Opportunistic Scheduling

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					                    Opportunistic Scheduling: An Illustration of Cross-layer Design
         Xin Liu                            Ness B. Shroff                            Edwin K. P. Chong
Computer Science Dept.       Dept. of Electrical & Computer Engineering   Dept. of Electrical & Computer Engineering
Univ. of California, Davis                Purdue University                        Colorado State University

 Abstract                                                       losses. 2) Transport layer and network layer, e.g.,
                                                                TCP modifications for energy efficiency. 3) Network
                                                                layer and data link layer. For instance, link reliability
  The      unique      characteristics    of     wireless
                                                                information can facilitate routing decisions. 4) MAC
  communication systems - namely, timing-varying
                                                                and physical layer, which is the focus of this paper.
  channel conditions and multiuser diversity - call for
  specifically tailored system designs. In this paper, we       The basic idea of ―opportunistic scheduling‖ that will
  overview a cross-layer design method, named                   be overviewed in this paper is allocating resources to
  opportunistic scheduling, for exploiting the time-            links when they experience good channel conditions
  varying nature of the radio environment to increase           while avoiding allocating resources to links when
  the overall performance of the system under certain           they experience poor channel conditions, thus
  QoS/fairness requirements of users. We first discuss          efficiently utilizing radio resources. This is also
  a general system model, and the fairness and QoS              referred to as channel-aware scheduling or exploiting
  constraints being considered. Then, we present                multi-user diversity. We use the term opportunistic
  various optimal index policies and discuss their              to denote the ability to schedule users based on
  properties. We also outline a methodology to                  favorable channel conditions. On the other hand, the
  implement these opportunistic scheduling solutions            potential to transmit at higher data rates
  in practice. Lastly, we discuss the advantages and            opportunistically (i.e., when channel conditions
  costs associated with opportunistic scheduling, and           permit) also introduces an important tradeoff between
  identify possible future research directions.                 wireless resource efficiency and the level of
                                                                satisfaction among different users. For example,
                                                                allowing only users close to the base station to
 1. Introduction and Literature Review
                                                                transmit at high transmission power may result in
 Built upon the success of the second generation                very high throughput, but sacrifice the transmissions
 cellular services, tremendous amounts of resources             of other users. Such a scheme cannot satisfy the
 have been invested into 3G systems, especially in              increasing demand for quality of service (QoS)
 Asia, by companies such as SK Telcom and NTT                   provisioning in the emerging high-rate data wireless
 DoCoMo. Furthermore, other wireless systems,                   networks. To address this problem, we present here a
 including WLAN, ad-hoc networks, and wireless                  framework for scheduling users in an opportunistic
 sensor networks, are rapidly proliferating. In                 way. The objective is to improve wireless resource
 summary, the demand for wireless data services is              efficiency by exploiting time-varying channel
 skyrocketing and there is a real need to optimally             conditions while at the same time controlling the
 design and engineer wireless systems.A major                   level of fairness and QoS among users.
 challenge is that wireless communication requires
                                                                1.1 Literature Review
 sharing a limited natural resource: the radio
 frequency spectrum. The data-rate capacity that a              Wireless scheduling schemes have attracted a lot of
 radio frequency channel can support is limited by              recent attention. First, the scheduling policies of
 Shannon’s capacity laws. Hence, in the wireless                wireline networks are extended to wireless networks.
 environment, one has to engineer the network very              The proposed wireless scheduling schemes provide
 carefully so that little, if any, wireless spectrum is         various degrees of performance guarantees, including
 wasted.                                                        short-term and long-term fairness, as well as short-
                                                                term and long-term throughput bounds. A survey of
 Compared with the wired counterpart, wireless
                                                                these algorithms can be found in [1]. However, these
 systems have some unique characteristics, namely
                                                                efforts model a channel as being either ―good‖ or
 scarce resource, time-varying and location dependent
                                                                ―bad,‖ which may be too simple to effectively
 channel conditions. The nature of the wireless
                                                                characterize realistic wireless channels, especially for
 medium makes it particularly attractive for cross-
                                                                data services.
 layer design. Different layers have been considered,
 as illustrated by the following examples. 1)                   In [2, 3], the authors present a scheduling scheme for
 Application layer and network layer. For instance, a           the Qualcomm/HDR system. Their scheduling
 multimedia server can adjust the coding scheme                 scheme exploits time-varying channel conditions
 based on available bandwidth by tracking packet                while maintaining ―proportional fairness,‖ as defined
in [3, 4].                                                  conclusion in Section 5. All the results are presented
                                                            without proofs. We refer readers to the corresponding
In [5, 6, 7, 8], the authors study scheduling
                                                            papers for reference.
