Delay Optimal Control Algorithm for a Multiaccess Fading Channel with Peak Power Constraint Munish Goyal, Anurag Kumar, Vinod Sharma Department of Electrical Communication Engg Indian Institute of Science, Bangalore, India Email: munish, anurag,
[email protected] Abstract— We consider an optimal power and rate scheduling problem for a multiaccess fading wireless channel with the objective of minimising a weighted sum of mean packet transmission delay subject to a peak power constraint. The base station acts as a controller which, depending upon the buffer lengths and the channel state of each user, allocates transmission rate and power to individual users. We assume perfect channel state information at the transmitter and the receiver. We also assume a Markov model for the fading and packet arrival processes. The policy obtained represents a form of Indexability.
I. I NTRODUCTION We address an optimal resource allocation problem in wireless networks. Several challenging analytical problems arise because of the special limitations of a wireless link such as time varying nature of the multipath channel, and limited battery power available at a typical wireless handset. It is desirable to allocate transmission rates to a user such that the energy used to transmit the information is minimized while keeping errors under control. Most applications, however, also have quality of service (QoS) objectives such as mean delay, delay jitter, and throughput. Thus there is a need for optimal allocation of wireless resources which provides such QoS guarantees subject to the above said error and energy constraints. Various methods for allocating transmission resources are part of most wireless standards. In this work we address a special case where the transmission is peak power constrained and the receiver has multiuser detection capability. The objective is to obtain a resource allocation strategy that minimizes a weighted sum of mean transmission delays. Tse and Hanly [6], [2] obtained the optimal power allocation policies for a multiaccess fading wireless channel. In [6], the objective was to maximize the weighted throughput while in [2], the objective was to provide deterministic delay guarantees. Each transmission was average power constrained. These works looked at the physical layer performance and ignored the network layer aspects such as queueing delay. In a recent work, Berry and Gallager [1] have considered a problem similar to ours. They obtained structural results exhibiting a tradeoff between the network layer and physical layer performance, i.e. the optimal power and mean delay. Recently Yeh and Cohen [8] showed that given a power allocation policy that is independent of the queue length process, a longest queue highest possible rate policy minimizes the mean queue length. In this work (a part of the PhD thesis [3]) we have
tried to obtain a joint rate and power allocation policy with an objective of minimizing mean delay. The paper is organized as follows. In Section II, we give the model of the system under consideration. We analyse a two user problem in Section III. Multiuser results are stated in Section VI. II. S YSTEM M ODEL We consider M user system. The system model in our work is shown in Fig. 1. Time is slotted into slots of length τ (time units) each, and the nth slot is the interval [nτ, (n + 1)τ ), n ≥ 0. Depending upon the channel bandwidth let N be the number of channel uses per slot. The packets to be transmitted arrive into the system from a higher layer at the end of each slot and are placed into a buffer of infinite capacity (See Figure 1). The packets are of length b bits each. The arrival process A[n] is assumed to be a finite state Markov chain; let Pa be the transition probability matrix. The channel power gain process h is assumed to remain fixed over a slot and the process H[n] is a finite state Markov chain; let Ph be the transition probability matrix. We further assume that the receiver can correctly estimate the channel gains h on the “uplink” using a pilot channel. According to our model, if in slot n, users transmit a signal ys [n], then the received signal, yr [n] =
M hi [n]ys,i [n] + ζ[n], i=1
where ζ[n] is the additive white Gaussian noise or the receiver noise. Let σ 2 be the receiver noise power. Over a mini-slot (shown via a shaded time in Fig. 1), the buffer length information is communicated to the receiver/controller. The receiver acts as a controller which, given the buffer state and the channel gain of each user, obtains an optimum transmission schedule (transmission rate and power) that minimizes the weighted sum of mean buffer delay subject to peak power constraint Pˆ . The decisions are taken within a mini-slot shown as shaded and conveyed back to the transmitter also in the same mini-slot. We assume that the transmitter can transmit at any rate and power level. The transmitter removes the scheduled amount of data from the buffer, encodes it at the allocated rate and transmits the encoded data over the channel at a scheduled power level. Let, for n ∈ {0, 1, 2, · · ·}, q[n] be the vector representing the number of packets in the buffers of different users,
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Data Arrives fromhigher layer
Exchange of Contol information Data transmission
HIGHER LAYER
USER 1
k+2
k+1
k TX BUFFER
q1[k]
CHANNEL
r1[k]
n
AWGN
η i[k]
h [K] 1 RECEIVER &
Controller USER M
qM [k]
rM[k]
Fig. 1.
