a single-cell scenario, any α-fair scheduling strategy achieves .... of the BS are dedicated to serve the user, we call this rate the feasible ...... cells in a city center.
Mobility-driven Scheduling in Wireless Networks S.C. Borst
N. Hegde
A. Proutiere
Alcatel-Lucent, Bell Labs, USA Eindhoven University of Technology The Netherlands
Orange Labs France
Microsoft Research Cambridge, UK
Abstract—The design of scheduling policies for wireless data systems has been driven by a compromise between the objectives of high overall system throughput and the degree of fairness among users, while exploiting multi-user diversity, i.e., fast-fading variations. These policies have been thoroughly investigated in the absence of user mobility, i.e., without slow fading variations. In the present paper, we examine the impact of intra- and inter-cell user mobility on the trade-off between throughput and fairness, and on the suitable choice of α-fair scheduling policies. We consider a dynamic setting where users come and go over time as governed by random finite-size data transfers, and explicitly allow for users to roam around. It is demonstrated that the overall performance improves as the fairness parameter α is reduced, and in particular, that proportional fair scheduling may yield relatively poor performance, in sharp contrast to the standard scenario with only fast fading. Since a lower α tends to affect short-term fairness, we explore how to set the fairness parameter so as to strike the right balance between overall performance and short-term fairness. It is further established that mobility tends to improve the performance, even when the network operates under a local fair scheduling policy as opposed to a globally optimal strategy. We present extensive simulation results to confirm and illustrate the analytical findings.
I. I NTRODUCTION Opportunistic scheduling strategies provide an effective mechanism for improving the throughput performance of wireless data systems by exploiting the channel variations induced by multi-user diversity [1], [2]. The performance of opportunistic scheduling strategies has been extensively studied in the context of a single transmitter with a static user population and channel conditions that fluctuate on a relatively fast time scale [3]–[12]. It is well known that there exists an intrinsic trade-off between the total system throughput and the degree of fairness among spatially diverse users in such a setting (even in the absence of any dynamic channel fluctuations). Specifically, the total throughput is maximized by allocating all the transmission resources to users in the proximity of the transmitter with typically strong channels, at the expense of starving users on the periphery of the coverage region with usually weaker channels. In contrast, providing equal throughputs to all users involves a potentially severe sacrifice in total throughput. The notion of throughput utility maximization provides a useful paradigm to capture the above-described trade-off [13], and has motivated the design of utility-based scheduling strategies [7], [14]–[17]. A convenient family of utility-based
scheduling strategies involve so-called α-fair utility functions [18], which cover several common fairness concepts as special cases. In particular, the values α = 0, α = 1 and α → ∞ correspond to resource allocations that achieve maximum total throughput, proportional fairness, and maxmin fairness, respectively. It is widely believed that proportional fairness strikes a good balance between total throughput objectives and fairness considerations, and the proportional fair (PF) scheduling strategy has been implemented in several commercial wireless data systems [19], [20]. The PF scheduling strategy tends to allocate roughly equal fractions of the transmission resources to the various users, even when the channel conditions differ among users or vary over time [21]– [23]. In a dynamic setting, where users come and go over time as governed by random finite-size data transfers, the trade-off between the total system throughput and fairness exhibits itself in a different way. Dedicating more transmission resources to users with strong channels then causes these users to complete their service and leave the system sooner, which reduces the future opportunities for selecting high-rate users. Indeed, in a single-cell scenario, any α-fair scheduling strategy achieves stability whenever feasible at all, as long as the value of α is strictly positive [24]. Instead, the above-mentioned tradeoff manifests itself in terms of the delay characteristics and user-perceived throughput performance. Granting preference to users with strong channels speeds up their transfers and enhances their experienced throughputs, which tends to improve the overall performance, but hurts the low-rate users. Just like in a static scenario, the PF scheduling strategy achieves a good trade-off between minimizing the overall transfer delay and avoiding excessive delays for low-rate users. In particular, the expected delays are proportional to the amounts of transmission resources required by the respective transfers. Moreover, the overall performance is well predictable as it is insensitive to the detailed traffic statistics and propagation characteristics, and only depends on simple load factors [23], [25]. All of the above observations rely on the assumption that the channel variations occur on a relatively fast time scale, and ‘average out’ over the typical duration of a transfer (which in particular is the case in the absence of any channel fluctuations). Besides the fast variations due to multi-path propagation, the channel conditions also vary however on slower time scales as governed by user mobility on a more
macroscopic level. In order to examine the performance tradeoffs associated with the value of α in the presence of such slow fading, we analyze in the present paper a network of several base stations, each operating according to a local αfair scheduling strategy. We consider a dynamic setting as described above, and explicitly allow for users to roam around within cell areas and to be handed off between cells over the course of their service. We show that, in contrast to the scenario without slow fading, the capacity region now does depend on the value of α, and in fact monotonically grows as the value of α decreases. Simulation experiments demonstrate that the overall performance in terms of throughputs experienced by users also improves as the value of α is reduced. This implies that the PF scheduling strategy may no longer offer a good compromise in the presence of slow fading, and creates an incentive for more aggressive schedulers with lower values of α. Lower values of α, however, affect the short-term fairness, as users receive a smaller fraction of the transmission resources when they reside in less favorable locations. This gives rise to the complex issue of how to set the value of α so as to achieve an optimal balance between the overall user-experienced throughput performance and short-term fairness. The optimal value of α depends of the time scale of the mobility relative to the the flow dynamics, which suggests potential scope for the design of adaptive, mobility-driven schedulers. In order to examine the capacity impact of user moiblity, we also compare the capacity region with that in a scenario where users remain stationary over the complete duration