Utility-based Radio Resource Allocation for
QoS Traffic in Wireless Networks
Wen-Hsing Kuo and Wanjiun Liao
Abstract—In this paper, we study utility-based resource al-location for soft QoS traffic in infrastructure-based wireless networks. Soft QoS traffic here refers to the traffic which demands certain amount of bandwidth for normal operation but allows some flexibility when the given bandwidth is close to the preferred value. The resource requirement of soft QoS traffic can be described with sigmoid utility function. Our objective is to maximize the total utility of all soft QoS flows without going through a wireless bidding process. We develop essential theorems as the design guidelines for this problem, and then propose a sub-optimal, polynomial time solution based on the developed theorems. We prove that the difference in the performance of our mechanism and the optimal solution is bounded. The performance of the proposed solution is evaluated via simulations. The results show that our solution can adapt to any types of soft QoS flows. Specifically, it acts like a hard QoS system and allocates resource in a fairness-oriented manner when the utility functions of flows are unit-step functions; on the other hand, when the utility functions are concave, it behaves like a best effort system and allocates resource in a throughput-oriented way.
Index Terms—Utility optimization, resource allocation, soft QoS, wireless networks.
I. INTRODUCTION
I
N this paper, we study the utility-based resource alloca-tion problem in infrastructure-based wireless networks. In wireless networks, radio resource is limited and scarce, and the channel quality of each user may vary over time. Given the channel condition of each user and the total available system resource, the amount of bandwidth assigned to each user may be guided by such performance metrics as throughput and fair-ness [1] or according to the type of traffic [2]. “Throughput” and “fairness,” however, are conflicting performance metrics in scheduling. In this paper, we avoid such a “throughput-fairness” trade-off dilemma, and focus on “user satisfaction” for radio resource allocation.The degree of user satisfaction can be described by the utility function of the traffic under consideration. A utility function U(r) is a non-decreasing function with respect to the amount of allocated resource r. The more the resource is
Manuscript received January 30, 2006; revised May 4, 2007; accepted June 22, 2007. The associate editor coordinating the review of this letter and approving it for publication was S. Shen. This work was supported by National Science Council (NSC), Taiwan, under a Center Excellence Grant NSC96-2752-E-002-006-PAE, and under Grant Number NSC96-2628-E-002-004-MY3.
W.-H. Kuo was with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan. He is now with the Department of Electrical Engineering, Yuan-Ze University (YZU), Taoyuan, Taiwan.
W. Liao is with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan (e-mail: wjliao@ntu.edu.tw).
Digital Object Identifier 10.1109/TWC.2008.070116.
allocated, the more the user is satisfied. The wireless resource
r here may refer to timeslots or radio frequency occupied
by a single user. The marginal utility function u(r) is the derivative of the utility function U(r) with respect to r, i.e.,
u(r) = dUdr(r). The exact expression of a utility function
may depend on the type of traffic, and can be derived from some metrics that reflect the user’s perception or the content quality, such as Perceived Signal-Noise Ratio (PSNR) [3] or Mean Opinion Score (MOS) [4] of a multimedia stream. In this paper, we leave the work of finding utility functions to psychologists and economists, and focus on maximizing the total utility of given utility functions in the system.
Utility-based solutions have been widely used in wireless resource management. For example, a utility-based bandwidth allocation scheme is proposed in [5-6]; utility-based fairness allocation schemes are presented in [7-8], and a utility-based scheduler is proposed in [9]. Utility-based approaches have also been widely used for bandwidth pricing in wireless networks [10-12]. The idea of these schemes is to associate a price with each unit of radio resource and let each player maximize its welfare based on a bidding process. Bidding schemes, while useful for Internet pricing and congestion con-trol [13], are less practical and feasible for centralized wireless networks [14]. This is because in wireless environments, the type of traffic, the number of flows, and channel conditions are time-varying. Consequently, the bidding process would be very costly as users would have to repeatedly exchange messages in a real-time bidding. We show in [14] that in the infrastructure-based wireless network, both flow information and channel condition are accessible at the base station. It follows that optimizing the total utility of all flows at the base station can achieve the same objective as in a bidding scheme but in a simpler way. Therefore, it is more desirable to implement an optimal radio resource allocation mechanism in centralized wireless schedulers without a bidding process.
In this paper, we attempt to maximize the total utility of all flows at the base station while eliminating the bidding process. We focus on soft QoS traffic. The soft QoS traffic here refers to the traffic which requires a preferred amount of bandwidth for normal operation but can tolerate certain flexibility when the given amount of resource is close to the traffic’s preferred value. The bandwidth requirement of soft QoS traffic can be described by the sigmoid utility function with respect to the bandwidth resource [15]. The sigmoid utility function is a utility function whose U(r) > 0 and u(r) > 0, for all r, as shown in Fig. 1, where rc denotes the preferable amount of resource for the soft QoS traffic and Uc denotes the achieved
utility value with a given amount of resource rc. We observe
U
rc
BandwidthU
c(a) Utility functionU(r).
u
rc
Bandwidth
(b) Marginal utility functionu(r). Fig. 1. The sigmoid utility function for soft QoS traffic.
that if r < rc, u(r) > 0; if r = rc, u(r) = 0; otherwise, u(r) < 0. It can be interpreted as follows. When the amount
of resource r is given insufficiently, it is useless for real-time traffic; as r approaches rc, the flow is gradually operational, and thus the marginal utility increases dramatically. Once the allocated resource r has exceeded rc, allocating more resource may not be helpful for operation, and thus the marginal utility drops hence forth.
