Patent Publication Number: US-6985462-B2

Title: System and method for user scheduling in a communication network

Description:
BACKGROUND OF THE INVENTION 
     The present invention generally relates to wireless communication networks, and particularly relates to user scheduling in such networks. 
     In some types of wireless communication networks, access terminals share the air interface between the terminals and the network. High data rate (HDR) networks configured according to the TIA/EIA/IS-856 standard exemplify this shared interface concept. With shared forward link resources, the network serves only one access terminal in a group at any given instant. By rapidly switching service between the terminals, the plurality of terminals is simultaneously, albeit discontinuously, served by the network. Deciding which terminal to serve in any given service interval is the function of a “scheduler.” The scheduler typically comprises program logic or software services intended to fulfill this function. Scheduling operations are typically governed by a service goal or constraint. 
     Selecting the “best” user for service at each scheduling decision point is an optimization problem. In optimization problems, utility functions are used to characterize the cost or benefit of a change. In scheduling air interface usage, utility functions define the utility of serving individual users. The optimization problem becomes one of maximizing an objective function whose value is dependent upon the set of utility functions. A scheduler repeatedly evaluates the objective function to move towards an optimum schedule for sharing the limited resources of the air interface between active users. 
     Past approaches to network scheduling the use of shared resources within a communication network include so-called “proportional fair” scheduling techniques. Proportional fair scheduling seeks to maximize the objective function, and is essentially a maximum dual-ascent approach that reduces to finding the direction of maximum gradient for the objective function at each scheduling decision point. 
     While proportional fair scheduling works well in some circumstances, its use places overly limiting restrictions on the nature of utility and objective functions that may be defined for serving users. These restrictions hamper a service provider&#39;s flexibility in terms of denying scheduling flexibility. Thus, alternative approaches that offer additional flexibility or impose fewer constraints on the type or types of scheduling functions that may be used are needed. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention comprises a system and method for scheduling use of the shared air interface in a communication network, preferably a HDR network, to maximize revenue or other service goal defined by one or more objective functions. For each service interval, there are a number of service scenarios equal to the number of users that are candidates for service in the current service interval. Each service scenario corresponds to serving one of the users to the exclusion of the remaining users. The “primal ascent” approach to scheduling described herein serves the user whose corresponding service scenario yields the greatest net gain in the objective function. 
     By evaluating net gain rather than gradient, for example, the primal ascent scheduler allows tremendous flexibility in the selection of the utility functions that describe the incremental benefit or change in the objective function for individual users. With gradient-based approaches, utility functions must be differentiable, but with primal ascent scheduling, this restriction is removed. Thus, the objective function may be modified by one or more non-differentiable barrier functions that bound or restrict it in consideration of upper and/or lower data rates. Using barrier functions allows a system operator to bias the scheduling of users such that users on average do not exceed an upper data rate bound, or do not fall below a lower data rate bound if radio conditions permit. 
     In addition to the flexibility of biasing average data rates through barrier function use, the primal ascent approach to scheduling allows system operators to simultaneously use different utility functions for different users. As the utility function describes the incremental cost or gain associated with serving a given user, this allows the service provider to offer users differing qualities of service. That is, a premium user might be assigned a utility function that causes that user to generally attain a higher average data rate while connected with the network than would be achieved by a non-premium user. 
     This flexibility regarding utility functions further allows the system operator to use different utility functions at different times of the day, or to define utility functions whose weighting changes over time. Primal ascent scheduling easily accommodates non-separable utility functions, so the system operator may construct types or combinations of utility functions that may be awkward to use with conventional gradient-based scheduling optimization. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of an exemplary HDR network. 
         FIG. 2  is a graph of a conventional, differentiable utility function and a bounded, non-differentiable utility function. 
         FIG. 3  is a graph of a non-monotonic objective function. 
         FIG. 4  is a diagram of exemplary logic for implementing the primal ascent scheduling technique of the present invention. 
         FIGS. 5A and 5B  are diagrams of exemplary scheduling scenarios. 