n−1 M
n
subject to, i∈S
ηM [k]
System model
at the nth decision epoch and h[n] be the corresponding channel gains in the nth slot (i.e., the interval [nτ, (n + 1)τ ]). Let the state of the system be represented as x[n] := (q[n], h[n], a[n]). At the nth decision instant, the controller decides upon the number of packets r[n] to be transmitted in the current slot and P[n] the transmitter power level depending on the entire history of state evolution, i.e., x[k] for k = {0, 1, 2, · · · , n}. Since the number of packets transmitted in a slot should be less than that in the buffer, i.e., r[n] ≤ q[n], for all n, the evolution equation for the buffer can be written as q[n + 1] = q[n] − r[n] + a[n]. Denote by X[n], Q[n], R[n], n ≥ 0, the corresponding random processes. The cost of transmitting R[n] packets is the transmitter power P[n] required for reliable reception. The vectors R and P are related according to the Shannon’s formula for the ergodic capacity for the AWGN multiple access channels, i.e., given the channel gain vector h, power vector P and for every S ⊂ {1, 2, · · · , M }, 1 j∈S hj Pj R(S) ≤ ln 1 + , (1) θ σ2 where R(S) = i∈S Ri and θ = 2 ln(2)b . N Since delay is related to the amount of data in the buffer by Little’s formula, the objective is equivalent to minimizing a weighted sum of the mean buffer lengths. Further, observe from Equation 1 that the power vector P can be replaced by ˆ Note that this does not imply that the power allocation P. ˆ The power allocation vector P[k] can be vector P equals P. evaluated using inverse relation to Equation 1 once the rate allocation vector R[k] is known. The controller’s objective is thus to obtain the optimal R[n] that minimizes lim sup
h [K] M
1 ωi Qi [k], E n i=1 k=0
ˆ 1 j∈S hj Pj , ∀S ⊂ {1, 2, · · · , M }. Ri [k] ≤ ln 1 + θ σ2
where weights ωi define priority among users. This is a constrained average cost Markov decision problem. III. A NALYSIS In order to motivate the approach that we will follow, first we present a two user example. Define c(x) = θ1 ln 1 + σx2 . The two user problem is, n−1
min lim sup n
1 E (ω1 Q1 [k] + ω2 Q2 [k]), n k=0
subject to Ri [k] ∈ {0, 1, · · · , Qi [k]}; ∀ k and i ∈ {1, 2, · · · M }, R1 [k] ≤ c(H1 [k]Pˆ1 ), R2 [k] ≤ c(H2 [k]Pˆ2 ), R1 [k] + R2 [k] ≤ c(H1 [k]Pˆ1 + H2 [k]Pˆ2 ). Observe that had the problem been a single transmitter communicating to a single receiver, the optimal solution would be to transmit as much as possible subject to the peak power constraint, R[n] = min Q[n], c(H[n]Pˆ ) . In two users setting, the problem is nontrivial. The coupling constraint R1 [k] + R2 [k] ≤ c(H1 [k]Pˆ1 + H2 [k]Pˆ2 ) is the main cause of concern. We wish to relax this coupling constraint by requiring it to be satisfied on a long run. Such constraint relaxation is not new and to our knowledge Whittle proposed such relaxation in 1970’s and obtained a heuristic control policy for restless bandit problems [7]. These policies were later shown to possess certain asymptotic optimality. Refer to [4] for a recent reference on such problems. It is also known that such a relaxation often results in a good policy. We now briefly discuss the restless bandit problems, Whittle relaxation and index heuristic policies. Consider the “restless bandits” problem of designing an optimal sequential resource allocation policy for a collection of stochastic projects (say M ), each of which is modeled as a Markov decision chain having two actions at each state with associated rewards; an active action, which corresponds to engaging the project,
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and a passive action, which corresponds to letting it go. The passive projects can change state, in general through a given transition rule and hence the word “restless”. A fixed number of resources needs to be allocated; i.e., at each time instant a fixed number of projects (say k) are active. The performance objective is to maximize the time-averaged reward rate. Whittle [7] presented a simple heuristic based on a tractable optimal solution to a relaxed version, where instead of requiring that k projects be active at any time, k projects are needed to be active on an average. This yielded an upper bound on the optimal reward. Further the heuristic policy is a priority index rule associated with each project, that engages the top k projects at any given point of time. Motivated by Whittle’s work on restless bandits, we introduce a relaxed version of the two user problem. We relax the coupling constraint by requiring it to be met on an average. By associating a Lagrange multiplier β with this average constraint, we obtain the objective functional,
Theorem 4.1: V (u, c, a) and hence H(u, c, a) is convex nondecreasing in u. The unconstrained minimizer u∗ (c, a) in (3) is the value of u that solves the following inequalities, H(u, c, a) − H(u − 1, c, a) ≤
Note that the unconstrained minimizer is not a function of q. The solution for the constrained problem (u ∈ U (x)) is, ∗ • r(x) = 0 for q < u (c, a), ∗ • r(x) = c for q > u (c, a) + c, ∗ • r(x) = q − u (c, a) otherwise.