In [14], we proposes a resource allocation algorithm which achieves utility maximization for both hard QoS and best effort traffic, where hard QoS traffic refers to the traffic with strict demand on resource requirement and best effort traffic refers to the traffic which does not have the minimum bandwidth requirement. In this paper, we extend the discussion to soft QoS traffic in centralized wireless networks (i.e., with base stations (BSs)). Allocating resource to soft QoS traffic is very challenging since unlike hard QoS traffic, which allocates resource discretely, soft QoS traffic demands certain amount of bandwidth for normal operation, but allows some flexibility in resource allocation, i.e., continuously. To better understand the characteristics of the optimal solution to resource allocation problem for soft QoS traffic, we develop some theoretical results as the design guidelines for this problem. Based on the developed theorems, we propose an efficient algorithm which assigns wireless resource to soft QoS traffic in polynomial time. The proposed algorithm considers the traffic type, the total available resource, and all users’ channel qualities, rather than just considering the channel quality or traffic type as in most existing work [5-9]. More importantly, our algorithm can adapt to any kind of traffic with sigmoid utility functions. Specifically, when the slope of the utility function is very
steep, it acts like a hard QoS system and allocates resource to flows in a fairness-oriented manner; on the other hand, when the slope of the utility function is relatively flat, it behaves like a best effort system and allocates resource in a throughput-oriented way. We also show that our algorithm can achieve a very tight bound to the optimum.
The rest of the paper is organized as follows. In Sec. II, the theoretical foundation for this problem is developed, the design guidelines for supporting soft QoS traffic are described, and a wireless resource allocation scheme is proposed. In Sec. III, the performance of the proposed scheme is evaluated via simulations. Finally, the paper is concluded in Sec. IV.
II. RADIORESOURCEALLOCATIONBASED ONUTILITY
MAXIMIZATION
A. System Model and Problem Statement
Consider a base station with a set of soft QoS flows, denoted by Γ, in the wireless network. The resource requirement of each flow is described with a sigmoid utility function U(·). Let rtotaldenote the total amount of radio resource available at
the base station; let ridenote the amount of resource allocated
to flow i, i ∈ Γ, and qi, channel quality of flow i, where 0 ≤ qi≤ 1. The smaller the value of qi, the worse the channel
quality. Radio resource here is defined as the resource used to transmit data, and is allocated by the base-station’s controller. Such resource can be the number of time-slots, the number of codes, and so on, depending on the type of the wireless network in use.
Advances in wireless networks allow the upper layers to access such information as user channel conditions, thus facilitating adaptive modulation and coding (AMC) schemes. As a result, it is reasonable to assume that 1) each base station knows about the channel conditions of all users and 2) users under different channel conditions can transmit data at different data rates. Let qi denote the channel quality
parameter, which refers to the ratio of the actual amount of resource received by the user to the amount of resource allocated by the system to the user, and thus it is in a range of [0, 1]. Given the channel quality qiof flow i, the amount of
resource actually beneficial to flow i is given by θi = ri· qi.
Therefore, the utility function of flow i can be expressed as
Ui(·) = U(riqi), where U(·) is the utility function of the
traffic under consideration and Ui(·) is the utility function for
the type of traffic described by U(·) but taking into account the channel quality of flow i. To better distinguish U(·) from
Ui(·), we refer to U(·) as the traffic utility function, and Ui(·), the flow utility function in the rest of the paper. The
marginal utility function of Ui(·), denoted by u(·), is defined
by ui(ri) = dUdr(riqi)
i = qi· u(qi· ri). Suppose that all flows
in the system have the same traffic utility function U(·). Considering the channel condition of each flow may not be identical, we denote the preferable amount of resource for flow i by rci. For each flow i, i∈ Γ, Ui(rci) = U(rc) = Uc,
and thus rci = rcqi. Since a user who is allocated more radio
resource is happier, any utility function must be an increasing function, i.e., U(ri) ≤ U(rj) holds when ri≤ rj or u(r) ≥ 0
In this paper, we propose a wireless resource allocation scheme which maximizes the total utility
i∈Γ
Ui(ri), subject
to
i∈Γ
ri ≤ rtotal and ∀ri ≥ 0, i ∈ Γ. We claim that
an allocation solution ∗ = {ri, i ∈ Γ} is optimal if for
any allocation = {rj, j ∈ Γ}, U(∗) ≥ U(
), where U(∗) = i∈Γ Ui(ri) and U( ) = j∈Γ Uj(rj). The optimal
solution may not be unique.
Note that in a wireless network, the channel condition of each mobile node may vary over time. To reflect the latest channel condition, each base station may execute this proposed algorithm periodically or when some significant change to the wireless channel is detected. Since this algorithm operates in polynomial time, its computational overhead is acceptable and can be executed frequently.
B. Design Guidelines
For each sigmoid function U(r), U(r) > 0 holds for all
r > 0. This implies that to maximize the total utility of
the system, the available radio resource at the base station must all be assigned to flows, i.e.,
i∈Γ
ri ≤ rtotal. Since
sigmoid utility functions are continuous, we prove that in an optimal allocation ∗ = {rf, f ∈ Γ}, every flow f whose rf >0 (called allocated flow) has an identical marginal utility
(Lemma 2.1). We further prove that in an optimal allocation
∗ = {rf, f ∈ Γ}, there is at most one flow, say, f which
satisifies rf >0 and uf(rf) > 0, where uf(rf) = duf(rf)
drf
(Lemma 2.2).
