Abstract- Opportunistic scheduling algorithms are effective in requirements of users at each schedulingepoch. As explained exploiting channel variations and ...
Two Phase Scheduling Algorithm for Maximizing the Number of Satisfied Users in Multi-Rate Wireless Systems Sourav Pal, Preetam Ghosh, Amin R. Mazloom, Sumantra R. Kundu and Sajal K. Das
Abstract- Opportunistic scheduling algorithms are effective in exploiting channel variations and maximizing system throughput in multi-rate wireless networks. However, most scheduling algo-
requirements of users at each scheduling epoch. As explained in Section I-A, it is not possible to provide such delaysensitive scheduling using existing scheduling techniques. It
and try to allocate resources (i.e., the time slots) among multiple users. This leads to a phenomenon commonly referred to as the exposure problem wherein the algorithms fail to satisfy the minimum slot requirements of the users due to substitutability and complementarity requirement of user slots. To eliminate this exposure problem, we propose a novel scheduling algorithm based on two phase combinatorial reverse auction with the primary objective to maximize the number of satisfied users in the system. We also consider maximizing the system throughput as a secondary objective. In the proposed scheme, multiple users bid to acquire the required number of time slots, and the allocations
should be noted here that the challenges associated with such delay-sensitive schedulinghave been extensively studied in the context of wired networks (see [5] and references within). However, the solutions applicable to wired networks cannot be directly ported to wireless networks because of the funda-
rithms ignore the per-user quality of service (QoS) requirements
are done to satisfy the two objectives in a sequential manner. We provide an approximate solution to the proposed scheduling problem which is a PP-complete problem. We prove that our proposed algorithm is (1 + log m) times the optimal solution, where m is the number of slots in a schedule cycle. We also present an extension to this algorithm which can support more satisfied users at the cost of additional complexity. Numerical
results are provided to compare the proposed scheduling algorithms with other competitive schemes.
mental difer physical layer
transmission behior thatstem from transmission characteristics. Also, the wireless data systems support incremental error-correction mechanisms, Medium Access Control (MAC) layer retransmission of lost packets, and multi-rate transmission capabilities; all
wiresch antl
o wireless channel.
A. Related Work Most of the existing opportunistic scheduling schemes suffer
from a syndrome popularly referred to as the exposure problem
[6] in auction theory. This refers to the phenomenon where a bidder who bids straightforwardly according to his demand I. INTRODUCTION schedule is exposed to the possibility that he may wind up Opportunistic scheduling in wireless networks first intro- winning a collection of slots that he does not want at the prices duced in [1] continuously monitors the uncertainty of the un- he has bid, because the complementary slots have become derlying wireless channel and takes decisions opportunistically too expensive. Such a situation arises when the minimum so as to optimize the objective functions under consideration. data requirement of the users are not met. Since opportunistic Extensive research has been conducted with varying objec- scheduling algorithms make their decisions on a slot-by-slot tives e.g. maximizing the system throughput [2], maintaining basis, they fail to provide the users with the minimum amount both long and short term fairness among users [3][4], and of requested data till the very end of the schedule cycle. maximizing the user utility [1]. In general, the goal is to Such limitations in scheduling decisions negatively impact maximize a concave utility function representing the above the performance of delay- sensitive applications. The schedulobjective functions. Unfortunately these concave functions fail ing algorithm is an important component that determines to capture the importance of the timelineness of decision the performance of multi-rate wireless systems supporting real-time data streams. The scheduler not only needs to be making in user scheduling. However, the next-generation multi-rate wireless data net- aware of the wireless channel conditions but also of the works such as Evolution-Data Optimized (1xEV-DO), High QoS requirements of the users. In the literature, significant Data Rate (HDR), Enhanced Data Rates for Global Evolution research focussing on the varied issues such as user fairness (EDGE) promise to provide data services and applications [3][7], throughput maximization [8][9], and efficiency [4]. with strict timing constraints. Examples of such applications Existing opportunistic scheduling algorithms exploiting timeinclude streaming multimedia, voice-over-IP (VoIP), Instant varying channel conditions concentrate mainly on throughput Messaging (IM), and real-time video conferencing; all of maximization while satisfying other QoS requirements. In which demand that packets be delivered within certain delay [8][9], the authors have shown how CDMA-based HDR sysbounds so as to comply with the application level Quality of tems can maximize the system throughput while maintaining Service (QoS). Thus time constraint scheduling is a necessity "proportional fairness" amongst users. Similarly, in [4] it for delay sensitive applications, has been shown how the opportunistic problem for multiWe advocate that the objective of scheduling is not only channel scenario with resource constraints can be formulated to improve the throughput of the system and enforce fairness along with a scheduling scheme that aim to provide fairness among participating users, but also to meet the minimum data among users. The work in [1] considered techniques that exploit the wireless channel conditions while guaranteeing Sourav Pal, Preetam Ghosh, Amin R. Mazloom, Sumantra R. Kundu and
each user a predetermined time share in a schedule cycle.