algorithms where both delay and channel conditions
are taken into account. Throughput optimality is            2. System Model and Constraints
defined in [6] as follows: a scheduling algorithm is
                                                            2.1 System Model
throughput optimal if it is able to keep all queues
stable if this is at all feasible to do with any            We consider a time-slotted system where time is the
scheduling algorithm. Furthermore, the authors of [9]       resource to be shared among all users. The system
investigate a scheduling algorithm to maximize the          can have more than one channel (frequency band),
minimum weighted throughput of users. We discuss            but at any given time, only one user can occupy a
these schemes in more detail later.                         given channel within a cell. Here, we focus on the
                                                            scheduling problem for a single given channel. Such
In [10, 11], we present a framework for opportunistic
                                                            a system model includes TDMA systems as well as
scheduling. In particular, the overall performance of
                                                            time-slotted CDMA systems. An example of the
the system is maximized under certain QoS/fairness
                                                            latter is the IS-856 system, also known as HDR,
requirements of users.
                                                            where each time slot is 1.67ms.
Opportunistic scheduling exploits the channel
                                                            Channel conditions in wireless networks are time-
fluctuations of users. Thus a natural question to ask is
                                                            varying, and thus users experience time-varying
what we should do in environments with little
                                                            performance. We use a stochastic model to capture
scattering and/or slow fading. In [12], the authors
                                                            the time-varying and channel-condition-dependent
present a scheme that uses multiple transmission
                                                            performance of each user. Specifically, following the
antennas to ―induce‖ channel fluctuations, and thus                                    k
exploit multi-user diversity. Further, such a scheme        approach of [12], let {U i } be a stochastic process
can also be used opportunistically to null inter-cell
                                                            associated with user    i , where U ik is the level of
                                                            performance that would be experienced by user i if
In [13], scheduling problems for real-time traffic are                                                             k
studied. The authors show that the greedy algorithm         it is scheduled to transmit at time k. The value of U i
is 1/2 competitive against the offline optimal              measures the ―worth‖ or ―utility‖ of time-slot k to the
algorithm. Furthermore, they show that no                   user i, and is in general a function of its channel
deterministic online algorithm can achieve a                condition. The better the channel condition of user i,
competitive ratio higher than 1/2.                                                    k
                                                            the larger the value of U i . Examples of the value of
Opportunistic scheduling has also been studied under
various scenarios, including distributed systems,
                                                            U ik are throughput, the value of throughput minus
(e.g., [14, 15, 16],) systems with multiple input and       the cost of power consumption, etc. We assume that
multiple output (MIMO) antenna arrays, (e.g., [17,          U ik is nonnegative and bounded. For simplicity, we
18, 12],) multi-carrier systems, (e.g., [19],) sensor                         k
networks, (e.g., [20],) multi-hop wireless systems,         assume that {U i } is stationary and ergodic (this
(e.g., [21, 22],) and with power and rate control, (e.g.,   assumption can be removed in some case, e.g., see
[23, 24, 25]). A cautionary note on cross-layer design      [11]). For convenience, we use the notation
is presented in [26]. A thorough analysis on user –         
level performance is given in [27].                         U  {U1 ,,U N }, where U i is a random
                                                            variable representing the performance value of user i
1.2 Organization                                            at a generic time-slot, and N is the number of users.
This paper overviews the concept of opportunistic           Note that the stationary assumption does not preclude
scheduling and how it can be used to improve system         correlations across users or across time and moreover
efficiency as well as provide adequate QoS. In              this assumption can be relaxed, as in [11].
Section 2, we provide a general system model, and           Wireless spectrum is a scarce resource, hence
the fairness and QoS constraints being considered.          improving the efficiency of spectrum utilization is
Then, in Section 3, we present various optimal index        important, especially to provide affordable high-rate-
policies and discuss their properties. In Section 3, we     data service. However, a scheme designed only to
also outline a methodology of how to implement              maximize the overall throughput could be unfairly
these opportunistic scheduling solutions in practice.       biased, especially when there are users with widely
In Section 4 we provide a discussion of various issues      disparate distances from the base station. To address
as well as some precautionary notes and possible            this    problem,     we     introduce    QoS/fairness
future research directions. This is followed by a           requirements into the framework of opportunistic
scheduling— our goal is to maximize the system                 objective is to maximize the overall system
performance (defined later) by exploiting time-                throughput regardless the performance of each
varying channel conditions while maintaining certain           individual user. In other words, each user is
user-oriented constraints. Examples of such                    guaranteed zero percent of the system throughput. In
constraints include long-term and/or short-term                this case, a small number of users with very good
fairness constraints or (direct) performance                   channel conditions may consume all the resource and
constraints.                                                   starve other users.
