Price-Based Distributed Algorithms for Rate-Reliability Tradeoff in Network
Utility Maximization
Jang-Won Lee, Member, IEEE, Mung Chiang, Member, IEEE, and A. Robert Calderbank, Fellow, IEEE
Abstract—The current framework of network utility maximiza- tion for rate allocation and its price-based algorithms assumes that each link provides a fixed-size transmission “pipe” and each user’s utility is a function of transmission rate only. These assumptions break down in many practical systems, where, by adapting the physical layer channel coding or transmission diversity, different tradeoffs between rate and reliability can be achieved. In network utility maximization problems formulated in this paper, the utility for each user depends on both transmission rate and signal quality, with an intrinsic tradeoff between the two. Each link may also pro- vide a higher (or lower) rate on the transmission “pipes” by al- lowing a higher (or lower) decoding error probability. Despite non- separability and nonconvexity of these optimization problems, we propose new price-based distributed algorithms and prove their convergence to the globally optimal rate-reliability tradeoff under readily-verifiable sufficient conditions.
We first consider networks in which the rate-reliability tradeoff is controlled by adapting channel code rates in each link’s phys- ical-layer error correction codes, and propose two distributed algorithms based on pricing, which respectively implement the “in- tegrated” and “differentiated” policies of dynamic rate-reliability adjustment. In contrast to the classical price-based rate control algorithms, in our algorithms, each user provides an offered price for its own reliability to the network, while the network provides congestion prices to users. The proposed algorithms converge to a tradeoff point between rate and reliability, which we prove to be a globally optimal one for channel codes with sufficiently large coding length and utilities whose curvatures are sufficiently negative. Under these conditions, the proposed algorithms can thus generate the Pareto optimal tradeoff curves between rate and reliability for all the users. In addition, the distributed algorithms and convergence proofs are extended for wireless mul- tiple-inpit–multiple-output multihop networks, in which diversity and multiplexing gains of each link are controlled to achieve the optimal rate-reliability tradeoff. Numerical examples confirm that there can be significant enhancement of the network utility by distributively trading-off rate and reliability, even when only some of the links can implement dynamic reliability.
Index Terms—Mathematical programming/optimization, network control by pricing, network utility maximization, phys- ical-layer channel coding, rate allocation.
Manuscript received February 15, 2005; revised January 15, 2006. This work was supported in part by Yonsei University Research Fund of 2005 and in part by the National Science Foundation (NSF) under Grant CCF-0440443, Grant CNS-0417607, Grant CNS-0427677, and Grant CCF-0448012. This paper was presented in part at IEEE Infocom 2006.
J.-W. Lee is with the Center for Information Technology, Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]).
M. Chiang and A. R. Calderbank are with the Department of Electrical En- gineering, Princeton University, Princeton, NJ 08544 USA (e-mail: chiangm@
princeton.edu; [email protected]).
Digital Object Identifier 10.1109/JSAC.2006.872877
I. INTRODUCTION
S
INCE the publication of the seminal paper [1] by Kelly et al.in 1998, the framework of Network Utility Maximization (NUM) has found many applications in network rate allocation algorithms, Internet congestion control protocols, user behavior models, and network efficiency-fairness characterization. By allowing nonlinear, concave utility objective functions, NUM substantially expands the scope of the classical Network Flow Problem based on linear programming. Moreover, there is an elegant economics interpretation of the dual-based distributed algorithm, in which the Lagrange dual variables can be in- terpreted as shadow prices for resource allocation, and each end user and the network maximize their net utilities and net revenue, respectively.
Consider a communication network with logical links, wired or wireless, each with a fixed capacity of b/s, and sources (i.e., end users), each transmitting at a source rate of
b/s. Each source emits one flow, using a fixed set of links in its path, and has a utility function . Each link is shared by a set of sources. NUM, in its basic version, is the following problem of maximizing the network utility , over the source rates , subject to linear flow constraints for all links
(1) Making the standard assumption on concavity of the utility functions, problem (1) is a simple concave maximization of decoupled terms under linear constraints, which has long been studied in optimization theory as a monotropic program [2].
Among many of its recent applications to communication networks, the basic NUM (1) has been extensively studied as a model for distributed network rate allocation (e.g., [1], [3], and [4]), TCP congestion control protocol analysis (e.g., [5]
and [6]), and design (e.g., FAST TCP [7]). Primal and dual- based distributed algorithms have been proposed to solve for the global optimum of (1) (e.g., [8]–[12]). Various extensions of the basic NUM problem (1) have been recently investigated, e.g., for joint congestion control and power control [13], and joint opportunistic scheduling and congestion control [14] in wire- less networks, for TCP/IP interactions [15], for medium access
0733-8716/$20.00 © 2006 IEEE
control [16]–[20], for joint congestion control and medium ac- cess [21], for joint multicommodity flow and resource allocation [22], for nonconcave utility functions [23]–[25], and for hetero- geneous congestion control protocols [26].