Networking (CReWMaN), The University of Texas at Arlington, Arlington, TX 76019. E-mail:{spal, ghosh, mazloom, kundu, das}@cse.uta.edu.
In [10], a bandwidth pricing mechanism was proposed that solves congestion related problems in wireless networks. This
Sajal K. Das are with the Center for Research in Wireless Mobility and.
1-4244-0992-6/07/$25.00 © 2007 I EEE
..
scheme, is based on second-price auctions and shows how the allocation of resources maximizes social welfare. This work was subsequently extended in [11] for designing a pricing mechanism for the downlink transmission power in a CDMA based wireless system. In [12], an auction-based algorithm was proposed that allowed users to compete for time slots in a fading wireless channel. Using the second-price auction mechanism, the users in the system were allocated channel slots; the existence of a Nash equilibrium for such a strategy was proved by the authors. Later in [13], the Nash equilibrium strategy was found when the channels for two users are uniformly distributed. Thus existing opportunistic scheduling algorithms aim at maximizing the overall system throughput and do not focus on delay-sensitive requirements of the applications.
. Using mathematical analysis, we demonstrate that the proposed scheme is capable of supporting more users with hard real-time requirements than the existing scheduling schemes. There also occurs significant gain in the system throughput achieved using our approach. . We prove that the worst case performance of the proposed algorithm is bounded and is (1 + log m) times the optimal solution, where m denotes the number of slots in a schedule cycle. We also have derived the time complexity. . We present an extended algorithm that increases the number of satisfied users at the cost of additional complexity. . We propose a design parameter, a, that determines the trade-off between guaranteeing user utility (a measure for user satisfaction defined in Section III) level and system throughput. The variation of system capacity with the number of satisfied users for different scheduling algorithms is also shown. The rest of the paper is organized as follows. In Section II, we formulate the scheduling problem for a multi-rate, multiuser time-slotted system. Section III models the user utility, identifies the exposure problem and studies its implications on user utility. The mapping of combinatorial auction to multirate scheduling and the scheduling algorithm is presented in Section IV. We analyze the proposed algorithm in terms of algorithmic complexity and compare its performance with existing schemes in Section V. Numerical results are presented in Section VI followed by conclusions in the last section.
B. Contributions In this paper, we take a fresh approach to delay-sensitive scheduling problems by borrowing techniques from [14] auction theory. We consider a cellular network with one base station and multiple users. The resources available to the base station (e.g., time slots, frequency bands, codes) form the goods which are sold to the users in a market-like environment. The users value these goods distinctively and express the values in terms of a common transaction unit called money'. In order to satisfy the minimum data requirement, each user demands a certain number of slots (a bundle which is equivalent to the number of slots satisfying minimum data II. PROBLEm FORMULATION requirement) which correspond to the minimum data requireIn this section, we describe the system model under conment that must be satisfied within a specific schedule cycle. The number of such slots required depends on the condition of sideration and qualitatively formulate the scheduling problem. the underlying wireless channel. Since the market has multiple We also define the objective functions for optimal scheduling. indivisible goods and each user's individual valuation of the Prior to that, auction theory basics are briefly described. goods depends on the bundle of goods received, we formulate the scheduling problem as a specific case of combinatorial . . auction. This is due the fact that a single item transaction of Auction is the process of buying and selling goods by the goods do not suffice since the user is more interested in the sum total of the data received. This underlying condition offering them up for bid (i.e., an offered price), accepting is exactly the reason why single slot allocation approach bids, and then selling the item to the highest bidder [14]. In is not appropriate for delay-sensitive applications in multi- economics, an auction is a method for determining the value rate wireless systems. Consequently, schedulers based on the of a commodity that has an undetermined or variable price. principles of opportunistic scheduling are unable to satisfy In some cases, there is a minimum or reserve price; and if the minimum data rate constraints demanded by the users. the bidding does not reach the minimum price, no transaction In contrast, our proposed scheduling scheme which is based between buyers and sellers is executed. Most of the auctions on combinatorial auction deals with multi-slot allocation and are primarily forward auctions which involve a single seller is capable of satisfing the minimum data rate constraint of and multiple buyers. The buyers compete among themselves individual users. We formulate a combinatorial reverse auction in order to procure the goods of their choice by placing an based multiple slot scheduling scheme that guarantees the initial bid that they feel is an appropriate price for the item minimum requirements of the users. The main contributions under consideration. However, in reverse auctions, the role of the buyers and seller are reversed. A buyer places a request of this paper are as follows: *We show that most of the existing scheduling algorithms to purchase a particular item and multiple sellers bid to sell have the exposure problem and hence fail to guarantee the requested item. The winner of a reverse auction is the the minimum data requirements of admitted users. seller who offers the lowest price. Sometimes, the bidders are . ~~~~~~~~~interested in bidding for multiple items at the same time. In . We use combinatorial* reverse auctions to formulate theg scheduling problem with two different objectives: (i) to such a combinatorial bid, the bidder offers a price for the collection ofgoods according to the choice of the bidder rather guaante mnium te atarat ofth than placing a bid on each individual items separately. This maximize the overall system throughput. results in combinatorial auction where the auctioneer selects a
usrs,and(ii.t
1We use the concept of "money" as a tool for designing the resource allocation problem and as such has no significance in real life,
set of combinatorial bids that provides the maximum return in revenue without assigning any item to more than one bidder.