c
r c
n−1 2 1 lim sup E ( (ωi Qi [k] + βRi [k])). n n i=1
u*
k=0
This can be seen as a form of Whittle relaxation and by choosing an appropriate value of β we can satisfy the coupling constraint at all times. Note that the objective functional is an additive function and the remaining constraints are not coupled. Thus we can solve the decoupled problem for each i. We drop the subscript i. Define a decoupled Markov decision problem as n−1
min lim sup n
1 E (Q[k] + βR[k])), n
(2)
k=0
subject to R[k] ≤ c(H[k]Pˆ ); R[k] ∈ {0, 1, · · · , Q[k]}. The capacity function c(H[k]Pˆ ) defines a Markov chain, say C[k], since H[k] is a Markov chain. Thus we use c as a state variable instead of h. This is an average cost Markov decision problem. We first analyse the corresponding discounted cost problem and then show that the results for this average cost problem can be obtained as a limit of the discounted cost solution. IV. T WO U SER R ELAXED P ROBLEM : D ISCOUNTED C OST The state x is the triplet (q, c, a). The relaxed problem satisfies the nominal conditions [5] required for the existence of the discount optimal stationary policy, and the value function V (x) is obtained as a solution to the following dynamic programming optimality equation. Define u = q − r and U (x) = {u integer : (q − c)+ ≤ u ≤ q}. The variable u is the residual number in the queue after the policy has acted in an interval. Then the discounted cost optimality equation corresponding to the Markov decision problem (Equation 2) is, V (q,c,a)=minu∈U (x) {q(1+β)−βu+αEa,c V (u+A,C,A)}.
(3)
Define H(u, c, a) = Ea,c V (u + A, C, A). We state the following result. The proof is given in Appendix.
β ≤ H(u + 1, c, a) − H(u, c, a). α
q
Remark 4.1: Observe that u∗ (c, a) = q is the break point that will be used to define the indices [7] as it is the boundary between not sending anything from the queue and sending something. Given c and a , if q ≤ u∗ (c, a), no packet is transmitted and the number of packets transmitted increases with q justifying the intuition that the number of packets transmitted should increase with the queue length. 1) An Algorithm for Computing u∗ (·): Consider the discounted cost value iteration algorithm corresponding to the relaxed problem (3), Vn (q,c,a)=minu∈U (q,c,a) {q(1+β)−βu+αEa,c Vn−1 (u+A,C,A)}.
