schemes for wireless packet services (see e.g., [11, 12, 13]), the Gilbert-Elliot model [14, 15] is used. The Gilbert-Elliot model is a two state Markov model, where in one state the channel is extremely bad corresponding with the case that bit-error-rate (BER) is 0.5, and in the other state the channel is very good corresponding with the case that BER is zero. Practically in most cases, the channel BER is a function of signalto-noise ratio (SNR) at the receiver where SNR takes a continuous range of values. Therefore, discretizing the channel state into two regions corresponding to only two states may lead to a highly conservative allocation strategy. In [16], information theoretic arguments are used to show that the Gilbert-Elliot model conservatively underestimates the channel capacity. Lastly, it is shown in [17] that by employing a specific SNR partitioning approach, the network capacity increases dramatically as the number of states of the employed Markov model is increased. 3. The Fairness Concept in Cellular Wireless Communication. The fairness issue arises whenever a given amount of “resource” is required to be shared by a number of “users”. There are two main approaches to model fairness in resource allocation: max-min fairness and proportional fairness. Max-min fairness is achieved by allocating available resources to the maximum extent possible to the disadvantaged users, while not unnecessarily wasting resources. Proportional fairness however approaches the resource allocation problem from another perspective; proportional fairness is realized when an objective function is maximized that represents the overall utilization of all users while respecting the total available resource constraint. Both proportional fairness and max-min fairness possess optimality properties and are Pareto optimal1 and can be implemented under the convex optimization [18]. In wireless communication the resource required to be shared between the mobile users is the radio resource, including, e.g., frequency spectrum. In a wireless environment, because of the random channel variations, we must distinguish between effort (radio resource given to a user) and outcome (actual useful throughput achieved by the user). While “effort” equals “outcome” in a wireline environment, they can be substantially different in a wireless environment. In a broader sense, fairness in a wireless network can be defined based on either effort or outcome. Both max-min and proportional fairness concepts can be defined based on either effort or outcome. In the followings we define the concepts of max-min and proportional fairness for resource allocation in wireless cellular communication and then review the effort1 An assignment is Pareto optimal if one cannot increase the assignment to one source i without strictly decreasing an assignment to another source j.

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outcome discussions for wireless communications. 3.1. Max-Min Fairness. As a concept, max-min fairness has been much discussed by political philosophers (see e.g., [19]). Max-min fairness is recognized as an optimal throughput-fairness definition [20] and is used to define fairness in data networks [1]. Here is the max-min fairness definition: In the followings, we denote S as the set of active users. PResource allocation mechanism then assigns to a users s, a resource Rs , such that s∈S Rs ≤ R, where R is the total available system resource. Definition (Max-Min Fairness)[20]: A feasible allocation of resource to the users in set S, is max-min fair if for each user s, Rs cannot be increased (while maintaining feasibility) without decreasing Rs´ where Rs´ ≤ Rs . In order to realize max-min fairness over a shared capacity constraint communications medium, fair queueing is used [6]. The algorithms for computing the max-min fair allocation are based on the notion of bottlenecks, [1], [20]. In [21] it is shown that finding max-min fair schedules requires global state and timing information of all the nodes in the network. From the application point of view, max-min fairness is used in window flow control protocols; it is also very popular in the context of bandwidth sharing policies in various types of networks. It is now widely used in various areas of networking [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In spite of this wide applicability, there are examples, in the context of mobile or peer-to-peer networks, where the existing definitions and methods or theories cannot be directly applicable [20]. 3.2. Proportional Fairness. The concept of proportional fairness for resource allocation in data networks, is derived in [33]. Here is the proportional fairness definition: Definition (Proportional Fairness)[34]: A feasible allocation of resource to the users in set S, Rs , is proportionally fair if for any other feasible resource allocation Rs∗ , X R∗ − Rs s

s∈S

Rs

≤ 0.