Lemma 2.1: For an optimal allocation ∗ = {rf, f ∈ Γ},
if ri>0 and rj>0, i, j ∈ Γ, then ui(ri) = uj(rj).
Proof: We can rewrite the optimization problem
into the Lagrange multiplier form by letting constraint
g(r1, r2, ...rn) = n i=1 ri and objective f(r1, r2, ...rn) = n i=1
Ui(ri). In optimum, the gradients of [f(r1, r2, ...rn) +
λ(g(r1, r2, ...rn) − c)] = 0. Thus, for all i ∈ Γ,
d
dri[f(r1, r2, ...rn) + λ(g(r1, r2, ...rn) − c)] = ui(ri) + λ = 0.
Since λ= −ui(ri) for all i ∈ Γ, all allocated users’ ui(ri)
are identical to−λ.
Lemma 2.2: For an optimal allocation ∗ = {rf, f ∈ Γ},
there is at most one flow, say, f whose rf >0 and uf(rf) >
0, where uf(rf) =duf(rf) drf .
Proof: This lemma can be proved by contradiction.
Assume that more than one flow f satisfies rf > 0 in ∗
and u
f(rf) > 0. Then, we can find two flows i and j in ∗
whose u
i(ri) > 0 and uj(rj) > 0. Since u(r) is
continu-ous, we can always find a real number Δr, which satisfies both ri
ri−Δrui(r)dr < ui(r) · Δr and
rj+Δr
rj uj(r)dr > uj(rj) · Δr. Consider another resource allocation solution
= {rf, f ∈ Γ} in which flows i and j are allocated an
amount of ri−Δr and rj+Δr, respectively. Since the sigmoid
function is continuous and increasing, the difference in the total utilities of and∗ can be expressed by:
U() − U(∗) = [Ui(ri− Δr) + Uj(rj+ Δr)] − [Ui(ri) + Uj(rj)] = [Uj(rj+ Δr) − Uj(rj)] − [Ui(ri) − Ui(ri− Δr)] =rj+Δr rj uj(r)dr − ri ri−Δrui(r)dr Sincerj+Δr rj uj(r)dr > uj(rj)· Δr > ri ri−Δrui(r)dr, we
obtain U() − U(∗) > 0, violating the requirement that ∗
is optimal. Hence, for all allocated flows in ∗, there exists
at most one such flow f whose uf(rf) > 0.
Lemma 2.3: Consider two soft QoS flows i and j whose
traffic utility functions are identical (i.e., with the same U(.)). For an optimal allocation ∗ = {rf, f ∈ Γ}, if qi ≥ qj,
i, j∈ Γ, then inequality “θi≥ θj” must always hold.
Proof: This lemma can be proved by contradiction. We
assume that there exists two flows i, j in∗satisfying qi≥ qj
and θi < θj. Then, θj can be expressed by θi+ Δθ, where
Δr > 0. It follows that the total amount of resource given to these two flows, i.e., ri+ rj, can be expressed in terms of θ
and q by ri+ rj= θqii+ θj qj = θi qi+ θi+Δθ qj = θi( qi+qj qiqj ) + Δθ qj. We can find another allocation= {r
f, f ∈ Γ} in which
the only difference from∗is that flows i and j are allocated
resource differently. In , we let θ
i = θi + Δθ and θ
j =
θi+ (1 − qj
qi)Δθ. The total amount of resource allocated to flows i and j in is given by θi
qi + θj qj = θi( qi+qj qiqj ) + Δθ qj, equal to ri+ rj in∗.
The aggregate utility contributed by flows i and j in is
given by U(θi)+ U(θ
j) = U(θi+ Δθ)+ U(θi+ Δθ(1 − qj qi)),
which is larger than U(ri)+U(rj) (i.e., U(θi+Δθ)+U(θi)).
This leads to U() > U(∗), violating the assumption that ∗ is optimal. Thus, for any optimal allocation, if qi ≥ qj,
inequality θi≥ θj must always hold.
Theorem 2.4: An optimal solution to this soft QoS
alloca-tion problem, denoted by∗= {rf, f ∈ Γ}, must satisfy (1)
to (4). ∀i∈Γri= rtotal ; (1) ∀i, j ∈ Γ, ui(ri) = uj(rj), if ri>0 and rj >0; (2) ∀i, j ∈ Γ, qi≥ qj, if ri >0 and rj= 0; (3)
∀i ∈ Γ, there is at most one flow i whose ri>0 and ui(ri) >
0. (4)
Since an optimal solution to this problem is very hard to find and may be dependent on the channel qualities and utility functions of flows, we relax the constraint (4) as follows to further reduce the computational complexity.
Definition 2.5: A solution which satisfies (1), (2), (3), and
(5), denoted byα= {rf, f ∈ Γ}, is a sub-optimal solution
to this allocation problem.
∀i ∈ Γ, u
i(ri) < 0 if ri>0
(5) Since each allocated flow i in α satisfies u
i(ri) < 0,
this implies that the amount of resource ri allocated to flow i in α must exceed its preferable amount rc
i. Therefore,
highest among all sub-optimal solutions. This obtained solu-tion, while sub-optimal, can greatly reduce the computational complexity. We will prove in the next section that this sub-optimal approach is tightly bounded to the optimum.
Next, we develop some lemmas to help the design of the algorithm. For simplicity, we assume that all flows in the queue are sorted in decreasing order of their channel qualities. Let
α
j denote a sub-optimal allocation which allocates resource
to a total of j flows. We claim that the solutionα
j is unique.
This leads to the following lemma.
