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Multi-hop wireless networks use two or more wireless hops to transmit data from a source to a destination. Wireless net- works provide great advantages that are ...
Joint Routing, Scheduling and Power Control Providing QoS for Wireless Multihop Networks∗ Satya Kumar V and Vinod Sharma Department of Electrical Communications Engineering, Indian Institute of Science, Bangalore Email: [email protected], [email protected] Abstract—We consider the problem of joint routing, scheduling and power control in a multihop wireless network ensuring Quality of Service (QoS) to different users. The QoS to a user may be the stability of its queues in the network or a minimum rate guarantee. Transmission channels may experience fading. Power is consumed only for transmission of data. We consider the case when the power required is a linear function of the transmission rate. Our policies minimize power, are also easy to implement and computationally very efficient. We also provide explicit mean end-to-end delay of individual flows for the optimal policy. Index Terms—Multihop wireless network, routing, scheduling, end-to-end QoS, minimizing power.

I. I NTRODUCTION Multi-hop wireless networks use two or more wireless hops to transmit data from a source to a destination. Wireless networks provide great advantages that are not available through their wired counterparts such as flexibility, ease of deployment, cost reduction, convenience, and allow mobility of the users. However, these advantages come at the expense of some drawbacks, such as limitation in transmission range due to the characteristics inherent in wireless communication such as time varying multipath fading, broadcast nature, frequency reuse, noise, electro-magnetic interference and receiver sensitivity. As a result, a wireless node can only communicate directly with other nodes which are in its transmission range and needs to share channels with neighbouring nodes. In order to communicate with out-of-range nodes when wireless nodes are deployed in an ad-hoc setup, a wireless node has to depend on other intermediate nodes for relaying its messages until they reach the intended destination. In a wireless multi-hop network, one of the important challenges is how to route data packets efficiently. Choosing an optimal path from a source to a destination can be done by optimizing one or more routing metrics (such as number of hops, distance, mean delay, packet loss rate, and energy consumption). This depends on the application requirements such as delay-sensitivity and/or on constraints such as limited energy or frequent topology changes. Also, because of broadcast nature of the wireless channels, scheduling link transmissions and fixing transmit power also need to be considered along the routing. This joint optimization problem *This work was partly supported by a funding from ANRC. c 2015 IEEE 978-1-4799-6619-6/15/$31.00

is computationally hard and not scalable even for a centralized algorithm. Problem of minimizing average queue length under average power constraint or vice-versa, has been well studied in [3], [2], [16], [21] for a singlehop wireless network, where some bounds on approximations for the mean queue length and structural results for the optimal solution are also obtained. However the average queue length expression and an optimal policy for single hop for the above problems are presented in [17], [18] which are computationally efficient. Also, for guaranteeing end-to-end mean delay for a flow, approximate optimal solutions are provided in [17], [18]. Multihop QoS problem can be solved in either a distributed or a centralized manner. Pioneering work on joint routing, scheduling and power control was provided in [6] which maximizes a utility function under average power constraint. As this problem is intractable they provided a heuristic suboptimal algorithm. [13] considered the problem of ensuring a fair utilization of network resources by jointly optimizing routing, scheduling and power control and obtained an efficient sub-optimal solution. [10] extended the solution in [6] to a multihop network where different nodes have multiple antennas and presented efficient, fair algorithms. In [9], [14] Lyapunov drift approach is applied on a multihop network which uses the channel and queue state information at different nodes. [11] uses quadratic Lyapunov functions to provide novel back-pressure algorithms. In [12], using the above approach upper bounds on average delay are presented. These back-pressure algorithms provide stability of the network if the load is within the capacity region. But under high load, the end-to-end delay will be large and may violate any mean delay constraints. Due to complex coupled queue dynamics in multihop networks, Markov Decision Process (MDP) or approximated MDP technique has a large state space and the computations become unrealistic even for smaller networks [7]. Furthermore, MDP techniques may not provide any insights in the structure of the optimal policy and require huge signaling overheads. A. Problem Statement We consider the problem of jointly optimizing routing, link scheduling and transmit power control while ensuring end-toend QoS to individual data users. Within these constraints we try to optimize the overall average transmit power of all the

nodes. The QoS to a user may be simply stability of its queues at all the nodes in the network or a minimum rate guarantee.

