Optimal rate allocation in wireless networks with delay constraints

Ad Hoc Networks 13 (2014) 282–295

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Optimal rate allocation in wireless networks with delay constraints Fan Qiu ⇑, Jia Bai, Yuan Xue Institute for Software Integrated Systems, Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA

a r t i c l e

i n f o

Article history: Received 17 May 2012 Received in revised form 11 August 2013 Accepted 14 August 2013 Available online 3 September 2013 Keywords: Rate allocation End-to-end delay Wireless networks

a b s t r a c t The optimal rate allocation problem with end-to-end delay constraints in wireless networks is highly challenging due to the dynamics incurred by delay related factors, such as traffic arrival pattern, queuing process, and wireless resource sharing mechanism. In this paper, we solve the problem through a utility maximization framework with two sets of constraints: (1) capacity and schedulability constraints and (2) end-to-end delay constraints. The end-to-end delay of a flow can be adjusted by controlling the per-link delays via a novel parameter called Virtual Link Capacity Margin (VLCM), which is the measurement of the gap between the schedulable link capacity and the maximum allowable flow rate over a link. This optimization problem can be solved via its dual decomposition through two prices derived with regard to the constraints: the link congestion price that reflects the relationship between the traffic load and the capacity of a link, and the flow delay price that reflects the margin between the average packet delay and the delay constraint of a flow. We prove that the algorithm converges to the optimal solution under the M/M/1 queuing model. We have also discussed using our algorithm under generalized traffic patterns, where we use link delay functions to formulate the relationship between VLCM and the average link delay. The algorithm is implemented distributedly via joint wireless link scheduling, VLCM adjustment and congestion control. Extensive experiments are conducted over diverse network topologies and traffic arrival models. The experimental results show that our algorithm is able to achieve optimal rate allocation while satisfying the end-to-end delay constraints.  2013 Elsevier B.V. All rights reserved.

1. Introduction End-to-end delay is highlighted as a critical performance requirement for numerous network applications, such as multimedia live streaming [1] and networked control systems [2]. Providing delay assurance, which is notably difficult even in wireline networks [3,4], has been a long lasting challenge for wireless networks. The main difficulty comes from the complex interactions between traf⇑ Corresponding author. Tel.: +1 615 3436216. E-mail addresses: [email protected] (F. Qiu), jia.bai@vanderbilt. edu (J. Bai), [email protected] (Y. Xue). 1570-8705/$ - see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.adhoc.2013.08.010

fic arrival and departure, shaped by the scheduling mechanisms that deal with the location-dependent interference of wireless networks. The studies of [5,6] have focused on the delay analysis of wireless networks, where delay bounds are derived for given data flows. Understanding the delay bounds and minimizing the end-to-end delay, however, are barely enough for real world applications, which have strict delay requirements. On the other hand, traffic elasticity is exhibited by many networking applications, in which the flow rates can be adjusted to operate at multiple quality levels [7]. Thus, identifying an operating point, where wireless network resource is efficiently and fairly allocated to flows while satisfying their

F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295

end-to-end delay constraints, carries practical value, especially in resource-constrained wireless networks. In this paper we investigate the optimal rate allocation problem in wireless networks with delay constraints. We adopt a utility maximization framework [8–11], where the utility of a flow is defined as a concave function of its rate. The objective is to maximize the aggregate utility of all flows subject to two constraints: the schedulability constraint and the end-to-end delay constraint. We introduce a novel parameter called Virtual Link Capacity Margin (VLCM), which is the gap between the schedulable link capacity and the maximum allowable rate over a link. The end-to-end delay of a flow can be tuned via the per-link delays along its route by adjusting the VLCMs. Through the dual decomposition of the optimization problem, we employ a doubleprice scheme, where the link congestion price reflects the relationship between the traffic load and the capacity of a link, and the flow delay price reflects the margin between the average packet delay and the delay constraint of a flow. This naturally leads to a cross-layer design consisting of VLCM adjustment, wireless link scheduling and congestion control. The main contributions of this paper are summarized as follows: (1) By introducing a novel parameter Virtual Link Capacity Margin (VLCM) that can be adjusted to jointly control the delay and the rate of a wireless link, this paper presents a tractable optimization formulation for maximizing the network utility under the end-to-end delay constraints. (2) We present a double-price based algorithm which converges to the optimal solution in a distributed way. (3) Our approach can be extended to handle generalized network models with unknown packet arrival patterns. Since this approach only requires the first order derivative of the delay-margin function, it can be applied to a large range of application domains. The remainder of the paper is organized as follows: In Section 2, we briefly overview the related works. Section 3 introduces the system model. Section 4 describes the delay control mechanism. In Section 5 we propose a cross-layer rate allocation algorithm for wireless networks with endto-end delay constraints, under the M/M/1 queuing model. Section 6 presents a distributed implementation of the optimal rate allocation algorithm. We introduce an extended algorithm to deal with generalized packet arrival patterns in Section 7. Section 8 presents the simulation results. Finally, Section 9 concludes the paper.

2. Related work Considerable research efforts have recently been devoted to resource management in wireless networks, for instance, scheduling, rate allocation and routing, under delay constraints. One common objective of these works is to achieve performance optimality based on certain metrics, such as throughput, system utility and service disruption probability, while satisfying the delay requirements. A large class of existing studies seek to solve the delay constrained QoS problems in wireless networks via scheduling. For example, the delay based scheduling algorithms

283

in [12–14] use the delay information from the head-of-line packets as link weight, to achieve throughput optimality or network utility maximization in single-hop wireless networks. The studies of [15,16] make further attempts to solve the delay constrained QoS using game theory. The approach in [15] addresses the utility maximization problem under packet level delay constraints in wireless networks consisting of one access point and multiple clients. The optimization problem is decoupled into an access point sub-problem and a client sub-problem, which are jointly coordinated by a bidding game framework. In addition, a weight based scheduling policy is proposed to solve the access point sub-problem, where the weight of a client is related to its history of scheduling. The algorithm proposed by [16] maximizes network utility with respect to service rate under time varying channels, the central idea behind their solution is max-weight scheduling and VCG-based auction. The study in [17] is another example of delay constrained scheduling, which employs a dynamic program for scheduling decision with packet arrival deadline. Because the aforementioned works mainly investigate the delay problem in single-hop wireless networks, their techniques cannot be easily applied to multi-hop networks, in which the delay problem is further complicated by the process at the intermediate nodes. The work of [18] extends the above works by proposing a new delay metric, which is the delay difference between a queue’s head-of-line packet and the head-of-line packet of its previous hop. The throughput optimality is achieved by using a back-pressure scheduling in multi-hop wireless networks. The study of [6] investigates the impact of scheduling policies on delay performance in multi-hop wireless networks. Nevertheless, these works do not investigate how to satisfy the given delay requirements by adjusting flow rates. In addition to the scheduling oriented delay solutions, algorithms based on rate allocation or even cross-layer resource management have also been proposed. The work of [4] investigates the delay-aware cross-layer algorithm of rate allocation, routing and scheduling. Especially, they propose delay-aware routing and back-pressure scheduling to jointly improve the end-to-end delay performance. However, this work does not provide any assurance of satisfying explicit delay requirements. In great contrast to the existing works of delay control in wireless networks, one salient feature of our approach is that it only requires the knowledge of the derivative of link delay with respect to link capacity margin, which can be either derived analytically for known arrival processes (e.g., Poisson process), or measured online over each link for unknown processes. Therefore it is applicable to a wide range of applications. This work extends our preliminary works [19,20] with a local packet scheduling algorithm and a rate allocation algorithm that handles generalized traffic arrival models. Moreover, we present simulation study using a new packet-level simulator. We conduct extensive simulations with more realistic and sophisticated network settings to validate our theoretical claims.