(3.1)

In other words, a resource allocation scheme is proportionally fair if another resource allocation scheme is used to increase the throughput of a specific user by x% of what that user receives under the proportional fair algorithm, the summation of all percentages of throughput decreases suffered by all other users in the new algorithm will be more than x%. Such a vector is also the Nash bargaining solution which satisfies certain axioms of fairness [34], and, as such, has been advocated in the context of telecommunications in [35]. In [3] it has been shown that there is a general equivalence between maximizing concave utility functions, which indicates the level of satisfaction of each user, and achieving some system-wide notion of fairness. Utility functions are formally defined in microeconomics and can be intuitively understood as a quantitative description of the users satisfaction. The achieved fairness depends on the utility function used in the allocation algorithm. Maximization of the total throughput is one special case of this general formulation, where the utility function is simply the throughout. Although there has been no consensus on the exact form of the users utility as a function of its throughput, it is widely recognized that for a delay tolerant data user, or an “elastic user”, such a function should be increasing, concave and continuously differentiable

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[3]. In general it is also shown in [36] that proportional fairness scheduling can be the output of a utility maximization problem. 3.3. Effort Fairness vs. Outcome Fairness. Assume that for each user i, P there is a generic resource share φi such that s∈S φi ≤ 1, a resource allocation is effort fair, if the allocated resource to each user is proportional to its generic resource’ shares. In other words, this means the difference between the normalized resources the system allocates to any two users, i, j, is bounded as follows [37]: ¯ R (t , t ) R (t , t ) ¯ ¯ i 1 2 j 1 2 ¯ − ¯ ¯ < ², φi φj

(3.2)

where Ri (t1 , t2 ) denotes the allocated resource to a certain session i during time interval [t1 , t2 ], and ² is a finite constant shows the tightness of fairness constraint. An effort fair resource allocation guarantees the effort expended on the user is fair. Therefore, the fairness in actual throughput achieved by different users is not guaranteed. In the other hand, a resource allocation is outcome fair, if the difference between the normalized amount of realized throughput of any two users i, j is bounded as follows [38]: ¯ T (t , t ) T (t , t ) ¯ ¯ i 1 2 j 1 2 ¯ − ¯ ¯ < ², φi φj

(3.3)

where Ti (t1 , t2 ) denotes the actual throughput that a user achieves during the time interval [t1 , t2 ]. Note that the the definition in (3.2) is a soft equivalent to GPS in R (t ,t ) which | Ri (tφ1i,t2 ) − j φ1j 2 | = 0. An outcome fair resource allocation guarantees the fairness in actual throughput achieved by different sessions, however the fairness in the allocated resource to the sessions is not guaranteed. As a matter of fact, the effective resources allocated to mobile users is directly related to the useful data (outcome) sent/received, and not to the bandwidth provided (effort). Therefore, guaranteeing only effort-fairness without considering the actual outcome is therefore not meaningful. On the other hand, guaranteeing only outcome-fairness irrespective of the link quality differences may result in very low bandwidth efficiency. Since the exact amount of any short-term outcome depends on the fast-changing and instantaneous link status, which is not within the control of the scheduler, it is very difficult, if not impossible, to maintain short-term fairness based outcome. To incorporate the wireless channel variations into resource allocation, a similar fairness notion, channel-adaptive fairness (CAF), is defined in [39]. A resource allocation is channel-adaptive fair if in the short term the difference between the normalized throughput (normalized with respect to the channel capacity) of any two users i, j is bounded as follows [39]: ¯ T (t , t ) T (t , t ) ¯ ¯ i 1 2 j 1 2 ¯ − ¯ ¯<² φi f (Θi ) φj f (Θj )

(3.4)

where Θi denotes the channel state, and f (Θi ) = M (Θi )η , in which 0 ≤ M (Θi ) ≤ 1 is the effective throughput factor and η > 0. In this fairness notion, the throughput of a session is proportional to its channel quality. Therefore, in the long term, outcome fairness is maintained among all sessions. The channel-adaptive fairness is appropriate