         FIG. 6  is a graph of a utility function modified by use of one or more barrier functions. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In many types of communication systems, users compete for or otherwise share one or more limited resources. Scheduling techniques or algorithms decide how these limited resources are shared or allocated between users. Understandably, not all scheduling techniques are equally suitable in all circumstances. The scheduling technique of the present invention, referred to as “primal ascent” scheduling, has particular suitability for use in HDR networks, such as those networks based on the TIA/EIA/IS-856 standard. Reasons why the primal ascent approach has particular advantages within the framework of HDR networks are thoroughly discussed below, but it should be understood that the scheduling techniques and supporting ideas disclosed herein have utility in other types of communication networks, and for other types of shared resources. 
     Turning now to the drawings,  FIG. 1  illustrates an exemplary HDR network generally referred to by the numeral  10 . The network  10  comprises one or more base station controllers (BSCs)  12 , a plurality of radio base stations (RBSs)  14  for communicating with wireless access terminals (ATs)  18 , one or more packet control functions (PCFs)  20 , an IP network  22 , and one or more packet data serving nodes (PDSNs)  24  for communicating with external public data networks (PDNs)  26 , such as the Internet. 
     In operation, the BSC  12  controls one or more RBSs  14  that provide the RF-based air interface  16  used to communicate with the users. The term “user” herein connotes an AT  18  or other wireless device, unless otherwise stated. For HDR and certain other types of communication networks, the air interface  16  represents a resource that is shared by the users of the network  10  (i.e., shared between ATs  18 ). Data to and from the ATs  18  is routed to the appropriate BSC  12  by the PCF  20 . The PCF  20  communicates through the IP network  22  with the PDSN  24 , which is communicatively coupled to the Internet or other PDNs  26 . Thus, data passes between one or more PDNs  26  and individual ones of the ATs  18  through the network  10 . 
     The scheduler of the present invention may be implemented in software, hardware, other types of program logic, or in some combination thereof. For example, a processor  30  (or processors) in the BSC  12 , or in the RBSs  14 , may perform primal ascent scheduling based on execution of program code or software. In this manner, the BSC  12  or RBSs  14  can control use of the air interface  16  by users (i.e., ATs  18 ) in accordance with the present invention. However, it should be understood that other network entities, whether or not illustrated in  FIG. 1 , might be responsible for scheduling. 
     To understand some of the advantages associated with scheduling use of the air interface  16  based on the primal ascent technique of the present invention, one must first understand something about the air interface as defined in HDR networks. At any given time, a plurality of users (e.g., ATs  18 ) is simultaneously connected with the network  10 . However, the network  10  delivers data to only one user at a time. That is, the whole group of users shares the air interface  16 , but the network  10  feeds data to only one user at any given instant. The trick comes in determining which particular user candidate to serve at any given instant. Making that service decision depends on any number of constraints, as well as on the service provider&#39;s objectives. 
     HDR networks adopt a time division multiple access (TDMA) approach to the forward link air interface  16  used to transfer data from the network  10  to individual ones of the users. Thus, discussing the scheduling of users or user candidates practically means scheduling the times at which the particular users (i.e., ATs  18 ) are served within the time multiplexing scheme of the HDR air interface. 
     HDR networks use rate control rather than power control on the forward link and each AT  18  sends updated rate requests to the network  10  indicating the rate at which the AT  18  desires service. Thus, an AT  18  generally requests data at the highest rate that is feasible for the current radio conditions at the terminal. One measure of radio conditions is expressed as the carrier-to-interference (C/I) ratio. Higher C/I ratios indicate favorable reception conditions at the AT  18  and support higher data rates. 
     Each user (AT  18 ) updates its requested rate by periodically transmitting Data Rate Control (DRC) symbols to the network  10  on reverse link DRC channels. The DRC symbol value corresponds to one of a defined number of forward link data rates. At any given instant in time, some ATs  18  will request relatively high data rates, and some will request relatively low data rates, reflecting the differing reception conditions at various ones of the active terminals. The TIA/EIA/IS-856 standard provides for DRC symbol rates at up to 600 Hz, meaning that individual ATs  18  may request very rapid adjustments in the forward link data rate. 