A scheduling policy is a rule that specifies which user        Max-min fairness The objective of the throughput
is scheduled at each time-slot. For simplicity of              max-min fairness is to maximize the minimum
                                                              throughput of all users. Let N be the number of users
notation, let Q (U ) be the decision of a stationary
                                                              in the system. Each user is guaranteed 1/N portion of
policy Q at a general time-slot where U is the                 the system throughput. This objective is ―absolutely‖
performance value of users. At a generic time-slot, if         fair. However, when there are users with very poor
                                                              channel conditions, to achieve max-min throughput
a policy Q schedules user i  Q(U )  {1,..., N }
                                                               fairness will cause significant system performance
to transmit, then the system receives a ―reward‖ of            penalty.
U i . Note that E(UQ (U ) ) is the average system
                                                               Proportional fairness: Proportional fairness (PF)
performance value associated with policy Q, and it is          scheduler maximizes the product of the throughput
the sum of all users’ average performance values               delivered to all the users. In other words, the set of
(where we reap a reward of U i only if user i is               throughput achieved by different users is
                                                               proportionally fair if increasing the throughput of one
scheduled). The objective is to find a policy Q that           user from the current level by x% requires a
maximizes the average system performance value                 cumulative percentage decrease in all the users of
 E(UQ(U ) ) under the constraints. In this paper, we
                                                              more than x%. Proportional fairness presents a
focus on stationary policies. Extensions on more               tradeoff between the overall throughput and each
general policies, including non-stationary and non-            user’s throughput.
causal policies, can be found in [11].                         Temporal resource-sharing fairness: In this case,
We are interested only in policies that satisfy specific       each user is guaranteed a certain portion of the
QoS/fairness requirements. We say that a policy Q is           resource, i.e., time-slots. Note that temporal resource-
feasible if it satisfies the constraints for all users. Our    sharing fairness is different from the utilitarian
goal is to find a feasible policy Q that maximizes the         fairness. In wireline networks, when a certain amount
system performance, which may be defined                       of resource is assigned to a user, it is equivalent to
differently under different assumptions.                       granting the user a certain amount of
                                                               throughput/performance value. However, the
We focus on the downlink of a wireless network. The            situation is different in wireless networks, where the
base station serves as the scheduling agent. The               amount of resource and the performance value are
scheduling scheme does the following: at the                   not directly related (though closely correlated). By
beginning of a time-slot, the scheduler (i.e., the base        limiting the resource of each individual user, a user is
station) decides which user should be assigned the             guaranteed a certain throughput (based on its channel
time-slot based on the performance values of the               conditions). Resource consumed by a user can be
users at that time-slot. If a user is assigned a time-         directly connected with the price the user should pay.
slot, then the base station will transmit to the user in       Premium users will obtain better services in a
that time-slot. In general, downlink transmission is           stochastic sense.
more important for data traffic because of the highly
asymmetric nature of the data service.                         Minimum data-rate constraints: In this case, each
                                                               user is guaranteed a minimum data-rate. This type of
2.2 Fairness and QoS Constraints                               QoS constraint is desirable for users, but difficult for
                                                               the system where feasibility is a major concern.
We first summarize the fairness and QoS constraints
discussed in the paper.                                        3. Optimal Scheduling Policies
Utilitarian fairness: In this case, each            user is    A common objective of opportunistic scheduling is to
guaranteed a certain portion of the total            system    maximize the system performance given the
throughput. Two extreme cases of utilitarian        fairness   fairness/QoS constraints. To maximize the system
is the max-min throughput fairness and               system    performance is formally presented as
throughput maximization.
                                                                                max Q E(U Q(U ) ) .
Throughput maximization In this case, the only
    Ti (Q)  E(Ui 1{Q(U )  i}) be the throughput             : i 1 ri  1 is a tuning parameter such that the
of user i under policy Q where 1{.} is the indicator         smaller the value of  , the less restrictive the
function. We have                                            fairness constraint, and the greater the opportunity to
                                                             improve the system performance.
         T (Q)  i 1 Ti (Q)  E (U Q (U ) ).
                                                             An optimal policy Q is defined as follows:
Of course, the values of Ti (Q ) depends on the                             
                                                                       Q* (U )  arg maxi (ui*  Ui ) ,
distribution of U , which is ignored in the notation
for simplicity. In this section, we summarize some           where the u i s are real parameters satisfying
results in opportunistic scheduling. It is interesting
                                                             a) mi n i (ui )  0 ;
that optimal policies turn out to be simple, easily                          *

implementable index policies under various fairness                                 
and QoS constraints.                                         b) for all i,  P{Q* (U )  i}  ri .
We should note that the problem formulations are                                      
                                                             c) for all i, if P{Q (U )  i}  ri then u i =0.