Despite the above extensions, the current NUM framework for rate control and its price-based algorithms still assumes that each link provides a fixed coding and modulation scheme in the physical layer, and that each user’s utility is only a function of local source rate. These assumptions break down in many practical systems. On some communication links, physical-layer’s adaptive channel coding (i.e., error correc- tion coding) can change the information “pipe” sizes and decoding error probability, e.g., through adaptive channel coding in Digital Subscriber Loop (DSL) broadband access networks or adaptive diversity-multiplexing control in mul- tiple-input–multiple-output (MIMO) wireless systems. Then, each link capacity is no longer a fixed number but a function of the signal quality (i.e., decoding reliability) attained on that link.1 A higher throughput can be obtained on a link at the expense of lower decoding reliability, which in turn lowers the end-to-end signal quality for sources traversing the link and reduces users’ utilities, thus leading to an intrinsic tradeoff between rate and reliability. This tradeoff also provides an additional degree-of-freedom for improving each user’s utility, as well as system efficiency. For example, if we allow lower decoding reliability, thus higher information capacity, on the more congested links, and higher decoding reliability, thus lower information capacity, on the less congested links, we may improve the end-to-end rate and reliability performance of each user. Clearly, rate-reliability tradeoff is globally coupled across the links and users.
How can we provide a price-based framework to exploit the degree of freedom in utility maximization offered by rate-relia- bility tradeoff and make the above intuitions rigorous? The basic NUM framework in (1) is inadequate because it completely ig- nores the concept of communication reliability. In particular, it takes link capacity as a fixed number rather than a function of decoding error probabilities, does not discriminate the transmis- sion data rate from the information data rate,2and formulates the utility function as a function of rate only.
In this paper, we study the rate-reliability tradeoff by ex- tending the NUM framework. In the basic NUM, convexity and separability properties of the optimization problem readily lead to a distributed algorithm that converges to the globally optimal rate allocation. However, our new optimization formulations for the rate-reliability tradeoff are neither separable problems nor convex optimization. The constraints become nonlinear, nonconvex, and globally coupled. Despite such difficulties, we develop simple and distributed algorithms based on network pricing that converge to the optimal rate-reliability tradeoff
1Some recent work like [13] studies how adaptive transmit power control in the physical layer interacts with congestion control. Note that the dimension of end-to-end signal reliability is still missing in such work. However, this paper can indeed be viewed as a new case in the general framework of “layering as optimization decomposition,” as discussed in Section VII.
2In this paper, the information data rate refers to the data rate before adding the error control code and the transmission data rate refers to the data rate after adding the error control code.
with readily verifiable sufficient conditions. In contrast to stan- dard price-based rate control algorithms for the basic NUM, in which each link provides the same congestion price to each of its users and/or each user provides its willingness to pay for rate allocation to the network, in our algorithms each link provides a possibly different congestion price to each of its users and each user also provides its willingness to pay for its own reliability to the network.
In addition to extending the approach of price-based dis- tributed algorithms, we also consider the specifics of controlling the rate-reliability tradeoff in different types of networks. We first study the case where the rate-reliability tradeoff is con- trolled through the code rate of each source on each link.
We proposes a new optimization framework and price-based distributed algorithms by considering two policies: integrated dynamic reliability policy and differentiated dynamic reliability policy. In the integrated policy, a link provides the same error probability (i.e., the same code rate) to each of the sources traversing it. In this policy, since a link provides the same code rate to each of its sources, it must provide the lowest code rate that satisfies the requirement of the source with the highest reliability. This motivates a more general approach called the differentiated policy that we propose to fully exploit the rate-reliability tradeoff when there exist multiclass sources (i.e., sources with different reliability requirements) in the network. Under the differentiated dynamic reliability policy, a link can provide a different error probability (i.e., a different code rate) to each of the sources using this link. We also prove convergence of the proposed algorithms to the globally optimal rate-reliability tradeoff for sufficiently strong channel codes and sufficiently elastic applications.
We then consider wireless MIMO multihop networks. In MIMO networks, each logical link has multiple transmit antennas and multiple receive antennas, which can provide multiple independent wireless links between a transceiver pair.
If independent information streams are transmitted in parallel, data rate can be increased. This is often referred to as the multiplexing gain. On the other hand, if the same information streams are transmitted in parallel, decoding error probability can be lowered. This is often referred to as the diversity gain.
A larger value of the diversity gain implies a lower decoding error probability and a larger value of the multiplexing gain implies a higher information transmission rate. As in adaptive channel coding, there exists a tradeoff between information data rate and error probability on each MIMO link [27]–[29].
By appropriately controlling diversity and multiplexing gains on each individual link, we can achieve the optimal end-to-end rate-reliability tradeoff for all sources in the network. We show that our techniques developed for networks with adaptive channel coding can be extended to wireless MIMO networks with adaptive diversity-multiplexing gain.
The rest of this paper is organized as follows. In Section II, we provide the system model for the adaptive channel coding problem that we consider in this paper. In Sections III and IV, we present algorithms for the integrated dynamic reliability policy and the differentiated dynamic reliability policy, respectively.
We provide numerical examples for the proposed algorithms in Section V to illustrate how our algorithms can trace the Pareto
optimal curves between rate and reliability for all users, and highlight the enhancement to the network utility through dy- namic rate-reliability adjustments, even when only some of the links can implement dynamic reliability. In Section VI, we show how to control the diversity-multiplexing gains to attain the op- timal rate-reliability tradeoff in wireless MIMO multihop net- works. Finally, we conclude in Section VII.
II. SYSTEMMODEL
In this section, we present the system model for the adaptive channel coding problem. Introducing the concept of reliability into the NUM framework, each source has a utility function , where is an information data rate and is the reliability of source . We assume that the utility function is a continuous, increasing, and concave function3of and . Each source has its information data rate bounded between a minimum and a maximum: and , and has a minimum reliability requirement . The reliability of source is de- fined as
where is the end-to-end error probability of source . Each logical link has its maximum transmission capacity . After link receives the data of source from the upstream link, it first decodes it to extract the information data of the source and encodes it again with its own code rate , where the code rate is defined by the ratio of the information data rate at the input of the encoder to the transmission data rate at the output of the encoder [30], [31]. This allows a link to adjust the transmission rate and the error probability of the sources, since the transmission rate of source at link can be defined as
and the error probability of source at link can be defined as a function of by
which is assumed to be an increasing function of . Rarely is there an analytic formula for , and we will use various upper bounds on this function in this paper. The end-to-end error probability for each source is
Assuming that the error probability of each link is small (i.e., ), we can approximate the end-to-end error probability of source as
3Throughout this paper, a concave (convex) function means a strictly concave (convex) function.