F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295 Table 1 Notations.

3. Network model We consider a wireless network consisting of V nodes. The node set is denoted as V. The nodes communicate with each other via L directed wireless links, collectively denoted as L. Let S be the set of end-to-end flows. Each flow s 2 S is associated with a utility function Us(xs), where xs is the flow rate. The link set LðsÞ represents all the links along the route of a flow s. The flow set SðlÞ is used to denote all the flows that traverse link l. Let link delay dl be the average packet delay of link l. Then the average packet delay of flow s is the sum of all link delays along its route. Here we consider the delay constraint such that the average end-to-end packet delay of s cannot exceed a fixed value Ds. In addition, we assume that the packet sizes in the network are exponentially distributed. Wireless communication is subject to location dependent interference. We adopt the concept of conflict graph [21] to model wireless interference. Each vertex in the conflict graph represents a wireless link of the original network. There is an edge between two vertices if their corresponding wireless links interfere with each other. The communications along wireless links are scheduled on a slotted time basis. In each time slot, one independent set I of the conflict graph is selected and only the links corresponding to the vertices in I are allowed to transmit because they are interference free. Let cl represent the amount of bits that can be transmitted by the channel per second. A L-dimensional column vector rI is used to represent the capacity vector of I, where rIl ¼ cl if l 2 I, and rIl ¼ 0 otherwise. We adopt the model of feasible capacity region K as introduced in [22] to model the feasible link capacity allocation. The feasible capacity region is a convex hull, which is defined as P P K :¼ I aI rI , where I aI ¼ 1 and aI P 0. Scheduling essenc ¼ ð^cl ; l 2 LÞ of tially determines the capacity allocation ^ the links, where ^cl is the average capacity over time based c 2 K. In this study, we mainly on the scheduling. Obviously, ^ consider the traffic arrival patterns that follow a Poisson distribution. We collect the notations used in this paper in Table 1. 4. Virtual link capacity margin The delay at a wireless link is caused by traffic burstiness and the waiting time incurred by the wireless link scheduling algorithm. To control the delay at a link l, we can limit the maximum allowable link rate yl, which satisP fies yl P s2SðlÞ xs . We introduce a parameter Virtual Link Capacity Margin (VLCM) rl at link l for per-link delay control, which is defined as follows

rl ¼ ^cl  yl ; with yl < ^cl ; 8l 2 L

ð1Þ

It is the difference between the feasible link capacity ^cl and the maximum allowable flow rate yl. We regard the link delay dl of link l as a function of the virtual link capacity margin rl: dl = ul(rl). ul() can be different, under different queuing models. For example, if the packet arrivals follow a Poisson distribution and the service time follows an exponential distribution (M/M/1 queue), the delay function can be described by

Notation

Definition

v 2 V ¼ f1; 2; . . . ; V g

Node set End-to-end flow set Wireless link set Links the flow s traverses Flows incident to the link l Feasible link capacity vector Maximum allowable link rate vector Flow rate vector Virtual link capacity margin vector End-to-end delay constraint vector Link congestion price vector Flow congestion price vector Flow delay price vector Link delay price vector Delay pressure vector Delay function of link l

s 2 S ¼ f1; 2; . . . ; Sg l 2 L ¼ f1; 2; . . . ; Lg LðsÞ SðlÞ ^ c ¼ ð^cl ; l 2 LÞ 2 K y ¼ ðyl ; l 2 LÞ x ¼ ðxs ; s 2 SÞ r ¼ ðrl ; l 2 LÞ D ¼ ðDs ; s 2 SÞ p ¼ ðpl ; l 2 LÞ q ¼ ðqs ; s 2 SÞ l ¼ ðls ; s 2 SÞ g ¼ ðgl ; l 2 LÞ k ¼ ðks ; s 2 SÞ ul(rl)

1.2

10 Mbps 8 Mbps 6 Mbps 4 Mbps

1

Average link delay (ms)

284

0.8 0.6 0.4 0.2 0 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Normalized VLCM Fig. 1. Average link delays under different normalized VLCMs. A normalized VLCM is the ratio of the VLCM of a link to its capacity. The network contains one single link with a data flow over the link.

1 1 ¼ cl  yl rl

ul ðrl Þ ¼ ^

ð2Þ

Fig. 1 presents a simple illustration of the relationship between the average link delay and the normalized VLCM (the ratio of VLCM to link capacity). We collect the average link delays of a single link under different wireless channel capacities. The packet arrivals at the source node follow a Poisson distribution. Obviously, applying VLCM to link rate allocation effectively reduces the average link delay.

5. Optimal rate allocation with delay constraints 5.1. Problem formulation The end-to-end delay of a packet is the accumulative effect of all the link delays along the route traversed by the

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packet. The average per-link delay can be effectively controlled by the VLCM of the link, as we discussed in Section 4. Here we first consider the close form of VLCM under the M/M/1 queuing model, which means the arrivals of packets follow a Poisson distribution and the service time at each node has an exponential distribution. To this end, we formulate the optimal rate allocation problem with end-to-end delay constraints as the following nonlinear optimization problem:

W:

X max U s ðxs Þ

2 3 X X X Lðx; p; r; l; ^c Þ ¼ U s ðxs Þ  ls 4 ul ðrl Þ  Ds 5 s2S

s:t:

xs 6 ^cl  rl ;