Lemma 2.6: α
j is the one and the only one sub-optimal
allocation in which a total of j flows are allocated resource.
Proof: We prove this lemma by contradiction. Assume
that there are more than one allocation, say j1 and j2,
which satisfies the same constraints asα
j. According to (5),
for any allocated flow i, ui(ri) < 0. Therefore, for all i, if its
has a different marginal value from the others, it must has a different value of ri. From (2), all allocated flows inj1(or in
j2) must have the same marginal value; from (3), the sets of
allocated flows inj1 and inj2 are identical since they both
have the same number of allocated flows, i.e., j. These facts dictate that given a marginal value and a total of j allocated flows, there exists only one allocation and thus only one total amount of allocated resource, because if these two allocations were different, they would have different marginal values. Sincej1 andj2 are different by assumption, they each have
a different marginal value for their respective allocated flows. However, a different marginal value leads to a different total amount of allocated resource. This violates the fact that the total amount of resource allocated to flows byj1 orj2 is
equal to rtotal, since both of them must satisfy (1). Therefore,
it can be concluded thatj1 and j2 are identical, and that
j1 = j2 = αj.
Next we discuss the amount of resource allocated to each flow inα
j. From (5), we obtain that inαj, j = 1, 2, · · · , n,
inequality “ri > rci” holds for each allocated flow i, i = 1, 2, · · · , j, where rci is given and determined by the
traffic type (i.e., its traffic utility function U(.)) of flow i. Consider the marginal sigmoid utility function ui(ri) of flow i, i= 1, 2, · · · , j. Let ˆui(r) = ui(r − rci), r > rci. ˆui(.) is
a decreasing function with respect to the amount of resource
r. Letˆu−1i (.) be the inverse function of ˆui(.). We sum up the
inverse functions ˆu−1
i (.) of all flows i, i = 1, 2, · · · , j, i.e.,
ˆu−1 Σj(.) = j i=1 ˆu−1
i (.), and find the aggregate marginal utility
functionˆuΣj(.) by inversing ˆu−1Σj(.). Denote the residual band-width after j allocations by rrj, i.e., rrj = rtotal−
j
i=1
rci. We
then find the aggregate utility ua,j= ˆuΣj(rrj). Based on ua,j
and ˆu−1
j (.), we can obtain the amount of resource allocated
to each flow, which is given by ri = rci+ ˆu−1i (ua,j) for i= 1, 2, · · · , j, and ri= 0 otherwise. We claim that all flows
i inαj = {ri, i∈ Γ} must be allocated in such a way that
ri= rci+ ˆu−1i (ua,j), 0 ≤ i ≤ j, and ri= 0 otherwise.
Theorem 2.7: Inα
j = {ri, i∈ Γ}, all flows i,∀i ∈ Γ, must
satisfy (6). ri= ⎧ ⎨ ⎩rci+ ˆu −1 i (ua,j), 0 ≤ i ≤ j 0, otherwise (6)
Proof: We prove that an allocation (say j) in which
the amount of resource allocated to each flow i satisfies (6) must beα
j. We first show that the allocation
j satisfies (1), (2), (3), and (5). Since j i=1 ri = j i=1 [rci+ ˆu−1i (ˆuΣj(rrj))] = j i=1
(rcj) + rrj = rtotal, it satisfies (1); since for i =
1, 2, · · · , j, ui(ri) = ˆui(ˆu−1i (uΣj(rrj))) = uΣj(rrj) = ua,j,
all allocated flows must have the same marginal value (i.e.,
ua,j), satisfying (2); since flows are sorted by their channel
qualities and only the first j flows with the best channel qualities are allocated, we obtain (3); for all allocated flows
i, the amount of allocated resource ri is larger than rci, and
therefore u
i(ri) ≤ 0, satisfying (5). Thus, we can conclude
that this allocationj must be a sub-optimal allocation. Since
such sub-optimal allocation is unique (from Lemma 2.6),j
must be equal toα
j. Thus, the resource allocated to each flow i inαj must satisfy (6).
LetΔUα
j denote the difference between the total utilities of
α j and α j−1, i.e.,ΔU α j = U( α j)−U( α j−1). We show next
that the value ofΔUα
j decreases as the value of j increases.
Lemma 2.8: For j = 1, 2, · · · , n, ΔUα
j > ΔUjα+1, where ΔUα j = U(αj) − U(αj−1). Proof: Let α j = {r α,j i , i ∈ Γ}, j = 1, 2, · · · , n. ΔU α j
can be expressed as follows. ΔUα j = U(αj) − U(αj−1) = j i=1 Ui(riα,j) − j−1 i=1 Ui(rα,ji −1) = Uj(r α,j j ) + j−1 i=1 Ui(rα,ji ) − j−1 i=1 Ui(riα,j−1) = Uj(rα,jj ) − [ j−1 i=1 Ui(riα,j−1) − j−1 i=1 Ui(rα,ji )] = Uj(rα,jj ) − {(j − 1)Uc+ ˆUΣ(j−1)(rrjα−1) − [(j − 1)Uc+ ˆ UΣ(j−1)(rrα j−1− r α,j j )]} = Uj(rα,jj ) − rrαj−1 rrαj−1−rjα,jˆuΣ(j−1)(r)dr , where rrα
j is the residual bandwidth after j allocations by
α j, and ˆUΣj(rr α j) + j · Uc= j i=1 Ui(rα,ji ).