Hence, the reliability of source can be expressed as
Since each link has a maximum transmission capacity , the sum of transmission rates of sources that are traversing the link cannot exceed , i.e.,
III. INTEGRATEDDYNAMICRELIABILITYPOLICY
We first investigate a simpler, more restrictive policy where a link provides the same code rate to each of the sources that are using it, i.e.,
and an extended NUM problem is formulated as follows:
(2) In this problem, the first constraint states that the network must provide reliability to each source that equals to its desired re- liability. Notice that the inequality constraint will be satisfied with equality at optimality since the objective function is an in- creasing function of . The second constraint states that the aggregate transmission rate of the sources at each link must be smaller than or equal to its maximum transmission capacity. The rest are simple, decoupled constraints on the minimum and max- imum values for the data rate, reliability for each source, and code rate for each link.
In order to derive a distributed algorithm to solve problem (2) and to prove convergence to global optimum, the critical prop- erties of separability and convexity need to be carefully exam- ined. Because of the physical layer coding and the rate-relia- bility tradeoff that we introduce into the problem formulation, these two properties are no longer trivially held as in the basic NUM (1).
First, the integrated policy naturally leads to a decomposition of problem (2) across the links, since the second constraint can be written as
(3)
The more complicated issue is the convexity of function . As an example, if random coding based on -ary binary coded signals is used, a standard upper bound on the error probability is [31]
where is the block length and is the cutoff rate. Hence, if we let
then is a convex function for given and . A more general approach is to use the random code ensemble error ex- ponent [30] that upper bounds the decoding error probability
where is the codeword block length and is the random coding exponent function, which is defined for discrete memo- ryless channels as [30]
where
is the size of input alphabet, is the size of output alphabet, is the probability that input letter is chosen, and is the probability that output letter is received given that input letter is transmitted.
In general, may not be convex,
even though it is known [30] that is a convex function.
However, the following lemma provides a sufficient condition for its convexity.
Lemma 1: If the absolute value of the first derivatives of is bounded away from 0 and absolute value of the second derivative of is upper bounded, then for a large enough codeword block length , is a convex function.
Proof: Assume that there exist positive constants and
such that and .
We have
If , then and convexity is
proved.
Throughout this paper, we assume that is a convex function, e.g., the conditions in Lemma 1 are satisfied if the random code ensemble error exponent is used. Hence, problem (2) is turned into a convex and separable optimization problem.
To solve problem (2), we use a dual decomposition approach.
We write down the Lagrangian associated with problem (2) as follows, where and are the Lagrange multipliers on link
with an interpretation of “congestion price” and on source with an interpretation of “reliability price,” respectively
where and with interpreta-
tions of “end-to-end congestion price” on source and “aggre- gate reliability price” on link . The Lagrange dual function is
(4)
where and are vectors whose elements are all zeros and ones, respectively. The dual problem is formulated as
(5) To solve the dual problem, we first consider problem (4).
Since the Lagrangian is separable, this maximization of the Lagrangian over can be conducted in parallel at each source
(6) and on each link
(7) Then, dual problem (5) can be solved by using the gradient pro- jection algorithm as
Fig. 1. Diagram for the distributed algorithm of the integrated dynamic reliability policy. Each link needs the aggregate information rate x and reliability price from the sources using it, and in turn provides the same congestion price and code rate r to these sources. Note that the source does not need to tell its desired reliabilityR to links.
and
where , is step size, and are
solutions of problem (6), and is a solution of problem (7)
for a given .
We now describe the following distributed algorithm, where each source and each link solve their own problems with only local information, as in Fig. 1. In this case, we can interpret as the price per unit rate to use link , and as the price per unit reliability that the source must pay to the network.
Algorithm 1 for Integrated Dynamic Reliability Policy Source problem and reliability price update at source :
• Source problem
(8)
where is the end-to-end conges-
tion price at iteration .
• Price update
(9)
where is the end-to-end
reliability at iteration .
Link problem and congestion price update at link :
• Link problem
(10)
where is the aggregate reliability
price paid by sources using link at iteration .
• Price update
(11)
where is the aggregate informa-
tion rate on link at iteration .
In each iteration , by locally solving the following problem (8) over , each source determines its information data rate and desired reliability [i.e., and ] that maximize its net utility based on the prices in the current iteration. Furthermore, by price update (9), the source adjusts its offered price per unit reliability for the next iteration. Intuitively, from the end-to-end principle of network architecture design, reliability prices should indeed be updated at the sources.
Concurrently, in each iteration , by locally solving problem (10) over , each link determines its code rate [i.e., ] that maximizes the “net revenue” of the network based on the prices in the current iteration.
This pricing interpretation holds because maximizing over , , i.e., maximizing over , is equivalent to maximizing
over . Since is the available (anticipated) aggregate information data rate at link , we can interpret that
as the revenue obtained at link through the congestion price.
We can also interpret as the revenue
obtained from source through the reliability price. In addition, by price update (11), the link adjusts its congestion price per unit rate for the next iteration.