                                                                                  *                     *
expressed in terms of expectation, which is a long-
term performance measure. Short-term fairness                                                                   *
                                                             Let us explain the heuristics of the policy Q , which
depends on the time-correlation of the performance
values and can be tuned through parameter                    is helpful to understand the intuition of opportunistic
estimations. It will be discussed in Section 4.2.            scheduling. For a proof of the optimality of Q , we
3.1. Temporal Fairness                                       refer readers to [11]. We can think of the parameter
                                                             u i* above as an ―offset‖ used to satisfy the fairness
Because time is the resource shared among users, a
natural fairness criterion is to give each user at least a   requirement. To elaborate, consider the case where
                                                             we want to maximize the overall performance
certain share of the entire resource, i.e., time. Let ri     without any QoS requirements. It is straightforward
denote the minimum time-fraction that should be              to show that we should always choose the ―best‖ user
assigned to user i, where ri  0 ,            r  1 , and
                                          i 1 i
                                                             (i.e., the user with the maximum performance value)
N is the number of users in the cell. Here, we assume
                                                             to transmit. In other words,      Q* (U )  arg maxi Ui .
                                                             However, such a scheme may be unfair to certain
that the ri s are predetermined and serve as pre-
                                                             users. Hence, to satisfy the fairness requirement, the
specified fairness constraints. The value of ri              scheduling policy schedules the ―relatively-best‖ user
                                                             to transmit. User i is ―relatively-best‖ if
dictates the minimum fraction of time that a user
should transmit on the channel, which is typically           ui*  Ui  u*  U j for all j. If u i* >0, then user i
determined by the user's class, the price the user is        is an ―unfortunate‖ user, i.e., the channel condition it
willing to pay for the wireless service, or the user's       experiences is relatively poor. Hence, it has to take
current channel conditions.         The scheduling           advantage of some other users (e.g., users with
algorithm then decides which time-slot should be
assigned to which user, given the minimum time-              u *j =0) to satisfy its fairness requirement. Thus, to
fraction requirement. Our goal is to develop a               maximize the overall system performance, we can
scheduling policy Q that exploits the time-varying           only give the ―unfortunate‖ users the amount of
                                                             resource equivalent to their minimum requirements.
channel conditions to maximize the total expected                                      
system performance while satisfying the resource-            Last, when          P{Q* (U )  i}  ri for user i, the user
sharing constraint. The problem can be stated                gets more than its minimum requirement---this user
formally as follows:                                                                                                *
                                                             cannot take advantage of other users, i.e., u i =0. In
      max Q E(U Q(U ) )
                                                             summary, condition (a) above on                 u i* is for
                                                            normalization and condition (b) is the feasibility
      subject to P{Q(U )  i}  ri , i  1,2,...N            requirement. Condition (c) is important to the
In words, the problem is to find an optimal policy           optimality of Q . Its heuristic interpretation is that a
among all policies such that the total system                good user that gets more than its minimum
throughput is maximized where each user’s resource-          requirement cannot take advantage of other users.
sharing requirement is satisfied. Note that                  This condition can also be explained in terms of
complementary slackness: if the constraint is not
                                                                        b  1  i 1 ai vi* , and the v i* s are real
active (i.e., the average performance of user i is
greater than its minimum requirement), then the               parameters satisfying
corresponding u                                               a) mi n i ( vi )  0 ;
                        (which can be interpreted as a                     *
Lagrange multiplier) is zero. In a special case where
                                                              b) for all i, Ti (Q )  ai T (Q );
                                                                                       *       *
users have independent and identically distributed
                                                              c) for all i, if Ti (Q )  ai T (Q ); then v i =0.
performance values and the same requirements, we                                           *       *       *

have   ui*  u * and Ti (Q* )  T j (Q* ) for all i and j.
               j                                              The authors of [9] consider a special case of the
                                                              above      opportunistic     scheduling     problem.
Property:      If   the   performance       values   U i s,   Specifically, they consider maximizing the minimum
i  1,2,...N , are independent, then                          weighted performance of users. This is a special case
                                                              of the utilitarian fairness problem defined in this
 Ti {Q}  P{Q(U )  i}U i  riU i , i  1,2,...N.             section.
                                                               This problem setting requires fairness in terms of
This property makes a strong statement about the              performance values, which, to some extent, parallels
individual performance of each user. If users’                the concept of weighted fair queueing used in
performance values are independent, the average               wireline networks. The difference is that the overall
performance of every user in our opportunistic                capacity here is not fixed; it depends on channel
scheduling scheme will be at least that of any non-
opportunistic scheduling scheme. In this sense, the           conditions, the values of a i , and the scheduling
opportunistic scheduling policy does not sacrifice any        policy.
user’s performance to improve the overall system                                   *
performance. Of course, different users may                   The parameter v i can be considered a ―scaling‖ used
experience different amounts of improvement. This             to satisfy the utilitarian fairness constraint. The
property can also be explored to provide direct               optimal scheduling policy always chooses the
performance guarantees to users with simple resource          relatively-best user to transmit. In this case, a user is
allocation schemes.                                           relatively-best if

                                                                        (b  v* )U j  arg maxi (b  vi* )Ui .
