Adaptive Modulation with Smoothed Flow Utility
© E. Akuiyibo and S. Boyd. 2010
Received: 6 May 2010
Accepted: 14 September 2010
Published: 21 September 2010
We consider the problem of choosing the data flow rate on a wireless link with randomly varying channel gain, to optimally trade off average transmit power and the average utility of the smoothed data flow rate. The smoothing allows us to model the demands of an application that can tolerate variations in flow over a certain time interval; we will see that this smoothing leads to a substantially different optimal data flow rate policy than without smoothing. We pose the problem as a convex stochastic control problem. For the case of a single flow, the optimal data flow rate policy can be numerically computed using stochastic dynamic programming. For the case of multiple flows on a single link, we propose an approximate dynamic programming approach to obtain suboptimal data flow rate policies. We illustrate, through numerical examples, that these approximate policies can perform very well.
We consider the flow rate assignment problem on a wireless link with randomly varying channel gain, to optimally trade off average transmit power and the average utility of the smoothed flow data rate. We pose the multiperiod problem as an infinite-horizon stochastic control problem with linear dynamics and convex objective. For the case of a single flow, the optimal policy is easily found using stochastic dynamic programming (DP) and gridding. For the case of multiple flows, DP becomes intractable, and we propose instead an approximate dynamic programming approach using suboptimal policies developed in the single-flow case. Simulations show that these suboptimal policies perform very well.
In the wireless communications literature, varying a link's transmit rate (and power) depending on channel conditions is called adaptive modulation (AM); see, for example, [1–5]. One drawback of AM is that it is a physical layer optimization technique with no knowledge of upper layer optimization protocols. Maximizing a total utility function is also very common in various communications and networking problem formulations, where it is referred to as network utility maximization (NUM); see, for example, [6–10]. In the NUM framework, performance of an upper layer protocol (e.g., TCP) is determined by utility of flow attributes, for example, utility of link flow rate.
Our setup involves both adaptive modulation and utility maximization but is nonstandard in several respects. We consider the utility of the smoothed flows, and we consider multiple flows over the same wireless link .
2. Problem Setup
2.1. Average Smoothed Flow Utility
A wireless communication link supports data flows in a channel that varies with time, which we model using discrete-time intervals . We let be the data flow rate vector on the link, where , , is the th flow's data rate at time and denotes the set of nonnegative numbers. We let denote the total flow rate over all flows, where is the vector with all entries one. The flows, and the total flow rate, will depend on the random channel gain (through the flow policy, described below) and so are random variables.
where at time , each smoothed flow rate is the exponentially weighted average of previous flow rates.
The smoothing parameter determines the level of smoothing on flow . Small smoothing parameter values ( close to zero) correspond to light smoothing; large values ( close to one) correspond to heavy smoothing. (Note that means that flow is not smoothed; we have .) The level of smoothing can be related to the time scale over which the smoothing occurs. We define to be the smoothing time associated with flow . Roughly speaking, the smoothing time is the time interval over which the effect of a flow on the smoothed flow decays by a factor . Light smoothing corresponds to short smoothing times, while heavy smoothing corresponds to longer smoothing times.
where . Here, the expectation is over the smoothed flows , and we are assuming that the expectations and limit above exist.
parameterized by and . The parameter sets the curvature (or risk aversion), while sets the overall weight of the utility. (For small values of , approaches a log utility.)
So the time smoothing step does affect our average utility; we will see later that it has a dramatic effect on the optimal flow policy.
2.2. Average Power
where is increasing and strictly convex in for each value of ( is the set of positive numbers).
where, again, we are assuming that the expectations and limit exist.
2.3. Flow Rate Control Problem
where is used to trade off average utility and power.
where . In other words, the policy depends only on the current smoothed flows and the current channel gain value.
The flow rate control problem is to choose the flow rate policy to maximize the overall objective in (9). This is a standard convex stochastic control problem, with linear dynamics.