(4)
It is easy to verify conditions (refer [5]) required for the existence of a stationary discount optimal policy and that the value iteration indeed converges to the discount value function. It can be shown along the lines of the result stated in Theorem 4.1 that the functions Hn (u, c, a) are convex in u for each n. Let u∗n (c, a) be the value of u that solves the following inequalities, Hn (u, c, a)−Hn (u−1, c, a) ≤
β ≤ Hn (u+1, c, a)−Hn (u, c, a). α
Based on the above said constrained solution, we have, If q ≤ u∗n (c, a), Vn+1 (q, c, a) − Vn+1 (q − 1, c, a) = 1 + α(Hn (q, c, a) − Hn (q − 1, c, a)) ∗ ∗ • If un (c, a) < q ≤ c+un (c, a), Vn+1 (q, c, a)−Vn+1 (q− 1, c, a) = 1 + β ∗ • If q > un (c, a) + c, Vn+1 (q, c, a) − Vn+1 (q − 1, c, a) = 1 + α(Hn (q − c, c, a) − Hn (q − c − 1, c, a)) Define Wn (q, c, a) = Vn (q, c, a) − Vn (q − 1, c, a). Thus Hn (q, c, a) − Hn (q − 1, c, a) = Ea,c Wn (q + A, C, A). The iterative algorithm to compute u∗ (c, a) is as follows. Initialize •
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W0 (q, c, a) = 0. Let u∗n (c, a) be the value of u that solves the following inequalities,
Ea,c Wn (u + A, C, A) ≤ αβ ≤ Ea,c Wn (u + 1 + A, C, A).
(5)
The following procedure then obtains Wn+1 (·) from Wn (·) and un (·): ∗ • If q ≤ un (c, a), Wn+1 (q, c, a) = 1 + αEa,c Wn (q + A, C, A). ∗ ∗ • If un (c, a) < q ≤ c + un (c, a), Wn+1 (q, c, a) = 1 + β. ∗ • If q > un (c, a) + c, Wn+1 (q, c, a) = 1 + αEa,c Wn (q − c + A, C, A). Further, u∗n+1 (·) is then calculated from Wn+1 (·) using Equation 5. The convergence of the value iteration algorithm (4) ensures that this algorithm converges and u∗n (c, a) converges to the optimal solution u∗ (c, a). Note that though the numerical recursion is over a countably infinite state space, the algorithm possess a very nice structure that helps in recursing over infinite space as well. In particular for each n there is a finite number say Dn (c, a) such that for q > Dn (c, a), Wn+1 (q, c, a) is constant. We have the following result showing the structural properties of the policy in terms of the penalty β. The proofs are given in the Appendix. Theorem 4.2: As β → 0, the solution u∗ (c, a) → 0 and α ∗ u (c, a) = ∞ for β > 1−α . Further, the unconstrained minimizer u∗ (c, a) is monotonically nondecreasing in β. Given a state x = (q, c, a) with q > 0, the number of packets served r(x) decreases to zero as β increases and r(x) = 0 for α β > 1−α . This is natural to expect since the larger is β, the higher is penalty for transmitting. Moreover, r(x) decreases monotonically with β. A. Mean Delay Criterion Thus far we looked at the discounted cost problem corresponding to the two user relaxed problem. We now consider the average cost problem (Equation 2). n−1
1 π min lim sup Ex (Q[k] + βR[k]) n n k=0
subject to R[k] ∈ {0, 1, · · · , Q[k]}, R[k] ≤ C[k], ∀k. First we verify the conditions for the existence of a stationary average cost optimal policy {R[k]} [5]. We first state a result from the paper which would imply that supα∈(0,1) wα (x) < ∞ where wα (x) is as defined in [5]. This will also verify the necessary conditions for the average optimal policy to be a limit of discount optimal policies. Theorem 4.3: [5] Given a constant η > 0, define a stopping time ς as, ς = inf{n ≥ 0, Vα (X[n]) ≤ Vα (x0 ) + η}, where Vα (X[n]) is the discount value function for discount α and x0 is a state for which Vα (x) is minimum. Define
wα (x) = Vα (x) − Vα (x0 ). Then for x in the state space, we have ς−1
π wα (x) ≤ η + inf Ex c(X[n], A[n]) . π
n=0
Note that the stability of the system ensures that the mean arrival rate E[A] for a user is strictly less than the average service obtainable when the user is the only one competing for the service, i.e., E[C]. Define a policy R[k] = min{Q[k], C[k]}. Note that the system is stable under this policy. Thus stability results imply that the system would be able to clear the buffer in finite mean time. Let ς be a stopping time as defined in Theorem 4.3. Given x = (q, c, a), we have for some η > 0, (see Theorem 4.3), ς−1