t then the data Rij (f ) transmitted to node j for flow f is, t Rij (f ) =

B. Our Contribution We present a computationally efficient algorithm, for optimal routing, scheduling and power control for a multihop network which minimizes total average power while providing end-to-end QoS to individual users. We assume that the transmit rate is a linear function of the power invested at each node. This can be a good approximation to a practical system at low SNR (sensor networks) or under high bandwidth ([1], [8]). Our solution is explicit and we even provide closed form end-to-end mean delays for our optimal policy. We have used these results to guarantee end-to-end mean delay to individual users, which will be reported in a future work. We are not aware of any other work on multihop wireless networks that ensures QoS except stability, which again does not minimize power. The paper is organised as follows. In Section II, we describe the system model and our notation and assumptions. Section III develops the algorithms to ensure stability. Section IV modifies these algorithms to ensure a minimum rate to each flow. Section V concludes the paper. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a network which is a connected, directed graph G(N , L), where N = {1, 2, . . . , N } is the set of nodes and L = {1, 2, . . . , L} is the set of directed links. A subset of nodes (called source nodes) in the network transmits data to another subset of nodes (called destination nodes). Each source has one destination. The time axis is slotted. The stream of packets transmitted from a source node to its respective destination node is called a flow. The set of user flows {1, 2, . . . , M } is denoted by F. A flow carries data for which we may only need to ensure the stability of all the queues in the network and/or need to ensure that a flow gets a minimum end-to-end rate. Let Aft be the number of packets generated by flow f in slot t at its source. All packets will be assumed to be of same length. We assume {Aft , t ≥ 0} to be iid, independent for different flows. We assume that the links are half- duplex. (i.e., when a node is transmitting to some other node, it cannot receive data from any other node(s)). Similarly, when a node is receiving from a node, it cannot transmit data to any other nodes. Let the channel gain in slot t from node i to node j be t Hij which is available to the node at the beginning of the t slot. We assume that the channel gain Hij remains constant during one slot. We also assume that the channel gain process t {Hij , t ≥ 0} is iid on all links and independent for different t links. The channel gain Hij takes values on a finite set. It can be a good approximation for continuous distributions, e.g., Rayleigh, Rician, Nakagami, by taking the finite set arbitrarily large. Let (i, j) be the link which connects node i to node j. If in time slot t, the power spent by node i for flow f is Pijt (f )

1 t 2 log2 (1 + Gij Pijt (f )Hij (f )/σij ). 2

(1)

2 where σij is the receiver noise variance and Gij is a constant that depends on the modulation and coding used. We consider the case, when (1) can be approximated by t t Rij (f ) = αij Pijt (f )Hij (f ),

(2) t (f ) Rij

is in where αij > 0. We will scale it so that number of packets. This is a good approximation of Shannon formula (1) at low SNR [18] and also at high bandwidth ([1], [8]). However as shown in [20], even when (1) cannot be approximated well by a linear function, (2) can still hold in a practical system. The rate (2), also depends on transmissions on other links. However we partition the set of links into independent sets such that the links with in a set do not interfere much with each other. In a slot only the transmissions from one independent set will be allowed. Our problem is to compute the routing of all flows f , scheduling of links and power allocation to the links such that the average sum power needed in the system is minimized. We consider a centralized setup, i.e., system has all the information including channel gain statistics and average external arrival rates. III. OPTIMAL POLICIES: STABILITY CONSTRAINT In this section, we consider joint routing, scheduling and power allocation policies which minimize, n−1 M

lim sup n→∞

1 XXX t E Pi,j (f ), n t=0 (i,j)

(3)

f =1

such that the long term average queue length E[qi,j ] < ∞, ∀(i, j) ∈ L.