l2L

ð4Þ

ul ðrl Þ 6 Ds ; 8s 2 S

ð5Þ

s2S

l2L

0



s2SðlÞ

1

X X X X xs @ pl A þ ls Ds þ pl ^cl s2S

s2S

l2LðsÞ

ð8Þ

l2L

The dual problem is

s2SðlÞ

X

s2SðlÞ

2 3 X X X 4pl rl þ ¼ U s ðxs Þ  ul ðrl Þls 5

ð3Þ

8l 2 L

l2LðsÞ

1 X X  pl @ xs  ^cl þ rl A

s2S

X

s2S

0

Dðp; lÞ ¼ min Dðp; lÞ

ð9Þ

pP0;lP0

l2LðsÞ

1 1 ¼ c l  y l rl

ul ðrl Þ ¼ ^ over ^c 2 K

ð6Þ ð7Þ

The objective of the problem is to maximize the aggregate utility of all flows. The inequality (4) is the constraint of link capacity utilization, which requires that the aggregate rate of the flows over a link cannot be greater than the maximum allowable rate. Here ^cl actually determines the service rate, which is the time-averaged result based on the scheduling algorithm. Since the packet size is exponentially distributed, the service time at each node follows an exponential distribution during each scheduling cycle. As we introduced in Section 4, a non-negative margin rl between the feasible link capacity and the maximum allowable rate is imposed on a link l to control the link delay, so the rate allocation is actually operated under a reduced capacity region. The inequality (5) states that the sum of all link delays along a flow route is not greater than its end-to-end delay constraint. When the queuing process at each link follows the M/M/1 model, the constraints in (5) form a convex set. The constraint (7) requires the capacity space generated by the scheduling mechanism is within the feasible capacity region. Eq. (6) defines the delay function of each link under the M/M/1 model, which is derived from Eq. (2). Since each constraint in (4) and (5) is a convex function over (x, r) with an upper bound, W is a convex optimization problem that aims at maximizing a concave objective function over a convex set. This framework is based on a fluid model. However, the implementation of our algorithm actually provides a discrete solution based on a time-slotted system, which will be introduced in Section 6. 5.2. Double-price algorithm Direct solution to W requires a global coordination of all network components, such as flows and links. This is computationally expensive and hence we consider its dual decomposition. Let p ¼ fpl ; l 2 Lg and l ¼ fls ; s 2 Sg be the Lagrange multipliers with respect to constraints (4) and (5) respectively. The Lagrangian of W is

where

Dðp; lÞ ¼ max Lðx; p; r; l; ^c Þ x;r;^c 8 9 > > > < X = > X pl rl þ u ðrl Þls ¼ max  s2SðlÞ l r > > > : l2L |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> ; Hðrl Þ 8 9 > > > X U s ðxs Þ  xs l2LðsÞ pl þ max x > > > : s2S |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> ; Wðxs Þ

( ) X X ^ pl cl þ ls Ds þ max ^c

l2L

ð10Þ

s2S

Here the multiplier pl can be understood as the implicit congestion price [10] of link l, which represents the cost of delivering a unit of data through link l. The multiplier ls can be interpreted as the implicit delay price of flow s, which represents the cost of imposing a unit of delay on flow s. The solution ðx ; r ; ^ c Þ to Eq. (10) should satisfy:

8 0 19
^cl 2K

ð11Þ

ð12Þ

ð13Þ

l2L

Now it is clear that we need to solve three sub-problems: the congestion control problem (11), the VLCM assignment problem (12) and the scheduling problem (13). If p and l are given, the maximizers xs and rl can be obtained by taking the derivatives of L() with respect to xs and rl respectively.

X dWðxs Þ ¼ U 0s ðxs Þ  pl ¼ 0 dxs l2LðsÞ 0 1 X dHðrl Þ 0 @ ¼  pl þ ul ðrl Þls A ¼ 0 drl s2SðlÞ

ð14Þ

ð15Þ

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Hence the optimal solution is:

xs ðqs Þ ¼ U 01 s ðqs Þ;

rl ðgl ; pl Þ ¼ u01 l

with qs ¼

X

pl ;

8s 2 S

ð16Þ

l2LðsÞ

  pl ;

with gl ¼

gl

X

ls ; 8l 2 L

ð17Þ

s2SðlÞ

Here qs is the flow congestion price of flow s, which is the sum of all link congestion prices along the route of s. gl is the link delay price of link l, which is the sum of all flow delay prices over l. Eq. (16) implies that the optimal rate of a flow is determined by its flow congestion price, because a flow rate is affected by the cost of every link along its route. Eq. (17) implies that the optimal VLCM of a link is relevant to its congestion price and link delay price. The intuition behind this is: (1) the congestion price determines the available capacity margin that can be used for VLCM adjustment and (2) the link delay price implicitly reflects the overall delay requirement (from all of its incident flows) on its VLCM value. The maximizer ^cl can be generated from a maximum weight based scheduling policy, which will be discussed later. Intuitively, we need to find a VLCM assignment r and a rate allocation x that satisfy the delay constraints and maximize the network utility simultaneously. In other words, we need the optimal solutions for problem (11) and problem (12). The central idea is to control the flow rate and VLCM via updating the link congestion price p and the flow delay price l iteratively, from the opposite direction to the gradient rL(p, l) (gradient projection algorithm) [23]:

2

0

pl ðt þ 1Þ ¼ 4pl ðtÞ þ b@

X

13þ

xs  ^cl þ rl A5

ð18Þ

s2SðlÞ

2

2

ls ðt þ 1Þ ¼ 4ls ðtÞ þ c4

X

33þ

ul ðrl Þ  Ds 55

ð19Þ

l2LðsÞ

Eqs. (18) and (19) reflect the demand–supply interactions of network resources: if the flow demand over a link exceeds the maximum allowable flow rate, the link congestion price pl will increase. Similarly, if the sum of link delays of a flow exceeds its delay constraint, the flow delay price ls will increase. The two prices are updated iteratively so the network resources can be properly allocated. It is noticeable that x and r interact closely with each other: the VLCMs cause direct impact on link capacities and therefore determine the flow rates that can be allocated; the flow rates are involved in link price computations and thus pose effect on VLCMs. The properties of this algorithm is formally characterized in Propositions 1 and 2. The detailed proofs of the two propositions are provided in the Appendix. Proposition 1. There is no duality gap between (3) and (9). For any (p⁄,l⁄) that minimizes (10), if (x⁄,r⁄) solves (11) and (12), then (x⁄,r⁄) is the unique maximizer of (3). Proposition 2. If kbk2 and kck2 are sufficiently small, starting from any initial values x(0),r(0) and prices p(0) P 0, l(0) P 0, the cross-layer algorithm converges to the optimal solution (x⁄,r⁄,p⁄,l⁄).

Let b = mb0 and c = mc0 , where m > 0 is a tuning parameter of the step sizes b and c. b0 and c0 can be regarded as the baseline step sizes. Consider the following function Y(t):

YðtÞ ¼

    X pl ðtÞ  p 2 X ls ðtÞ  ls 2 l þ c0 b0 s2S l2L

We prove that the following result holds for any

ð20Þ

 > 0:

pffiffiffiffiffiffiffiffiffi lim YðtÞ < 

ð21Þ

t>1

0

Hence it is clear to conclude from the continuity of U 1() that x(t) converges when t ? 1. Please refer to the Appendixfor more details. 5.3. Feasibility check Note that real world network systems usually impose requirements on the minimum rates of data flows. The vector of minimum flow rates is denoted as x. Our method can be straightforwardly extended to handle additional flow rate requirements:

xs P xs ;

8s 2 S

ð22Þ

The rate requirement constraint (22) naturally leads to the third price variable qs in the dual formulation of W, which can be calculated via the gradient projection method:

qs ðt þ 1Þ ¼ ½qs ðtÞ þ xðxs  xs ðtÞÞþ

ð23Þ

This variable reflects the supply and demand relationship between the minimum rate requirement and the allocated rate of a flow. It is easy to show that there may not always exist a feasible r which satisfies a given delay constraint vector, therefore a feasibility check is necessary to verify that a feasible solution exists. The feasibility check can be performed by using qs, in which the price qs and the price ls for each flow are monitored. Once the two prices evolve towards the same direction, the problem is declared infeasible, because in such a situation the delay constraint and the minimum rate requirement cannot be satisfied at the same time. 6. Implementation of optimal rate allocation algorithm 6.1. Cross-layer implementation To implement link scheduling according to Eq. (13) so P ^ the aggregated link weight l2L pl cl is maximized, we adopt the maximum weighted matching algorithm introduced in [9,22]. This algorithm guarantees the total weight across the network is at least 12 of the theoretical upper bound. In our implementation, we measure the instantaneous end-to-end delay of flow s for Eq. (19) and we denote it as Ds(t). Therefore, the flow delay price is ls(t + 1) = [ls(t) + c(Ds(t)  Ds)]+. Our algorithm can be implemented via a cross-layer design, which consists of congestion control (11), VLCM adjustment (12) and link scheduling (13). Many existing works have discussed the implementation of cross-layer

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F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295 Table 2 Distributed implementation of double-price algorithm.