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for wireless environment because it explicitly considers different channel states. The factor η can help to decide between making use of the bandwidth more efficiently and treating every session more fairly. 4. Wireless Fair Queueing, and Wireless Scheduling. Scheduler is the main resource allocation entity in a wireless cellular packet network that realizes the fairness resource allocation concept for the users. In wireline networks, fair scheduling has long been a popular paradigm for providing fairness [12]. The earliest known fair scheduling algorithm is Wireless Fair Queueing (WFQ) [2]. WFQ is a packet approximation algorithm that is designed to implement Generalized Processor Sharing (GPS) [2]. Since WFQ does not take any feedback from the channel capacity and it is only based on the queue statistics, for wireless communication with bursty errors and location-dependent wireless channel capacity and errors, WFQ is not a right choice [38]. For wireless fair queueing the basic idea is to extend scheduling policies for wireline networks to wireless networks, which provide various degrees of performance guarantees, including short-term and long-term fairness, as well as short-term and long-term throughput bounds(see e.g., [11, 12, 13, 37, 38, 40, 41, 42, 43, 44]). For such algorithms the general idea is to model wireless channel with two states: a good channel state corresponds to a transmission without error and a bad state corresponds to transmission that contains error. The scheduler, then simulates an error-free system running a wireline packet scheduling algorithm when the sessions (i.e., communication between a user and base-station) have good (or perfect) channel states where the effective throughput is maximum [45]. In adapting fluid fair queueing to the wireless domain, the following critical issues need to be addressed [45]: 1) the failure of traditional fluid fair queueing in the presence of location-dependent channel error; 2) the compensation model for flows that perceive channel error: how transparent should wireless channel errors be to the user? and 3) the trade-off between full separation and compensation, and its impact on fairness of channel access. In addition to these issues, other important problem are handling inaccuracies in channel state prediction, discovery of uplink flow state by the base station, and coordination of scheduling and medium access. When the session that is scheduled to transmit data encounters a bad channel state, it will give up its transmit opportunity to other error-free sessions; in return, these error-free sessions will give their transmit rights back to the erroneous session when it escapes from a bad channel state. Thus, in fact, the scheduler tries to swap the allocated time slots between error-free sessions and error-prone sessions when some sessions are encountering error. The goal is to hide the short-term channel error bursts from the end users. Note that in this approach the resource allocation maintains long-term fairness at the expense of instantaneous fairness between sessions. In [45], a wireless fair service model that captures the scheduling requirements of wireless scheduling algorithms is presented. This model is based on the fluid fair queueing that can provide fairness among backlogged flows even over infinitesimal time windows. In order to capture the behavior of flows in a wireless environment while incorporating the constraints of the channel, the error-free service of a flow is defined as the service that it would have received at the same time instant if all channels had been error-free, under identical offered load. A flow is said to be leading if it has received channel allocation in excess of its error-free service. A flow is said to be lagging if it has received channel allocation less than its error-free service. A flow that is neither leading nor lagging is said to be in sync.