In Algorithm 1, to solve problem (8), source needs to know , the sum of ’s of links that are along its path . This can be obtained by the notification from the links, e.g., through the presence or timing of acknowledgment packets in TCP [5]. To carry out price update (9), the source needs to know the sum of error probabilities of the links that are along its path [i.e., its own reliability that are offered by the network ].
This can be obtained by the notification from the destination that can measure its end-to-end reliability. To solve link problem (10), each link needs to know , the sum of ’s from sources using this link . This can be obtained by the notification from these sources. To carry out price update (11), the link needs to know , the aggregate information data rate of the sources that are using it. This can be measured by the link itself.
After the above dual decomposition, the following result can be proved using standard techniques in distributed gradient al- gorithm’s convergence analysis.
Theorem 1: By Algorithm 1, dual variables and converge to the optimal dual solutions and , and the cor- responding primal variables , , and are the globally op- timal primal solutions of (2).
Outline of the Proof: Since strong duality holds for problem (2) and its Lagrange dual problem (5), we solve the dual problem through distributed gradient method and recover the primal op- timizers from the dual optimizers. By Danskin’s Theorem [32]
and
Hence, the algorithm in (9) and (11) is a gradient projection al- gorithm for dual problem (5). Since the dual objective function
is a convex function, there exists a step size that guarantees and to converge to the optimal dual solu- tions and [32]. Also, if satisfies a Lipschitz con- tinuity condition, i.e., there exists a constant such that
then and converges to the optimal dual solutions
and with a constant step size , [32].
The Lipschitz continuity condition is satisfied if the curvatures of the utility functions are bounded away from zero, see [10] for further details.
Furthermore, since problem (2) is a convex optimization problem and problems (8) and (10) have unique solutions, , , and are the globally optimal primal solutions of (2) [33].
IV. DIFFERENTIATEDDYNAMICRELIABILITYPOLICY
We now investigate the more general differentiated dynamic reliability policy in which a link may provide a different code rate to each of the sources traversing it.4To this end, the ex- tended NUM formulation in (2) needs to be further generalized to the following problem:
(12) The objective function and constraints of problem (12) are the same as those of problem (2), except that we have here in- stead of . Due to this critical difference, we may not modify the second constraint in this problem as in (3), and problem (12) in general is neither a convex problem nor a separable one, even when is assumed to be a convex function.
These issues can be resolved through a series of problem transformations. First, we modify problem (12) by introducing the auxiliary variables , which can be interpreted as the allo- cated transmission capacity to source at link . Then, the above problem can be reformulated as
4An example of such coding techniques recently proposed is coding with em- bedded diversity [34], which allows data streams with different rate-reliability tradeoffs be embedded within each other. An interesting next step is to study how to use this technique as a practical mechanism to implement the differenti- ated dynamic reliability policy. Other adaptive resource allocation in the phys- ical layer, such as adaptive modulation or interleaver depth, can also be used to the implement dynamic reliability policy.
(13)
The second constraint in problem (12) is now decomposed into two constraints in problem (13): the second and third con- straints. Here, the second constraint states that the transmission data rate of each source at each link must be smaller than or equal to its allocated transmission capacity at the link, and the third constraint states that the aggregate allocated transmission capacity to the sources at each link must be smaller than or equal to its maximum transmission capacity. In this formula- tion, each link explicitly allocates a transmission capacity to each of its sources. We can easily show that problem (13) is equivalent to problem (12), since at the optimal solution of problem (13), the second constraint must be satisfied with the equality.
Note that is a parameter in the link layer. Hence, even though we started from the TCP/PHY layer problem, here we introduce another layer, i.e., link layer, by using a vertical de- composition of the optimization problem. At the first sight, it may seem that this makes the problem more complicated. How- ever, it in fact enables us to implement a simple and distributed algorithm.
The next step of problem transformation is to take the log of both sides of the second constraint in problem (13) and a change of variable: (i.e., ). This reformulation turns the problem into
(14)
where , , and
.
Note that problem (14) is now separable but still may not be a convex optimization problem since the objective
may not be a concave function, even though is a con- cave function. However, the following lemma provides a suffi- cient condition for its concavity. Define
Lemma 2: If , , and
, then is a concave function of and .
Proof: This can be easily verified using the fact that is a concave function if and only if
where and is a Hessian matrix of ,
which is defined as
It can be readily verified that if the conditions in the Lemma are satisfied, then indeed
i.e., is negative definite and is a concave function
of .
Furthermore, if is additive in each variable, (i.e.,
), the following lemma provides the suffi- cient condition. Define
(15)
Lemma 3: Suppose and
is a concave function. Then, if , is a concave function of and is a concave function
of and .
Proof: Since is a concave function,
. Furthermore, since is additive, im-
plies , and . This im-
plies . Hence, the
conditions in Lemma 2 are all satisfied.
The conditions of and are
equivalent to
Since is positive and is
negative, the above inequalities state that the utility function needs to be more than just concave. In particular, if it is additive in and , its curvature needs to be not just negative but bounded away from 0 by as much as , i.e., the application represented by this utility function must be elastic enough.
For example, consider the following utility function parame- terized by [12]:
if otherwise where can be interpreted as the correctly decoded throughput of source . Then, we can show that
if if if Hence, in this type of utility functions, if , becomes a concave function.
Throughout this section, we will assume that the conditions in Lemma 2 are satisfied. Hence, problem (14) is turned into a convex and separable optimization problem.