(4)

We provide an optimal solution of this problem based on our previous work on single hop networks [18]. We briefly recall that algorithm in Section III-A and then extend it to our current problem in Sections III-B and III-C. A. Single user single hop From [18] we know that for a single user, single hop network, the optimal average power policy which provides stability of the queue is Rt = (q t + At )1{H t =hb } at time instant t, where hb is the best channel gain which is attained with probability pb , and Rt and q t are the transmit rate in slot t and the queue length at the beginning of time slot t. Then, q t+1 = (q t + At − Rt )+ .

(5)

For iid arrival traffic {At } and iid channel gains {H t }, the average queue length E[q] for this policy is   1 E[A] b − 1 , (6) p

where E[A] is the mean number of packets generated per slot by the source. Average power consumption E[W ] for this policy is E[A] , where α > 0 is the constant of (2) for this αhb system. B. Single user multihop network Now we extend the optimal policy of Section III-A to the case where a single flow traverses multiple hops to reach the destination. Also, we do not want to split the traffic on multiple routes. This ensures arrival of packets in order at the destination and avoids reordering delays. This is a common constraint [4]. From (6) we see that to minimize average power on a link, we should use it only when it has the highest channel gain and then should just clear the queue (due to linear function (2). Also, then for link (i, j) the average power consumption is E[A] . Thus, to minimize (3), we should use a route from the αij hbij source to the destination along the links which have minimum sum cost when the cost of link (i, j) is αij1hb ij The optimal route can be computed from Dijkstra’s algorithm [4]. Once the path is determined then our next objective is to schedule the links from the source to the destination along that path which satisfies the half duplex constraints, provides stability of queues and minimizes (3). In a wireless network, transmission from different nodes also interfere with each other. Thus we can generalize the transmission schedule of the links taking into account the interference from other nodes. We incorporate this by making independent sets Sk , k = 1, ..., C1 , of directional links where all links in Sk can be active at the same time. Given the constraints on simultaneous transmission of links, we can use [15] to develop independent sets. A link can be in more than one independent sets. We schedule the independent sets P such that each Sk is active for γk > 0 fraction of time and k γk ≤ 1. The allocation of slots to the different links will be done as follows. A central authority generates iid random variables Yt in the beginning of each slot t with probability P [Yt = k] = γk , k = 1, ..., C1 . If Yt = k then slot t is assigned to the independent set Sk . If in a slot a link has its best channel and if the link is in an independent set that is active in that slot then it will transmit all the data from its queue in that slot. The probability that link (i, j) will clear its queue in any slot is X pij = pbij [ γk ], (7) k:(i,j)∈Sk

where pbij = P [Hij = hbij ]. This policy is average power optimal which provides stability of all queues. Total average power consumption of this policy is X E[A] E[W ] = , (8) hbij αij where the summation is over all the links on the selected path. Interestingly, we also have a closed form expression for endto-end mean queue length of this policy, which we provide in Theorem 2 below.