of their service. In the latter case, transfers must be entirely served in the regions where they originate, and the capacity region no longer depends on the value of α. We prove that the mobility increases the capacity region, which is reminiscent of findings in the setting of mobile ad-hoc networks [26]. A crucial distinction, however, is that in the latter context the scheduling strategy is flexible, so that the mobility provides additional opportunities for users to be served in the most favorable positions, and hence naturally increases the capacity. In contrast, an α-fair scheduling strategy grants users a dedicated portion of the resources, even in adverse conditions, and possibly a larger fraction in fact than in favorable locations for high values of α, so the above explanation why mobility increases the capacity region no longer applies. The key mechanism at work appears to be that even for high values of α, an arbitrary bit of a mobile user is still more likely to be transmitted while the user resides in a relatively favorable position compared to a user in a random yet fixed location. The remainder of the paper is organized as follows. In Section II we present a detailed model description and introduce some important preliminary results. In Section III we derive a characterization of the capacity region of a wide family of α-fair schedulers and consider some special cases of interest and an illustrative example. Section IV examines the impact of the fairness parameter α on the capacity region, and provides a comparison of the capacity region with that in a scenario with stationary users so as to investigate the capacity impact
of mobility. Section V presents the numerical experiments that we conducted to explore the flow-level performance measures and confirm the analytical findings. Motivated by the above considerations, we discuss in Section VI the scope for adaptive, mobility-driven schedulers. II. M OBILITY, SCHEDULERS AND NETWORK DYNAMICS We consider a network of BS’s, indexed by the set N , which provide service to the users located in the corresponding cells through a shared downlink. a) Radio conditions and fading: The service rate of a user attached to a given BS depends on its radio conditions to the serving BS, but also to the other BS’s (because of inter-cell interference). In addition, the rate depends on the fraction of the BS resources allocated to the user. When all the resources of the BS are dedicated to serve the user, we call this rate the feasible rate of the user. We distinguish fast and slow variations of the radio conditions. To capture slow channel variations, typically induced by user mobility, it is convenient to use the notion of state. At any time, each user resides in one of several possible states indexed by a set I; this state characterizes the radio conditions to the various BS’s averaged over the fast fading variations. The feasible rate of a user in state i averaged over fast fading is denoted by Ci . We assume that for a given user, fast fading is represented by a stationary ergodic Markovian process whose statistics depend on the user state only. The fast fading processes are independent across users. Due to fast fading, the time evolution of the feasible rate of a user in state i is a process whose stationary distribution is the same as that of a generic process denoted by (Ci (t), t ≥ 0). We have Ci = E[Ci (t)]. We further assume that the process Ci (t) may take a finite number of values, and denote by C i = max{x : P r[Ci (t) = x] > 0}. b) α-fair schedulers: To simplify the analysis, we assume throughout the paper that fast fading variations occur at a much smaller time scale than the rest of the system dynamics (user mobility, flow arrivals and departures). Consider a particular cell in the network, and suppose for now that the population M and the states i1 , . . . , iM of the users are fixed. The corresponding BS n shares its downlink radio resources according to some scheduling scheme. Here we consider the family of α-fair schedulers. These schemes are able to take advantage of fast fading variations, and to exploit multi-user diversity. For α > 0, the α-fair scheduler is designed so as to maximize the total cell utility defined by PM (1−α) (T ) /(1−α), where Tj denotes the average service j=1 j rate of user j during a time that is much greater than the time scale of fast-fading variations. A way of implementing the αfair scheme is to use a gradient algorithm [17]: at time u, schedule user j ⋆ with: j ⋆ = arg max Cj (u) × Tj (u)−α , j
(1)
where Cj (u) denotes the feasible rate of user j at time u and Tj (u) is the average service rate received by user j between
times u − G and u (G is typically large). Again we assume that the schedulers considered converge instantaneously when observing the network dynamics at the time scale of user mobility or at that of the change in the population of active users. Each BS runs its scheduling scheme independently of the other BS’s. c) Traffic demand and mobility: The network is offered traffic from K classes of users. Class-k users generate flows according to a Poisson process of intensity λk . Denote by Fkj the size of the j-th arriving class-k flow, and by ikj (t) ∈ I the state of the corresponding user at time t, which for conciseness will be referred to as the state of the flow. Fkj and ikj (t), j = 1, 2, . . ., are i.i.d. copies of an exponential random variable Fk with mean 1/µk and a stationary ergodic Markov process ik (t), respectively. Denote by ρk = λk /µk the traffic intensity associated with class-k flows. The stationary distribution of the mobility process ik (t) is πi,k = P r[ik (t) = i]. Define Jk = {i ∈ I : πi,k > 0} as the set of states visited byP class-k flows, In as the set of states served by BS n, pk,n = i∈In πi,k as the stationary probability for a class-k user to be served by BS n, and Nk = {n ∈ N : pk,n > 0} as the set of cells visited by class-k users. d) Flow-level network dynamics and capacity: We are interested in the flow-level dynamics of the system (since the users perceive performance at flow level, e.g., they are sensitive to the mean flow delay). The network state at time t is then defined by Ni,k (t), the numbers of active classk flows in state i. These flows are served in the various cells according to an α-fair scheduling scheme, and at the completion of a flow transfer, the corresponding user leaves the system. Denote by φi,k (t) the service rate of class-k flows in state i. As explained earlier, since we assume that fast fading and the scheduling decisions evolve very rapidly compared to the flow-level dynamics, the rates φi,k (t), i ∈ In , of class-k flows served by BS n are such that the total cell utility is maximized over all possible scheduling decisions exploiting fast-fading variations: max
X
K X
Ni,k (t)α
i∈In k=1
φi,k (t)1−α . 1−α
(2)
Note that each class-k flow in state i is served at rate φi,k (t)/Ni,k (t) by symmetry. In general it is difficult to analytically characterize these rates, because of the strong dependence on the statistical properties of fast fading. However, we can compute them explicitly (i) in the absence of fast fading or (ii) when the flow population becomes very large. (i) In the absence of fast fading, denote by τi,k the proportion of resources allocated to class-k flows in state i. Then the convex problem (2) is equivalent to: K P P (C τ (t))1−α , Ni,k (t)α i i,k max 1−α s.t.