To solve problem (14), we use a dual decomposition approach again. We first write down the Lagrangian function associated with problem (14) as
where . As in the previous case, we can inter- pret , , and as “congestion price” on link , end-to-end congestion price on source , and “reliability price” on source , respectively. Note that in this Lagrangian function, we do not relax the third constraint in problem (14). The Lagrange dual function is
(16)
where
. The dual problem is formulated as
(17) To solve the dual problem, we first consider problem (16).
This maximization of the Lagrangian over , , , can be conducted in parallel at each source
(18) and on each link
(19) Then, dual problem (17) can be solved by using the gradient projection algorithm as
and
where and are solutions to problem (18), and and are solutions to problem (19) for a given .
We now describe the following distributed algorithm where each source and each link solve their own problems with only local information, as in Fig. 2. As before, we can interpret as the price per unit rate (in terms of ) for source
to use link , and as the price per unit reliability that the source must pay to the network.
Fig. 2. Diagram for the distributed algorithm of the differentiated dynamic reliability policy. Each link needs the individual information rate x and reliability price from each of the sources using it, and in turn provides a different congestion price and code rater to each of these sources. Note that the source does not need to tell its desired reliabilityR to links, and the link does not need to tell the allocated capacityc to sources.
Algorithm 2 for Differentiated Dynamic Reliability Policy Source problem and reliability price update at source :
• Source problem
(20)
where is the end-to-end conges-
tion price at iteration .
• Price update
(21)
where is the end-to-end
reliability at iteration .
Link problems and congestion price update at link :
• Link problems Link-layer problem
(22) Physical-layer problem for source ,
(23)
• Price update
(24)
In each iteration , by locally solving (20) over , each source determines its information data rate and desired relia-
bility [i.e., , or equivalently, , and ]
that maximize its net utility based on the prices in the current iteration. Furthermore, by price update (21), the source adjusts its offered price per unit reliability for the next iteration.
Concurrently, in each iteration , by locally solving problems
(22) over , and (23) , for each , which
are decomposed from (19), each link determines the allocated transmission capacity and the code rate , respec- tively, of each of the sources using the link so as to maximize the “net revenue” of the network based on the prices in the cur- rent iteration. In addition, by price update (24), the link adjusts its congestion price per unit rate for source during the next iteration.
Even though Algorithm 2 (i.e., the differentiated dynamic re- liability policy) can be implemented and interpreted in a sim- ilar way to Algorithm 1 (i.e., the integrated dynamic reliability policy), there are important differences. In Algorithm 2, the link differentiates each of its sources and provides a different code rate and a different price , while in Algorithm 1, the link provides the same code rate and the same price to all of its sources. In Algorithm 2, a link provides an explicit capacity allocation to each of its sources and the price is deter- mined based on the expected information data rate and the actual information data rate of each individual source, while in Algorithm 1, a link does not explicitly allocated the ca- pacity to each individual source, and the price is determined based on the expected aggregate information rate and the actual aggregate information rate of its sources.
The advantages of allowing different code rates for different sources on a shared link and more message passing overhead in- clude a more dynamic rate-reliability tradeoff and a higher net- work utility, as confirmed through numerical examples below.
However, this performance improvement is achieved at the ex- pense of keeping per-flow states on the links and more restric- tive conditions on utility functions for convergence proof (i.e., the conditions in Lemma 2).
After the above decomposition and by Lemma 2, the fol- lowing result can be proved similar to Theorem 1.
Theorem 2: By Algorithm 2, dual variables and converge to the optimal dual solutions and and the corre- sponding primal variables , , , and are the globally optimal primal solutions of problem (14), i.e., , , , and are the globally optimal primal solutions of problem (12).
V. NUMERICALEXAMPLES
A. Basic Examples
In this section, we present numerical examples for the proposed algorithms by considering a simple network, shown in Fig. 3, with a linear topology consisting of four links and eight users. Utility function for user is in the following standard form of concave utility parameter-
ized by [12], shifted such that and
, and with utility on rate and utility on
Fig. 3. Network topology and flow routes for numerical examples.
Fig. 4. Optimal tradeoff between rate and reliability for different users under the integrated dynamic reliability (Algorithm 1).
reliability summed up with a given weight between rate and reliability utilities
The decoding error probability on each link is assumed to be of the following form:
where is the channel code block length and the code rate for link .
We trace the globally optimal tradeoff curve between rate and reliability using Algorithms 1 and 2, and then compare the net- work utility achieved by the following three schemes.
• Static reliability (by the standard dual-based algorithm in the literature e.g., [5], [10]): Each link provides a fixed error probability 0.025. Only rate control is performed to maximize the network utility.
• Integrated dynamic reliability (by Algorithm 1): Each link provides the same adjustable error probability to each of its users.
• Differentiated dynamic reliability (by Algorithm 2): Each link provides a possibly different error probability to each of its users.
In Figs. 4–9, the constant parameters have the following
values: , Mb/s , Mb/s ,
Mb/s , , and . Each point
on the curves in all the graphs represents the result of running a corresponding distributed algorithm until convergence to a provably global optimum.
We first investigate the case where all the users have the same , and vary the value of from 0 to 1 in step size of
Fig. 5. Aggregate information data rate on each link if the integrated dynamic reliability is used (Algorithm 1).
Fig. 6. Error probability on each link if the integrated dynamic reliability is used (Algorithm 1).
Fig. 7. Comparison of the achieved network utility attained by the differentiated dynamic policy (Algorithm 2), the integrated dynamic policy (Algorithm 1), and the static policy (current practice).