Table 3 Priority packet scheduling algorithm.

Double-Price Based Cross Layer Rate Allocation 1: Alg.D

Priority Packet Scheduling: Alg.Q

0) Feasibility check If this is a feasible problem, Start timer, set t = 0, repeat 1)-7) on a slotted time basis else exit end if 1) If timer expires (t = = T) stop timer and go to 2) else If a link is scheduled send packets from the uplink node to the downlink node end if t=t+1 end if 2) Link congestion price update Link l updates its link congestion price h P iþ ^ pl ðt þ 1Þ ¼ pl ðtÞ þ b s2SðlÞ xs  c l ðtÞ þ rl

Initialization Each flow s initializes its delay pressure ks(t) Local Packet Scheduling

 is scheduled 1) If link l v up ; v dn l l

Add price pl(t) to the flow price qs if s traverses l 3) Delay price update Flow s updates its delay price ls(t + 1) = [ls(t) + c(Ds(t)  Ds)]+ Update ls in the data packets of s to notify all related links 4) Rate adaption Each source node adjusts its sending data rate when it receives an ACK packet from the destination node: xs ðt þ 1Þ ¼ U 01 s ðqs Þ 5) VLCM adjustment Each link adjusts its VLCM when it receives a data packet   pl rl ðt þ 1Þ ¼ u01 g l l

6) Scheduling Node v starts scheduling at the beginning of each scheduling cycle. 6.1) If v has been scheduled by any of its neighbor node, send messages to notify all the nodes in its interference set and stop scheduling end if 6.2) Each incident directional link l is assigned a weight wl = plcl 6.3) Find a link with the maximum weight among all incident links. If the link is found, Schedule this link and update the effective capacity ^cl Notify all neighbors and all links in the interference set else Stop scheduling end if 6.4) If timeout occurs (end of the scheduling cycle) stop scheduling. else repeat 6.1) to 6.4) end if 7) Restart timer, set t = 0 and go to (1)

node v up conducts the following steps: l a) Re-order the packets in the outgoing queue Qo, according to their delay pressure values b) Send the packets in Qo to node v dn l end if 2) Each flow s periodically updates its delay pressure ks(t)

Table 4 Distributed implementation of double-price algorithm with generalized traffic. Double-Price Based Cross Layer Rate Allocation 2: Alg.Dg 0) Link Capacity Generation Generate capacity region ^ c ¼ ð^cl ; l 2 LÞ, under max-weight based scheduling. 1) Feasibility check If this is a feasible problem Start timer, set t = 0, repeat 3)-9) on a slotted time basis else exit end if 2) Link Delay Function Profiling Profile the link delay functions under the fixed capacity region ^ c 3) If timer expires (t = = T) stop timer and go to 4) else If a link is scheduled send packets from the uplink node to the downlink node; t=t+1 end if end if 4) Link congestion price update Link l updates its link congestion price h P iþ ^ pl ðt þ 1Þ ¼ pl ðtÞ þ b s2SðlÞ xs  c l ðtÞ þ rl Add price pl(t) to the flow price qs, if s travels through l 5) Delay price update Flow s updates its delay price ls(t + 1) = [ls(t) + c(Ds(t)  Ds)]+ Update ls in the data packets of s to notify all related links 6) Rate adaption Each source node adjusts its sending data rate when it receives an ACK packet from the destination node: xs ðt þ 1Þ ¼ U 01 s ðqs Þ 7) VLCM adjustment Each link adjusts its VLCM when it receives a data packet   pl rl ðt þ 1Þ ¼ u01 g l l

rate allocation [9,22,24–27]. Our cross-layer design differs from these works in that we implement a VLCM adjustment mechanism for delay control. The distributed implementation of our cross-layer algorithm (Alg.D) is given in Table 2. 6.2. Enhancement with local packet scheduling We further extend the algorithm Alg.D by adding a local packet scheduling process. In this scheduling process, packets with more ‘‘urgent’’ delay requirements are assigned with higher priorities. In particular, we define a delay pressure variable ks for each flow s to reflect the gap

8) Scheduling Schedule the links and maintain a capacity vector ^ c 9) Restart timer, set t = 0 and go to (3)

between its end-to-end delay at time t and its delay requirement:

ks ðtÞ ¼ Ds ðtÞ  Ds

ð24Þ

The local packet scheduling algorithm, denoted as Alg.Q, dn is listed in Table 3. When a link lðv up l ; v l Þ with uplink node dn v up and downlink node v is scheduled, the uplink node l l v up re-orders the packets in its outgoing queue according l to the delay pressure values of the packets.

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(P) Parallel flows

(O) Overlapped flows

500

500 400

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1

2

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100

100 0 -100 0

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0

nodes links f1: 0->1->2->3 f2: 4->5->6->7 f3: 8->9->10->11

7 nodes links f1: 0->3->4->5 f2: 1->3->4->6->8 f3: 2->3->4->7 f4: 1->3->4

-100 600

700

0

100

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(R) Random flows 1000

32

2 25

37 18

22

17 23

800 28 20

5

0

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600 3

12 6

400

24 29 11

8

9 21

38

1

36 7

200

14

3327 26

30

16 13 10

4 1531

19

nodes 39 links f1: 2->22->5->33->29->11->16->10->39 f2: 19->7->38->11->8->35 f3: 32->23->14->9 f4: 2->22->28->34->6->38->30 f5: 25->5->27->24->11 f6: 15->1->9->35

0

-200 0

200

400

600

800

1000

1200

1400

3000

2400

2500

2000

Flow rates (kbps)

Flow rates (kbps)

Fig. 2. Experimental wireless networks with (P) parallel flows (top left), (O) overlapped flows (top right) and (R) random flows (bottom).

2000 1500 1000 500

f1 f2 f3

0 0

1000 2000 3000 4000 5000 6000 7000 8000

Time slots

f1 f2 f3

1600 1200 800 400 0 0

15000 30000 45000 60000 75000 90000

Time slots

Fig. 3. Flow rate allocation using the cross-layer algorithm without delay control (left) and flow rate allocation with delay control (right): parallel flows.

The local packet scheduling algorithm can ‘‘balance’’ the delay pressures of flows: the packets of the flows with larger delay pressure values are processed first. This process actually slightly ‘‘reshapes’’ the delay functions at the wireless links, and eventually helps satisfy the delay constraint.