Let |P| denotes the number of links on the path P and let qt (l) be the queue length of the lth link on P in the beginning of slot t, l = 1, ..., |P|. Let S(l) be the independent set to which l belongs (if l belongs to more than one set, pick one of them). Let qt = (qt (l), l = 1, ..., |P|). Theorem 1. The process {qt , t ≥ 1} has a unique stationary distribution π. Starting from any initial state, qt converges in total variation to π exponentially. If E[Aα k ] < ∞ for any α > 1 then Eπ [(qt (l))α ] < ∞ for all l = 1, ..., |P|. Proof: We omit the proof due to lack of space. The following theorem provides the mean queue lengths under stationarity at the different queues in the network. These will be used later to guarantee end-to-end mean delays to individual flows in the network. In the proof we also obtain computable stationary distributions of the queue lengths. Theorem 2. The mean queue length under stationarity at each node i on P is E[A1 ]( p1ij − 1) where (i, j) is the outgoing link from node i. Proof: The proof of the theorem is some what different at the source queue qt (1) and at the other queues qt (l), l > 1. We first prove the result for {qt (1)}. Although for queue1 we can use the result (6) but we provide a different proof which can be extended to other queues. We have proved in Theorem 1 that all queues {q(t)} have a stationary distribution. Consider qt (1) under stationarity. Let p(1) be the probability that this queue transmits in any given slot t. The interval N between two such transmissions is geometric with parameter p(1) and is independent of {Ak }. Under stationarity, by key renewal theory [19], at any slot t, the time t − N 0 when the last transmission took place, has distribution, l

P [N 0 ≤ l] =

1 X (1 − p(1))k−1 , E[N ]

(9)

k=1

Thus N 0 also has a geometric distriPN −1 bution with parameter p(1). Hence qt (1) = k=1 Ak and PN −1 Pπ [qt (1) = n] = P [ k=1 Ak = n] which can be computed if we know the distribution of {Ak }. Also, Eπ [q(1)] = 1 − 1). (E[N ] − 1)E[A] = E[A]( p(1) We extend the above proof for the later queues qt (l), l ≥ 2. Now the argument will remain same and hence we prove it for l = 2. Let this queue be cleared in any slot with probability p(2). Consider qt (2) under stationarity. Let N (2) denote the interval between two transmissions, which is geometrically distributed with parameter p(2). Thus at any time t, by (9), Q2 transmitted the last time at t−N 0 (2), where N 0 (2) has the distribution of N (2). During [t − N 0 (2), t), Q1 transmits with inter-transmission intervals N1 , N2 , ... which are iid, independent of N 0 (2) and {Ak }, and geometrically distributed with PN1 p(1) parameter 1−p(2) , q. Let X = k=1 Ak and {Xk , k ≥ 1} iid with the distribution of X. Then, when Q1 transmits for the kth time, during [t − N 0 (2), t), we can denote the number of packets it transmits as Xk . Let Q1 transmit M times during where E[N ] =

1 p(1) .

[t − N 0 (2), t). Then, qt (2) =

M X

Xk .

(10)

k=1

Therefore, Pπ [q(2) = 0]

=

∞ X

P [N1 > k|N 0 (2) = k]P (N 0 (2) = k)

k=1

=

For a multihop route we can have multiple values P of γk which provides optimal (3); in fact any γk > 0 with k γk ≤ 1 will do. Thus, we can consider such a solution that also minimizes end-to-end mean delay among such policies. From (14) for that we consider the convex optimization problem,  X E[A]  1 X (15) min pbij γk (i,j) k:(i,j)∈Sk

p(2)(1 − q) . 1 − (1 − q)(1 − p(2))

subject to,

Similarly, qp(2) Pπ [q(2) = X1 ] = (p(2) + q − qp(2))2

C1 X

(11)

and we can compute Pπ [q(2) =

l X

Xk ]

(12)

k=1

for any l. Of course it is cumbersome, but from (10), Eπ [qt (2)] = E[M ]E[X1 ]

(13)

1] and E[M ] = p(1) where E[X1 ] = E[N1 ]E[A1 ] = E[A q p(2) . E[A1 ](1−p(2)) p(1) 1 Thus Eπ [qt (2)] = . p(2) = E[A1 ]( p(2) − 1). p(1)

is,

Therefore, the end-to-end average queue length expression X  1 E[q] = E[A] − β1 , (14) pij (i,j)