m slots in the Schedule Vector / / ithSlot
Schedule Decision
thSlot. \I
TABLE I NOTATIONS USED IN AUCTION BASED SCHEDULING
n scheduled users in the schedule cycle
Notation Meaning m Number of slots that define the schedule window Set of slots available for auction M Set of users admitted by the system nt IV~~~~~~~~~~~~~7 Set ot satisfied users * 0 u * CriJ /Possible transmission bit rate for slot i supported by user j
00
"/'#""'#UserlUser\Useri
Sched le Cycle Schee CYcle
Rate r {i,1} for User 1 Rate r_{i,2} for User 2
\~ ,(x + y) (10) Opportunistic scheduling mechanisms concentrate on the current slot to be scheduled and base their decision on an objective function. Such schemes do not consider prior or future allocations and are thus unable to capture the complementarity and substitutability effect between the
slots,
IV. MULTIPLE SLOT SCHEDULING AND COMBINATORIAL REVERSE AUCTIONS In this section, we first highlight the equivalence between the optimal resource allocation problem in multi-rate wireless systems and combinatorial auctions and then derive the mapping between the two. When the objective in a market is achieved, such as value minimization for buyer in reverse auctions, the market is said to be in an equilibrium state. The market equilibrium corresponds to the optimal schedule as defined in Section II-D where the goods map to the time slots and the objective is to satisfy equation (8). Under such circumstances, the combinatorial auction problem can be formulated as follows: We model the wireless base-station as the buyer who wants to procure m slots (let M ={1, 2,... ,m} denote the set of slots available for auction) and the N users are the sellers each having m slots of different values (a.k.a. data rates). The prices (pj(S)) that the users quote for the bundle of slots depend on the utility (uj(S)) that is derived by the user when the base-station procures those slots. We formulate the utility as follows:
us(S
S 1 such that >ZScM W(S, j) < 1
lw(S,j) ={o i} ..
(13)
J
Vi C M Vj C N VS CM, Vj C N
(14)
condition ensures that overlapping sets of items are never assigned as where the second one ensures that no The first
bidder receives more than one subset. A careful observation of Equation (14) reveals that the formulation is identical
to
the
obective roblem defined in equation (8) (8. The solution ective eqwin tysystem the problemis qefing in the winner to the above problem nothing but determination
detsolution
other to sell the set of slot slots to the base-station. They ar are oer o sete se of to the base-station deprived of some value if they cannot get the base-station to
buy the slots from them. Identifying the deprivation function of slots for any user. Thus, is essential for deciding the best set function the quoted price for any bundle is of the deprivation fucIo.
function.
l
a
A. Deprivation Function Th oectiof the
t
obtain the desired utility. The buyer (base-station) buys the set of y slots only if the slots honor the specific minimum value of Dmiri. Failure to sell the slots deprives the user of the
minimum utility. Hence, a deprivation value is associated with the set of slots not acquired by the base-station from that user. The deprivation function depends on two factors: (i) The utility derived by the user by giving the bundle to
the base-station. (ii) The throughput loss the base-station may experience while procuring the bundle. The utility that user gets from a bundle of slots, S, depends on the type of applications. Followig equation (9), the utility function can be defined: Viu = V(Ui), where V(.) is a value function mapping the utility to equivalent money metric. Similarly, the throughput loss that the system experiences by acquiring slot i from a user j is calculated using equation (6). Therefore, we can define the deprivation function for a slotbundle S as: (15) P(S)= /3VU(S) + (1 - )VL(S) Here /3 is a control or tunable design parameter that controls the relative weight of the two attributes. For /3 = 1, the deprivation function basically boils down to only user satisfaction guarantee whereas for /3 = 0, the system considers only throughput maximization.