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In general, due to location-dependent channel errors, service allocations that are designed to be fair over one time interval may be inconsistent with fairness over a different time interval, even though both time intervals have the same backlogged set. Wireless fair queueing must distinguish between a non-backlogged flow, for which no compensation is provided in fair queueing, from a backlogged flow that perceives channel error. However, compensating for the latter will void the separation property of fair queueing. Most wireless fair queueing algorithms differ in terms of how the swapping takes place, which flows are to be swapped, and how the compensation model is structured. However, all these algorithms can be thought of as instances of a unified wireless fair queueing architecture, which consists of the following five components [45]: 1. The error-free service, which defines an ideal fair service model assuming no channel errors. 2. The lead and lag model in wireless service, which determines which flows are leading or lagging their error free service, and by how much. 3. The compensation model, which compensates lagging flows that perceive an error-free channel at the expense of leading flows, and thus addresses the key issues of bursty and location-dependent channel error. 4. Slot queues and packet queues, which allow for the support of both delay sensitive and error sensitive flows in a single framework and also decouple connection level and link-level packet management policies. 5. Channel monitoring and prediction, which provides a reliable and accurate estimation of the channel state at any time instant for each backlogged flow. It is worth noting that, in this model, using lead-lag concept the scheduler keeps track of the effort allocated to each user and then through compensation model, makes “effort” and “outcome” equal. This is based on the fundamental assumption that the wireless channel is either available or not unavailable. Note that in general case this equality happens in longer time scales. In [45], the most important wireless fair queueing algorithms are shown to be simply mapped into this basic model: Channel State Dependent Packet Scheduling algorithm (CSDPS) [46], class based Channel State Dependent Packet Scheduling algorithm (CBQ-CSDPS) [47], Idealized Wireless Fair Queueing algorithm (IWFQ) [41], Channel Independent Fair Queueing algorithm (CIF-Q) [11], Server Based Fairness algorithm (SBFA) [44], Wireless Fair Service algorithm (WFS) [38], and Wireless Packet Scheduling algorithm (WPS) [41]. In the above model it is assumed that the wireless channel have two states corresponding to good and bad transmission. In [48] a scheduling algorithm, an N -state Markov, N > 2 model is used to characterize the wireless channel. This model is then incorporated into the compensation model of the scheduler through the use of future channel estimates, making the scheduler more immune to channel variations. This scheme allows for adaptive forward error correction (FEC), whereby the code rate varies according to the forecasted channel state. It is shown in [48] that using this method higher throughput and lower average delay can be achieved. Generalized sharing fairness model in [6] has been extended for wireless scheduling in [49] in which fairness is modelled as outcome fairness in a hybrid TimeDomain/Code-Division-Multiple-Access (TD/CDMA) based wireless networks. The proposed algorithm tries to realize fairness in the received bit-rate for wireless users. GPS concept is also used in [50] where a rate-scheduling approach based on GPS is applied to the CDMA downlink. In this scheme, for a given total downlink transmission

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power, the rate-scheduling scheme allocates the downlink power and rates according to weights assigned to the users. For each scheduling period, the user weights are optimized to guarantee the minimum channel rates required by the users, adapting to the time-varying channel condition. Although a high complexity optimization of weights is considered, in this method, short-term fairness cannot be guaranteed due to the frequent adjustment of weights. To support multimedia traffic in wide-band direct-sequence code-division-multipleaccess cellular network, a class of dynamic fair scheduling schemes based on the GPS, named code-division GPS (CDGPS) is proposed in [51] and [52]. CDGPS scheduler utilizes both the traffic characteristics in the link layer and the adaptivity of the wide-band CDMA physical layer to perform fair scheduling on a time-slot by timeslot basis, by using a dynamic rate-scheduling approach rather than the conventional time-scheduling approach. Further improvement of this method is presented in [51] as credit-based CDGPS (C-CDGPS) in which soft-capacity is exploited; however they show that this method results in short-term unfairness that can be bounded while long-term weighted fairness for all users can still be satisfied. For uplink GPS has been used to develop a fair resource allocation in [53]. This scheme takes into account the characteristics of channel fading and inter-cell interference to support realtime and non-realtime multimedia traffic with guaranteed statistical QoS in the uplink of wide-band CDMA cellular networks. In wireless scheduling model, using an intelligent compensation model along with a more practical channel model with the ability of modelling multiple level of transmission capacities2 , it is possible to trade-off between total network throughput and fairness. This trade-off can be made by letting users with good channel condition to lead, and keeping the users with bad channel condition until the occurrence of a better channel condition. In more general case the compensation model can be converted to an optimization problem. The objective of this optimization process can be, e.g., maximizing total network throughput, minimizing latency per user, etc.. This general idea is the basis of the opportunistic scheduling. 5. Fairness and Opportunistic Scheduling. A wireless scheduling scheme should be able to adaptively exploit the time-varying channel conditions of users to achieve higher utilization of wireless resources. The potential to exploit higher data throughputs in an opportunistic way is called multi-user diversity. Multi-user diversity is a form of diversity provided by independent time-varying wireless channels across different users [54]. The diversity gain is then exploited by tracking the wireless channel variations and scheduling transmissions to the users with better corresponding instantaneous channel quality. The diversity gain increases with the dynamic range of the fluctuations. Since in exploiting multi-user diversity, the “effort” is assigned opportunistically based on expected “outcome”, the efficiency-fairness tradeoff [55] is an important issue. In [56], [57], [58] using the utility-based model in [33], an optimal opportunistic transmission scheduling policy is presented. In a time domain scheduling mode, for each user a concave utility function for a given time-fraction is considered to model its level of satisfaction. Total utility is then defined as an aggregate of the utility functions of the users. The main objective of this opportunistic scheduling is to maximize total network throughput. They then give a heuristic solutions for this maximization 2 Multiple levels of channel capacity can be realized by adaptive rate allocation through power adjustment and adaptive modulation and coding schemes.