where β1 is the number of hops on the optimal path and pij is the probability with which link (i, j) clears all the data in the queue. The scheduling of independent sets can also be done via time division multiple access (TDMA) instead of randomly allocating slots as mentioned above. Then accordingly, each link can be assigned one or more slots in a TDMA frame. If in a slot assigned to a link, it has its best gain, it will transmit all its data; otherwise not. Thus, a node needs to know only its own channel gains and the slots alloted to each of its links. An advantage of this scheme is that a central authority need not inform an independent set about its allocation in each slot. Also, this leads to less randomness as compared to the random allocation scheme, reducing end-to-end mean delays. However now the mean delay expression becomes more complicated and changes from node to node on the path and on the ordering the slots are allocated in a TDMA frame. Furthermore, from limited simulations we have observed that the gain in mean delays for the TDMA scheme over (14) may not be substantial. For multiuser case to be dealt with in the next subsection it will become intractable while as we will see (14) will continue to hold. Thus, we will allocate channels using the random scheme. If the optimal path from the source to the destination is connected by a single hop then transmitting all the data whenever that link has the best channel is mean power optimal.

γk = 1,

(16)

γk ≥ , ∀k.

(17)

k=1

where  > 0 is a small constant. The objective function is a sum of convex functions with a convex, closed constraint set. Hence it has a unique global optimal which can be easily computed via standard algorithms [5]. From Little’s law [4], we can compute end-to-end average delay E[D] from end-to-end total average queue length expression (14). We can also use the optimal γk ’s from (15) and then use TDMA scheduling with these. As mentioned above, the TDMA setup will only reduce the mean delays further but we can have the implementation advantage mentioned above. 

0 0  7  0  0  0  0  W = 0 0  2  5  2  0  7 0

7 0 0 5 9 0 0 0 0 0 0 10 0 0 0

4 0 0 10 3 0 0 0 0 0 0 7 5 2 4

6 0 0 0 6 0 8 0 7 6 0 5 0 0 6

3 0 0 0 0 0 0 1 5 1 6 0 0 0 0

0 10 9 10 8 0 0 0 0 0 3 0 0 9 3

8 5 9 8 0 0 0 8 3 8 3 6 0 7 5

0 0 0 9 5 10 5 0 2 6 10 7 9 0 6

0 0 0 0 8 0 8 1 0 1 0 0 6 0 1

1 0 0 1 0 6 1 0 0 0 0 8 0 3 10

5 1 0 4 0 1 0 4 7 0 0 0 0 0 0

0 0 1 8 3 0 0 0 2 0 0 0 0 0 2

9 0 0 7 9 8 0 9 5 0 0 0 0 0 1

0 8 10 0 0 7 10 7 2 9 10 0 0 0 0

7 1 0 0 2 0 5 0 3 9 0 0 6 0 0

                         

We solve this optimization problem on a single sourcedestination, multihop network of 15 nodes. We generated a random binary matrix A of size 15×15. Its element (i, j) is 1 if there is a link from node i to node j. If it is 0 then we assume that the wireless link is very bad and it is decided not to use it. If aij = 0 then αij = 0 and hbi,j = 0. The source node is 1 and the destination node is 15. We take the process {Ak } with Poisson distribution, with mean arrival rate 1. The weight wij of link (i, j) is αij1hb and is given below in matrix W . We ij find the optimal path 1 → 10 → 5 → 15 by using Dijkstra’s algorithm. The probabilities of the best channel gains for the optimal path {(1, 10),(10, 5),(5, 15)} are {0.25, 0.18, 0.3}.