2.4. Our Results
We let be the optimal overall objective value and let be an optimal policy. We will show that in the general (multiple-flow) case, the optimal policy includes a "no-transmit" zone, that is, a region in the space in which the optimal flow rate is zero. Not surprisingly, the optimal flow policy can be roughly described as waiting until the channel gain is large, or until the smoothed flow has fallen to a low level, at which point we transmit (i.e., choose nonzero ). Roughly speaking, the higher the level of smoothing, the longer we can afford to wait for a large channel gain before transmitting. The average power required to support a given utility level decreases, sometimes dramatically, as the level of smoothing increases.
We show that the optimal policy for the case of a single flow is readily computed numerically, working from Bellman's characterization of the optimal policy, and is not particularly sensitive to the details of the utility functions, smoothing levels, or power functions.
For the case of multiple flows, we cannot easily compute (or even represent) the optimal policy. For this case we propose an approximate policy, based on approximate dynamic programming [18, 19]. By computing an upper bound on , by allowing the flow control policy to use future values of channel gain (i.e., relaxing the causality requirement ), we show in numerical experiments that such policies are nearly optimal.
3. Optimal Policy Characterization
3.1. No Smoothing
which does not depend on . A simple and effective approach is to presolve this problem for a suitably large set of values of the channel gain and store the resulting tables of individual flow rates versus ; online we can interpolate between points in the table to find the (nearly) optimal policy. Another option is to fit a simple function to the optimal flow rate data and use this function as our (nearly) optimal policy.
(Each of these can be expressed in terms of conjugate functions; (see, e.g., [21,Section ].) We then adjust (say, using bisection) so that . An alternative is to carry out bisection on , defining in terms of as above, until , where refers to the derivative with respect to .
where the flow values come from the equation above. (The left-hand side is decreasing in , while the right-hand side is increasing.)
3.2. General Case
where the expectation is over . The fixed point equation and Bellman operator are invariant under adding a constant; that is, we have , for any constant (function) , and, similarly, satisfies the fixed point equation if and only if does. So without loss of generality we can assume that .
(apply Bellman operator).
(estimate optimal value).
For technical conditions under which the value function exists and can be obtained via value iteration, see, for example, [27–29]. We will simply assume here that the value function exists, and and converge to and , respectively.
The iterations above preserve several attributes of the iterates, which we can then conclude holds for . First of all, concavity of is preserved; that is, if is concave, so is . It is clear that normalization does not affect concavity, since we simply add a constant to the function. The Bellman operator preserves concavity since partial maximization of a function concave in two sets of variables results in a concave function (see, i.e.,[21, Section ]) and expectation over a family of concave functions yields a concave function; finally, addition (of ) preserves concavity. So we can conclude that is concave.
Another attribute that is preserved in value iteration is monotonicity; if is monotone increasing (in each component of its argument), then so is . We conclude that is monotone increasing.
3.3. No-Transmit Region
as the necessary and sufficient condition under which . Since is decreasing (by concavity of ), we can interpret (24) roughly as follows: do not transmit if the channel is bad ( small) or if the smoothed flows are large ( large).
4. Single-Flow Case
4.1. Optimal Policy
In the case of a single flow (i.e., ) we can easily carry out value iteration numerically, by discretizing the argument and values of and computing the expectation and maximization numerically. For the single-flow case, then we can compute the optimal policy and optimal performance (up to small numerical integration errors).
4.2. Power Law Suboptimal Policy
where is an approximation of the value function.
where are the discretized values of , with associated value function values . We do this by bisection on .
Experiments show that these power law approximate functions are, in general, reasonable approximations for the value function. For our power law utilities, these approximations yield very good matches to the true value function. For other concave utilities, the approximation is not as accurate, but experiments show that the associated approximate policies still yield nearly optimal performance.
and is the Lambert function; that is, is the solution of .
Note that this suboptimal policy is not needed in the single-flow case since we can obtain the optimal policy numerically. However, we found that the difference between our power law policy and the optimal policy (see the example of value functions below) is small enough that in practice they are virtually the same. This approximate policy is needed in the case of multiple flows.