7. Extension to handle generalized traffic arrival model 7.1. Rate allocation with generalized traffic arrival model In general, the average packet delay at link l depends on not only the capacity margin rl, but also the scheduled link

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1400

1000

Flow rates (kbps)

Flow rates (kbps)

1200 1000 800 600 400 f1 f2 f3 f5

200 0 0

f1 f2 f3 f4

800 600 400 200 0

1000 2000 3000 4000 5000 6000 7000 8000

0

8000

16000

Time slots

24000

32000

40000

Time slots

Fig. 4. Flow rate allocation using the cross-layer algorithm without delay control (left) and flow rate allocation with delay control (right): overlapped flows.

1800

2500

f1 f2 f3 f4 f5 f6

1400

Flow rates (kbps)

Flow rates (kbps)

1600 1200 1000 800 600

f1 f2 f3 f4 f5 f6

400 200 0

0

1000 2000 3000 4000 5000 6000 7000 8000

2000 1500 1000 500 0

0

5000

Time slots

10000

15000

20000

Time slots

Fig. 5. Flow rate allocation using the cross-layer algorithm without delay control (left) and flow rate allocation with delay control (right): random flows.

capacity ^cl . If the traffic arrival pattern is unknown, then the delay function of link l is a function of the capacity margin rl and the link capacity ^cl , so the rate allocation problem becomes

Wg :

max

X U s ðxs Þ s2S

s:t:

X

xs 6 ^cl  rl ;

8l 2 L

s2SðlÞ

X

ð25Þ

ul ðrl ; ^cl Þ 6 Ds ; 8s 2 S

which is complicated by the indeterministic interactions among traffic generation, scheduling and queuing. Deriving the close form of the delay function under wireless interference and scheduling is highly complex and beyond the scope of this paper. Here we present a practical online profiling method to estimate the link delay function with respect to VLCM based on empirical measurements. For a link l, we collect its average link delay values under different VLCMs, and use a polynomial function whose curve accurately fits these points in a VLCM-Delay coordinate

l2LðsÞ

over ^c 2 K

250 200 150 100 50 0

7.2. Link Delay function profiling The delay at a link is largely determined by the higher order dynamics of the traffic arrival and departure [4],

Delay(original) Delay(optimal)

300

Delay (ms)

Wg differs significantly from W due to the delay constraint (25). Now the new constraint poses a twofold challenge: (a) it is non-trivial to obtain a close form of the link delay function ul(rl, ^cl ) and (b) the feasible rate region may not be convex due to the constraint (25). Here, we attempt to find an approximate solution by fixing the feasible link capacity ^cl . It includes two steps: (1) capacity region generation: each link obtains its throughput-optimal link capacity ^cl generated from the max-weight based scheduling and c, the links seek their optimal (2) margin estimation: under ^ VLCMs under the capacity region.

f1

f2

f3

f4

f5

f6

Flow Fig. 6. The average packet delays of the flows in the network with random flows, before/after applying VLCM.

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1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

CDF

CDF

f4 1

0.5

f6

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

0.05

0.1

0.15

0.2

0.25

0

0.01

0.02

Delay (s)

0.03

0.04

0.05

0.06

Delay (s)

Fig. 7. The CDFs of the average packet delays of f4 and f6 in the network with random flows, after applying VLCMs.

8.1. Experimental setup

f1

f2

f3

f4

f5

The experiments are conducted under three network configurations: (P) grid network with parallel data flows, (O) network with overlapped data flows, and (R) random network with multiple data flows, as illustrated in Fig. 2. The transmission range and interference range are both 250 m. The wireless channel capacity is 10 Mbps. The wireless links are scheduled based on a slotted time scale, in which each slot represents 1 ms. We consider the Poisson traffic arrival model. In our experiments, the maximum queue size length is set to be large enough so that we can only consider the delays incurred by scheduling and queuing. The delay constraints of the flows are:

f6

Flows Fig. 8. Impact of local packet scheduling on flow throughput in the network with random flows.

system as its delay function ul(). For example, a link delay function generated from a degree n polynomial fitting is

ul ðrl Þ ¼ kl;n rnl þ kl;n1 rn1 þ    þ kl;1 rl þ kl;0 l

 Delay constraint set 1: Dp = (D1, D2, D3) = (12 ms, 20 ms, 12 ms), Do = (D1, D2, D3, D4) = (24 ms, 30 ms, 24 ms, 7 ms), Dr = (D1, D2, D3, D4, D5, D6) = (200 ms, 120 ms, 32 ms, 160 ms, 50 ms, 30 ms).  Delay constraint set 2: Dp = (D1, D2, D3) = (26 ms, 45 ms, 26 ms), Do = (D1, D2, D3, D4) = (36 ms, 46 ms, 36 ms, 12 ms), Dr = (D1, D2, D3, D4, D5, D6) = (300 ms, 130 ms, 45 ms, 200 ms, 40 ms, 50 ms).

ð26Þ

Throughput (kbps)

2500

Delay-Priority Optimal

2000 1500 1000 500

f1

f2

Flow

f3

Throughput (kbps)

where kl,n, kl,n1, . . . , kl,1, kl,0 are the polynomial coefficients corresponding to link l, which are obtained from the least square polynomial fitting method. Here we only use convex polynomial fitting functions. The profiling process is performed after the capacity region ^ c is obtained. We summarize the extended algorithm that handles general traffic arrival model in Table 4.

1600 1400 1200 1000 800 600 400 200 0

Delay-Priority Optimal

f1

f2

Flow

f3

f4

1800

Throughput (kbps)

Throughput difference (kbps)

8. Simulation study 100 80 60 40 20 0 −20 −40 −60 −80

Delay-Priority Optimal

1600 1400 1200 1000 800 600 400

f1

f2

f3

f4

f5

f6

Flow

Fig. 9. Throughput comparison between the optimal rate allocation scheme with the delay priority scheduling (DPS) based rate allocation algorithm.

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F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295 Table 5 Aggregate network utility 1.

Table 6 Aggregate network utility 2.

Parallel Overlapped Random

DPS

Optimal

62.349 78.865 118.915

62.656 79.124 119.833

Parallel Overlapped Random

UMA

RMA

Optimal

60.740 73.489 118.356

59.792 75.272 119.460

60.929 75.546 119.675

8.4. Evaluation of local packet scheduling Where Dp, Do and Dr represent the delay constraint vectors of the grid network, overlapped network and random network respectively. We mainly use the first set to evaluate the performance of our algorithm unless otherwise specified. The second delay constraint set is used for the comparison between our algorithm and a delay–priority– scheduling based algorithm, which will be introduced later in this section.

To evaluate the impact of packet scheduling, we measure the flow throughput values in the network with random flows (R) under two scenarios: our original algorithm without local packet scheduling and the algorithm enhanced with local packet scheduling. The box-plot of the throughput differences (throughput with scheduling  throughput without scheduling) is presented in Fig. 8, with the upper edge of each box representing the 25th quantile and the bottom edge representing the 75th quantile. The results imply that the throughput performance with local packet scheduling outperforms the throughput performance without local packet scheduling.