Since for all j, both ˆuΣj(·) and ˆuj(·) are decreasing
functions, we know that for all a > b, ˆuΣj(a) · (a − b) ≥
a
b ˆuΣj(r)dr ≥ ˆuΣj(b) · (a − b), and
Uj(a) − Uj(b) =
a
b uj(r)dr ≥ uj(b) · (a − b).
We can further obtain that ΔUα j = U(αj) − U(αj−1) = Uj(r α,j j ) − rrαj−1 rrαj−1−rjα,jˆuΣj−1(r)dr ≥ Uj(rα,j j )−ˆuΣj−1(rr α j−1)·(r α,j j ) ≥ Uj(rα,jj )−ua,j−1·rα,jj ≥ Uj(rα,j j ) − ua,j· rjα,j = Uj(rα,jj+1+1) − ua,j· rα,jj − (Uj(r α,j+1 j+1 ) − Uj(rα,jj )) ≥ Uj(rα,j+1 j+1 ) − ua,j· r α,j j − Uj(r α,j j ) · (r α,j+1 j+1 − r α,j j ) = Uj(rα,jj+1+1) − ua,j· rα,jj − ua,j· (r α,j+1 j+1 − r α,j j ) = Uj(rα,jj+1+1) − ua,j· rα,jj+1+1
= Uj(rα,jj+1+1) − ˆuΣj(rr α j) · r α,j+1 j+1 ≥ Uj(r α,j+1 j+1 ) − rrαj rrαj−rα,j+1j+1 ˆuΣj(r)dr = ΔUα j+1.
Therefore, for j= 1, 2, · · · , n, ΔUα
j >ΔUjα+1. C. Resource Allocation Algorithm
Based on the lemmas and theorems developed in Sec. II-B, we propose a heuristic, called Utility-based allocation for Soft QoS (USQ), which finds an allocation with the highest total utility among all sub-optimal allocations (i.e., satisfying Definition 2.5). Suppose that there are n flows sorted in decreasing order of their channel qualities in the system, i.e.,
|Γ| = n. We determine the amount of resource to be allocated
to each flow in the queue such that the total utility of the target allocation sigmoid is maximized. Since an optimal solution to this problem is difficult to find, the target allocation
sigmoidwe determine may be sub-optimal in order to reduce
the computational overhead. In other words, we attempt to find all sub-optimal allocationsα
1,α2,· · · , αn, each of which is
unique (Lemma 2.6), and then determine α
K as the target
allocationsigmoid, where U(α
k) = maxx=1,···,nU( α
x). We will
prove shortly that the solutionsigmoid is tightly bounded to the optimum.
In Theorem 2.7, we have shown that the amount of resource allocated to each flow i byα
j must satisfy (6). Accounting for
the number of flows being allocated resource in the system, we can determine the sub-optimal allocationα
j for j flows, j = 1, 2, · · · , n. In Lemma 2.8, we have further shown that
ΔUα
j decreases as j increases given that all flows’ traffic
utility functions are identical. It follows that whenΔUα j <0 and ΔUα j−1 > 0 (i.e., 0 < ΔU α j−1 < · · · < ΔU α 2 <ΔU1α and ΔUα
n < · · · < ΔUjα+1 < ΔUjα < 0 ), αj−1 has
the highest total utility among all sub-optimal allocations
α
1,α2,· · · , αn. In summary, our algorithm first sorts all
flows by their channel qualities, and then finds all α j and
ΔUα
j, j = 1, 2, · · · , n. Once “ΔUj < 0” is determined,
the algorithm stops and returnsα
j−1 as the target allocation
sigmoid. Fig. 2(a) summarizes howα
j is obtained and Fig.
2(b) illustrates the relation betweenα
j andΔUjα.
The detailed algorithm is summarized in Table I. Note that in Table I, since some steps are repeated as j increases from1 to n, rrα
j ofαj can be obtained more easily by subtracting rj
from the rcj−1ofαj−1 (Step 3.1.1). Similarly, we can obtain
ˆu−1Σj by addingˆu−1j (.) into the last ˆu−1Σj−1of α
j−1(Step 3.1.2).
Thus, the complexity of the algorithm can be further reduced. The worst-case complexity of the USQ algorithm is given by
O(n2), where n is the number of flows in the system.
D. The Performance Bound to Optimum
Next we prove that the proposed USQ allocation algorithm is bounded to the optimal solution. In the previous sub-section, we have shown in Theorem 2.4 that an optimal solution must satisfy conditions (1), (2), (3) and (4). Then, we can classify all these optimal solutions into two mutually exclusive sets: α and β. All allocations in set α satisfy conditions (1), (2), (3) and (5) (i.e., Definition 2.5), while those in set β satisfies (1),
i i i i i ΞΞΞΞ a,i a,i a,i i i
rr
D iu
iu
6 i rD 2 rD 1 rD ir
D(a) The procedure of derivingα j
(b)α j andΔU
α j
Fig. 2. An illustration of the proposed algorithm USQ
(2), (3) and (6), where condition (6) is the complement set of (5) in (4), i.e.,
∀i ∈ Γ, there is one and only one flow i whose ui(ri) > 0 if
ri>0.
(7) Let Sβ
j be a subset of β that has j allocated flows. Based on
Input: For each flow i with channel qualities qi in the queue,
i = 1, 2, · · · , n, a sigmoid utility function U(r), and total available resource isrtotal;
Output:sigmoid= {r1, r2, · · · rn};
Algorithm:
(1) Initializeri= 0, i = 1, 2, · · · , n, rr0= rtotal, andU(0) =
0.
(2) Sort all flowsi in descending order of qi, and store them in
the queue. (3) Forj = 1 to n,
(3.1) Derive the allocation (α
j) as follows.