i∈In k=1 K P P
τi,k ≤ 1,
i∈In k=1
and φi,k (t) = τi,k Ci . Solving (3) yields: 1/α−1
Ni,k (t)Ci . K P P 1/α−1 N (t)C j,l j j∈In
φi,k (t) = Ci ×
(4)
l=1
(ii) When the population of flows is large, meaning that either Ni,k (t) is 0 or very large, then by the law of large numbers, for any i, k such that Ni,k (t) > 0, there exists a flow whose corresponding user enjoys the maximum possible feasible rate over fast-fading variations. This flow would be served at rate C i if scheduled. Should the BS decide to schedule a classk flow in state i, it would schedule the flow whose feasible rate is C i . Hence, the service rates of the various flows would approximately be given by: 1/α−1
Ni,k (t)C i φi,k (t) = C i × . K P P 1/α−1 N (t)C j,l j j∈In
(5)
l=1
To summarize the state of the system N (t) = (Ni,k (t)) evolves due to flow arrivals (class-k flows arrive as a Poisson process, and the initial states of the corresponding users are i.i.d. across flows with distribution πi,k , i ∈ I), due to flow departures, and due to user mobility (along i.i.d. copies of the mobility process ik (t) for class-k flows). (N (t), t ≥ 0) is a Markov process. We define the capacity region as the set of traffic vectors (ρ1 , . . . , ρK ) compatible with some specific flow-level performance requirements. A natural requirement is that the mean flow delay remain finite for all classes. This happens if and only if the Markov process (N (t), t ≥ 0) is ergodic (we also use the term stable). In the present paper, we exactly characterize the capacity region when defined through stability, but also explore via simulations refined performance metrics such as the mean flow delay, or equivalently, the mean flow throughput. III. C APACITY REGION OF α- FAIR SCHEDULERS In this section, we derive an explicit expression for the capacity region achieved by α-fair schedulers. In other words, we analyze the flow-level stability of the network under these schedulers. We first state the result in the absence of fast fading, and then generalize it to the case of fast fading. A. Without fast fading Define: n K Rα := (r1 , . . . , rK ) ∈ RK + : ∃θ ∈ R+ such that rk ≤
o X θk Aα k,n , ∀k = 1, . . . , K , (6) K P α n∈N θl Bl,n l=1
where (3)
Aα k,n =
X
i∈In
1/α
πi,k Ci
α , Bk,n =
X
i∈In
1/α−1
πi,k Ci
.
