Efficient Low Complexity Power Allocation Policies for Wireless Communication Systems Guaranteeing QoS

Efficient Low Complexity Power Allocation Policies for Wireless Communication Systems

Guaranteeing QoS ^∗

Satya Kumar V, Vinod Sharma

Department of Electrical Communications Engineering, Indian Institute of Science, Bangalore

Email: satyakumar@ece.iisc.ernet.in, vinod@ece.iisc.ernet.in Abstract—We consider near-optimal policies for a single user

transmitting on a wireless channel which minimize average queue length under average power constraint. The power is consumed in transmission of data only. We consider the case when the power used in transmission is a linear function of the data transmitted. The transmission channel may experience multipath fading. Later, we also extend these results to the multiuser case.

We show that our policies can be used in a system with energy harvesting sources at the transmitter. Next we consider data users which require minimum rate guarantees. Finally we consider the system which has both data and real time users. Our policies have low computational complexity, closed form expression for mean delays and require only the mean arrival rate with no queue length information.

Index Terms—Fading channels, Average Power Constraint, Average Rate guarantee, Optimal mean delays.

I. INTRODUCTION

Number of mobile users is growing at a rapid pace across the globe. Information and communications technology (ICT) consumes 3%of world power consumption. Furthermore, any reduction in power consumption in the ICT will reduce the ever increasing carbon footprint from this sector and also reduce the operating cost of the service providers. Other benefits will be smaller diesel generators and batteries with longer life time. Thus, one of the primary challenges for Next Generation Networks (NGN) is to reduce energy consumption.

In a BS the power is consumed mainly in the air conditioner (25%), Radio equipment (60%), DC power (11%) and RF load and Feeder (2%) [14]. In this paper we will concentrate on reducing power in RF transmission, while providing quality of service (QoS) to the users. This indirectly reduces the power consumed in the other components also.

Goldsmith and Varaiya [8] obtained a power allocation policy for a single fading link which optimizes the rate. Energy efficient scheduling under average delay constraint was first addressed in [3]. Goyal et al. [9] considered the same problem, proved existence of an optimal policy and obtained some structural results for the optimal policy. [18] proposed delay and power optimal policies in a single server queue when the power used in transmission is a linear function of rate. In [19] low complexity, power efficient algorithms are proposed

*Partially funded by a project from ANRC.

which ensure stability or satisfy mean delay constraints when power is a linear/general function of rate. In [13] existence and structural results are proved for optimal power policies which satisfy certain mean delay constraints. [5] proposed an energy efficient scheduling policy with individual packet delay constraints. [6] presented a policy which minimizes energy for sending a fixed number of packets under given delay constraints. Neely [15] proposed a new innovative al- gorithm which optimizes power by dropping a small number of packets when there is no knowledge of arrival rate and channel probabilities. [25] proposed a suboptimal policy which minimizes the average power under average queue constraint when there is an upper bound on packet loss also. [24]

obtained asymptotic lower bounds for average queue length and average power consumption. Neely [16] presented an algorithm for multiuser and multi channel scenario which requires only channel gain in the present slot and does not require channel statistics.

In [23] new throughput optimal energy management policies were developed when the transmitter is powered by energy harvesting sources. Explicit throughput optimal policies which also minimize mean delay were also obtained when linear power rate relation holds. Algorithms for multihop energy harvesting nodes are developed in [12]. [17] implemented an online algorithm by modifying value iteration equation which provides QoS by minimizing the average power for a single user.

The above policies either use Markov Decision Process (MDP), where it is computationally very expensive to obtain optimal policies or do not provide explicit quality of service (QoS) guarantees. The policies which use learning are often very slow in convergence. MDP and learning based algorithms also may not provide any insight in the optimal policies.

A. Our Contribution

Our objective is to minimize the average queue length under an average power constraint for a single user when power is a linear function of rate. This is extended to multiuser scenario also. We then also show how these algorithms can be used when the transmitter is powered by an energy harvesting source. We also obtain optimal policies for data users which require minimum rate guarantees and finally, the policies

when some users have mean delay constraints and others data constraint.

As against the above literature, our policies are in closed form or very easy to compute, and are very efficient. Also, one can explicitly compute the mean delay of each user under these policies which is very rare in a multiuser fading environment.

This paper is organized as follows. In Section II we present the system model. In Section III we present a near-optimal policy which minimize the average queue length under average power constraint. In Section IV we compare our policy with the optimal policy via simulations. In Section V we consider the multiuser scenario. In Section VI we present optimal policies for maximizing the average sum rate under average power and rate constraints. In Section VII consider a system with delay sensitive users and minimum rate guaranteed users.

Finally, Section VIII concludes the paper.

II. SYSTEM MODEL

We consider a single server, discrete time queue with M users. A slot is of duration 1. Let A^j_k bits arrive in slot k representing interval[k, (k+1)),k≥0for userj. The data to be transmitted arrives into the system from higher layers at the beginning of each slot and placed into an infinite buffer queue till transmission. The arrival process{A^j_k} is an independent and identically distributed (iid) sequence with mean E[A^j].