The values of the u i s are determined by the                                 j
distribution of U and the values of the ri . In                                *
                                                             As before, if v i >0, then the user is an ―unfortunate‖
practice, the distribution of U is unknown, and               user, and its average performance value equals the
                                             *                minimum requirement.
hence we need to estimate the parameter u i .
Similarly, in the opportunistic scheduling schemes            Utilitarian scheduling schemes have certain notable
discussed in other sections, there are also parameters        features. First, the utilitarian fairness constraint
that need to be estimated. Estimations of the                 controls the maximum discrepancy of performance
parameters are discussed in Section 3.5.                      values among users. Second, the constraint given
                                                              ensures that a user is given at least a certain share of
3.2. Utilitarian Fairness
                                                              the total performance, and is hence more suitable in
The problem formulation of the utilitarian fairness is        some situations than the temporal fairness constraint.
presented as                                                  However, there is also a significant disadvantage of a
                                                              utilitarian scheduling scheme: a user experiencing
       max Q T (Q )                                           poor channel conditions could have a detrimental
                                                              impact on the overall system performance because a
       subject to Ti (Q)  aiT (Q), i  1,2,... N ,           substantial portion of the total time-slots may have to
                                                              be allocated to this user in order to meet its fairness
where ai  0, and                 ai  1.                     requirement. To alleviate this potential problem, one
                           i 1                               can devise an adaptive threshold strategy [11].
An optimal policy is defined as follows:                      3.3 Minimum-performance Guarantee
           Q* (U )  arg maxi (b  vi* )Ui                    Thus far, we have discussed two optimal scheduling
                                                              schemes that provide users with different fairness
                                                              guarantees. From a user’s viewpoint, a more direct
                                                              QoS is defined in terms of minimum-performance
guarantees.    To elaborate, the objective is to           always provides ―no-worse‖ performance values for
maximize the average system performance subject to         each user relative to that of the non-opportunistic
meeting     each    user's  minimum-performance            scheduling policy, assuming that the signaling cost is
requirement, formulated as:                                negligible. Thus, the opportunistic scheduling policy
                                                           dominates non-opportunistic policies.
           max Q T (Q )
                                                           3.4 Index Policies
           subject to Ti (Q)  Ci , i  1,2,... N          It turns out that a scheduling policy of the form of
where C i is the minimum-throughput requirement of                        Q* (U )  arg maxi i*Ui
user i. Consideration of this problem raises two
                                                           is optimal for many types of scheduling problems.
questions: (i) Is the requirements feasible, i.e., does
                                                           For example, the optimal utilitarian policy defined
there exist a policy such that Ti (Q )  Ci , for all i?
                                                           earlier is in this form. If    i*  1 for all i , then the
(ii) If it is feasible, which policy maximizes the
                                                           policy maximizes the total system throughput. On the
overall performance under the given QoS
requirement?                                               other hand, if     i*  1 / Ti (Q * ) for all i , then the
Compared to fairness requirements, the formulation         policy is proportional fair [2].
here offers users a more ―direct‖ service guarantee.       In [5,6], the authors study scheduling algorithms
For example, if the performance measure is defined         where both delay and channel conditions are taken
as the data-rate, then each user is guaranteed a           into account. Roughly speaking, the algorithm is:
minimum data-rate, which may be more important to                          
a user than knowing that a minimum amount of                           Q* (U )  arg maxi i*UiWi ,
resource will be assigned to it. While appealing to
users, providing minimum-performance guarantees            where Wi is the head-of-the-line packet delay for
can be quite difficult in practice because of the
feasibility issue---can the system satisfy the             queue i, and      i* is some constant. The policy is
performance requirements for all users? (Note that         throughput optimal.
feasibility is not a concern in the fairness-based
constraints.) More discussion on feasibility can be        We have presented a framework for opportunistic
found in [11]. There are, however, some natural            scheduling and studied different scheduling
settings where feasibility is not a problem. For           problems. These scheduling problems share a
example, the requirements can be set as the average        common goal: to improve the spectrum efficiency
data rates in a non-opportunistic round-robin              while maintaining certain levels of QoS for each user
scheduling scheme. Then, it is guaranteed to be            using opportunistic scheduling algorithms.          The
feasible for opportunistic scheduling policies.            solutions to these scheduling problems turn out to be
                                                           index policies---all the schemes choose the
An optimal policy is defined as follows:                   ―relatively-best‖ user to transmit. Although
                                                          ―relatively-best‖ has a different meaning for each
                 Q* (U )  arg maxi i*Ui                  scheduling policy, the basic idea is to use an offset or
                                                           a scaling to satisfy the QoS requirements for users. In
where    i* s are real parameters satisfying              general, the larger the number of users sharing the
a) mi n i ( )  1 ;
             *                                             same channel, or the larger the variance of U , the
                                                           larger the ―opportunistic‖ scheduling gain compared
b) for all i, Ti (Q )  Ci ;
                                                           with     non-opportunistic    scheduling     policies.