plicit dependence on t is suppressed. Then:
For a given cell n, the values of
τi,k =
1/α−1 πi,k θk Ci K P P
l=1
j∈In
,
i ∈ In ,
1/α−1 πj,l Cj
l=1
= λk∗ − µk∗
l=1
α θl Bl,n
< λk∗ − µk∗
= λk ∗ − < ǫ,
Theorem 1: If (ρ1 , . . . , ρK ) ∈ R′α , then the system is stable, where R′α is the largest open subset of Rα . If ρ ∈ / Rα , then the system is unstable. Proof The proof relies on the consideration of fluid limits [27], where systems with a large population of flows are considered. In such limiting systems, the total service rate received by class-k flows is given by: ¯k (t)Aα X N k,n , K P α ¯ n∈N Nl (t)Bl,n
X
¯k (t) > 0. In the above formula, N ¯k (t) denotes the for N number of class-k flows at time t. Intuitively, one can justify this formula by observing that when the number of class-k flows is very large then at any instant, the number of such ¯k (t). This statement can be flows in state i should be πi,k N formally justified as in [28], but this goes beyond the scope of the present paper. The evolution of the fluid limit is characterized by the following set of differential equations: d ¯ Nk (t) = λk − µk × rk (t). dt Sufficient stability condition Assume that (ρ1 , . . . , ρK ) ∈ R′α . Then there exists a vector P θk Aαk,n K − ǫ for (θ1 , . . . , θK ) ∈ R+ such that λk < µk K P n∈N
l=1
α θl Bl,n
ǫ > 0 sufficiently small for all k = 1, . . . , K. We now consider ¯k (t)/θk and show that it the quantity y(t) := maxk=1,...,K N will continuously decrease at a strictly negative rate. ¯k (t)/θk , where the imDenote k ∗ := arg maxk=1,...,K N
K P
l=1
α N ¯l (t)/N ¯k∗ (t) Bl,n
Aα k∗ ,n
K P
¯k∗ (t) > 0. We conclude that the fluid limit reaches whenever N 0 in finite time, and hence the system is stable [27]. Necessary stability condition Denote by ∂Rα the boundary of R, i.e., r ∈ ∂Rα if r ∈ Rα and there exists a k such that ∀ǫ > 0, r + ǫ.ek ∈ / Rα . This boundary is the union of the surfaces ∂RL over all non-empty α subsets L of {1, . . . , K}, with: ¯ |L| |L| L ′ ∂Rα = r ∈ RK + : ∃θ ∈ R+ , ∃θ ∈ R+ , θk Aα k,n α , θl Bl,n l∈L n∈N X θk′ Aα k,n ¯ P ∀k ∈ L, rk ≤ , ′ α ¯ θl Bl,n l∈L L
∀k ∈ L, θk > 0, rk =
l=1
Aα k∗ ,n
α θ /θ ∗ Bl,n l k l=1 X Aα k∗ ,n θk∗ µk∗ K P α θ n∈N Bl,n l l=1 n∈N
class-k flows. The next theorem shows that Rα is the capacity region of the α-fair scheduler.
rk (t) =
X
n∈N
may be interpreted as the fraction of resources allocated to class-k flows in state i. The components of the vector (θ1 , . . . , θK ) may be interpreted as the numbers of flows of the various classes. With that interpretation, the quantity P θk Aαk,n represents the total service rate received by K P n∈N
¯k∗ (t)Aα∗ X N d ¯ k ,n Nk∗ (t) = λk∗ − µk∗ K P dt ¯l (t)B α n∈N N l,n
X
P
n∈N
where L¯ = {1, . . . , K} \ L and N L denotes the set of cells that are not visited by users of classes in the set L. The surface ∂RL α is the set of points of Rα parametrized by θ such that ¯ θk ≫ θl . It is worth remarking that ∂RL ∀k ∈ L, ∀l ∈ L, α is part of a cylinder with directions parallel to the components ¯ i.e., if r is a point of this surface, then r′ = (rk , k ∈ in L, ¯ is also a point of this surface provided that for L, rl′ , l ∈ L) ¯ all l ∈ L, 0 ≤ rl′ ≤ rl . It is also important to remark that in |L| the definition of ∂RL α , we can choose θ ∈ R+ with strictly positive components; this is because the points obtained when some of the components of this θ are equal to 0, are included in some other surfaces with a different set L. Now we prove that the system is unstable by induction on the number K of flow classes. The result holds for K = 1. Assume that it is true for all systems with at most K − 1 classes. Let us prove it in the case of K-class systems. Assume that ρ = (ρ1 , . . . , ρK ) ∈ / Rα . Without loss of generality, we can assume that ρk > 0 for all k. Now define γ as the maximum real number such that γ × ρ ∈ Rα . By assumption, γ < 1. Of course we have γ × ρ ∈ ∂Rα , and there exists a set of classes L such that γ × ρ ∈ ∂RL α . We consider two cases: (i) If L = {1, . . . , K}, then we deduce that there exists
θ ∈ RK + such that for all k, θk > 0 and: X θk Aα k,n . K P α n∈N θl Bl,n l=1
ρk >
We deduce that the system is unstable. Indeed one can easily show that the fluid limit grows at least linearly to ∞. (ii) Otherwise, we consider the restricted system where the classes in L¯ have no traffic. Note that the restricted system provides a stochastic lower bound of the actual system (this is due to the fact that all the systems considered are monotonic [29]). Hence we just need to prove that the restricted system is unstable. Note that the projection of ∂RL α on the sub-space of L components is actually the boundary of the set RL α defined by: RL α =
|L|
|L|
r ∈ R+ : ∃θ ∈ R+ , ∀k ∈ L, rk ≤
X
n∈N
θk Aα k,n P
α l∈L θl Bl,n
ff .