4.3. Numerical Example
In this section we give simple numerical examples to illustrate the effect of smoothing on the resulting flow rate policy in the single-flow case. We consider two examples, with different levels of smoothing. The first flow is lightly smoothed ( ; ), while the second flow is heavily smoothed ( ; ). We use utility function , that is, , in our utility (4). The channel gains are IID exponential variables with mean . We use the power function (7), with .
Average Power versus Average Utility.
Comparing Average Power.
Average power required for target , lightly smoothed flow , heavily smoothed flow .
5. A Suboptimal Policy for the Multiple-Flow Case
5.1. Approximate Dynamic Programming (ADP) Policy
This approximate value function is separable, that is, a sum of functions of the individual flows, whereas the exact value function is (in general) not. The approximate policy, however, is not separable; the optimization problem solving to assign flow rates couples the different flow rates.
In the literature on approximate dynamic programming, would be considered basis functions [32–34]; however, we fix the coefficients of the basis functions as one. (We have found that very little improvement in the policy is obtained by optimizing over the coefficients.)
with optimization variables , . This is a convex optimization problem; its special structure allows it to be solved extremely efficiently, via waterfilling.
5.2. Solution via Waterfilling
Since our surrogate value function is only approximate, there is no reason to solve this to great accuracy; experiments show that around 5–10 bisection iterations are more than enough.
Each iteration of the waterfilling algorithm has a cost that is which means that we can solve (31) very fast. An interior point method that exploits the structure would also yield a very efficient method; see, for example, .
5.3. Upper-Bound Policies
In this section we describe two heuristic data flow rate policies: a steady-state flow policy and a prescient flow policy. We show that both policies result in upper bounds on (the optimal objective value). These upper bounds give us a way to measure the performance of our suboptimal flow policy : if we obtain a from that is close to an upper bound, then we know that our suboptimal flow policy is nearly optimal.
5.3.1. Steady-State Policy
with optimization variable , and being known. Let be our steady-state upper bound on obtained using the policy (35) to solve (9). Note that in the above optimization problem, we ignore time (and hence, smoothing) and variations in channel gains, and so, for each , is the optimal (steady-state) flow vector. (This is sometimes called the certainty equivalent problem associated with the stochastic programming problem [36, 37].)
By Jensen's inequality (and convexity of the max) it is easy to see that is an upper bound on . Note that once is determined, we can evaluate (35) using the waterfilling algorithm described earlier.
5.3.2. Prescient Policy
where the optimization variables are the flow rates , , and smoothed flow rates , , . (The problem data are and , , .) The optimal value of (37) is a random variable parameterized by . Let denote our prescient upper bound on . We obtain by using Monte Carlo simulation: we take large and solve (37) for independent realizations of the channel gains. The mean is our prescient upper bound.
5.4. Numerical Example
In this section we compare the performance of our ADP policy to the above prescient policy using a numerical example.
(Note that this is easily extended to a problem with more than two flows.)
Let denote the objective obtained using our ADP policy. Each obtains an ADP controller, a point in the plane. Using the same , we can compute the corresponding prescient bound giving the point . (Every feasible controller must lie on or below the line, with slope , that passes through .)
We carried out Monte Carlo simulation (100 realizations, each with time steps) for several values of , computing as described in Section 5.2 and our prescient upper bound as described above.
In this paper we present a variation on a multiperiod stochastic network utility maximization problem as a constrained convex stochastic control problem. We show that judging flow utilities dynamically, that is, with a utility function and a smoothing time scale, is a good way to account for network applications with heterogenous rate demands.
For the case of a single flow, our numerically computed value functions obtain flow policies that optimally trad off average utility and average power. We show that simple power law functions are reasonable approximations of the optimal value functions and that these simple functions obtain near optimal performance.
For the case of multiple flows on a single link (where the value function is not practically computable using dynamic programming), we approximate the value function with a combination of the simple one-dimensional power law functions. Simulations, and comparison with upper bounds on the optimal value, show that the resulting ADP policy can obtain very good performance.
This material is based upon work supported by AFOSR Grant FA9550-09-0130 and by Army contract W911NF-07-1-0029. The authors thank Yang Wang and Dan O'Neill for helpful discussions.
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