                                                           Furthermore, the more restrictive the QoS constraint,
c) for all i, if Ti (Q )  Ci , then
                                        i* =0.            the less the flexibility for opportunistic scheduling
                                                           decisions, and the lower the system performance
The parameter       i* ―scales‖ the performance values    gain.
of users, and the scheduling policy schedules the
relatively-best user, where a user is relatively-best if   3.5 Implementation
 *U j  arg maxi i*Ui . If the scaling factor for a
                                                           Figure 1 shows a block diagram of a practical
                                                           scheduling procedure that incorporates on-line
user is larger than 1, then the user is an ―unfortunate‖
                                                           parameters estimation. In our scheduling policy, the
user, and it is granted only an average performance        base station needs to obtain information of each
value that equals its minimum-performance                  user’s performance value at a given time slot to make
requirement. The opportunistic scheduling policy
the scheduling decision. At a time slot, a user could
                                                           gik  (uik  min j u k )(1{Qk (U )  i}  ri ) ,
measure the received signal power level (from the
user’s base station) and the interference power level.     where i=1,2,…N. The observation error in this case is
Based on the estimated SINR, the user can then
obtain its performance value. The information is sent      eik  g ik  f i (u k )
back to the base station, which can be accomplished                                                     
in several ways. For example, each user could               (uik  min j u k )(1{Q k (U )  i}  P{Q k (U )  i})
maintain a small signaling channel with the base
station. Such signaling/control channels have been
standardized in the third generation cellular networks.    which is an unbiased estimate. Hence, we can use a
Then the base station makes the scheduling decision        stochastic approximation algorithm of the form
based on the scheduling policy and transmits to the
selected user. Last, parameters used in the scheduling                             uik 1  uik   k g ik ,
policy are updated, which is discussed next.
                                                           where, e.g.,    k =1/k.
                                                           When    uik  min j u k , we also need to ensure that
                                                                                              
                                                           P{Q (U )  i}  ri . If P{Q k (U )  i}  ri , then

                                                           u k is an infeasible parameter vector, which causes
Figure 1. Block diagram of the scheduling policy           some fairness constraint to be violated. To ensure
                                                                 k                                       
with on-line parameter estimation.                         that u converges to   u * , we should project u k
The opportunistic scheduling policies described in                              u s. However, because we do
                                                           onto the feasible set of
previous sections all involve some parameters that         not have knowledge of the distribution of U , it is
need to be estimated online. Such parameters are           very difficult to find the exact projection. Hence, we
determined by the values of the QoS requirements           use the following intuitive algorithm as a projection.
and the distribution of the utility values. In practice,                                                 
such a distribution is a priori unknown, and hence we      It is easy to see that P{Q (U )  i} is an increasing
need to estimate the parameters. In this section, we
use the temporal fairness scheduling scheme as an
                                                           function of u i . Hence, if                       uik  min j u k , and
example to describe briefly how to estimate these                   
parameters efficiently via stochastic approximation        P{Q k (U )  i}  ri , then we increase the value of
techniques. Similar estimations can be applied for         u ik to increase the value of P{Q k (U )  i} , as a
other scheduling policies.
                                                           projection to the feasible set. Although we do not
Recall that the parameters are chosen to satisfy the                                                 
                                                           know the value of P{Q (U )  i} , we can estimate
following     requirement:    for    all     i,    if
                                                                                                            k
P{Q(U )  i}  ri then u i* =0. Hence, we can write        it by a moving average. Let p i be the estimate of
                                                                 
as a root of the equation f (u )  0 , where the ith       P{Q k (U )  i} . We update pik in each time-slot
component is given by                                      as follows:
                                                                                       
   fi (u )  (ui  min j u j )(P{Q(U )  i}  ri ) ,       pik  (1  w) pik 1  w1{Qk (U )  i},
where i=1,2,…N. Next, we use a stochastic                  where w is a constant, indicating how fast p i tracks
approximation algorithm to generate a sequence of                 
                                                        P{Q k (U )  i} .                         If pi  ri ,
iterates u 1 , u 2 , u 3 , … that represent estimates of                                                                          and
               
u * . Each u k defines a policy Q k given by                u  min j u , then we increase the value of u by
                                                                                                     k
Q k (U )  arg maxi (Ui  uik ) . To construct the         a small constant. By doing this, we push u towards
stochastic approximation algorithm, we need an             the feasible set of u s. Simulations indicate that this
estimate of    f (u k ) . Although we cannot obtain        approach works well.