     The scheduler, in at least some embodiments uses the requested data rates to determine which user or users to serve in any given service interval. The historic or past average data rates of the individual users may also be considered. For this discussion, let r i  represent the average throughput for the i th  user, and let d i  represent the currently requested data rate for serving the i th  user. Average throughput and requested rate are generally expressed in Kbits per second (kbps). With this notation, the entire sets of requested rates and average throughputs for all users are expressed in vector form as,  (n) and  (n), respectively. Here, the “n” generally indicates the n th  time slot. It is assumed that user scheduling entails selecting the user or users to be served in each of a succession of service intervals. That is, it is assumed that user scheduling is an ongoing operation for so long as one or more users have connections to the network. 
     For example, scheduling intervals might be based on the HDR frame rate of 26.6 ms, wherein the DRC information associated with each AT  18  determines the number of bits per frame and the number of time slots used to deliver data to the AT  18 , if that terminal is to be served. For a detailed treatment of the air interface  16  and its frame/slot timing in HDR embodiments, the reader is referred to standard document 3GPP2 C.S0024, Version 2.0, dated Oct. 27, 2000, as promulgated by the 3rd Generation Partnership Project 2 (3GPP2). This standards document is entitled “cdma2000 High Rate Packet Data Air Interface Specification,” and is incorporated in its entirety herein by reference. 
     In HDR networks, if the scheduler chooses to serve a given terminal, the amount of service given to that terminal within that service interval is set by the DRC information associated with the terminal. For example, choosing to serve a given user might entail serving that user for a total of eight time slots, although these time slots may be interleaved with time slots dedicated to one or more other users. Overall, the scheduler might try to ensure that each user is selected for service often enough to insure that the average throughput for that user does not fall below some desired limit. However, this strategy reflects just one of many possible considerations that might bear on the scheduling strategy. 
     A service provider (e.g., a network operator) may have a particular goal or goals in mind with regard to scheduling service to users. For example, the service provider may want to maximize its revenue, where revenue may be a function of maintaining the highest aggregate average data throughput, or by insuring that at least a subset of users achieve relatively high average throughput, perhaps at the expense of other users. Scheduling thus becomes an optimization problem wherein the goal is to control scheduling in a manner that attempts to optimize one or more service objectives. 
     In optimization problems, a utility function is used to describe the measure of cost or benefit associated with a contemplated change in a variable or quantity bearing on the function to be optimized. The scheduler defines a utility function for each user, describing the benefit associated with serving that user.  FIG. 2  plots a typical utility function U 1 (r) for the i th  user as a function of data rate r in kbps (solid line), and further plots a bounded utility function U 2 (r) (dashed line) that is non-differentiable because of the upper limit placed on U 2 (r). By bounding U 2 (r), the gain in revenue or benefit as generated by evaluating U 2 (r) for individual users is capped at a given upper threshold. This has, in this example, the effect of reducing the incentive to serve users having an already high average served rate r. 
     In a conventional approach to scheduling, such as in a proportional fair approach, the same U(r) function is associated with all users, and certain restrictions are placed on U(r). For example, because proportional fair scheduling is essentially a gradient-based approach, the utility function U(r) must be differentiable at all points. Thus, use of the bounded U 2 (r) would not be possible with any of the common gradient-based scheduling approaches. Use of bounded utility functions is discussed more fully below in the context of “barrier functions.” 
     If the objective function is non-separable, partial differentiation of the objective function as required in gradient-based approaches may require overly complex calculations. Avoiding this complexity limits the opportunity to use different utility functions to support groups of users at different scheduling priorities, and constrains the types of utility functions that may be used. This separability limitation is discussed more fully below, particularly in the context of scheduling users based on differing user classes and in time-of-day based scheduling. 
     A further drawback to proportional fair scheduling in HDR networks arises from the essential operation of that type of scheduling. With proportional fair scheduling, an objective function F is based on the set of utility functions U(r) associated with the users. For N users, the objective function F exists in N-dimensional space. Thus, at each point along the objective function F, one can move in any one of N directions. At each scheduling interval, a proportional fair scheduler serves the user that moves the objective function in the direction of the steepest gradient. Because the user corresponding to that direction must be served by the amount requested in accordance with the DRC information for that user, the magnitude of movement in the chosen direction is set by the AT  18  and not by the scheduler. If the objective function F is not monotonic, moving in the direction of steepest gradient may actually reduce rather than increase the objective function. 