wα (x) ≤ η + inf Ex,π (Q[n] + βR[n]) π
n=0
≤ η + E[ς(q + amax ς) + ςβ E[C]] Note that the last equation does not involve α and hence we can take supremum over α < 1 to obtain the desired results. Further, since the average optimal policy is a limit of discount optimal policies, the average cost optimal policy also possess the structural properties of discount optimal policies. This implies that the unconstrained solution for the average cost problem again defined as u∗ (c, a) monotonically nondecreasing in β and thus the solution r(q, c, a) is monotone nonincreasing in β. To show the dependence of β, we denote the solution by r(q, c, a, β). The solution when the weight is ω is r(q, c, a, ωβ ). V. T WO U SER H EURISTIC P OLICY Now we fix the relaxed coupling constraint. Let ri (qi , hi , ai , ωβi ) be the optimal rate policy for user i when in state (qi , hi , ai ). We need to find a β that satisfies the constraint that the sum rates r1 + r2 ≤ c(H1 [k]Pˆ1 + H2 [k]Pˆ2 ). We know that as β increases, ri (·) decreases. Thus the two user solution would correspond to the smallest value of β for which the constraint is satisfied. Formally, find a β ∗ such that β ∗ = min {β > 0 : 2 β ˆ ˆ ≤ c(H1 [k]P1 + H2 [k]P2 ) . ri Qi [k], Hi [k], Ai [k], ωi i=1 Note that β ∗ is a function of (q, h, a). The policy corresponding to β ∗ defines the two user solution. VI. G ENERAL M
USER
P ROBLEM
The M user rate constraint set is ˆ ∀J ⊂ {1, 2, · · · , M }. Ri [k] ≤ c Hi [k]Pi ; i∈J
i∈J
The constraints that introduce coupling correspond to those sets J which have cardinality more than one. Define a Lagrange multiplier corresponding all such sets, say λ(J ) corresponding to the constraint involving set J . Define βi = λ(J ). Observe that βi is the sum of all the J :i∈J ;|J |>1
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Lagrange multipliers associated with the constraints involving ri . The new objective functional is lim sup n
n−1 M 1 ( (ωi Qi [k] + βi Ri [k])). E n i=1 k=0
Decoupling results in Equation 2. Thus as for the two user problem, we have the solution ri (qi , hi , ai , ωβi ) which is monotone nonincreasing in β. We need to find the values of β(J ) for J such that | J |> 1 and satisfies the constraints. The procedure would just be the same as that for the two user problem. Note that monotonicity of r(·) in β is essential. VII. C ONCLUSION We obtained an M user heuristic policy for minimizing mean packet transmission delay over multiaccess fading channels subject to a peak power constraint. The policy obtained exhibits nice structural properties. Note that the policies are simple in the sense that ri (qi , hi , ai , ωβi ) is independent of the number of users present in the system. Also the priorities ωi are easily incorporated into the policy. Policies obtained via such a relaxation are known to perform well [4]. R EFERENCES
where the inequality (∗) follows from the fact that the policies 2 2 u1 +u and u1 +u are feasible for the state (q, c, a). That 2 2 the functions are nondecreasing can also be proved along similar lines. Proof of Theorem 4.2: We need to find the behaviour of α . As r(x) as β decreases to zero or it increases beyond 1−α β → 0, Equation 3 implies that the cost of serving decreases to zero except that the constraint should be satisfied. Thus the solution would be to serve as much as possible, i.e., r(x) → min(q, c). Thus the action is active in any state where it is possible to do so. To show the other part, it is enough to show 1 α that Wn (q, c, a) ≤ 1−α . Since W0 (q, c, a) = 