 f (u k ) directly, we have a noisy observation of its     4. Discussion
components:                                                Different schemes may be suitable for different
                                                           scenarios. For example, if the service provider wants
to build a simple wireless network with pricing, the      Opportunistic scheduling exploits the fluctuation of
temporal fairness scheduling scheme is a reasonable       channel conditions, and thus scheduling gain
choice. The temporal fairness scheduling scheme is        inherently depends on the amplitude of the variations
simple and flexible without feasibility concerns. The     of channels. In general, the greater the fluctuation of
amount of resource consumed by a user determines          channel conditions, the larger the number of users,
the minimum performance the user gets (with               the better the performance gain.
technical assumptions). The resource consumed by a
                                                          Another concern in opportunistic scheduling is the
user can be connected directly with the price the user
                                                          time scale of fluctuation. The fluctuation of channels
should pay. On the other hand, the minimum-
                                                          should be slow enough for users to estimate and
performance guarantee scheme provides users a
                                                          exploit it. On the other hand, the fluctuation should
direct performance assurance, but involves the
                                                          be fast enough, so that users won’t experience
additional complication of feasibility. If the service
                                                          extreme long delays. Though many data users are
provider wants to build a network that provides data-
                                                          delay-tolerant, extreme delays may cause upper-layer
rate guarantees, then this scheme is an appropriate
                                                          problems such as TCP timeout.
choice. However, in practice, the feasibility issue
may be difficult to handle, especially in a wireless      There is a tradeoff between scheduling gain and
setting, and providing service performance                short-term performance. In general, the stronger the
guarantees is challenging in both wireless and            time-correlation of channel conditions (i.e., the
wireline networks.                                        slower the channel fluctuation), the worse the short-
                                                          term performance, and the greater the improvement
It should be noted that the framework for
                                                          in the short-term performance, the less the scheduling
opportunistic scheduling that we have described here
can also cover cases where there are different
constraints from different users. For example, some       In general, scheduling gain increases as the number
users may have resource requirements while other          of users increases. However, the normalized
users can have a minimum-data-rate requirement. In        scheduling gain (scheduling gain over number of
such scenarios, similar optimal solutions can be          users) decreases with the increase of the number of
provided under this framework using similar               users, while the signaling cost per user remains the
optimization techniques.                                  same. Hence, it is a question of practical importance
                                                          to decide the number of users sharing the same
4.1 Precautionary Notes
Opportunistic scheduling schemes, as an illustration
                                                          In summary, opportunistic scheduling presents a new
of the cross-layer design of wireless systems, exploit
                                                          design approach, especially for delay-tolerant data
time-varying channel conditions of users with the
                                                          traffic. It has its own advantages and limitations. It is
objective to improve the system throughput.
                                                          thus important that the system designer to take a
However, nothing comes for free. Opportunistic
                                                          holistic view of the cross-layer design in order to
scheduling also has its own costs and limitations
                                                          avoid potential negative system-wide impacts.
discussed as follows.
                                                          4.2 Possible Research Directions
There are signaling costs involved in all opportunistic
scheduling schemes because scheduling decisions           Many interesting problems are yet to be resolved in
inherently depend on channel. Users need to               opportunistic scheduling. We discuss some possible
constantly estimate their channel conditions and          research problems next.
report to the base station. Hence, the actual
                                                          Short-term Fairness As mentioned earlier, the
scheduling gain should take into account the
                                                          scheduling problems are expressed in terms of
signaling costs.
                                                          expectation in this paper, which is a long-term
Because users need to estimate the channel                performance measure. There is no guarantee of short-
conditions, estimation errors occur in all scheduling     term performance. In [10], an extension is provided
schemes. There are various sources of estimation          to improve short-term performance. The basic idea is
errors: errors of estimations of channels, errors of      to increase a user's probability of transmission when
estimations of parameters involved in scheduling          it is behind in its share. There is a need for general
schemes, and errors caused by various delays such as      short-term fairness criteria tailored to wireless
transmission delay, estimation delay, and restriction     networks and dealing with the short-term
of time-slots, etc. In general, if the variation of       performance in depth. We also refer interested
channel conditions is relatively slow, then the           readers to [5-8] where queueing delays are
estimation is good. We recommend a rigorous study         considered, [13] where real-time scheduling is
on this problem, especially in the case of fast fading.   discussed, and [27] where user-level performance is
studied.                                                    layer-breaking designs can be potentially beneficial.