(3.1.1)rrα
j = rrαj−1− rcj;
(3.1.2)ˆu−1Σj(.) = ˆu−1Σj−1(.) + ˆu−1j (.);
(3.1.3) Invertˆu−1Σj to obtain ˆuΣj; (3.1.4)ua,j= uΣj(rrαj); (3.1.5) Fori = 1, 2, · · · , j, rα,j i = rci+ ˆu−1i (ua,j) ; Fori = j + 1, · · · , n, rα,j i = 0; (3.1.6) Returnα j
(3.2) CalculateΔUj= U(αj) − U(αj−1) as follows.
(3.2.1) IfΔUj< 0, set αj−1assigmoid;
exit this loop; (4) Returnsigmoid.
TABLE I
TABLEI. THE PROPOSED ALGORITHMUSQFOR SOFTQOSTRAFFIC
Fig. 3. An illustration ofβ j inS
β j
in Sβ
j satisfy the condition that the jth flow (which is the one
with the least amount of allocated resource among all allocated flows) has u(rj) > 0. Let βj be the solution with the
largest utility value in Sβ
j. Fig. 3 gives an example allocation
in Sβ
j. We claim that if β
j is the solution obtained by the
proposed USQ allocation algorithm, the optimal solutions to this soft QoS allocation problem must only fall in the set
{α j, β j, β j+1}.
Lemma 2.9: The optimal solution to this soft QoS problem
must fall in the set{α j, β j, β j+1} , where α k is the solution
obtained by the USQ allocation algorithm.
Proof: We classify all possible optimal solutions in two
mutually exclusive sets α and β. If an optimal solution falls in α, it must be α k = {r α,k f , f ∈ Γ}, where U( α k) = max x=1,···,nU( α
x), i.e., the solution obtained by our proposed
USQ algorithm. If it is in β, the optimal allocation, say,
β
k = {r β,k
f , f ∈ Γ} must satisfy U(
β k) = maxx=1,···,nU( β x). If β j is optimal, U( β j) ≥ U( α
j−1) holds, it follows that Uj(rjβ,j) + (k − 1) · Uc+ ˆUΣ(j−1)(rr β j) ≥ (k − 1) · Uc+ ˆUΣ(j−1)(rrα j−1), which lead to Uj(r β,j j ) ≥ ˆUΣ(j−1)(rrα j−1) − ˆUΣ(j−1)(rr β j) =rrαj−1 rrβj ˆuΣ(j−1)(r)dr ≥ ˆuΣ(j−1)(rr α j−1) · (rrjα−1− rr β j) = ua,j−1· (rjβ,j).
Note that here we let rβ,j
j denote the amount of
re-source allocated to the jth flow in β
j, and let rr β j denote rtotal − j−1 i=1
rci − rβ,jj , which is the amount of residual
resource allocated to the first (j− 1) flows (i.e., flows with ui(ri) < 0, i = 1, 2, · · · , j − 1) in βj.
Similar to Lemma 2.8, for r < rci, i = 1, 2, · · · , n, Ui(·)
is increasing, leading to Uj(rcj) − Uj(rjβ,j) =rcj rβj uj(r)dr ≥ uj(rβ,j j ) · (rcj− r β,j j ) = uΣ(j−1)(rr β j) · (rcj− r β,j j ) ≥ ua,j−1· (rcj− rβ,jj ). Note that uΣ(j−1)(rrβ j) ≥ uΣ(j−1)(rr α j−1) = ua,j−1 since uΣ(j−1(.) is a decreasing function and rrβj ≤ rr
α j.
Adding the two functions above, we obtain Uj(rβ,jj ) +
(Uj(rcj) − Uj(r β,j
j )) = Uc≥ ua,j−1· rcj.
Then from Lemma 2.8, we have
U(αj−1) − U( α j−2) = Uj−1(r α,j−1 j−1 ) − rrj−2 rrj−1 ˆuΣj−2(r)dr ≥ (Uc +rj−1α,j−1 rcj−1 ˆuj−1(r)dr) − ˆuΣj−2(rr α j−1) · (rr α j−2 − rrα j−1) ≥ [Uc+ ˆuj−1(rα,jj−1−1) · (r α,j−1 j−1 − rc α j−1)] − ˆuΣj−2(rr α j−1) · (rrα j−2− rr α j−1) ≥ [Uc+ ˆuj−1(r α,j−1 j−1 ) · (r α,j−1 j−1 − rc α j−1)] − ˆuΣj−1(rrjα−1) · rα,jj−1−1 = [Uc+ ua,j−1· (r α,j−1 j−1 − rc α j−1)] − ua,j−1· r α,j−1 j−1
= Uc− ua,j−1· rcαj−1 ≥ [ua,j−1· rcαj − ua,j−1· rcαj−1]
≥ 0. Since β j is optimal, U( β j) ≥ U( α j) holds. Then
from Lemma 2.8, it follows that U(α
j−1) > U(αj−2) >
· · · > U(α
1) if U(βj) ≥ U( α
j−1). Similarly, given that U(βj) ≥ U(αj), we can prove U(αj) ≥ U(αj+1), and
that U(α n) < · · · < U( α j+1) < U( α j). Thus, if β j is an
optimal allocation, eitherα j−1or
α
j is the solution obtained
by USQ, i.e., α k whose U( α k) = maxx=1,···,nU( α x). On the other hand, if α
k is the solution obtained by our algorithm,
onlyβ k and
β
k+1 can be optimal in set β. Hence, given that
α
k is the solution of our algorithm, the optimal solution must
fall in the set{α j,
β j,
β j+1}.