       FIG. 3  illustrates an objective function F that is not monotonic, and is used to illustrate the consequence of considering only the gradient when selecting the user to be served. Here, moving in the direction of the steepest gradient by the amount required to meet the selected user&#39;s DRC request actually moves the objective function F(r) past a peak value, and on downwards on its reverse slope. Because the network  10  must serve the selected user by the requested amount, the change in objective function F(r) goes from a starting point P 1  to an ending point P 2 . Here, with a non-monotonic objective function F(r), the change ΔF in the objective function F is actually negative. That is, the net change in the objective function F arising from the decision to serve the i th  user in the particular service interval illustrated resulted in an overall decrease of F. While this i th  user might correspond to the steepest gradient, the user does not correspond to the optimum service selection. Here, the primal ascent approach to scheduling would identify the undesirability of serving this user, where a gradient-based approach would not. 
     Thus, the primal ascent technique in the present invention evaluates the net change in an objective function F that would result from serving each user by the required amount for a given service interval. The starting value of the objective function (point P 1 ) is the same for all users before the next user is chosen for service, so the primal ascent approach evaluates the ending points (points P 2 ) for all users and picks the user corresponding to the most favorable ending point. 
     Essentially, then, primal ascent scheduling assigns one or more types of utility functions to users that are competing for the shared resource (e.g., the HDR air interface  16 ). These users may be thought of as candidates for service. By defining the utility function or functions assigned to the candidates as benefit functions, these functions define benefit curves associated with serving the candidates. As noted, all candidates may use the same curve, or different candidates may use different curves (functions). 
     The average served data rate of a given candidate defines a current point (value) on that candidate&#39;s benefit curve. Selecting that candidate for service in a given service interval moves that candidate positively along the benefit curve, but also causes the other candidates to incrementally move negatively along their respective benefit curves. This latter result arises from not serving these other candidates during the service interval. Note that “positive” and “negative” as used here are relative terms and their absolute sense depends on the nature of the particular utility functions involved. 
     Because the objective function depends on the overall set of utility functions associated with the candidates, its value changes as the aggregate of changes in the utility functions. At a given scheduling decision point, the value of the objective function F is based on the set of average throughputs  (n) from the most recent time slot n. This objective function value may be considered a starting point. Then, for each user, the primal ascent approach determines the overall change in the objective function F for each user by comparing the ending point that the objective function F will move to if that user is served. 
     Therefore, as noted, primal ascent considers the net or overall benefit that would be gained by serving each of the candidates at the expense of the other candidates for at least a portion of a given service interval, and serves the candidate having the most favorable or desirable net change. The net change may be considered a “selection metric.” That is, selecting different candidates for service results in different values of net change in the objective function. The process or step of selecting a user depends on identifying the best or most favorable net change hence this may be thought of as considering each candidate&#39;s selection metric. 
     By evaluating net or aggregate change, the primal ascent approach works to move the objective function F towards a maximum value with each service decision made, and removes the restrictions otherwise placed on the utility and objective functions. That is, by evaluating net changes in the objective function F, the objective function need not be monotonic, and the underlying user utility functions need not differentiable, and need not be separable. A further advantage detailed later is that the utility functions assigned to the users may be modified using so-called barrier functions that place constraints on upper or lower throughput such that users or selected groups of users may be biased toward desired average throughput values. 
     Consider the HDR case of a separable objective function F comprising a sum of strictly concave differentiable utility functions. The change in the objective function F if user j is chosen may be denoted as, 
               Δ   ⁢           ⁢       F   j     (       r   ρ     ⁡     (   n   )       )       =       [         U   j     ⁡     (         (     1   -     1     τ   c         )     ⁢       r   j     ⁡     (   n   )         +       d   j       τ   c         )       -       U   j     ⁡     (       r   j     ⁡     (   n   )       )         ]     +       ∑       i   =   1     ,     i   ≠   j       k     ⁢     [         U   i     ⁡     (       (     1   -     1     τ   c         )     ⁢       r   i     ⁡     (   n   )         )       -       U   i     ⁡     (       r   i     ⁡     (   n   )       )         ]                 (   1   )             
 