Independent sets over the links on the minimum cost path are computed using existing algorithms [15]. Using only the half duplex constraint, we have two independent sets {(1, 10), (5, 15)} and {(10, 5)}. We solve the above optimization problem and obtain the optimal γ1 = 0.5346. The end-to-end average queue length is computed from (14) and equals to 22.65. The optimal average power consumption is 2.5. C. Multiuser multihop Now we consider the multiuser (M ≥ 2) multihop network, where the stability of all the queues in the network is required. Let the mean arrival rate of flow f be E[Af ]. We consider the policies which minimize (3) subjected to (4). We extend our single user algorithm to the multiuser network. We find the optimal cost path Rf for each flow f from its source to its destination using Dijkstra’s algorithm, when the weight of link (i, j) is αij1hb . ij We find independent sets Sk , k = 1, ..., C1 for the links which are on the optimal paths of at least one flow, using the algorithm from [15]. The independent set Sk will transmit in any given slot with probability γk , independently, from slot to slot. A node i will transmit in slot t all packets of all the sources for link (i, j), if in this slot (i, j) has the best channel and it belongs to a set Sk which is scheduled to transmit in that slot. This makes the transmission and movement of packets through the network decoupled for different flows. Thus, the mean end-to-end queue length of each flow is as in Theorem 2. Therefore, we solve the following convex optimization problem to find the optimal probability γk , that the set Sk can be active,  X X E[Af ]  1 X min (18) pbij γk f (i,j)∈Rf k:(i,j)∈Sk

subject to,

C1 X

γk = 1, γk ≥ , ∀i.

(19)

k=1

Here, again the objective function is a sum of strictly convex functions and the constraints are linear. Thus the unique solution of the above problem provides γk , k = 1, ..., C1 . The sum average queue length for the above optimal solution is E[q] =  X X  X M 1 E[Af ] X − E[Af ]βf . (20) b p γk ij f =1 f (i,j)∈Rf k:(i,j)∈Sk

where βf is the number of hops on the optimal route selected for flow f . From the Little’s law, we can compute end-to-end average delay for all flows. The corresponding total average power consumption is X X E[Af ] (i,j) ∀f

αij hbij

.

(21)

For M flows, N nodes and L links in the network, the complexity of finding the optimal paths from M source nodes to their destinations via Dijkstra’s algorithm in the worst case is O(M |L|+M N log|N |). Complexity of finding independent sets via the algorithm in [15] is O(L2 ). We demonstrate the above algorithm on a network of 20 nodes, which has three source-destination pairs (8, 9), (20, 10) and (14, 16). We generated random weight matrix of 20×20 for the links, we are not mentioning matrix due to space constraint. the optimal path from node 8 to 9 via Dijkstra’s algorithm is 8 → 13 → 7 → 9. The optimal path from node 20 to node 10 is 20 → 1 → 7 → 2 → 10 and the optimal path from node 14 to node 16 is 14 → 2 → 3 → 16. For the above set of links, for half duplex constraints we have 4 independent sets, S1 ={(20, 1),(7, 2),(8, 13),(3, 16)}, S2 ={(1, 7),(2, 10),(8, 13),(3, 16)}, S3 ={(13, 7),(2, 3),(20, 1)} and S4 ={(7, 9),(14, 2),(3, 16),(8, 13), (20, 1)}. Best channel probabilities on all links are not required and we are mentioning the best channel probabilities on the optimal paths only due to space constraints. The probabilities of the best channel gains for the sets S1 , S2 , S3 , S4 are {0.4, 0.25, 0.1, 0.35}, {0.5, 0.6, 0.1, 0.35}, {0.14, 0.8, 0.4}, {0.59, 0.22, 0.35, 0.1, 0.4}. The average arrival rates for the 3 users are 3, 1, 2 packets/slot. After solving the optimization (18), we get (γ1 , γ2 , γ3 , γ4 ) = (0.1747, 0.1672, 0.3359, 0.3222). The total optimal average power consumed for the three users is 12 and the total end-to-end average mean delay for the 3 users are (121.75, 44.51, 41.26). IV. RATE GUARANTEE CONSTRAINT In this section, we assume that all the sources in the network are saturated (i.e., all the sources have always sufficient packets to transmit). We encounter such a scenario when a large data file is being carried by TCP connections. Our objective is to minimize the total average power (3) consumed by all nodes such that the data can be carried from all the sources to their respective destinations such that each user f meets its average rate demand rf . The solution in this section is similar to that in Section III-C. For each link (i, j), we take the link cost αij1hb . We find ij optimal paths of all the users from their sources to destinations using Dijkstra’s algorithm. We find independent sets Sk for all the links which are on at least one optimal path. A node i will transmit all its data (from all flows except for the flow it is the source node) for link (i, j) in a slot whenever (i, j) has the best channel hbij in that slot and it belongs to an independent set that is allowed to transmit in that slot. If a node is source node for flow f then it will transmit r P f (22) b pij [ k:(i0 ,j)∈Sk γk ] packets from its source whenever it is allowed to transmit by the above rule. Sk will be active for probability γk > 0, where PSet C1 γ i=1 k = 1. As in Section III-C we can compute the γk