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problem that realized utility-fairness through a closed loop control system. In general, it is possible to show that an opportunistic scheduling problem with short term fairness constraints can be formulated as an optimization problem [56], [57]. In [59] the solution of this optimization problem is given for some special cases. The model in [56], [57], [58] is based on the assumption that in each time instant only one user would be transmitted. An important drawback of these algorithm is the requirement of having system-wide channel and service information that may create huge signalling overhead, and computational complexity. A similar approach to utility based resource allocation is considered in [60] and [61]. In this credit based method the fact that users contending for the wireless medium, depending on their current channel condition, will have different “costs” of transmission is used to define a cost function. The cost function reflects the channel quality and the fact that a user with a high-quality channel can transmit at a higher throughput or lower resource consumption. Obviously transmission given to the users with lower valued cost function may results in larger network throughput but worse fairness condition. To address these conflicting objectives, a credit abstraction is considered along with a general “cost function”. In particular, the credit abstraction is utilized to balance a users cost of accessing a channel with the elapsed time since its prior transmission. However, rather than selecting the user with the largest number of credits, the credit counts are compared with the cost of selecting a particular user. An algorithm that implements opportunistic scheduling is presented in [54], in which its scheduler implements proportional fairness concept. In this algorithm presented for the downlink of a cellular CDMA system, the scheduling algorithm keeps track of the average throughput of each user in a past window of time with a given length. In each time slot, the scheduling algorithm simply transmits to the user with the largest fraction of current channel data rate to its average throughput, among all active users in the system. If the scheduling time-scale is much larger than the correlation time-scale of the channel dynamics, then it is simple to show that each user’s throughput converges to the same quantity. In this scheduling algorithm, users compete for resources not directly based on their requested rates but only after being normalized by their respective average throughputs. Multiuser diversity gain has its root in independent fluctuation of channels of different users so that if there is a sufficient number of users in the system, there will likely be a user near its peak at any one time. It has been shown in [7, 54] that the above algorithm guarantees proportional fairness. In the context of wireless networks, proportional fair scheduling principle has been adopted in the forward link scheduler design in the newly emerged CDMA2000 1x-EV-DO high speed packet data system [7]. A similar idea has been applied to the CDMA2000 reverse link in [62] but it requires that the multiple users be TimeDivision-Multiplexed (TDM) rather than Code- Division-Multiplexed (CDM). Thus significant signaling overhead must be added and backward compatibility may not be preserved. 1x-EV-DO system [63] adopts dynamic data rate transmission technique to support efficient data service with proportional fairness scheduler. It is shown that under simplifying assumptions proportional fairness scheduling gives equal power and time to users who have the same fading characteristics but only differ in their distance from the base-station [7]. In [64] it is shown that for the two classes of users with different fading characteristics, the user class with more fading variability gets more throughput with a lower (but not much lower) fraction of time transmitting. Furthermore, in [65] it has been shown that the proportional fair