Where U j (r) and U i (r) are utility functions for the j th  and i th  users, respectively, r j (n) and r i (n) are average throughputs of the n th  time slot or service interval for the j th  and i th  users, respectively, and τ c  is a throughput filter time constant.
 
     The primal ascent approach selects the user j* that maximizes the net change in the objective function F. This selection may be expressed mathematically as follows, 
               j   *     =           a   ⁢           ⁢   r   ⁢           ⁢   g   ⁢           ⁢   m   ⁢           ⁢   a   ⁢           ⁢   x     j     ⁢     {     Δ   ⁢           ⁢       F   j     (       r   ρ     ⁡     (   n   )       )       }       =         a   ⁢           ⁢   r   ⁢           ⁢   g   ⁢           ⁢   m   ⁢           ⁢   a   ⁢           ⁢   x     j     ⁢     {         U   j     ⁡     (         (     1   -     1   /     τ   c         )     ⁢       r   j     ⁡     (   n   )         +         d   j     ⁡     (   n   )       /     τ   c         )       -       U   j     ⁡     (       (     1   -     1   /     τ   c         )     ⁢       r   j     ⁡     (   n   )         )         }                 (   2   )             
 
where j* corresponds to the user that yields the greatest net change in the objective function F.
 
     If proportional fair scheduling were applied to (2), with the utility function defined as U(r)=1/r, the result would be, 
                 j   *     =         a   ⁢           ⁢   r   ⁢           ⁢   g   ⁢           ⁢   m   ⁢           ⁢   a   ⁢           ⁢   x     j     ⁢     {         d   j     ⁡     (   n   )           r   j     ⁡     (   n   )         }         ,           (   3   )             
 
which represents choosing the user having the largest ratio of requested data rate to past average data rate (i.e. requested rate to average throughput). For this particular U(r), primal ascent scheduling yields the same result (i.e., selects the same user j) as when the proportional fair scheduler is used. However, primal ascent scheduling arrives at this result by considering the actual changes associated with the different service scenarios, rather than simply choosing the user with the steepest gradient.
 
     Primal ascent scheduling also yields an equivalent result to the so-called maximum C/I scheduling algorithm when the maximum C/I utility function U(r)=r is used. 
     However, primal ascent may be used where neither proportional fair nor maximum C/I scheduling is suitable. For example, consider the more general problem,
 
               maximize   ⁢           ⁢     F   (     r   ρ     )       ,           (   4   )                   subject   ⁢           ⁢   to   ⁢           ⁢       ∑     i   =   1     k     ⁢     r   i         &lt;   C     ,           (   5   )             
 
over  r   i   εS,   (6)
 
where C represents an aggregate throughput capability or overall throughput limit for the portion of the network  10  supporting the set of users, and S represents the set of possible or defined data rates supported by the network  10 .
 