0, if β > 1−α , ∗ then u0 (c, a) = ∞ and W1 (q, c, a) = 1. Let Wn (q, c, a) ≤ 1 α ∗ 1−α . Then un (c, a) = ∞ and Wn+1 (q, c, a) ≤ 1 + 1−α . By 1 induction hypothesis it follows that W (q, c, a) ≤ 1−α and u∗ (c, a) = ∞. Thus all actions are passive. Need to show that the u∗ (q, c, a) is monotone nondecreasing in β. We introduce the parameter β as a variable in the functions defined earlier. Observe that the recursive algorithm stated for Wn (q, c, a) in the previous section is equivalent to the following recursion (obtained by dividing throughout by β as β > 0). Initialize W0 (q, c, a, β) = 0. Let u∗n (c, a, β) be the value of u that solves the following inequalities,
αEa,c Wn (u+A, C, A, β) ≤ 1 ≤ αEa,c Wn (u+1+A, C, A, β). (6) Furthermore, 1 ∗ • If q ≤ un (c, a, β), Wn+1 (q, c, a, β) = β + αEa,c Wn (q + A, C, A, β). ∗ ∗ • If un (c, a, β) < q ≤ c + un (c, a, β), Wn+1 (q, c, a, β) = 1 β + 1. 1 ∗ • If q > un (c, a, β) + c, Wn+1 (q, c, a, β) = β + αEa,c Wn (q − c + A, C, A, β). In order to show that u∗n (c, a, β) is monotonically nondecreasing in β, it is enough to show that the function Wn (q, c, a, β) is nonincreasing in β for all n. We show this by induction. The function W0 (u, c, a, β) = 0. Let Wn (q, c, a, β) be nonincreasing in β. This implies Ea,c Wn (q+A, C, A, β) is nonincreasing in β and u∗n (c, a, β) is monotone nondecreasing in β. Now, given (q, c, a), the above recursion seen as a function of β is, APPENDIX ∗ • For β where un (c, a, β) + c < q, Wn+1 (q, c, a, β) = 1 Proof of Theorem 4.1: It suffices to show that V (q, c, a) is β + αEa,c Wn (q + A − c, C, A, β). • For β where u∗n (c, a, β) < q ≤ c + u∗n (c, a, β), convex in q. Consider Equation 4. For n = 0, V0 (q, c, a) = 0 Wn+1 (q, c, a, β) = β1 + 1. hence convex. Let Vn−1 (q, c, a) be convex in q. Fix q. Let u1 1 ∗ • For β where un (c, a, β) ≥ q, Wn+1 (q, c, a, β) = β + and u2 be the optimal policy for q − 1 and q + 1. αEa,c Wn (q + A, C, A, β). Vn (q + 1, c, a) + Vn (q − 1, c, a) It follows from the definition of the minimizer and (6) that for = 2q(1 + β) − β(u1 + u2 ) + αEa,c [Vn−1 (u1 + A, C, A) the domain of β where the first item holds, αEa,c Wn (q + A − c, C, A, β) ≥ 1 and for the domain of β where the third item +Vn−1 (u2 + A, C, A)], holds αEa,c Wn (q+A, C, A, β) ≤ 1. Thus combining this with ≥ 2q(1 + β) − β(u1 + u2 ) the hypothesis that Ea,c Wn (q + A, C, A, β) is nonincreasing u1 + u2 +αEa,c Vn−1 ( + A, C, A) in β implies that Wn+1 (q, c, a, β) is nonincreasing in β and 2 the result follows. u1 + u2 +αEa,c Vn−1 (
+ A, C, A, ), 2 ≥∗ 2Vn (q, c, a) [1] Berry Randall A., “Power and Delay Trade-offs in Fading Channels,” MIT Thesis, June 2000. [2] Hanly Stephen V. and David N. C. Tse , “Multi-access fading channels: Part II: Delay limited capacities,” IEEE Trans. on Info. Theory, 44(7):2816-2831, 1998. [3] Goyal Munish,“Stochastic Control of Transmissions over Multiaccess Fading Channels,”PhD Thesis, IISc, Bangalore, 2004. [4] Nino-Mora J.,“ Restless bandits, partial conservation laws and indexability,” Advances in Applied Probability, 33(1):76-98, 2001. [5] Schal M.,“ Average Optimality in Dynamic Programming with General State Space,” Mathematics of Operations Research, 18(1):163-172, 1993. [6] Tse David N. C. and Stephen V. Hanly, “Multi-access fading channels: Part I: Polymatroid structure, optimal resource allocation and throughput capacities ,” IEEE Trans. on Info. Theory, 44(7):2796-2815, 1998. [7] Whittle Peter,“ Restless bandits: Activity allocation in a changing world,” In A Celebration of Applied Probability, J. Gani (Ed.), J. Appl. Prob., 25A:287-298, 1988. [8] Yeh E. M., Aaron S. Cohen,“ Throughput and Delay Optimal Resource Allocation in Multiaccess Fading Channels,” Proc. of ISIT, Yokohama, Japan, 2003.
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