(11) B. Mechanism for Reverse Auction
where X is the set of factors determining the overall utility of a bundle, cg is the weight for agiven factorx, and xxox 1. The term V>, (5) is the value of the factor x by allocating the bundle S to buyer. We define w(S, j) as:
wS, j) ,f1
The problem can be formally stated as:
(12)
We use the simple single-round sealed-bidfirst price combinatorial reverse auction mechanism. All the "asks" (or quotes) are submitted prior to a deadline and the slot allocation is achieved based on the set of "asks" received. Throughput would have been drastically penalized had the auction been non-incentive based. However, the equilibrium is not guaranteed for non-incentive combinatorial auctions [6]. In general, the price pj;(5), quoted by a user j quotes for the bundle
S is a function of the deprivation function Pj, whether the mechanism is incentive compatible or not. Thus:
ps(S)
=
fj(S)Pj(S)
(16)
6: 7: 8:
9:
Find j* such that p3 (S) is minimum and Aj C PERMITTEDSET if A* has a slot i, for which Ai = 1, but Oi O then Remove A from the PERMITTEDSET else
where, fj (S) is the price mapping function that defines the 10: Add A* to ACQUIREDSET Remove A* from the PERMITTEDSET relationship between the price and the deprivation function. 11: For all i C A*, make Oi = ° Since we have assumed incentive compatible auction mecha- 12: 13. Add j* to SATISFIEDSET nism, fj(S) =-I,Vj and VS. Increment k by 1 The solution to the winner determination provides the 15: desired schedule vector. Hastad et.al [18] has shown that 16: Construct Ak from j C (N - SATISFIEDSET) in reverse auction, approximate solutions can be developed 17: end while inspite of the problem being JVFP-complete. We develop our 18: end while slot procurement algorithm along similar lines. However, due 19: Return ACQUIREDSET to primary and secondary objectives with conflicting goals, we decouple the algorithm into two phases. In the first phase, the D. Unrestricted Phase restricted phase, we compute the set of users whose minimum The residual slots aid in achieving the secondary objective requirement is satisfied. That is slots from as many users as of maximizing the utility of allocated users as well as maxpossible are acquired while requiring that each user is able to imizing the system throughput during the unrestricted phase. get rid of the minimum deprivation value. During the second The allocated users try to further minimize the deprivation phase the unrestricted phase, we allocated the residual slots value by selling their slots. However, unlike the restricted which cannot satisfy the Dmin for any additional user. phase, no restriction on slot bundle size exists. Note that none of the users whose minimum utility,( i.e., the minimum C. Restricted Phase deprivation value) have not been satisfied, are allowed to In this phase, multiple single round reverse auctions are compete in this phase. Also, in this phase, the slots may held till no additional user is able to get rid of the minimum not exhibit complementary/substitution relationship. Consedeprivation value. In each round, the system considers "asks" quently, the exposure problem explained earlier will not occur. from the users on the remaining minimal unallocated bundles. Under such conditions when the utility is assumed to be linear, This means that the bundle should just be able to get rid scheduling the residual slots can be performed by employing of the minimum deprivation value. Each user is allowed to any one of the existing opportunistic scheduling algorithms. On the contrary, if complementary/substitution effect exists provide an "ask" for only one bundle of slots Aj. From these initial "asks", the initial feasible bundles (Ak) iS constructed between the residual slots, the allocation should be performed using combinatorial reverse auction so as to overcome the for round k of the restricted phase. For each round, the reverse auction takes the following exposure problem. For the unrestricted phase, the "asks" are based on the further reduction of the deprivation value. The formulation: objective for the buyer i.e. (the system), is now set to choose minimize E E Pj(S)w(S, j) (17) the "asks" from the users, which minimizes its total price. jeN SeAk This guarantees throughput maximization for both the system as well as the chosen users. The auction proceeds similar to Ak the restricted phase but continues till all the slots have been ViC W(S,j) 1 > {EZ3S EjeN C exhausted. The only difference for the unrestricted phase is j) 1 such that < N W(S, Vj C SeAk the type of "asks" possible and the set of slots which are part VS C k, Vj C N tW(S, j) = {0: if of the reverse auction. The ACQUIREDSET obtained in the Once the minimum deprivation value of a user has been previous algorithm is used in the Unrestricted phase. Thus, satisfied in a certain round, the user is barred from taking part after the execution of the unrestricted phase algorithm, the in subsequent rounds of the auction process in the restricted ACQUIREDSET is updated which provides the distribution phase. Let the ACQUIREDSET denote the set of accepted of the slots for the schedule cycle under consideration. "asks" and each "ask" Aj is represented by a set of vector < A1 2' * m >, where A- is 1 if the ith slot is in the "ask" V. PERFORMANCE ANALYSIS for user j otherwise it is 0. Let PERMITTEDSET be the set of In this section, we make some observations about the permitted "asks". Let us define Oi such that 0 = 1 if the "th Slot has not been acquired otherwise 0 = 0. Let SATISFIEDSET proposed algorithms and prove them. be the set of users whose minimum deprivation value has been Theorem 1: The worst case running time for the restricted satisfied. The algorithm for the restricted phase is enumerated phase slot procurement algorithm is 0 (m2m) where m is the below: number of slots in each schedule cycle and n is the number Restricted Phase: Algorithm for Winner Set Determination
1: Initialize ACQUIREDSET =SATISFIEDSET =5, the current M andfro jE (N Consruc N-STIFEST 2: Ak X do 4: Initialize PERMITTEDSET =Ak 5: while PERMITTEDSET 7& X do
r:oundtruct foroal
OTIFE1ST
of users.