Theorem 2.10: The difference in performance of the
pro-posed USQ algorithm and the optimal allocation is bounded by the value Uc.
Proof: From Lemma 2.7, given that the proposed
al-gorithm finds a solution of α
k, the solutions that could be
optimal include α k, β k and α k+1. If α k+1 is optimal, the
difference in total utilities of the optimal and the one obtained by our algorithm is zero. If β
k+1 is optimal, the difference
betweenβ k+1 and α k is given by U(βk+1) − U(α k) =k+1 i=1 ui(riα,k+1) − k i=1 ui(riα,k)
= uk+1(rα,kk+1+1) + [ k i=1 ui(rα,ki +1) − k i=1 ui(riα,k)] ≤ uk+1(rα,kk+1+1) < uk+1(rck+1) = Uc. Since α
j is the one with the largest utility value among
all allocations in α, we have j
i=1 ui(rα,ji ) ≥ j i=1 ui(rα,ji +1). From (6), we obtain rα,j+1 j+1 ≤ rcj+1, and thus uj+1(r α,j+1 j+1 ) ≤ uj+1(rcj+1) = Uc.
Similarly, it can be easily proved that ifβ
k is an optimal
so-lution, then the difference in total utilities can also be bounded by Uc due to U( β k) − U( α k) ≤ U( β k) − U( α k−1) ≤ Uc.
Therefore, the difference in the performance of our algorithm and the optimal solution is bounded by Uc, no matter if the
optimal solution isα k, β k or β k+1.
In Table I, we observe that our proposed sub-optimal algorithm has a polynomial time computational complexity, and is feasible for a network BS. However, we cannot find an approach to solving the original optimal problem because it is difficult to findβ
j from S β j.
III. SIMULATIONRESULT
In this section, we conduct simulations to evaluate the performance of the proposed USQ allocation algorithm. In the simulations, we have 16 user flows, all with the same sigmoid traffic utility function U(r). The channel quality q of each flow is randomly selected from the range [0, 1]. The sigmoid utility function is composed of two exponential functions:
(1)U(r) = q · ep(r−rc), if r < rc;
(2)U(r) = 1 − (1 − q) · e−p(r−rc), otherwise,
where q denotes the utility value when r = rc, and p determines the slope of the utility function. We fix q at0.5 in the simulations. When r < rc, we let the utility increase with
r. When r > rc, we let the utility saturate to1.
In the first simulation, we vary the slope of the utility function by tuning the value of p so as to observe how the value of p affects the resource distribution. The value of rtotal is fixed at 800 in this simulation. Fig. 4 plots the
impact of p on the utility function U(r). We observe that a different value of p results in a different function. The larger the value of p, the more similar it is to the hard QoS’s unit-step function; when the value of p is0.1, the shape is similar to the best effort concave utility function. The resource distribution among flows is depicted in Fig. 5. The result shows that when p is small, meaning that the marginal utility saturates more slowly as the given resource increases, the system favors flows with better channel qualities; on the other hand, when
pbecomes larger (i.e., the slope is sharper), meaning that the
marginal utility saturates faster and the amount of resource each flow obtains is no more than rci, the system tends to give
more resource to flows with bad channel qualities so as to let all flows i have an identical θi (i.e., θi = ri· qi). Therefore,
by tuning the value of p, we can adjust the wireless system to behave in a throughput-oriented (i.e., p is small) or in a fairness-oriented (i.e., p is large) manner.
Next, we compare the resource distribution obtained by USQ with the algorithm proposed in our previous work [14]. In [14], we proposed a mechanism which allocates resource to
0.5 0.6 0.7 0.8 0.9 1 tility p=0.1 p=0.2 0 0.1 0.2 0.3 0.4 0 5 10 15 20 U t Resource p=0.4 p=0.8 p=3.2 p=12.8
Fig. 4. Relationship betweenp and utility function U(.)
20 30 40 θi p=0.1 p=0.2 p=0.4 0 10 0.01 0.05 0.07 0.07 0.16 0.22 0.29 0.43 0.45 0.51 0.53 0.54 0.61 0.63 0.72 0.89 qi p=0.8 p=3.2 p=12.8
Fig. 5. Relationship betweenp and θiof each flow
both hard QoS and best effort traffic by maximizing the total utility. The utility functions of hard QoS and best effort traffic are based on a unit-step function and a concave function, respectively, as reported in [15]. We decrease the value of
p in the proposed system (i.e., from fairness-oriented to
throughput-oriented) and then compare the results obtained by USQ with the hard QoS traffic allocation and the best effort traffic allocation proposed in [12].