     As before, unlike proportional fair or maximum C/I scheduling, the objective function F need not be differentiable or monotonic.  FIG. 4  presents exemplary logic for the primal ascent approach to the scheduling problem framed by Eqs. 4–6. 
     Processing begins (step  100 ) with initialization of the relevant variables  ,  ,   (step  102 ). Here,   represents the set of average throughputs for the users (i.e., ATs  18 ) being scheduled. Thus,  (n) represents the value of the average served rate vector set in the n th  time slot. If there are N users being scheduled,  (n) comprises the individual average throughput value r j (n) for j=1 to N. Similarly,  (n) represents the set of requested data rates for serving individual ones of the users in the n th  time slot, and is referred to as the user request rate vector. The vector    j (n) represents the served rate vector at the start of time slot n+1 if the j th  user is served in slot n. That is, the conditional served rate vector    j (n) represents the set of vectors over which the objective function F(   j (n)) is to be maximized. In this example, the primal ascent approach evaluates net changes to determine which user j should be served to achieve the greatest gain in the objective function F(   j (n)). 
     After initialization, looping begins (step  104 ), with the algorithm repeatedly executing in an essentially infinite loop, repeating its objective function maximization over successive scheduling intervals by selecting the best user to serve at each scheduling decision point. In this context, a scheduling decision point is any point in time at which the primal ascent scheduler is required to make a scheduling decision. As will become clearer later, the scheduling intervals between scheduling decision points variably span a single and multiple HDR forward link time slots. Thus, in some circumstances, the scheduler makes scheduling decisions at each time slot, and in other circumstances, the scheduler makes scheduling decisions at multiples of individual time slots. With the dynamic nature of the radio environments associated with the users being scheduled and with changes in the constituency of the candidate set of users, the span of scheduling intervals is generally an ever-changing value. 
     In HDR networks, the number of time slots required to serve a given user depends on the requested data rate associated with that user. Thus, the decision to serve a given user commits the network  10  to serving that user for the required number of time slots. For example, at a given scheduling decision point, the primal ascent scheduler may select user x for service. Given user x&#39;s requested data rate, this decision requires the allocation of a given number of future time slots for serving this user. If past users previously selected for service have remaining allocated time slots, the scheduler may interleave user x&#39;s needed time slots with those previously allocated. 
     Further complicating this scenario, any one of the users which are currently allocated future time slots may report that some number of those future time slots is no longer needed as a result of successfully receiving data in a fewer number of time slots than was originally allocated. This circumstance might arise from radio conditions improving at the user&#39;s location. 
       FIGS. 5A and 5B  reflect some of these scheduling complexities. In  FIG. 5A , the scheduler identifies the best (most favorable outcome in F) candidate for service (User  1 ), and the three next-best candidates for service (Users  2 – 4 ) at a given scheduling decision point. Here, all of the users selected for service require the same number y of time slots for delivering the associated data. This fact allows the scheduler to allocate 4y time slots in interleaved fashion. The next scheduling decision point is then fixed 4y time slots in the future, unless circumstances change, such as an early completion of data delivery to one or more of the users being served. 
       FIG. 5B  illustrates a circumstance where time slots for a newly scheduled user x must be interleaved with time slots previously allocated to a set of earlier scheduled users. At an earlier scheduling decision point, the scheduler allocated a given number of future time slots to each of Users  1 – 4 . In this example, User  4  needed fewer time slots than Users  1 – 3 , and therefore the scheduler had an opportunity to interleave time slots for user x with the remaining time slots allocated to Users  1 – 3  at the point where scheduled service to User  4  ended. This example illustrates that the scheduler may have multiple, overlapping scheduling intervals involving different users or groups of users. Thus, depending on the staggering of these overlapping intervals, the scheduler may make scheduling decisions on a per-time slot basis, or less frequently, depending on exigencies of the moment. 
     With the above scheduling subtleties in mind, the discussion returns to the logic of  FIG. 4 . Within each time slot n, the average served rate vector and user request rate vector,  (n) and  (n) respectively, may be updated for the current time slot (step  106 ). Updating the average served rate vector  (n) comprises calculating the average throughputs associated with the users being scheduled in light of which user was served in the last slot, which calculation may be expressed in exemplary form as, 
                 r   i     ⁡     (     n   +   1     )       =     {               ⁢         (     1   -     1     τ   c         )     ⁢       r   i     ⁡     (   n   )         +         d   i     ⁡     (   n   )         τ   c                     ⁢       if   ⁢           ⁢   user   ⁢           ⁢   i   ⁢           ⁢   was   ⁢           ⁢   served   ⁢           ⁢   in   ⁢           ⁢   the   ⁢           ⁢     n   ⁢   th     ⁢           ⁢   time   ⁢           ⁢   slot     ,                     ⁢       (     1   -     1     τ   c         )     ⁢       r   i     ⁡     (   n   )                     ⁢     otherwise   ,                       (   7   )             
 
where τ c  is a time constant associated with filtering the computed value of the average served rate vector  . Updating the user request rate vector   comprises updating requested rate values based on DRC information received from or associated with each of the users in time slot n. Provisions may be made to accommodate instances where new DRC information is unavailable for one or more users, such as by using a past value or a default value.
 