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scheduling in [54] may be unfair and unable to fully exploit multi-user diversity for the case that the users don’t experience identical channel statistics. A new algorithm called score-based scheduling is then presented in [65] and shown that it overcomes this important shortcoming. Instead of selecting a user when its transmission rate is high relative to its own average throughput, the score-based scheduler selects a user when its transmission rate is high relative to its own rate statistics: a user i(t) is selected at slotted time t from a set of n active if it has the best score defined as i(t) = arg minj=1,...,n sj (t), where score sj (t) of a user j at time t is the rank of the current transmission rate rj (t) among the past values of the transmission rates observed over a window with a given length. This algorithm has higher complexity compare to the original proportional fairness scheduling algorithm in [54]. The proportional fairness algorithm in [54] only considers the fairness in throughput of the users. In [66], different utility functions have been studied that they can model multiple service classes as well as other quality-of-service metrics e.g., experienced delay for an opportunistic scheduler. Such utility function acts as a selection rule that is used by scheduler to make the trade-off between different service quality metrics and network efficiency factors. A natural extension of the one-by-one opportunistic resource allocation would be to consider the transmission to a subset of users in an opportunistic way such that this subset of users can be served simultaneously. That would be the case in some conditions such as having multiple wireless channels. For opportunistic scheduling over multiple wireless channels, [67] develops a framework in which with a realistic channel model, any subset of users can be selected opportunistically for data transmission at any time, with different throughputs and system resource requirements such that long-term deterministic or probabilistic fairness constraints are satisfied. For a power-controlled CDMA system, a multiuser opportunistic fair transmission scheduling is introduced in [68]. In this multiuser opportunistic fair transmission scheduling, for each time instant a subset of users is selected to maximize instantaneous system throughput subject to some fairness constraint. A discrete stochastic approximation algorithm is then given in [68] to adaptively select user subsets based on the timevarying channel to maximize system throughput. Stability of opportunistic scheduling problem for simultaneous transmission to a subset of users is studied in [69]. The opportunistic resource allocation is usually considered for the downlink. This is based on the logic that the amount of data transmitted in the downlink is much more than that of the uplink. This is because of the asymmetry of the majority of data applications e.g., web-surfing and file downloading. However, the uplink scheduling in wireless systems is gaining more importance due to the increasing uplink intensive data services (ftp, image uploads etc.). For uplink scheduling, in contrasts with the downlink, it is shown in [70] that it is advantageous to schedule the users with good channel condition in time domain scheduling fashion (i.e., one-at-a time). However, for the users with bad channel condition, it is better for the network to transmit simultaneously to a large number of users who are experiencing bad channel condition. Based on this result, they design a scheduler that uses proportional fairness scheduling algorithm when there is a user with “strong” channel condition and transmits simultaneously to the multiple users, for the case when there is no major user. 6. Conclusions and Remarks. In this Chapter, the fairness concept for resource allocation in wireless cellular packet networks has been examined. We briefly reviewed the characteristics of wireless cellular medium and investigated the fairness concepts for wireless cellular communications. In doing so, we concentrated on the