We first set p to 12.8, i.e., a utility function with a very steep slope, and then compare the result with the hard QoS allocation. The utility function of hard QoS traffic is given by fu(r − 10), where fu(·) is a unit-step function. Since the
step function for hard QoS traffic and the sigmoid function with p= 12.8 for soft QoS traffic are very similar (see. Fig. 6), and both mechanisms allocate resource based on utility maximization, the rciof each allocated flow i in the proposed
USQ algorithm (Fig. 6(a)) is very close to that in the hard QoS allocation in [14] (Fig. 6(b)). However, there are still some subtle differences. In the hard QoS allocation, the rci of each
allocated flow i is guaranteed to be10/qi, but in the soft QoS
allocation, slightly more resources are assigned to flows. The reason is that with the sigmoid utility function, all available resources are allocated to flows for maximizing the total utility (i.e., (1)), while the resource assigned to hard QoS is allocated in a discrete way and only with the requested amount, leading to some residual resource left unused. By setting p to a very
0. 02 0. 11 0. 12 0. 19 0. 30 0. 37 0. 41 0. 51 0. 51 0. 70 0. 71 0. 80 0. 86 0. 87 0. 91 0. 92 0 250 500 750 1000 0 10 20 30 40
θ
iq
ir
total(a) Soft QoS allocation (p = 12.8)
0.02 0.09 0.10 0.17 0.20 0.30 0.34 0.43 0.55 0.56 0.62 0.76 0.86 0.90 0.99 1.00 ˃ ˅˃˃ ˇ˃˃ ˉ˃˃ ˋ˃˃ ˄˃˃˃ 0 10 20 30 40
θ
ir
totalq
i(b) Hard QoS allocation
Fig. 6. Resource distribution for the soft QoS allocation (p = 12.8) and the hard QoS allocation in [12].
large value (i.e., rendering a very steep function similar to a unit-step function), the proposed soft QoS algorithm act like the hard QoS algorithm in [14].
Next, we set p to 0.1, which results in a relatively flat utility function, and compare the allocation result with the best effort allocation in [14]. Again, both systems are given similar settings and the results are shown in Fig. 7. The utility function of best-effort traffic is based on a concave function UBE(r) = 1 − er/10. The figure shows that when p is small, the result of the soft QoS allocation (Fig. 7(a))
is very close to that of the best effort allocation (Fig. 7(b)). Unlike in Fig. 7, USQ gives much more resource to flows with better channel qualities as rtotal increases, i.e., being more
throughput-oriented, as shown in Fig. 7. However, in Fig. 7(a), each allocated flow’s minimal rate can be guaranteed not less than 10, while in Fig. 7(b), allocated flows cannot be ensured to obtain a minimal value.
IV. CONCLUSION
In this paper, we study the utility maximization problem for resource allocation to soft QoS traffic in infrastructure-based wireless networks. We describe the design guidelines for
0. 02 0. 11 0. 12 0. 19 0. 30 0. 37 0. 41 0. 51 0. 51 0. 70 0. 71 0. 80 0. 86 0. 87 0. 91 0. 92 0 250 500 750 1000 0 10 20 30 40 50
θ
iq
ir
total(a) Soft QoS allocation (p = 0.1)
0.02 0.11 0.12 0.19 0.30 0.37 0.41 0.51 0.51 0.70 0.71 0.80 0.86 0.87 0.91 0.92 ˃ ˅ˈ˃ ˈ˃˃ ˊˈ˃ ˄˃˃˃ 0 10 20 30 40 50
r
totalθ
iq
i(b) Best effort allocation
Fig. 7. Resource distribution for the soft QoS allocation (p = 0.1) and the best effort allocation in [12].
this problem, and propose a polynomial time algorithm which is proved to be tightly bounded to the optimal solution. The performance of the proposed USQ mechanism is evaluated by simulations. The results show that this mechanism can not only allocate resource according to both users’ channel qualities and total network resource, but also adapt to different traffic types (i.e., throughput-oriented or fairness oriented). We also show that the mechanism proposed in our previous work in [14], which considers the co-existence of hard QoS traffic (e.g., VoIP traffic) and best effort traffic in the network, is a special case of USQ. Thus, the proposed USQ mechanism is applicable to the scenarios in which traffic with QoS requirements (including hard QoS and soft QoS) and without QoS requirements (e.g., best effort traffic) co-exist in the system.
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Wen-Hsing Kuo received his BS and the PhD
degrees in Electrical Engineering from National Tai-wan University, Taipei, TaiTai-wan, in 2002 and 2008, respectively. He joined the Department of Electrical Engineering, Yuan-Ze University, Taoyuan, Taiwan, as an Assistant Professor in 2008. His research inter-ests include network resource management, network economics and wireless access networks.
Wanjiun Liao received the BS and MS degrees
from National Chiao Tung University, Taiwan, in 1990 and 1992, respectively, and the Ph.D. degree in Electrical Engineering from the University of Southern California, Los Angeles, California, USA, in 1997. She joined the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, as an Assistant Professor in 1997. Since August 2005, she has been a full professor. Her research interests include wireless networks, multimedia networks, and broadband access net-works.
Dr. Liao is currently an Associate Editor of IEEE Transactions on Wireless Communications and IEEE Transactions on Multimedia. She served as the Technical Program Committee (TPC) chairs/co-chairs of many international conferences, including the Tutorial Co-Chair of IEEE INFOCOM 2004, the Technical Program Vice Chair of IEEE Globecom 2005 Symposium on Au-tonomous Networks, and the Technical Program Co-Chair of IEEE Globecom 2007 General Symposium. Dr. Liao has received many research awards. Papers she co-authored with her students received the Best Student Paper Award at the First IEEE International Conferences on Multimedia and Expo (ICME) in 2000, and the Best Paper Award at the First IEEE International Conferences on Communications, Circuits and Systems (ICCCAS) in 2002. Dr. Liao was the recipient of K. T. Li Young Researcher Award honored by ACM in 2003, and the recipient of Distinguished Research Award from National Science Council in Taiwan in 2006. She is a Senior member of IEEE.
![Fig. 6. Resource distribution for the soft QoS allocation ( p = 12.8) and the hard QoS allocation in [12].](https://thumb-ap.123doks.com/thumbv2/9libinfo/8872004.248417/8.877.511.822.63.640/fig-resource-distribution-soft-qos-allocation-hard-allocation.webp)