     Next, the scheduler determines if a scheduling decision is required at this point in time (step  108 ). If not, the scheduler continues in accordance with the previously determined scheduling decisions and serves the appropriate one of the previously scheduled users. However, if a scheduling decision is required, the scheduler initializes its candidate user index variable j (step  112 ) in preparation for determining the best user to serve. 
     In making this determination, the scheduler evaluates F(   j (n)) for each one of N service scenarios, where there are N user candidates (step  114 ). The scheduler tracks and stores, or otherwise maintains the results of that evaluation (step  116 ) so that the user j* corresponding to the greatest desired change in the objective function F(   j (n)) can be identified after evaluating all candidates. If there are more candidates to be evaluated (step  118 ), the user index variable is incremented (step  120 ), and the evaluation process repeats for the next user (steps  114 – 116 ). 
     One subtlety in the process should be noted here. If there are Ntotal users and y users still have future allocated time slots arising from prior service selections, the actual number of users that are evaluated as possible candidates for service is N−y. Thus, it should be understood that users already being served are not considered in making the current scheduling decision. The index variable j can be manipulated such that it skips those values associated with any user already being served. 
     After evaluating all users eligible for scheduling, the scheduler then selects the best user (i.e. user j*) corresponding to the maximum desired change in the objective function F(   j (n)) (step  122 ). Processing then continues (step  124 ) by looping to the next time slot. 
     As users are selected for service by the scheduler, it was earlier noted that the scheduler might apply overall constraints to its scheduling operations. For example, the scheduler might keep all users above a minimum data rate, assuming acceptable radio conditions for individual users. Further, the service provider might wish to avoid “over-serving” individual users, and thus the benefit of serving a user at an already high average throughput might be made to fall off according to some desired barrier function. 
       FIG. 6  illustrates the use of upper and lower barrier functions as applied to an underlying utility function U(r). Here, it is desired to define an objective function U i (r i ) that maintains the rate of user i close to some target rate {tilde over (r)}, while still achieving some diversity gains by allowing fluctuations around this target value. 
     The following utility function may be used to accomplish this desired action and is expressed as,
 
 U   i ( r   i )= r   i −η( r   i   −{tilde over (r)}   i ) 2 ,  (8)
 
where {tilde over (r)} i  is the target rate for user i, and where η may be configured as desired. When η is large, and if r i &lt;&lt;{tilde over (r)} i , then an increase in r i  (i.e., serving user i) results in large increases of the objective function, thus encouraging service to that user. Conversely, if r i &gt;&gt;{tilde over (r)} i , then an increase in r i  results in large decreases of the objective function, thus discouraging service to that user.
 
     As evident from the graph, the original utility function U(r) (solid line) was defined as a straight linear function of average throughput r. However, with the application of lower and upper barrier functions, the utility function U(r) (dashed line) takes on pronounced curvatures at lower and upper throughput rate thresholds R 1  and R 2 , respectively. These curvatures modify the net change in the objective function associated with serving users that fall into the regions of U(r) modified by the barrier functions. That is, choosing to serve users with average throughputs r i  that fall below the lower threshold R 1  result in a relatively larger positive net change in the objective function. 
     Conversely, choosing to serve users with average throughputs r i  that are above the upper threshold R 2  result in a relatively smaller positive net change in the objective function. This behavior has a tendency to favor users with low average throughput and disfavor serving users with high average throughput, resulting in the scheduler attempting to bias at least some users toward an acceptable middle range of data throughput. 
     Because primal ascent allows the scheduler to use different utility functions U(r) for different sets of users, some users may be scheduled according to different priorities. For example, the service provider may define different utility functions corresponding to differing qualities of service (QoS), and then assign these different utility functions to different classes or groups of users. 
     As a further illustration of the application of barrier functions to utility functions, a utility function may be defined as, 
                 U   i     ⁡     (     r   i     )       =     {               ⁢     r   i                 ⁢         if   ⁢           ⁢     r   i       &gt;     r   i   min       ,                     ⁢       γ   ⁢           ⁢     r   i       +       r   i   min     ⁡     (     1   -   γ     )                     ⁢   otherwise                     (   9   )             
 