Since ∂RL / RL α is a cylinder, we deduce that (ρk , k ∈ L) ∈ α, and finally that the restricted system is unstable by induction. 2 B. With fast fading
o X θk A¯α k,n , ∀k = 1, . . . , K , rk ≤ K P ¯α n∈N θl B l,n
(7)
l=1
X
with Aα ∗n =
i∈In
1/α πi Ci ,
α = B∗n
P
i∈In
1/α−1
πi Ci
and πi ≡
πi,1 . Note that even in the case of a single user class, the capacity region Rα depends on the value of α in general, α unless Ci ≡ C∗n for all i ∈ In , n ∈ N , so that Aα ∗n /B∗n = C∗n for all α ≥ 0. The latter scenario in particular occurs in the absence of intra-cell mobility, i.e., |In | = 1 for all n ∈ N . 2) Networks with intra-cell mobility only: We now turn to the special case with intra-cell mobility only, i.e., each of the sets Nk is just a singleton, which in particular covers a singlecell scenario, i.e., |N | = 1. The capacity region then takes the form n o θk Aα K k∗ Rα = r ∈ R K , ∀k = 1, . . . , K , + : ∃θ ∈ R+ such that rk ≤ K P α θl Bl∗ l=1
n K Rα := (r1 , . . . , rK ) ∈ RK + : ∃θ ∈ R+ such that
A¯α k,n =
n∈N
P
P 1/α with Aα := and k∗ i∈Jk πi,k Ci P 1/α−1 , which may be represented i∈Jk πi,k Ci
Define:
where
1) Networks with a single user class: We first consider the special case with only a single user class, i.e., K = 1. The capacity region then reduces to # " X Aα ∗n , Rα = 0, α B∗n
1/α
πi,k C i
α ¯k,n , B =
i∈In
X
1/α−1
πi,k C i
.
i∈In
As in the case of systems without fast fading, we can show that Rα is the capacity region of the α-fair scheduler when fast fading is taken into account. ′
Theorem 2: If (ρ1 , . . . , ρK ) ∈ Rα , then the system is ′ stable, where Rα is the largest open subset of Rα . If ρ ∈ / Rα , then the system is unstable. Proof Again the proof relies on the use of fluid limits. Since the latter are obtained considering systems with large populations, the service rates of flows are given by (5). Thus, in the fluid limit, the system with fast fading is equivalent to a system without fast fading and the feasible rates Ci replaced by C i . The proof is completed exactly as that of Theorem 1. 2 In view of the clear analogy between the scenarios with and without fast fading, in what follows we only analyze systems without fast fading. C. Special cases and examples In the previous two subsections we obtained a characterization of the capacity region Rα for α-fair schedulers. In this subsection we focus on some special cases of interest and present an example of a three-cell three-class network to illustrate the rather complex shape of the capacity region.
compact manner as n Rα = r ∈ RK + :
X
k:Nk ={n}
α Bk∗ := in a more
o rk /Dkα ≤ 1, ∀n ∈ N ,
α with Dkα := Aα k∗ /Bk∗ . Again, observe that even in the case of intra-cell mobility only, the capacity region Rα depends on the value of α in general, unless Ci ≡ Ck∗ for all i ∈ Jk , k = 1, . . . , K, so that Dkα = Ck∗ for all α ≥ 0. 3) Three-cell three-class network: In scenarios with several flow classes and inter-cell mobility, the capacity region Rα has a non-linear boundary in general, and the typical shape turns out to be rather intricate. We present an example of a three-cell three-class network to illustrate how astonishing this shape can be. There are five states, with C1 = C4 = C5 = 2, C2 = C3 = 1, I1 = {1, 2}, I2 = {3, 4}, and I3 = {5}. Class-1 users oscillate between states 1 and 3 with probability 0.1 to be in state 1, class-2 users between states 2 and 4 with probability 0.9 to be in state 2, and class-3 users between states 2 and 5, with equal probability to be in each state. Figure 1 depicts R1 , i.e., the capacity region for the PF scheduling strategy, and indicates the various surfaces ∂RL 1 composing the boundary of R1 .
IV. I MPACT OF FAIRNESS INDEX α AND OF MOBILITY ON CAPACITY
In the previous section, we have characterized the capacity region Rα of the network under the α-fair scheduler. As already mentioned, Rα in general depends on the value of α, even in networks with a single user class or intra-cell mobility only. In the next subsection, we examine how the capacity region varies with α. We establish that Rα shrinks when the fairness index α increases. This result implies that schedulers
so that K X
ζk σk,n ≥
K XX
yil
i∈In l=1
k=1
κ !− 1+κ
K XX
yik xik
i∈In k=1
κ ! 1+κ
.
We also have K X
′ ζk σk,n =
k=1
K “ ′ ”κ P P θk θk
k=1
i∈In
K P P
i∈In l=1
K P P
1
πi,k θk′ Ciα 1 −1 ′
πi,l θl′ Ciα
i∈In l=1
Fig. 1. Capacity region R1 of a three-cell three-class network with mobility.
that realize max-min fairness (obtained when α → ∞) achieve the smallest capacity region among all α-fair schedulers. It also shows that the PF scheduler (α = 1) can be outperformed (in terms of capacity) by α-fair schedulers with fairness index α < 1. For a given scheduler, it is also clear that the capacity region Rα depends on the mobility of users. In the last part of this section, we show that for a given scheduler, i.e., for a fixed α, user mobility increases the capacity region. A. Smaller fairness index increases capacity The following proposition formally states that if the fairness index decreases, then the capacity region of the network increases. In Section V, we will provide numerical experiments to quantify how this region is impacted by α, and show that this impact can be significant depending on the degree of mobility of users. Proposition 1: If α′ ≥ α, then Rα′ ⊆ Rα . ′ ) ∈ RK Proof Let (θ1 , . . . , θK ), (θ1′ , . . . , θK + . Define
0
1
1+κ xil yil ≥ @
K X X
i∈In k=1
1
K P P
i∈In l=1
Because of convexity, K X X
κ 1+κ
yik A
0 @
yik xik
i∈In k=1
=
K X X
i∈In l=1
.