8.2. Flow rate convergence We now test the convergence of our algorithm. In this experiment, the packets from the source nodes follow Poisson distribution. It is important to note that the traffic arrival processes at the intermediate nodes in the network may not follow Poisson distribution and the related link delay functions need to be profiled. The sending rates of the flows with/without VLCMs are illustrated in Figs. 3–5. Obviously, starting from any initial values, the flow rates quickly converge in all the scenarios.

8.5. Performance comparison We further compare our algorithm with three other rate allocation algorithms.  DPS algorithm: Delay–Priority–Scheduling based rate allocation algorithm adopts a joint link scheduling and rate allocation design. During link scheduling, the links with greater delay pressure have higher priority to be scheduled.  UMA algorithm: Uniform Margin Assignment rate allocation is similar to our proposed algorithm, where flow rates are adapted based on link prices and link capacity margins are adjusted to control delays. The difference is that UMA algorithm assigns VLCMs uniformly among all the links. The value of VLCM is set to be the minimum value that satisfies the delay constraints.  RMA algorithm: Random Margin Assignment rate allocation randomly assigns VLCM to each link to satisfy the delay constraints.

8.3. Flow delay control

Uniform Random Optimal

1600 1400 1200 1000 800 600

f1

f2

Flow

f3

800 700 600 500 400 300 200 100 0

We first compare our algorithm with DPS using delay constraint set 2. The throughput values of the flows are plotted in Fig. 9 and the network utilities are listed in

1800

Uniform Random Optimal

Throughput (kbps)

Throughput (kbps)

1800

Throughput (kbps)

We next evaluate the performance of our algorithm with regard to end-to-end delay control. The experiments are conducted under the Poisson arrival model. We compare our algorithm with the original cross-layer rate allocation algorithm (equivalent to Alg.D without delay control). Fig. 6 presents the average packet delays in the network with random flows. The delay performance of our algorithm is significantly better than the performance of the original cross-layer algorithm. Then we collect the packet delay samples of f4 and f6 in the random network and plot the cumulative distribution function of the packet delay of each flow in Fig. 7. The plots indicate that the delays of over 80% of the packets are controlled under the delay constraint values.

f1

f2

f3

f4

1600

Uniform Random Optimal

1400 1200 1000 800 600 400

f1

f2

Flow

Fig. 10. Throughput comparison with different margin allocation algorithms (UMA and RMA).

f3

f4

Flow

f5

f6

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F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295

Table 5. Our algorithm outperforms the DPS algorithm in the aggregate network utility. Then we compare our algorithm with UMA and RMA using delay constraint set 1. The throughput comparison results are illustrated in Fig. 10, and the network utilities are listed in Table 6. The performance of the optimal algorithm is better than the other two algorithms.

max Lðx; p; r; l; ^c Þ ¼ x;r;^c

X  U s xs

We consider a new problem W0 (k), which is a relaxation of W:

W 0 ðkðu; v ÞÞ :

X U s ðxs Þ

max s:t:

xs 6 ^cl  rl þ ul ;

s2SðlÞ

Optimal rate allocation with end-to-end delay constraints in wireless networks is a challenging problem, considering the complexity arising from delay modeling and wireless resource contention. We present an algorithm based on a novel parameter Virtual Link Capacity Margin (VLCM), which reflects the gap between feasible link capacity and the maximum allowable rate over a link. By controlling VLCM, the per-link average delay can be adjusted at the cost of achievable link capacity. We formulate the problem in a double-price based optimization framework so that the flow rates and VLCMs are tuned simultaneously. The algorithm is implemented through a cross-layer joint scheduling and congestion control approach. We show by extensive experimental results that our solution can achieve greater network utility than the heuristic solutions, while satisfying the end-to-end delay constraints.

l2LðsÞ

This work is supported in part by the U.S. Army Research Office and Lockheed Martin under Cooperative Agreement W911NF-10-1-0005 and by the National Science Foundation under Grant CNS-1150169. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government.

ð28Þ

s2S

X

9. Conclusion

Acknowledgements

ð27Þ

s2S

X

8l 2 L

ul ðrl Þ 6 Ds þ v s ; 8s 2 S

over ^c 2 K where k is a column vector defined as

k ¼ ðu0 ; u1 ; . . . ; ujLj1 ; v 0 ; v 1 ; . . . ; v jSj1 ÞT P 0; 8l 2 L;

with ul

v s P 0; 8s 2 S

and

ð29Þ

cÞ as follows We next define a column vector dðx; r; ^ X

dðx; r; ^cÞ ¼

xs  ^c0 þ r0 ;

s2Sð0Þ

X

X

xs  ^c1 þ r1 ; . . . ;

s2Sð1Þ

ul ðrl Þ  D0 ;

l2Lð0Þ

X

X

xs  ^cjLj1 þ rjLj1 ;

s2SðjLj1Þ

X

ul ðrl Þ  D1 ; . . . ;

l2Lð1Þ

1T

ul ðrl Þ  DjSj1 A

l2LðjSj1Þ

Note that the optimization problem W is equivalent to W 0 ð0Þ, which implies

W 0 ð0Þ ¼

X  U s xs

ð30Þ

s2S 0

6W ðkÞ

ð31Þ 0

0

From the Hessian of W , we know that W (k) is a concave function with respect to k, therefore there exists a sub-gradient g0 = (p0, l0) of W0 at k = 0, and g0 satisfies inequality (32) by the concavity of W0 .

W 0 ðkÞ 6 W 0 ð0Þ þ g T0 k

ð32Þ

Appendix A

cÞ in W0 (), and it is easy to verify that Consider k ¼ dðx; r; ^ W0 (k) P W0 (0) from the definition of W0 , hence we can reach the following result:

A.1. Proof of Proposition 1

X X  U s ðxs Þ 6 U s xs s2S

By using the method introduced in [28] and following the proof of Proposition 1 in [29], we present the proof of Propositions 1 and 2 in this section. Suppose the controllable VLCM of a link is within the range [rm, rM], we define a function family called F functions. Definition 1. A delay function ul(rl) is called F function if it is a continuous convex function over [rm, rM]. Obviously, if the packet arrivals of a link follow a Poisson distribution, its delay function is a F function. Let x⁄ denote the optimal flow rate vector, and r⁄ denote the optimal link margin vector of the delay constrained rate allocation problem W. Let ^ c denote the feasible link capacity vector under the max-weight based scheduling. To prove there is no duality gap between (3) and (9), we need to show that there exist Lagrange multipliers p and l that satisfy

ð33Þ

s2S 0

6W ðdðx; r; ^c ÞÞ 6W 0 ð0Þ þ g T0 dðx; r; ^c Þ X  ¼ U s xs þ g T0 dðx; r; ^cÞ

ð34Þ

s2S

From (33) and (34) we have

g T0 dðx; r; ^c Þ P 0

ð35Þ ⁄

⁄

If we replace (x, r) with (x , r ), (35) becomes

g T0 dðx ; r ; ^cÞ P 0

ð36Þ

Note that, from (32), we have

W 0 ð0Þ PW 0 ðkÞ  g T0 k 0

PW ð0Þ 

ð37Þ

g T0 k g0Tk

ð38Þ T

which implies that P 0, "k P 0, so g0 P 0. Because c Þ 6 0, from (36), the following result must hold dðx ; r ; ^