0.1. The resulting tradeoff curve, which is globally optimal, is shown in Fig. 4.5As expected, a larger (i.e., rate utility is given a heavier weight) leads to a higher rate at the expense of lower reliability. The tradeoff curve has a steeper slope at a larger (i.e., compared with low rate regime, in high rate regime further rate increment is achieved at the expense of a larger drop in reli- ability). The more congested links a user’s flow passes through, the steeper the tradeoff curve becomes. For each user, the area
5Note that by symmetry, the tradeoff curves for users 1 and 2 are the same, those for users 5 and 6 are the same, and those for the other fours users are the same. Therefore, only users 1, 3, and 5 are shown in this figure.
Fig. 8. Comparison of optimal tradeoff between rate and reliability for user 2 in each policy, whena are changed according to (25).
Fig. 9. Comparison of the achieved network utility attained by the differentiated dynamic policy (Algorithm 2), the integrated dynamic policy (Algorithm 1), and the static policy (current practice), whena are changed according to (25).
to the left and below of the tradeoff curve is the achievable re- gion [i.e., every (rate, reliability) point in this region can be ob- tained], and the area to the right and above of the tradeoff curve is the infeasible region [i.e., it is impossible to have any combi- nation of (rate, reliability) represented by points in this region].
It is impossible to operate in the infeasible region. Moreover, we can always find a better operating point on the boundary of the achievable region than a operating point in the interior of the achievable region. Hence, operating on the boundary of the achievable region, i.e., the Pareto optimal tradeoff curve, is the best. In the Pareto optimal tradeoff curve, which point is better depends on the relative weight between rate utility and relia- bility utility given by each user.
Figs. 5 and 6 show the aggregate information data rate and decoding error probability on each link as the weight varies.6 As expected, as increases, data rates increase but decoding error probabilities also rise. Since link 2 is more congested than link 1, link 2 provides a higher data rate, also a higher error probability than link 1.
For the same experimental setup, Fig. 7 shows the network utility achieved (i.e., sum of utilities of all the users) as the weight varies. It is clear that the performance of the NUM algorithm that takes into account the rate-reliability tradeoff is significantly better than the standard distributed algorithms for
6By symmetry, link 1 and 4’s curves are the same, and link 2 and 3’s curves are the same. Hence, the figure shows only links 1 and 2.
Fig. 10. Network topology and flow routes for the partial dynamic reliability example.
the basic NUM that ignores the possibility of jointly optimizing rate and reliability.7Within the dynamic approach, the perfor- mance gap between the integrated and differentiated policies is small because all the users are set to have the same weight .
We now give different weights to the eight users indexed by as follows:
if is an odd number
if is an even number (25) and vary from 0 to 0.5 in step size of 0.05. No two users have the same utility function and flow route now. Fig. 8 shows the globally optimal tradeoff curves between rate and reliability for user 2, under the three policies of static reliability, integrated dy- namic reliability, and differentiated dynamic reliability, respec- tively. The differentiated scheme shows a much larger dynamic range of tradeoff than both the integrated and static schemes.
Fig. 9 shows the relative performance in terms of the net- work utility achieved by the three policies as changes. As expected, dynamic reliability policies are much better than the static reliability policy, and in this case where users have dif- ferent weights , the gap between differentiated and integrated policies is wider than that in Fig. 7.
B. Partially Dynamic Reliability
In some networks, only a subset of links have adaptive channel coding while other links have fixed code rates. For example, in the DSL access networks, edge links can provide adaptive channel coding while links in the backbone may not.
Hence, it is useful to know what happens if only a subset of links adopt the dynamic reliability policy. Here, we present the performance of the differentiated dynamic reliability policy for such situations considering the network in Fig. 10, in which each link is symmetric in terms of the demand from users.
We fix the capacity of links 1 and 4 as
(Mb/s) and compute the network utility while varying the ca-
pacity of links 2 and 3 as Mb/s ,
. Hence, if , links 2 and 3 are bot- tleneck links. Otherwise, links 1 and 4 are bottleneck links. We compare the performance of four scenarios.
• Dynamic: All links use the differentiated dynamic relia- bility policy.
• Partial1: Only links 2 and 3 use the differentiated dynamic reliability policy, while links 1 and 4 still use the static reliability policy.
• Partial2: Only links 1 and 4 use the differentiated dynamic reliability policy, while links 2 and 3 still use the static reliability policy.
• Static: All links still use the static reliability polity.
7In this example, this performance gap narrows as the weighta on rate utility increases. This happens since the error probability of each link in the static scheme is set to barely satisfy the minimum reliability requirement of the users.
With a different parameter setting, the performance gap may widen as param- etera increases.
Fig. 11. Comparison of the achieved network utility in each system with the partially differentiated dynamic reliability policy. Dynamic: all links have the differentiated dynamic reliability policy; Partial1: only links 2 and 3 have the differentiated reliability policy; Partial2: only links 1 and 4 have the differentiated reliability policy; Static: all links have the static reliability policy.
Fig. 11 shows that we can still obtain significant performance gain over the static policy even though only a subset of links have adaptive channel coding. As expected, this performance gain is reduced compared with the case where all links have the dynamic policy. It also shows that we can obtain a higher performance gain by applying dynamic reliability to more con- gested links (usually the access links, precisely where adaptive coding is possible) than less congested ones. Furthermore, in the static policy, increasing link capacities beyond a certain threshold does not provide any further increase in network utility, while with the dynamic policy we can continue to improve network utility (with a diminishing marginal return) by increasing link capacities.