Proof: Assume m > n. The complexity of the algorithm in the restricted phase depends on the "ask" construction (line 2) and the selection of appropriate j (line 6). In the worst case, line 2 of the algorithm takes (n-k) (m -k) operations where k is the current round. Line 6 takes n-k operations in the worst case. The maximum number of rounds possible is n.
So, the worst case complexity can be given by 0(Zi-O [(n * k)(m - k) + (n - k)]) O(n2m). Corollary 1: The worst case complexity of the unrestricted phase slot procurement algorithm is O(n2m). If 02T is Lemma 1: Let 2(o) be defined as 2(o) = the total cost that the base-station pays for the optimal solution, then 2T < 1 P(0k) k+ -
m is an ordering of the slots based on where {oi}, i = 1, in which the sequence they are acquired by the base-station P() denoestthe heeffectve and 2(o) and denotes effective prie price of sot slot o. Proof: Let °k be covered when the "ask" Aj was picked by the algorithm. After, Ok, there are at least 1' - k + 1 slots to be covered. Since the optimal cost 02T covers all the I slots, it can also cover the remaining i'-k + 1 slots. So, there must be at least one "ask" whose average cost of covering is at most As our algorithm chooses the slots from lowest * to highest average cost per slot, P(ok) < OPT Theorem 2: The restricted phase slot procurement algorithm finds a solution that is within a factor (1 + In m) of the optimal solution where m is the total number of slots to be procured. Proof: The proof is similar to the one presented in [17]. Let I be the total number of slots that could be covered by the optimal solution and 1' be the total number of slots that are covered byby our algorithm. From Lemma lemma 1, thetheproof for the;, that atrofwere fore therem covera asrgoms 2 can be ourtlgoithm. theorem outlined as follows: Let the "asks", picked in the restricted phase that are able to get rid of the minimum deprivation value be denoted by Aj , Aj2, ... . Ajs, where js, is the last user whose bundle of slots was chosen. Zk-1 P(0k)- Using The total cost is given by Es Pj Lemma 1, the total cost can be written as: It + ... -1->.027.Hl/
reverse auction based scheduling and denote N0 to indicate users who have been satisfied by the opportunistic scheme. Then N, > N0. Proof: From Lemma 2, hi > hi, for all i whose minimum utility has been satisfied. By contradiction, let us assume N, < No. Then N,
+ ki) + I
(
i=l
ki
N,.
L(hi + ki)
=
i=l
(18)
satisfying of opportunistic scheduling to users whose minimum can be could not be satisfied. The above
and ki are the extra slots given to user i after
minimum utility and I is the total number of slots given in o.the the.cas the case
lO'P1
2 k5k=~Q1~~.07( 1 where Hil is the l'th harmonic number. Since
Hil
< I + In I' < I + In I < 1 + In m,
* the cost is bounded by (1 + In m) of the optimal. Lemma 2: Let hi be the number of slots obtained by user i in order to satisfy the minimum utility using our scheme and hi be the number of slots obtained using an opportunistic scheme. Then hi > hi,Vi C N Proof: In our scheme the slot allocation always tries to give the best available slots to any user so as to satisfy the minimum utility requirement at the minimum cost. hi is the minimum number of slots that is required to satisfy the minimum utility. Now consider an opportunistic scheme where the decision is based on a slot by slot basis. Consider an user whose minimum utility has been satisfied. If the user's best available slots comes in descending order of their individual utility value, then the user will reach the minimum utility level with the smallest number of slots. In this case, ri = j. Otherwise, the user may get another slot which is not the user's available slot. So, to satisfy the minimum utility, the user will require more or equal number of slots, than the minimum possible. In either case: ij * > ni. Theorem 3: Let Nc be the maximum number of users whose minimum utility has been satisfied by the combinatorial
utility
rewritten as: N,
equation
N,
(i- hi) + E
i
i=N,+l i=1 But this would imply that: N,
+I+
No i=Nc
N,
i
(i-ki) (19)
i=l
N,
5 E ki > E
(20) i i=1 i=N, This is clearly not possible since it would mean that our auction based scheme would have been able to accommodated U at least one more user using the ki's. Hence N > N_ a
A. Increasing number ofsatisfied users in the Restricted Phase Note that the optimization problem tries to minimize the price to be paid by the base station in scheduling the slots for the users. Though the price function is formulated such that it intuitively maximizes the number of satisfied users, it might be possible to satisfy more users albeit at a higher price to be paid by the base-station. The discrepancy arises because we cannot guarantee an optimal solution to the minimization problem in time. However, we can non-exponential explicitly besides the number of satisfied usersintroduce the criteria of maximizing