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F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295

g T0 dðx ; r ; ^c Þ ¼ 0

ð39Þ

^c

Next we consider the Lagrangian with multiplier g0:

8
s2S

X  U s xs  g T0 dðx ; r ; ^cÞ

s2S

s2S

¼ Lðx ; p0 ; r ; l0 ; ^c Þ Therefore, we have found a Lagrange multiplier vector g0 = (p0, l0) that satisfies Eq. (40), which completes the proof.

max Lðx; p0 ; r; l0 ; ^c Þ ¼ Lðx ; p0 ; r ; l0 ; ^c Þ ¼ x;r;^c

X  U s xs

ð40Þ

s2S

A.2. Proof of Proposition 2 Let X be a set of vectors (p⁄, l⁄) such that each object in the set is a solution to the dual problem (9). Next we define mb0 = b and mc0 = c, where m is a tuning parameter. Now we expand Y(t + 1) by using (18) and (19):

Yðt þ 1Þ ¼

  X pl ðt þ 1Þ  p 2 l b0 l2L   X ls ðt þ 1Þ  ls 2 þ 0

0

b

þ

X

h

P

ls ðtÞ þ c

¼

l2LðsÞ

l2L

@

ul ðrl ðtÞÞ  Ds  ls c0

X

s2S

þ m2

X

X

ls ðtÞ  l

þ m2

s2S

 s

0 1  X @ u ðrl ðtÞÞ  Ds A

ð42Þ

l

s2S

l2LðsÞ

From (41) and (42) and the definition of Y(t), the expression of Y(t + 1) is

Yðt þ 1Þ 6 YðtÞ þ 2m½Dðp ; l Þ  DðpðtÞ; lðtÞÞ 0 12 X 0 X 2 @ þm b xs ðtÞ  ^cl ðtÞ þ rl ðtÞA l2L

s2SðlÞ

0

2

þm

X

c

s2S

l

0@

X

12

ul ðrl ðtÞÞ  Ds A

l2LðsÞ

l2LðsÞ

0 b0 @

l2L

X



xs ðtÞ  ^cl ðtÞ þ rl ðtÞA

0 1 X X

 þ 2m l ðtÞ  l @ u ðrl ðtÞÞ  Ds A s

pl ðtÞ  pl

l2L

s2SðlÞ

s

X

1

s2SðlÞ

i2



 X pl ðtÞ  p 2 X ls ðtÞ  ls 2 l þ c0 b0 s2S l2L 0 1 X

 X  @ þ 2m pl ðtÞ  pl xs ðtÞ  ^cl ðtÞ þ rl ðtÞA l2L

2 39 = X X 4 DðpðtÞ; lðtÞÞ ¼ l ðtÞDs  pl ðtÞrl ðtÞ þ ul ðrl ðtÞÞls ðtÞ5 : s2S s ; l2L s2SðlÞ 8 2 39
þ 

s2S

8
0

P  i2  ^ X pl ðtÞ þ b s2SðlÞ xs ðtÞ  c l ðtÞ þ rl ðtÞ  pl l2L

The dual with time-varying multipliers under the crosslayer algorithm is:

Dðp ; l Þ  DðpðtÞ; lðtÞÞ P

h

6

 s Ds 

Hence we have

c

s2S

l2L

2 39 = X X   4p rl ðtÞ þ l ul ðrl ðtÞÞls 5 P l : s2S ; l2L s2SðlÞ 8 2 39 ( )
X X Lðx; p0 ; r; l0 ; ^cÞ ¼ U s ðxs Þ  g T0 dðx; r; ^c Þ 6 U s ðxs Þ ¼

( ) X pl ^cl

þ max

cðtÞ; D and r(t) are bounded and We assume that xðtÞ; ^ hence the following expression is bounded by a constant value W:

12

X

xs ðtÞ  ^cl ðtÞ þ rl ðtÞA

s2SðlÞ

0

c0 @

X

0 12 X 0 X b@ xs ðtÞ  ^cl ðtÞ þ rl ðtÞA

12

ul ðrl ðtÞÞ  Ds A

l2L

l2LðsÞ

ð41Þ Next we investigate the dual of W with the optimal multipliers:

8 2 39 =
þ

s2SðlÞ

X

0

c0 @

s2S

X

12

ul ðrl ðtÞÞ  Ds A

l2LðsÞ

6W

ð43Þ

so Y(t + 1) satisfies

Yðt þ 1Þ 6 YðtÞ þ 2m½Dðp ; l Þ  DðpðtÞ; lðtÞÞ þ m2 W We further define a superset of X: Xd, with (p⁄, l⁄) 2 X and d > 0, which is



Xd ¼ ðp; lÞjDðp ; l Þ 6 Dðp; lÞ 6 Dðp ; l Þ þ d

ð44Þ

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F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295

Now the we can investigate Y() under two situations: (a) (p(t), l(t)) R Xd Under this assumption, D(p, l) > D(p⁄, l⁄) + d, and we can pick a tuning parameter m such that m 6 d/W. Therefore, from (44) we have

Yðt þ 1Þ 6 YðtÞ  2

d2

W

þ

d2

W

6 YðtÞ 

d2

W

ð45Þ

2

Since dW > 0, it is obvious that the vector (p(t), l(t)) approaches (p⁄, l⁄) and it will belong to the set Xd after a certain time point. (b) (p(t), l(t)) 2 Xd Under this assumption, D(p⁄, l⁄)  D(p, l) 6 0, and we can a tuning parameter m such that pffiffiffiffipick ffi m 6 d= W, Therefore, from (44) we have

Yðt þ 1Þ 6 YðtÞ þ d2

ð46Þ

We define the following variable G(p, l), where

Gðp; lÞ ¼

    X pl  p 2 X ls  ls 2 l þ ; c0 b0 s2S l2L

2 Xd

ðp; lÞ ð47Þ

For a given d, it is reasonable to assume G(p, l) is upper bounded by a value C(d, X), therefore

YðtÞ 6 Cðd; XÞ þ d2

ð48Þ 2

Obviously, when d ? 0, C(d, X) + d ? 0. From (a) and (b) we know that for any  > 0, we can always find sufficiently small m and d, such that there is a time point t(0), when t P t(0)

YðtÞ 6 

ð49Þ

In other words, the price vector will converge to a small neighborhood of the optimal price vector [29] when t ? 1. References [1] S. Wee, W. tian Tan, J. Apostolopoulos, M. Etoh, Optimized video streaming for networks with varying delay, in: Proceedings of IEEE International Conference on Multimedia and Expo (ICME), vol. 2, 2002, pp. 89–92. [2] M. Branicky, S. Phillips, W. Zhang, Stability of networked control systems: explicit analysis of delay, in: Proceedings of American Control Conference, vol. 4, 2000, pp. 2352–2357. [3] M. Saad, A. Leon-Garcia, W. Yu, Optimal network rate allocation under end-to-end quality-of-service requirements, IEEE Transactions on Network and Service Management 4 (3) (2007) 40–49. [4] H. Xiong, R. Li, A. Eryilmaz, E. Ekici, Delay-aware cross-layer design for network utility maximization in multi-hop networks, IEEE Journal on Selected Areas in Communications 29 (5) (2011) 951– 959. [5] G. Gupta, N. Shroff, Delay analysis for wireless networks with single hop traffic and general interference constraints, IEEE/ACM Transactions on Networking 18 (2) (2010) 393–405. [6] G.R. Gupta, N.B. Shroff, Delay analysis and optimality of scheduling policies for multihop wireless networks, IEEE/ACM Transactions on Networking 19 (2011) 129–141. [7] A.T. Al-Hammouri, M.S. Branicky, V. Liberatore, S.M. Phillips, Decentralized and dynamic bandwidth allocation in networked control systems, in: Proceedings of the 20th International Conference on Parallel and Distributed Processing, IPDPS’06, IEEE Computer Society, Washington, DC,USA, 2006, p. 167.