VI. RATE-RELIABILITY TRADEOFF INWIRELESS
MIMO MULTIHOP NETWORKS THROUGH
DIVERSITY-MULTIPLEXINGGAINCONTROL
In the previous sections, we have developed the optimization framework and distributed algorithms for the optimal rate-relia- bility tradeoff by controlling adaptive channel coding. They can be applied to other types of networks such as wireless MIMO multihop networks, where the optimal rate-reliability tradeoff can be achieved by appropriately controlling diversity and mul- tiplexing gains on each MIMO link [27]–[29]. In this section, we briefly present how our framework can be extended to wire- less MIMO multihop networks. The detailed algorithms are then seen to be readily derived from Algorithms 1 and 2, and their de- tails are skipped here for brevity.
In MIMO networks, each logical link has multiple transmit and multiple receive antennas, and , respectively, enabling both diversity and spatial multiplexing gains, which are defined as follows [29].
Definition 1: Link has diversity gain and multiplexing gain if its data rate (b/s/Hz) and average error proba- bility satisfy the following conditions:
(26) and
(27) where is signal-to-noise ratio (SNR).
Hence, for a given (high) SNR, we can achieve a higher data rate with a larger multiplexing gain or a lower error probability with a larger diversity gain. There exists a tradeoff between these two gains, and the optimal tradeoff curve is proved for a point-to-point link in [29]. For given multiplexing gain , de- fine to be the supremum of the diversity gain that can be achieved over all schemes at link .
Lemma 4: [29] Suppose that the channel gain is constant for a duration of symbols. If ,8 is given by the piecewise-linear function connecting the point ,
, where
(28) By using the result in Lemma 4, we study the end-to-end rate- reliability tradeoff in wireless MIMO multihop networks, i.e., we assume that each link uses a multiple-antenna coding scheme that achieves the optimal diversity-multiplexing tradeoff given in Lemma 4, and that there is no joint coding across point-to- point logical links. To turn the piecewise linear tradeoff curve into a differentiable one, we approximate with a differ- entiable function as
(29) The approximated tradeoff curve (29) is always below the op- timum one (28), thus more conservative than the optimum curve.
As the number of antennas increases, the gap between (29) and (28) decreases. From (26) and (27), we assume that for each source traversing link , its data rate (b/s) and error probability are given by
and
where and are positive constants, and and are mul- tiplexing and diversity gains of source at link , respectively.
We assume that each link transmits data from each of its sources in a round-robin manner and the fraction of the trans- mission time of source on link is given by . Hence, the average data rate of source at link is obtained by
and the reliability of source is obtained by
SNR at each link is fixed, and as before, each source has its utility function .
We consider two policies: static and dynamic scheduling poli- cies. In the static scheduling policy, each link transmits data of each of its sources for a fixed fraction of time, i.e., is fixed
(e.g., , ). In the dynamic scheduling
8For the case ofi < m + n 0 1, the upper and lower bounds of d (r) are also proved in [29]. For the brevity of the paper, we only consider the case that i m + n 0 1 here, and the other case can be studied in a similar manner.
policy, can be adjusted based on network conditions under the following constraint:
where is the maximum fraction of the time that link can transmit data.9
A. Static Scheduling Policy
The NUM problem for the static scheduling policy is formu- lated as
(30) In the above problem, is a constant, and if
were a convex function, it would be a separable and convex problem. In general, is not convex. How- ever, the following lemma shows that in most cases that are in- teresting in practice, it is a convex function or can be closely approximated with a convex function. Define
Lemma 5: If , then is a
convex function, i.e., ,
. Otherwise, it has an inflection point such that
for and
for .
Proof:
Hence, is convex if and only if
Since , . There-
fore, is convex if
which completes the proof.
Since if , and
if , the above lemma implies that converges to a convex function if one of the following conditions is satisfied: 1) ;
9Ideally,t = 1, but if we considered collisions under random-access MAC protocols, it would be less than one.
Fig. 12. Error probability function(p (r )) with m = n = 2 and k = 0:5.
Fig. 13. Error probability function(p (r )) with = 10 (dB) and k = 0:5.
2) ; and 3) . Furthermore, from the
above lemma, we have the following.
Corollary 1: If and (dB)
[ 8.69 (dB)], then is a convex function.
This corollary implies that if the number of transmit antennas is different from that of receive antennas, is a convex func- tion even when SNR is not too high. Also, the required SNR for the convexity decreases as the difference between the num- bers of transmit and receive antennas increases. In Figs. 12 and
13, with is plotted when the number of
transmit antennas is the same as that of receive antennas, varying SNR and the number of antennas, respectively. As indicated in Lemma 5, the figures show that as either SNR or the number of antennas increases, can be approximated by a convex function with higher and higher accuracy.
In the rest of this section, we assume that is a convex function. Hence, problem (30) is turned into a convex and sepa- rable problem. Moreover, since its structure is similar to that of problem (2), we can achieve the optimal rate-reliability tradeoff to problem (30) by using the same dual decomposition approach to that in Section III. Due to space limit, we do not repeat the details here.
B. Dynamic Scheduling Policy
The problem formulation for the dynamic scheduling policy is similar to problem (30). The crucial difference is that, in this more general policy, each link can dynamically adjust the fraction of the transmission time for each of its sources.