minimizing the price. The algorithm we propose next is similar
to the "Winner Set Determination" algorithm with a few modifications. We introduce the set of users affected by any potential slot assignments of the ith user in the kth round as AFFECTEDSETik. This set is computed as the intersection of Ajr (where j* is the user identified by the minimization criteria of Step 6 of the Restricted Phase algorithm) and Ak. The idea is for each existing (unallocated) user, we compute whether there are any slots in his demand list matching the slots to be allocated to user j* If so, the user will potentially be affected by the slot assignments to j* such that it might not be possible to allocate slots for that user later on. We also define a threshold T denoting the maximum number of affected users allowed at each step k for any slot assignment to user j*. When the number of affected users exceed T, we choose the next +j1ew that minimizes the price that the base station has to pay. We continue this process till we find a jlew that keeps the number of affected users < T. In the corresponding algorithm WSD-EXTn, it might not be possible to find a suitable jlew that keeps the number of affected users K T. Hence, we store the j* identified first and only overwrite it if we find a jnew having affected users < T. WSD-EXTn is exactly similar to the restricted phase algorithm with the following addition between steps 6 and 7.
Winner Set Determination (WSD-EXTn) 1: temp j* 2: for (i C N - SATISFIEDSET - j*) 3: Compute AFFECTEDSETki 4: ifl|AFFECTEDSET, >Tte 5: TEMPPERMSET = PERMITTEDSET - j* 6: Find j* such that p3 (S) is minimum and Aj C PERMITTEDSET 7: else 8: temp j*
9:
break
11: end for 12: j* = temp A similar analysis as before will actually show that WSDEXTn has run-time complexity of 0(n3m) and achieves results within (1 + In m) of the optimal as before. However, because it tries to explicitly increase the number of satisfied users, WSD-EXTn shows better performance than the previous algorithm at the cost of complexity.
TABLE II
Dmin vs. MAXimum NUMBER OF USERS 1[ Dmin(kbps) 1[ Max Satisfied Users 16 76-82 64 16-20 128 5-8
256
512
1-2
0-1
the case of opportunistic scheduling with temporal fairness, the
throughput is penalized. The throughput performance of the proposed auction based scheme is better than both the roundrobin and opportunistic scheduling with temporal fairness and is very close to the throughput maximization scheme. In order to visualize the working of the proposed scheduling scheme a temporal snapshot of 4 successive scheduling decision cycles is presented in Figure 6. With Dmin 128 Kbps and 15 users in the system, the first 3 schedule cycles yield the schedule vector as [1, 2, 4, 5, 6, 7,10]. But for the 4th cycle, user 7 is replaced by user 9. Although the allocated users are receiving the variation of the slot distribution between users is Dmin, du tthvayncanecodin. Th sceuigshm
VI. SIMULATION STUDY In this section, we present simulation results to study the effectiveness of our proposed scheduling scheme. In the process, we compare how the auction based scheme fares with respect judiciously distributes the residual slots after the restricted phase and does not allocate any more slots if Dmax is attained, to the two extremes scheduling discipline: round- robin and as is the case with user 10 in this example. Careful observation reveals that though all the allocated users were receiving maximization. throughputd Thystemthrveghpasbasispforeco W or more data, user 7 was receiving lesser slots in each Dmin study how each scheme performs in terms of the number of succeeding schedule cycle such that in the 4th cycle, user satisfied users and global system throughput. 