which minimize the average end-to-end delay by solving the optimization problem (18) when E[Af ] now is rf . For this solution the sum power consumed in the network is (21) and the total mean end-to-end delay is (20) with E[Af ] = rf for each f and we do not include the delay of flow f at its source. We consider a network of 20 nodes. Due to space constraint we are not mentioning weight matrix. We consider three source-destination pairs (1, 15), (3, 5) and (13, 7). The optimal path for the first source is 1 → 4 → 2 → 15, for the second user is 3 → 2 → 8 → 5 and for the third user is from 13 → 9 → 20 → 10 → 7. In this example we have 4 independent sets and γk0 s are computed by solving (18). For the above set of links, we get 4 independent sets, S1 = {(1, 4), (2, 15), (8, 5), (13, 9), (20, 10)}, S2 = {(4, 2), (8, 5), (9, 20), (10, 7)}, S3 = {(3, 2), (1, 4), (8, 5), (13, 9), (20, 10)} and S4 = {(2, 8), (1, 4), (9, 20), (10, 7)}. Probabilities of the best channel gains for the sets S1 , S2 , S3 , S4 are {0.2, 0.18,0.5, 0.4,0.35}, {0.25, 0.5,0.6, 0.1}, {0.8, 0.2,0.5, 0.4,0.35}, {0.65, 0.2,0.6,0.1}. The minimum mean rates required for the 3 users are 1, 3, 2. After solving optimization problem (18), we get (γ1 , γ2 , γ3 , γ4 ) = (0.2570, 0.2589, 0.2111, 0.2730). The optimal total average power consumed is 35. If a network has some users which require stability of the queues (as in Section III) and some users which require rate guarantee (as in this Section), our algorithm can be easily modified to handle this case. V. CONCLUSIONS We have considered the problem of joint routing, scheduling and power control in a multihop network. Our main objective is to minimize the total average power while providing endto-end Quality of Service (QoS) to all users. QoS can be the stability of the queues or a minimum rate guarantee per user. Power is consumed only during the transmission of data. We have considered the case when the power is a linear function of transmission rate. Our optimal polices are explicit, easy to implement and computationally very efficient. Also we can obtain mean end-to-end delay of each flow for the optimal scheme. R EFERENCES [1] F. Berggren and R. Jantti, “Asymptotically Fair Transmission Scheduling over Fading Channels”, IEEE Transactions on Wireless Communications, Vol. 3, No. 1, January 2004. [2] R. A. Berry, “Power and Delay Trade-offs in Fading Channels”, June 2000, Phd Thesis, MIT. [3] R. A. Berry and R. G. Gallager, “Communication over Fading Channels with Delay Constraints”, IEEE Transactions on Information Theory, vol. 48, no. 5, pp. 1135-1149, 2002. [4] D. P. Bertsekas and R. G. Gallager, “Data Networks”, Prentice Hall , Second edition, 1992. [5] S. Boyd and L. Vandenberghe, “Convex Optimization”, Cambridge University Press, First edition, 2004. [6] M. Cao, V. Raghunathan, S. Hanly, V. Sharma and P. R. Kumar, “Power control and Transmission scheduling for network utility maximization in wireless networks”, Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, December 2007.

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