The extended NUM problem is formulated as
(31) In contrast to the static scheduling policy where is a constant, it is now a variable to be determined in the above problem. Due to the second constraint, problem (31) is neither convex nor sep- arable. However, using the same technique as in Section IV, we can turn problem (31) into a convex and separable problem. We take the log of both sides of the second constraint in problem (31) and a log change of variable: . The problem is turned into
(32)
where . The structure of the above
problem is similar to that of problem (14). Hence, if the condi- tions on utility curvature in Lemma 2 are satisfied, it is convex and separable, and the optimal rate-reliability tradeoff can be achieved by using the same dual decomposition approach as in Section IV. Algorithm 2 in that section then readily extends to solve problem (31).
VII. CONCLUDINGREMARKS
Motivated by needs from the application-layer utilities and possibilities at the physical-layer coding, this paper removes the rate-dependency assumption on utility functions and allows
the physical-layer adaptive channel coding (or diversity-multi- plexing gain control) to tradeoff rate and reliability. The basic network utility maximization is thus extended to concave maxi- mization problems over nonlinear, nonconvex, and coupled con- straints, which are much more difficult problems to be solved by distributed and globally optimal algorithms. In particular, the standard price-based distributed algorithm cannot be applied since the entire dimension of reliability is absent from the orig- inal formulation of network utility maximization.
We present two new price-based distributed algorithms for two possible formulations of the adaptive channel coding problem: the integrated policy where each link maintains the same code rate for all flows traversing it, and the differentiated policy where each link can assign different code rates for each of the flows traversing it. We also provide sufficient conditions under which convergence to the globally optimal rate-relia- bility tradeoff can be proved. A key idea is that, in addition to link-updated congestion prices for distributed rate control, we need to introduce source-updated signal quality prices for distributed reliability control. We also show that the optimiza- tion framework and distributed algorithms of network-wide rate-reliability optimal tradeoff can be extended from tuning the “knobs” of adaptive channel coding to tuning the “knobs”
of diversity-multiplexing gain in wireless MIMO networks.
In the context of “layering as optimization decomposition,” as outlined in [13] (see also, e.g., [15], [21], [22], [35], and [36]), this paper shows that in the case of joint congestion control in the transport layer and channel coding or MIMO signal processing in the physical layer, the new reliability prices, in addition to the standard congestion prices, become the “layering variables”
controlling the optimal interaction between the two layers. The technique of introducing auxiliary flow variables, using a log change of variables, and proving convergence under additional conditions on utility curvatures may also be applicable (e.g., [37]) to solve other nonconvex and coupled NUM problems in distributed network resource allocation.
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Jang-Won Lee (S’02–M’04) received the B.S.
degree in electronic engineering from Yonsei Uni- versity, Seoul, Korea, in 1994, the M.S. degree in electrical engineering from the Korea Advanced In- stitute of Science and Technology (KAIST), Taejon, Korea, in 1996, and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 2004.
From 1997 to 1998, he was with Dacom R&D Center, Taejon. From 2004 to 2005, he was a Post- doctoral Research Associate in the Department of Electrical Engineering, Princeton University, Princeton, NJ. Since September 2005, he has been an Assistant Professor in the School of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea. His research interests include resource allocation, quality-of-service (QoS) and pricing issues, optimization, and performance analysis in communication networks.
Mung Chiang (S’00–M’03) received the B.S.
(Hon.) degree in electrical engineering and mathe- matics, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1999, 2000, and 2003, respectively.
He is an Assistant Professor of Electrical En- gineering at Princeton University, Princeton, NJ.
He conducts research in the areas of nonlinear optimization of communication systems, distributed network control algorithms, broadband access network architectures, and information theory and stochastic analysis of communication systems.
Prof. Chiang has been awarded a Hertz Foundation Fellow. He received the Stanford University School of Engineering Terman Award, the SBC Communi- cations New Technology Introduction Contribution Award, the NSF CAREER Award, and the Princeton University Howard B. Wentz Junior Faculty Award.
He is the Lead Guest Editor of the IEEE JOURNAL ONSELECTEDAREAS IN COMMUNICATIONS(Special Issue on Nonlinear Optimization of Communica- tion Systems), a Guest Editor of the IEEE TRANSACTIONS ONINFORMATION THEORY, and the IEEE/ACM TRANSACTIONS ONNETWORKING(Special Issue on Networking and Information Theory). He is Program Co-Chair of the 38th Conference on Information Science and Systems.
A. Robert Calderbank (M’89–SM’97–F’98) re- ceived the B.Sc. degree from Warwick University, Warwick, U.K., in 1975, the M.Sc. degree from Oxford University, Oxford, U.K., in 1976, and the Ph.D. degree from the California Institute of Technology, Pasadena, in 1980, all in mathematics.
He is currently a Professor of Electrical Engi- neering and Mathematics at Princeton University, Princeton, NJ, where he directs the Program in Applied and Computational Mathematics. He joined Bell Telephone Laboratories as a Member of Tech- nical Staff in 1980, and retired from AT&T in 2003 as Vice President of Research. He has research interests that range from algebraic coding theory and quantum computing to the design of wireless and radar systems.
Dr. Calderbank was honored by the IEEE Information Theory Prize Paper Award in 1995 for his work on the Z4 linearity of Kerdock and Preparata Codes (joint with A. R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane, and P. Sole), and again in 1999 for the invention of space–time codes (joint with V. Tarokh and N. Seshadri). He is a recipient of the IEEE Millennium Medal, and was elected to the National Academy of Engineering in 2005. He served as Editor-in-Chief of the IEEE TRANSACTIONS ONINFORMATIONTHEORYfrom 1995 to 1998, and as Associate Editor for Coding Techniques from 1986 to 1989. He was a member of the Board of Governors of the IEEE Information Theory Society from 1991 to 1996.