1
xil1+κ yil
1
1
1 1+κ
κ+1 xil yil A
,
so that K X
′ ζk σk,n ≤
K XX
yil
i∈In l=1
k=1
Denoting σk :=
P
n∈N
κ !− 1+κ
σk,n σk′ :=
over n ∈ N , we obtain K X
k=1
ζk σk =
K XX
ζk σk,n ≥
n∈N k=1
K XX
P
n∈N
K XX
.
′ σk,n , and summing
′ ζk σk,n
n∈N k=1
which implies that σk′ ′ ) (θ1 , . . . , θK ), (θ1′ , . . . , θK
yik xik
i∈In k=1
κ ! 1+κ
=
K X
ζk σk′ ,
k=1
≤ σk for some k for all ∈ RK +. Now suppose that r 6∈ Rα . As in the proof of Theorem 1, define γ as the maximum real number such that γ × r ∈ Rα . There exists a set J of classes such that γ × r ∈ ∂RJ α . Using similar induction on the number of classes as in Theorem 1, we can just consider the case where J = {1, . . . , K}. Then, there exists (θ1 , . . . , θK ) ∈ RK + such that rk > σk for all k = 1, . . . , K. We deduce that rk > σk′ for some k for all ′ 2 ) ∈ RK (θ1′ , . . . , θK + , which means that r 6∈ Rα′ . B. Mobility increases capacity
1
P
πi,k θk Ciα
θk Aα i∈In k,n = , K K 1 −1 P P P α α πi,l θl Ci θl Bl,n
σk,n :=
i∈In l=1
l=1
′ and similarly, σk,n replacing α by α′ and θ by θ′ in the −1 ′ above expression. Further define ′ κκ:= [1/α − 1/α] , and θ for compactness denote ζk := θkk . Then K X
ζk σk,n =
k=1
K “ ′ ”κ P P θk θk
k=1
i∈In
K P P
i∈In l=1
1
πi,k θk Ciα 1 −1 α
=
πi,l θl Ci
K P P
i∈In k=1
κ
κ+1 yik xik
K P P
,
̺n :=
yil
K X X
i∈In k=1
κ/(1+κ) yik xik
≥@
K X X
i∈In l=1
1
yil A
1 1+κ
0 @
K X X
i∈In k=1
1
yik xik A
K X X
πi,k rk /Ci
k=1 i∈In
i∈In l=1
′ κ+1 1/κ+1 (1/α)−1 θ Ci and yik := πi,k θk Ci . with xik := θkk ′ Observe that α ≥ α implies that either 1/(1 + κ) ≤ 0 and κ/(1 + κ) ≥ 1 or κ/(1 + κ) ≤ 0 and 1/(1 + κ) ≥ 1. Because of convexity, 0
To examine the impact of mobility on capacity, we compare two scenarios where users are either mobile or completely stationary for the entire duration of their flow transfers. For a meaningful comparison, we assume that in both scenarios, the user positions have the same marginal distribution. In other words, in the absence of mobility, class-k flows are generated in state i with probability πi,k . In the absence of mobility, flows must be entirely served in the states where they are generated, and the load of cell n is
κ 1+κ
,
when the offered traffic from class-k flows is rk . Hence, the capacity region does not depend on the value of α and is given by Rno = {(r1 , . . . , rK ) ∈ RK + : ̺n =
K X X
k=1 i∈In
πi,k rk /Ci < 1, ∀n ∈ N }.
The next proposition shows that mobility increases the capacity region under any given α-scheduler. Proposition 2: For any α > 0, Rno ⊆ Rα . Proof Proposition 1 shows that the capacity region Rα shrinks when the value of α increases. Thus to prove Proposition 2, it suffices to show that Rno ⊆ Rmax-min , with
Capacity of a linear network
7.5
i∈In l=1
which is established in the next proposition.
i∈In l=1
thus θrkk > minm∈N ηm . Also, ̺n
=
K X X
>
min ηm
m∈N
K P P
i∈In k=1
#−1
, and
K X rk X πi,k θk /Ci θk
k=1 K XX
πi,l θl /Ci
i∈In l=1
πi,k rk /Ci =
k=1 i∈In
6 5.5
4.5 0
Proof Suppose that r 6∈ Rmax-min . As in the proof of Proposition 1, we can just consider the case where there exists (θ1 , . . . , θK ) ∈ RK + such that: P πi,k θk X X X i∈In = πi,k θk ηn , rk > K P P n∈N i∈I n∈N n πi,l θl /Ci for all k = 1, . . . , K, with ηn =
7 6.5
5
2
Proposition 3: Rno ⊆ Rmax-min .
"
Linear network with 6 states in each class.