[8] F.P. Kelly, A.K. Maulloo, D.K.H. Tan, Rate control for communication networks: shadow prices, proportional fairness and stability, The Journal of the Operational Research Society 49 (3) (1998) 237–252. [9] X. Lin, N. Shroff, The impact of imperfect scheduling on cross-layer rate control in wireless networks, in: Proceedings of IEEE INFOCOM, vol. 3, 2005, pp. 1804–1814. [10] S. Low, D. Lapsley, Optimization flow control. I. Basic algorithm and convergence, IEEE/ACM Transactions on Networking 7 (6) (1999) 861–874. [11] Y. Xue, B. Li, K. Nahrstedt, Optimal resource allocation in wireless ad hoc networks: a price-based approach, IEEE Transactions on Mobile Computing 5 (4) (2006) 347–364. [12] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, P. Whiting, R. Vijayakumar, Providing quality of service over a shared wireless link, IEEE Communications Magazine 39 (2) (2001) 150–154. [13] M.J. Neely, Delay-based network utility maximization, in: Proceedings of the 29th Conference on Information Communications, INFOCOM’10, IEEE Press, Piscataway, NJ, USA, 2010, pp. 2669–2677. [14] B. Sadiq, G. De Veciana, Throughput optimality of delay-driven maxweight scheduler for a wireless system with flow dynamics, in: Proceedings of the 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton ’09, IEEE Press, Piscataway, NJ, USA, 2009, pp. 1097–1102. [15] I.-H. Hou, P.R. Kumar, Utility maximization for delay constrained qos in wireless, in: Proceedings of the 29th Conference on Information Communications, INFOCOM’10, IEEE Press, Piscataway, NJ, USA, 2010, pp. 2426–2434. [16] I.-H. Hou, P.R. Kumar, Utility-optimal scheduling in time-varying wireless networks with delay constraints, in: Proceedings of the Eleventh ACM International Symposium on Mobile Ad hoc Networking and Computing, MobiHoc’10, ACM, New York, NY, USA, 2010, pp. 31–40. [17] J. Jaramillo, R. Srikant, L. Ying, Scheduling for optimal rate allocation in ad hoc networks with heterogeneous delay constraints, IEEE Journal on Selected Areas in Communications 29 (5) (2011) 979– 987. [18] B. Ji, C. Joo, N. Shroff, Delay-based back-pressure scheduling in multi-hop wireless networks, in: Proceedings of IEEE INFOCOM, 2011, pp. 2579–2587. [19] F. Qiu, J. Bai, Y. Xue, Towards optimal rate allocation in multi-hop wireless networks with delay constraints: a double-price approach, in: Proceedings of IEEE International Conference on Communications (ICC), 2012, pp. 5280–5285. [20] J. Bai, E. Eyisi, F. Qiu, Y. Xue, X. Koutsoukos, Optimal cross-layer design of sampling rate adaptation and network scheduling for wireless networked control systems, in: Proceedings of IEEE/ACM Third International Conference on Cyber-Physical Systems (ICCPS), 2012, pp. 107–116. [21] K. Jain, J. Padhye, V.N. Padmanabhan, L. Qiu, Impact of interference on multi-hop wireless network performance, in: Proceedings of the 9th Annual International Conference on Mobile Computing and Networking, MobiCom ’03, ACM, New York, NY, USA, 2003, pp. 66– 80. [22] L. Chen, S.H. Low, M. Chiang, J.C. Doyle, Cross-layer congestion control, routing and scheduling design in ad hoc wireless networks, in: Proceedings of IEEE INFOCOM, 2006. [23] D.P. Bertsekas, J.N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall Inc., Upper Saddle River, NJ, USA, 1989. [24] M. Chiang, Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power control, IEEE Journal on Selected Areas in Communications 23 (1) (2005) 104–116. [25] A. Eryilmaz, R. Srikant, Fair resource allocation in wireless networks using queue-length-based scheduling and congestion control, IEEE/ ACM Transactions on Networking 15 (6) (2007) 1333–1344. [26] U. Akyol, M. Andrews, P. Gupta, J. Hobby, I. Saniee, A. Stolyar, Joint scheduling and congestion control in mobile ad-hoc networks, in: Proceedings of IEEE INFOCOM, 2008, pp. 619–627. [27] A. Eryilmaz, R. Srikant, Joint congestion control, routing, and mac for stability and fairness in wireless networks, IEEE Journal on Selected Areas in Communications 24 (8) (2006) 1514–1524. [28] J. Borwein, A. Lewis, Convex analysis and Nonlinear Optimization: Theory and Examples, CMS Books in Mathematics, Springer, 2006. [29] X. Lin, N.B. Shroff, The Impact of Imperfect Scheduling on Cross-layer Rate Control in Multihop Wireless Networks, Tech. Rep., Purdue University, 2004. .

F. Qiu et al. / Ad Hoc Networks 13 (2014) 282–295 Fan Qiu received his B.E. degree in Electrical Engineering from Beijing Normal University, China in 2004. Currently he is a Ph.D. student at the Department of Electrical Engineering and Computer Science of Vanderbilt University. His research interests include distributed systems and computer networks with a focus on wireless mesh networks and peer-to-peer systems. He is a student member of IEEE.

Jia Bai received her B.S. degree in Department of Automation from Nankai University, China in 2006 and her M.S. in Electrical Engineering in 2008. Currently she is a Ph.D. candidate at the Department of Electrical Engineering and Computer Science of Vanderbilt University. Her research interests include distributed systems and computer networks with a focus on optimization and application of wireless networks and cyber physical systems.

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Yuan Xue received her B.S. in Computer Science from Harbin Institute of Technology, China in 1998 and her M.S. and Ph.D. in Computer Science from the University of Illinois at Urbana-Champaign in 2002, and 2005. Currently she is an assistant professor at the Department of Electrical Engineering and Computer Science of Vanderbilt University. Her research area includes networking and distributed systems with a focus on wireless andmobile systems, web applications and services, clinical information system and cloud computing. My research goal is to provide performance optimization, quality of service support, application and data integrity and privacy for next generation networking and distributed systems. She is a NSF CAREER Award winner.