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max-min as well as proportional fairness concepts. Subsequently we discussed the concepts of wireless fairness from the perspective of the applied efforts versus the outcome achieved by the wireless users. A basic model for wireless fair queuing together with a description of existing methods for its implementation were presented. The discussion was continued on the generalized processor sharing and opportunistic scheduling methods. Acknowledgment. The authors would like to express their gratitude to the reviewers the Editors and Prof. A. R. Sharafat for a wealth of comments and suggestions that improved significantly both the presentation and the content of this paper. The authors also, would like to thank Prof. Shahrokh Valaee and Prof. Ben Liang for their valuable comments and suggestions. REFERENCES [1] D. Bertsekas and R. Gallager, Data Networks, Second Edition. Prentice Hall, 1992. [2] H. Zhang, “Service disciplines for guaranteed performance service in packet-switching networks,” Proc. IEEE, vol. 83, pp. 1374–1396, October 1995. [3] S. Shenker, “Fundumental design issues for the future internet,” IEEEE JSAC, vol. 13, pp. 1176–1188, 1995. [4] H. Varian, “Equity, envy and efficiency,” J. Economic Theory, pp. 63–91, 1974. [5] L. Zhou, “Strictly fair allocations in large exchange economics,” J. Economic Theory, pp. 158–175, 1992. [6] A. K. Parekh and R. G. Gallager, “A generalized processor sharing approach to flow control the single node case,” in Proc. of IEEE INFOCOM’92, vol. 2, pp. 915–924, May 1992. [7] A. Jalali, R. Padovani, and R. Pankaj, “Data throughput of a CDMA/HDR a high efficiency high data rate personal communication wireless system,” in Proc. IEEE VTC’00, pp. 1854–1858, 2000. [8] P. Bender, P. Blake, M. Grob, R. Padovani, N. Sindhushyana, and A. J. Viterbi, “CDMA/HDR: a bandwidth-efficient high-speed wireless data service for nomadic users,” IEEE Comm. Mag., pp. 70–77, July 2000. [9] P. Viswanath, D. N. C. Tse, and V. Anantharam, “Asymptotically optimal waterfilling in vector multiple access channels,” IEEE Trans. on Info. Theory, vol. 47, 2001. [10] Y. Cao and V. Li, “Scheduling algorithms in broadband wireless networks,” Proc. of the IEEE, vol. 89, no. 1, pp. 76–87, 2001. [11] T. S. Ng, I. Stoica, and H. Zhang, “Packet fair queuing algorithms for wireless networks with location-dependent errors,” in Proc. of IEEE INFOCOM’98, March 1998. [12] S. Lu, V. Bharghavan, and T. Nandagopal, “Design and analysis of an algorithm for fair service in error-prone wireless channels,” ACM/Baltzer WL Net. J., vol. 6, pp. 323–43, August 2000. [13] T. Nandagopal, S. Lu, and V. Bharghavan, “A unified architecture for the design and evaluation of wireless fair queueing algorithms,” in Proc. of ACM MOBICOM’99, pp. 132–142, August 1999. [14] E. Gilbert, “Capacity of a burst-noise channel,” Bell System Technical Journal, vol. 39, pp. 1253–1266, Sept. 1960. [15] E. Elliot, “Estimates of error rates for codes on burst-noise channels,” Bell Systems Technical Journal, pp. 1977–1997, 1963. [16] H. Wang and N. Moayeri, “Finite-state markov channel - a useful model for radio communication channels,” IEEE Trans. on Vehicular Technology, vol. 44, no. 1, pp. 163–171, 1995. [17] M. Hassan, M. Krunz, and I. Matta, “Markov-based channel characterization for tractable performance analysis in wireless packet networks,” IEEE Trans. on Wireless Comm., vol. 3, no. 3, pp. 821–831, May 2003. [18] D. Julian, M. Chiang, D. O’Neill, and S. Boyd, “QoS and fairness constrained convex optimization of resource allocation for wireless cellular and ad hoc networks,” in Proc. of IEEE INFOCOM’03, vol. 2, pp. 477–486, 2003. [19] J. Rawls, A Theory of Justice. Cambridge: Harvard, 1971. [20] B. Radunovic and J.-Y. L. Boudec, “A unified framework for max-min and min-max fairness with applications,” in Proc. Allerton’02, 2002.

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