7 was eliminated by user 9 in the restricted phase. Thus, the scheduler is intelligent enough to identify and allocate the user A. System and Channel Model which achieves the system objective. In each schedule cycle, We consider a single cell wireless data network for our all the allocated users are guaranteed Dmin. simulation as the scheduling schemes under evaluation are Next, we investigate the variation of system throughput designed to work best in the presence of a single base-station. and the number of satisfied users with the tunable parameter We also assume that all the users under consideration are oa, as defined in equation (15). The value of a depends on receiving real-time streaming multimedia traffic. In order to the wireless service providers objective to either maximize support multimedia traffic (MPEG-4 or H.263) of various throughput or guarantee user utility or a combination of both. qualities (low, medium, and high), we consider three values Hence, we evaluate system throughput and number of satisfied for Dmin 16 Kbps, 64 Kbps, and 128 Kbps. We model users by varying oa from 0 to 1. As expected, the throughput our simulation based on a HDR system that is capable of maximizes for a 0 whereas the number of users satisfied is supporting 11 different data rates with each schedule cycle maximized for a 1 as shown in Figure 7. WSD-EXTn in general performs better than the previous consisting of 1000 slots. We assume user mobility is random (both speed and direction) and employ the path-loss model and algorithm but is very sensitive to the value of T. Figure 8 shows the dependence of the number of satisfied users the slow log-normal model as in [19] for wireless channels. on T for a sample 1000 user system where we find that a value of T < 70 improves the number of satisfied users. B. Simulation Results We observed that greater the variation in the cardinality of For our proposed auction baed scheduling scheme, the AFFECTEDSETkI, better the performance of WSD-EXTn variation of system throughput with the number of users for in comparison to the previous algorithm. However, further different values of Dmin, is shown in Figure 3. As expected, work is required to quantify this variation which would govern the system throughput initially increases but ultimately gets the effectiveness of using WSD-EXTn over the previous saturatedwith the increase in the number of users. algorithm. Note that, if this variation is small, WSD-EXTn Next, we identify the maximum system capacity in terms of reports its worst case performance as it conceptually falls back satisfied users by setting Dmin to different values (Figure 4). to the previous algorithm. For each value of Dmim, we obtained a range of users who are satisfied by the system. This is shown in Table II. It is VII. CONCLUSIONS logical that for smaller Dmin a greater number users can be satisfied. The comparison of the system throughput achieved In this paper, we have proposed auction based schedulby various schemes is shown in Figure 5. As expected, the ing algorithms for allocating the slots in a time-divisioned, system throughput is best for the throughput maximization multi-rate wireless system. We have proved that opportunisscheme and worst for the round-robin scheduling algorithm. In tic scheduling algorithms that aim to maximize the system
x
105o'o 2.3
70
-~2.1 o
2-U
AX
-mPRPOSEDbSCH60
c 50
C) 40-E ciD
2 30 -U
1.8 1.7
J
--70 -e- ~~~~~~~~~~~~~Dmin=l128 kbps
1.9
U8)
- - ROUND ROBIN~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ONDRBI THROUGHPUT MAXM~ ~ ~~~~~~~~~~~~~~~~-* TROUHPT - PRPPRDSPEME-E -----HEME -*- TEMP. FAIRNESS~~~~~-- EMP FIRES
80
2.2
0D
-1
-20
1.6
10 0
10
20
30
40
50
60
Number of Users
70
80
90
100
0
20
60
V80
1,0 120 Number of Users
140
160
180
2
I
10
20
30
IIb----0
40
50
60
Number of Users
70
80
90
Fig. 3. Throughput vs. Number of Users in the System. Fig. 4. Performance of each scheduling scheme mea- Fig. 5. Throughput vs. number of users in the Notice how the throughput decreases with increase in sured using satisfied users as a percentage of the total system for different scheduling algorithms. Dmin.
users.
168 138 0 u3 ul u2
SCHEDULE CYCLE 1 132 134 ~~~~~~~~136 0 132 u4 u5 u6 u7 u8
2.46
0 u9
10o'9
6 ulO
20
SCHEDLJLEC CLE2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
242
,
~~~~~~~~~~~70
............. 260
2.42
ul
u2
u3
u4
u5
u6
160138 01381221441000 0198 u7 u8 u9uO
.
..........60.-~
I3850-)a)*i..23.....
SCHEDULE C CCLE3 2
60 II~~~~~~~~~~~~~~~~~I 40E ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~22ystm hroghutII t itII:
NM
236,
234
-a
0
Saife
30
sr
4E 2or0
a' I I0II
7 ,.nmbe
..a.....
U)~~~~~~~~~~~~ It 19Ciprve
158 138 0150 120 116 0 0 140 178 ~~~~~~~~ ~~~~~~~~~~232820 u3 u4 02030. 506 0.9 80 10 ~~~CHEULEu u7uC9YCLE. 1CO 203006 U9 is scheduledin favor of U7 in Schedule Cycle 4 Variation of cx Threshold~~~C/ Fig. . Slt Ditribuion fUsrs inScheule ycle ig. Satisfiedusers . Snpshot4: SstemThrouhputand.o..ofFig.8..Nmber.f.Saisfid.uses.vs.Thrshol vs. a~~~~~~~~~~~~~~~~4 J---020
ulu2
l
144100 0 2.36 a market whermultiple uses bid for thenumber of slos thtlrereuie t stifythi mniumQo rqirmets 2.34
CYCLE 4 process, we have beenable to achieve the primary objectiveZ of mximzinth numer f uerswhoe miimu slt
SCHEDULE
ordssceroduedifalvwit the7 i real-tie
sycheduligVreuirment.