8
capacity (Mb/s)
K Rmax-min = R∞ = {(r1 , . . . , rK ) ∈ RK + : ∃θ ∈ R+ : P πi,k θk X i∈In rk ≤ , ∀k = 1, . . . , K}, K P P n∈N πi,l θl /Ci
Fig. 2.
i∈In
πi,k θk /Ci = min ηm /ηn , m∈N
implying that maxn∈N ̺n > minm∈N ηm maxn∈N 1/ηn = 1, which means r 6∈ Rno . 2 V. N UMERICAL EXPERIMENTS We now apply the theoretical results to numerical experiments on a few pertinent scenarios. We consider scenarios that, while simple, include the important features for this study such as user mobility. A. Radio environment As mentioned in Section II, a user’s feasible rate is a function of radio conditions. In the numerical experiments performed here these radio conditions are determined primarily by the user’s position in the cell, or serving area. We assume user feasible rates to be identical to those offered by the CDMA 1xEV-DO system [19], where each data rate corresponds to a target SINR (signal-to-interference-and-noise ratio). We classify the state of a user according to the highest rate corresponding to his SINR. In the following numerical experiments we assume an urban setting and consider linear and hexagonal networks. For each scenario we characterize the capacity region corresponding
Fig. 3.
0.2
0.4
0.6 α
0.8
1
max-min
Capacity of linear network with user mobility.
to various values of α. The interaction between classes and cells through user movement renders the exact analysis of performance metrics such as mean flow response time and throughput difficult. We therefore examine the mean flow throughput - defined as the ratio of flow size (in bits) and transfer delay (in seconds) - through simulations. B. Linear networks We first consider BS’s on a line, to represent roads, highways, or railways, as shown in Figure 2. The example shows the scenario implemented here, a linear network with three states (a,b,c in the figure) on either side of the base staton, with rates Ca = 2.4 Mb/s, Cb = 1.8 Mb/s, Cc = 1.2 Mb/s. These rates correspond to CDMA 1xEV-DO rates for cells of radius 1km. We consider file sizes of 75 KBytes or 150 KBytes and assume equal traffic intensity for all classes and user mobility with mean speeds from 30 km/h to 240 km/h. As mentioned in Section II, we assume Markovian user movements, here along the line between BS’s. For this network, we calculate the stability condition without user mobility to be ρ = λ/µ < 4.74 Mb/s. In the presence of mobility, Figure 3 shows that the limit of stability, or the capacity region, increases as α is decreased. By observing the capacity for the PF scheduler, about 5 Mb/s, we note that user mobility itself results in a slight gain. As α is decreased from 1 to 0, the capacity increases by about 60%. Note that with max-min fairness the capacity is decreased to the lowest value, identical to that in the absence of mobility, 4.74. We now examine the effect of the parameter α and mobility on the mean flow throughput. We first consider file transfers of mean size 75 KBytes. In the absence of mobility, we have seen above that the capacity of the system is unchanged as α varies. The value of α does however have an impact on flow throughput in the presence of mobility. The plot on the left in Figure 4 shows the flow throughput for varying values of α at a speed of 90 km/h. The throughput increases significantly as α is closer to 0. Indeed, when the PF scheduler has reached the capacity limit, the systems with a smaller value of α remain
Small α
At speed 90 km/h, file size 75 KBytes
0.25 0.2 0.15 0.1 0.05
0.2 speed: 0 30 km/h 60 km/h 90 km/h 120 km/h 240km/h
0.35 0.3 0.25 0.2 0.15 0.1 0.05
0 4.5
5
5.5 6 6.5 traffic intensity
7
7.5
8
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02
0 4
α ≈0 0.2 0.5 1 max-min
0.18 mean throughput (Mb/s)
α ≈0 0.2 0.5 1 max-min
0.3
At speed 90 km/h, file size 150 KBytes
0.4 mean throughput (Mb/s)
mean throughput (Mb/s)
0.35
0 4
4.5
5
5.5 6 6.5 traffic intensity
7
7.5
8
4
4.5
5
5.5 6 6.5 traffic intensity
7
7.5
8
Fig. 4. Mean throughput in a linear network- Left: mean user speed of 90 km/h, varying α - Center: small α, varying speed - Right: as on left, with file size 150 KBytes.
Capacity region for hexagonal network, classes 1 and 3 4
α ≈0 0.1 0.2 0.4 0.6 0.8 1 max-min
3.5 3
r3
2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
r1
Fig. 5. Left: Hexagonal network, edge of three cells with three classes. Right: The capacity region.