Delay A problem related to improving short-term             Admission Control The opportunistic scheduling
performance is to schedule traffic with deadlines, i.e.,    problems studied here have the net effect of increas-
real-time traffic. Specifically, upon arrival, each real-   ing the overall effective capacity of the wireless
time packet has a delay deadline, and packets that          network. This means that the network can now
cannot be transmitted before their deadlines are            accommodate more users or higher-data-rate users.
dropped/marked. Research on scheduling with                 Thus, we know that keeping all else fixed, the
deadlines in the wireline setting has led to various        admissible region of the wireless network will
approaches. The additional challenge in wireless            increase by using opportunistic scheduling schemes.
networks is due to the time-varying channel                 A challenging problem that still remains is how to
conditions. Approaches to these problems may                make intelligent admission control decisions on
include off-line optimal solutions with the                 whether or not to allow a new user into a cell.
assumption of entire traffic and channel information,       Although admission control is a difficult problem in
on-line model-based solutions, and heuristic/greedy         wireless systems whether or not opportunistic
algorithms. Heuristic algorithms play an important          scheduling is used, it is more challenging in the
role in real-time scheduling problems because               context of opportunistic scheduling because
(typically) the optimal scheduling problem is NP-           opportunistic scheduling increases the system
complete and simplicity is a desirable feature. In the      dynamics.
wireline world, it is sometimes the case that
                                                            Multi-hop Networks Most of the current research on
complicated scheduling schemes do not have
                                                            opportunistic scheduling focuses on the downlink of
significant performance gains over simple schemes,
                                                            a cellular system. In such a system, there exists a
such as static priority or earliest-deadline-first. A
                                                            natural central controller, the base station. An
similar situation may be expected to hold for wireless
                                                            interesting question is whether and how to exploit the
                                                            time-domain diversity in a distributed multi-hop
Another challenging problem is to minimize the              environment, such as an ad-hoc network
average packet delay. Although many schemes can             [15,20,24,28].
stabilize the queues, to control the average delay
                                                            5. Conclusion
performance is much more challenging.
                                                            To meet the increasing demand for wireless services,
Multi-carrier System Opportunistic scheduling is
                                                            especially affordable wireless data services, wireless
based on the premise that the wireless channel is
                                                            spectrum efficiency is becoming increasingly im-
time-varying, and we can schedule users to transmit
                                                            portant. In wireless networks, users experience
at those times that are opportunistically ―relatively
                                                            unreliable, location-dependent, and time-varying
good.‖ This idea can be extended to the frequency
                                                            channel conditions. Traditionally, the channel
domain: we opportunistically schedule users to
                                                            variation is considered as a negative factor for
frequencies (and time) that are relatively good [19].
                                                            reliable communication, and should be mitigated by
An example of such systems is an OFDM system. A
                                                            methods such as time interleaving, power control,
concern of opportunistic scheduling in such systems
                                                            and multiple antennas. On the other hand,
is the signaling cost. Because each sub-carrier is very
                                                            opportunistic scheduling is designed to exploit the
narrow in OFDM systems, signaling should be
                                                            variation of channel conditions to improve spectrum
carefully designed to ensure good channel estimation
                                                            efficiency. It adds an additional degree of freedom to
of users on different sub-carriers while avoiding
                                                            the system: time-domain diversity or also called
significant signaling overhead.
                                                            multi-user diversity. It improves spectrum efficiency,
Physical Layer The performance of opportunistic             especially for delay-tolerant data transmissions.
scheduling schemes is closely related to physical-          Various opportunistic scheduling schemes have been
layer designs. As explained earlier, estimation errors      studied. A common objective is to improve/maximize
occur in all opportunistic scheduling schemes. On           system performance (e.g., throughput) under various
one hand, we need a better understanding of the             fairness and QoS constraints. In many cases, the
effect of channel estimation errors on scheduling           optimal policies are given in a simple parametric
schemes. On the other hand, it calls for better channel     form, hence lending themselves to easy
estimation techniques and smart coding schemes              implementations. The advantages of opportunistic
(e.g., incremental redundancy transmission schemes          scheduling also include the ability to work with other
with turbo codes). Further, it is also important to         resource management mechanisms. A good example
study the performance of opportunistic scheduling in        of this is the joint scheduling and power-allocation
multiple antenna systems. In summary, a better              scheme [23]. In summary, opportunistic scheduling,
understanding of physical-layer technologies or even        with its own advantages and limitations, is an
excellent illustration of cross-layer design.                    [15] X. Qin and R. Berry, ―Exploiting multiuser diversity
                                                                 for medium access control in wireless networks,‖ in
6. Acknowledgement                                               Proceedings of IEEE Infocom 2003. IEEE, 2003.
This research is supported in part by NSF awards                 [16] ——, ―A distributed splitting algorithm for exploiting
ANI-0207728, ANI-0099137, EIA-0130599, ECS-                      multiuser diversity,‖ in Proceedings of IEEE International
0098089 and ANI-0207892, and the Indiana 21st                    Symposium on Information Theory. IEEE, 2003.
century center for wireless communications and                   [17] B. M. Hochwald, T. L. Marzetta, and V. Tarokh,
networking.                                                      ―Multiple-antenna channel hardening and its implications
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