E~~~~~~~~4453
[8] P. Bener, P. Black,M. Grob, R. Pdovani, N. Sidhushayana, ad A. Viterbi 'DA/DR ABadwdh-ffcintHihSpedWielssDtaSevie0o E~Nmai srs,IE CmuiatosMgaie ul 00 p.7-7 o IEEVeicla Tcnlgy Cofrne(VCSrn),20,vl 200-~~~Pocedng 3, pp. i
1854-1858.sfie users [0] P MalleandB. uffi, 'ult-bi Aucion fo Badwith Aloctio3inCom
oW [1]of
u,.Zegad.Mdao aWrees
Satisfied usutinelortm,rPoeeigsofAletoson
have dermaived theappoximloationraipoflmithe autonbased schedulin algorithmslthate usresls bin mortesatisfed ueslotha o that are equired o satisf their miimumQoSrequiremnts. With thehelpofcombinatrial
autions,
w
have
sown
how
ThaneshldAlctonUiga omunctin.Cnto,an
Computing, Apoacth200,volE24 Pp.on 1085-1096.Fb.199,vo. p Sunde,E.Moin Band, L. Zheng,RA NaovelN AuciondAlorihmyafor Fair Allocaterion ofDA/Wirles FAdn Channel,"h-ProcieedinghSpef3et AnnualeConferenceronineform ,
[13
P.
[14]V.aKisha AUction, ThEEory",unAcadeicn Press,zUSA,Juy2002.pp 0-7 "DaRCa A IntegratedoCDAdmissio An [15] H. Jln,M.Chatterjee,S.dK.Dasandk.asu,
Rate EfcentrolFramwor fora CDIVI Desoata NetorksniBasedon Non-Cooperastiem, the expsure prblem ca besucessfulleliminted.Inthe REFERENCES Games," nt on obie Cmpuing,andNetorkin 0 ~~~~~~Proceedings of I ehcl.rTcnlg Conference process, we havebeen able to achieve the prima(MobbCom),v2003,ppp.1326-338. and S.TffnK.ulDas,i 'uctoserSaifoBactwiontase DifferetiatednCer[1fX aiumE.K.P.Chng,th anduN.b.r Shoff 'Terasmisosio Sceuingifor Effciet [16] 5. Pal,l.Cateje WireessUtilzaton, Proeedngs f IEE NFOCM, 000,vol 3,~. 76-75.nicstfornieesDt ", Networks Proceedings of IEEE I nteOMMrnationa Cofrec (VT Fall),Orlado [17] T. Mandhlme,S.ucSun,n A.r Giwlpink TandsD.iLeionPoe,r'inneCDMCetermiatio ins ThreholgyCnferncelos C 203 vol. 3, pp. CombinatrorialAucinson Generaliztions", SyProeeings of theeFirsnterAnatyiona iin h sse1417-1421.. Flloritda[3]Octh Liu S. ruhl andE. Kighly, WCFQ An pporunisic Wrelss SheduerSJintlonfeenc oniels July 2002,pp.29Autnomu AgbiensadMtagt Systems,(S ) 69-6. withu statistica Faires Buneds"IE TranlsacTionso Wielerss Comunicatioin [18] J. HStad, 'LiquhenIs Hard to Appoximate"Withins ChannelActaMatheitnUica, 1999 Srepto 200,avl. 2,t pp. 1017-1028. [4] YLiu nd E Knihtly 'Oportuisti FairScheulin ove MulipleWireessiol.12 pp.itm" n 105-142.llronCn omniaio,Cotol 6 Channdels,"dProeedings of IEEEcton atonrai asNFCOCOM,,Oc 2003, vol.[92,4.10-15 Lu . .P hnn Np.B 10hroff, 'OpruistcTasisinShdln 13J Selete AEas MoianndCommunictin, Oct 2001, volt19, nlo.i10, pp. 2053-2064.tio 'Sheduin Reaul-tsinmeoreafficfineAt Nsetwrks, Pro[5]hTduling INFOCOM, ceedings 1996, ofIEEE 1, 198-205.n pp.nnal vol.WrelssFaingChnne,"Proeeing nfrm [6] A. Pekec and M. H. Rothkopf, 'Combinatorialrishn, Auction Design",cademc PreManagement02
WeqirelestsNtwrkats,"IEEiehicla
eulnrqirmnt.W
andgN.iShroff
Science, pp. 1485-1503.o), 2003,vol. 49,
203,pp.326338
100