stable, and still offer acceptable throughput. The center plot of Figure 4 plots the mean flow throughput against the traffic intensity for varying values of speed. As shown for a small value of α close to 0 in the plot, the mean flow throughput increases significantly as users move at higher speeds. We have plotted results for α close to 0 as an illustration, since the results are similar for other values of α, albeit with smaller gains for higher values of α. The plot on the right in Figure 4 plots the flow throughput under the impact of α for a mean file size of 150 KBytes. The throughput gains here are higher than those observed for file transfers of 75 KBytes. As the file size increases, mobility occurs at smaller time scales, with respect to the flow transfer time. This allows users even more opportunities to take advantage of passage through states with higher data rates. C. Hexagonal networks We now consider a hexagonal network that may represent cells in a city center. We assume cells of radius 200 m, with 6 states in each cell. We focus on the area near the cell edge as users in this area in general suffer the most from poor radio conditions. As shown on the left in Figure 5, we consider the junction of three cells, with 3 states per cell having rates Ca = 921 kb/s, Cb = 614 kb/s, and Cc = 307 kb/s. We assume user mobility with speeds of 3 km/h to 30 km/h. We consider a scenario of three classes, with their paths as shown in the figure. The right side of Figure 5 shows the capacity region for this
scenario with user mobility. Note that classes numbered 1 and 2 on the left are symmetric. We thus show results for classes 1 and 3, noting that class 3 is the more vulnerable class with its path mostly on areas with low data rates. As α is decreased we observe that the capacity region increases significantly with respect to that with PF and max-min scheduling. We first examine the mean flow throughput for file transfers of mean size 75 KBytes. The mean global throughput increases significantly as α is decreased, similar to the example of a linear network shown above. Here, we investigate the effect of the parameter α on the asymmetric classes by examining the mean throughput per class. The curve on the left in Figure 6 plots the mean flow throughput for various values of α, with mobility at pedestrian speed of 6 km/h. We note the important result that as α is reduced, the more vulnerable class does not suffer from reduced throughput, as may occur in systems without user mobility. The users’ mobility during the transfer of a flow allows them to experience high rates in good states, thus offsetting any reduced service received in the poor states. The poor classes, those with paths through less favorable conditions, also benefit from reduced values of α, albeit to a lesser extent than classes with better states. The center plot of Figure 6 shows the impact of mobility on flow throughput for α close to 0. An important observation here is that for user speeds as low as 3 km/h, decreasing α towards 0 increases the mean throughput even for the poorer class. The plot on the right in Figure 6 shows the impact of α for larger file sizes, with mean 150 KBytes. Here we note that even for a speed of 6 km/h, there are significant gains in mean throughput, for all classes of flows. As discussed for linear networks, the impact of α and mobility on the throughput is more significant as file transfers are larger. VI. D ISCUSSION : M OBILITY- DRIVEN S CHEDULER We have shown that under the family of α-fair scheduling schemes, user mobility increases capacity for a given value of α. As an aid to understand why this is the case, let us consider an implementation of an α-fair scheduler as in (1), the gradient-based algorithm [17]. We see here that for a given class of users, those in favorable states are more likely to be scheduled. The variations in users’ feasible rates due to
Small α, file size of 75 KBytes
At 6 km/h, file size of 75 KBytes
150 100 50 0
120 speed 0 3 km/h 6 km/h 12 km/h 18 km/h 30 km/h
200 150
mean throughput (kb/s)
α ≈0 0.2 0.5 1 max-min
200
At 6 km/h, file size of 150 KBytes
250 mean throughput (kb/s)
mean throughput (kb/s)
250
100 50 0
0
1
2
3 4 traffic intensity
5
6
α ≈0 0.2 0.5 1 max-min
100 80 60 40 20 0
0
1
2
3 4 traffic intensity
5
6
0
1
2
3 4 traffic intensity
5
6
Fig. 6. Mean throughput in a hexagonal network- Left: mean user speed of 6 km/h, varying α - Center: small α, varying speed - Right: as on left, with file size 150 KBytes.
mobility then allows users to enter states where they have a better chance of being scheduled, even when α is large. A more interesting result we have shown concerns the impact of α in the presence of user mobility. The capacity region of the system increases as α decreases. More importantly, less favorable flow classes (i.e. flows with low mean feasible rate over their entire path) do not suffer in mean throughput at low values of α. In fact they experience better performance, their gains being not as significant as those of more favorable classes. In the presence of user mobility then, a low value of α can increase the capacity of the system and have no adverse unfairness issues. Furthermore, our analysis holds when we allow α to vary across BS’s. This follows intuitively when we consider that each BS makes scheduling decisions largely independently of scheduling in neighboring BS’s. We then propose, especially for cells that include highly mobile traffic, such as cells on roads and railways, or city centers, that the scheduling parameter α be modified to take advantage of user mobility. The variations in users’ feasible rates due to mobility can be harnessed to schedule users as they move through favorable states. As we have seen in the above theoretical and numerical results, this offers benefits to all users. R EFERENCES [1] R. Knopp and P. Humblet, “Information theory and power control in single-cell multi-user communications,” in Proc. ICC ’95, 1995. [2] V. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inf. Theory, vol. 48, pp. 1277–1294, 2002. [3] R. Agrawal, V. G. Subramanian, and R. Berry, “Joint scheduling and resource allocation in CDMA systems,” Submitted for publication in IEEE Trans. Inf. Theory, 2008. [4] D. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar, and P. Whiting, “Scheduling in a queueing system with asynchronously varying service rates,” Prob. Eng. Inf. Sc., vol. 18, pp. 191–217, 2004. [5] A. Eryilmaz and R. Srikant, “Fair resource allocation in wireless networks using queue-length-based scheduling and congestion control,” in Proc. IEEE Infocom, 2005. [6] A. Eryilmaz, R. Srikant, and J. Perkins, “Stable scheduling policies for fading wireless channels,” IEEE/ACM Trans. Netw., vol. 13, pp. 411– 424, 2005. [7] L. Georgiadis, M. Neely, and L. Tassiulas, “Resource allocation and cross-layer control in wireless networks,” Found. Trends Netw., vol. 1, pp. 1–144, 2006. [8] M. Neely, E. Modiano, and C. Rohrs, “Power and server allocation in a multi-beam satellite with time-varying channels,” in Proc. IEEE Infocom, 2002. [9] ——, “Dynamic power allocation and routing for time-varying wireless networks,” IEEE J. Sel. Areas Commun., vol. 23, pp. 89–103, 2005.
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