where γ represents some biasing factor having a configured magnitude corresponding to the extent by which low-rate users are to be biased towards a higher data throughput. The service provider may thus set γ to a fixed value, make it configurable on a per-user basis, configure it to change as a function of time or other parameter, or any combination of these techniques. Of course, the utility function U i (r i ) may also have barrier function modification for r max  similar to that shown for r min  in Eq. 9. Note that the utility function of Eq. 9 is not differentiable at r i =r i   min  and gradient-based scheduling approaches could not be easily used with this function.
 
     In addition to the flexibility of handling non-differentiable utility functions, the primal ascent approach accommodates non-separable functions. Suppose that the service provider wishes to restrict the amount of bandwidth used by a particular class of users denoted by Q. One approach is to introduce a hard constraint where users in the class Q are not scheduled if their total average served rate exceeds a target value set by the service provider, or otherwise configured by the scheduler. However, suppose that only users of the class Q are using the network  10 , such that the possibility that there will be service intervals during which no user is served. 
     The primal ascent approach can handle this scenario by falling back to a more relaxed constraint governing the service of users. However, because primal ascent techniques relieve many constraints on allowable utility and objective functions, one might incorporate a barrier function into the objective function F as, 
                 F   b     ⁡     [       r   ρ     ⁡     (   n   )       ]       =     {               ⁢   0               ⁢       i   ⁢           ⁢   f   ⁢           ⁢       ∑     i   ∈   Q       ⁢     r   i         &lt;     r   max                       ⁢     -       (         ∑     i   ∈   Q       ⁢     r   i       -     r   max       )     2       )               ⁢   otherwise                     (   10   )             
 
The expression for the objective function given in Eq. 10 above has no effect on the objective function when the soft constraint is satisfied, but causes the objective function to decrease in value when the soft constraint is violated. If only users of the restricted class Q are using the network  10 , then the barrier function has negligible effect on the scheduling decisions made. Note that the above objective function (including the barrier function) is non-separable and therefore potentially difficult to use with gradient-based schedulers.
 
     A further point of flexibility involves time-dependent user scheduling. That is, the service provider may want to use different scheduling priorities at different times during the day. Suppose that a combination of M different functions (possibly including barrier, reward, utility, and revenue functions) are required or desired. This set of functions may be denoted by F( ). 
     Let T denote the time of day, and let 
       {             α   ρ     ⁡     (   T   )       ∈     R   m       |       α   i     &gt;     0   ⁢           ⁢     ∀     i   ⁢           ⁢   and   ⁢           ⁢       ∑     i   =   1     m     ⁢       α   i     ⁡     (   T   )                   =   1     }       
 
denote the weighting to be assigned to each function for each time of day T (e.g., for each hour). One may then use the following time-dependent objective function F to in user scheduling,
 
 F ( )=Σα i ( T ) F   i ( )  (11)
 
where α i  represents the weighting factor for F i , and   represents the average served rate vector as discussed in conjunction with the logic flow diagram of  FIG. 4 .
 
     The service provider or network operator might set different weighting factors α i  for different times of the day to effect changing scheduling priorities, and might further apply different sets of weighting factors to different classes of users. 
     In general, the primal ascent scheduling techniques of the present invention are advantageously used to schedule users sharing the air interface in a HDR network. By calculating the net change(s) in the desired objective function(s) that would arise from serving each user, the primal ascent approach identifies the user to whom service would yield the greatest or most desirable benefit. Because primal ascent evaluates the net change (i.e., it considers the affect of actually providing the user, if selected, the requested amount of data), primal ascent allows great flexibility in the types of underlying cost or benefit functions that may be optimized. 
     Of course, the above time-of-day details, like the earlier general developments of the primal ascent approach, are subject to significant variation. As such, the present invention is not limited by the foregoing details rather the present invention is limited only by the scope of the following claims, and the reasonable equivalents thereof.