For our policies in this paper, we can consider data in bits or packets. Packets need not be fragmented because in our policies all the data (bits/packets) in the queue will be transmitted within a slot. Thus, we will consider data in terms of bits only.

We assume that channel gain H_k^j is constant over the duration of slot k and {H_k^j, k ≥ 1} is iid , where H_k^j ∈ {h^j₁, h^j₂, ..., h^j_L

j},0< h^j_i < h^j_l <∞, fori < l, ∀1≤j ≤M andh^j_L

j <∞. Let H_k = (H_k¹, ..., H_k^M). We assume that the channel gainH_k is available at the transmitter at time k.

We assume that the arrival and channel gain processes of each user are independent of another. Theiidassumptions on {A^j_k}and{H_k^j} are commonly made (see [3], [17]). TheH_k^j taking values on a finite set can be a good approximation for continuous distribution, e.g., Rayleigh, Rician, Nakagami, by taking the finite set arbitrarily large. However we will generalize both of these assumptions in Section III-C.

Letp(h) =P(Hk=h). Letq_k^j denote the queue length in bits at time kfor userj. Then{q^j_k} evolves as,

q^j_k+1= (q^j_k+A^j_k−R^j_k)⁺, k≥0, ∀j, (1) where(x)⁺ = max(0, x) andR^j_k bits may be transmitted in slotkby userj.

In time slotk, the power required for userjto transmit data R^j_k is given by,

R^j_k= 1

2log₂(1 +β_jP_k^jH_k^j/σ²). (2) where βj is a constant that depends on the modulation and coding scheme used and σ² is receiver noise variance. We

consider the case, when (2) can be approximated by

R^j_k=αjP_k^jH_k^j, (3) where αj > 0. This is a good approximation of Shannon formula (2) at low SNR [18] and also at high bandwidth ([2], [7]). However as shown in [23], even when (2) cannot be approximated well by a linear function, (3) can still hold in a practical system.

III. OPTIMAL POLICIES: MINIMIZING MEAN DELAYS

In this section we consider the problem of minimizing, lim sup

n→∞

1 nE

M

X

j=1 n−1

X

k=0

q^j_kWj (4a) subject to,

lim sup

n→∞

1 nE

M

X

j=1 n−1

X

k=0

P_k^j≤P (4b) whereWj>0are chosen to provide priority to different users.

By Little’s law [1], minimizing stationary mean queue length of a user is equivalent to minimizing stationary mean delay for that user. Thus, (4a) can be useful for real time users. Of course it is useful even for data users. Hence (4a) is a commonly used objective function in various studies for example [16].

We will see in Section IV that such policies can also be used in the energy harvesting scenario.

We could use the MDP to obtain an optimal policy (say via value/policy iteration [4]). But this will be computationally expensive, will not provide any insights into the optimal policy and will become increasingly difficult to compute as M increases. Also it needs the queue length and channel gain information along with their statistics. Thus we will follow a different path.

A. Single User

We consider in this section a single user case. We remove the index j for simplicity. We consider the following policy.

We transmit all the data in the queue with probabilitypinde- pendent of everything else. Then{qk}becomes a regenerative process [1] with the epochs when the queue is cleared as regeneration epochs. Let a regeneration length be denoted by N. It is geometrically distributed with parameterpand hence E[N^a] < ∞ for all a > 0. Also N is aperiodic. Therefore, {q_k} has a unique stationary distributionπ and starting from any initial distribution,qkconverges in total variation toπ. The fact that E[N^a]<∞, for any a >1 provides the following rate of convergence kqk−πk ≤ bk^1−a, where k.k denotes total variation metric, andb is an appropriate constant [22].

Proposition. 1 If E[A^a0] < ∞ then Eπ[q^a] < ∞ under stationarity for any a≥1, where a⁰> a.

Proof: We have, ifq1= 0,

Eπ[q^a] = E[PN k=1q_k^a]

E[N] (5)

and qn ≤ PN

k=1Ak for 1 ≤ n ≤ N. Thus, by Holder’s inequality, from (5),

Eπ[q^a]≤E[N(PN

k=1Ak)^a]

E[N] ≤ E[N^p]^1pE[(PN

k=1Ak)^ar]^1r E[N]

(6) for anyp >1, ¹_p+¹_r = 1. SinceE[N^p]<∞for allp≥1, by taking plarge enough we can take ras close to 1 as needed and hence we need E[(PN

k=1Ak)^a0] <∞ for some a⁰ > a.

Since{Ak}isiidandN is independent of{Ak} (becauseN depends only on {Hk} which is independent of{Ak}), this condition is satisfied, ifE[N^a0]<∞andE[A^a₁⁰]<∞([10], Chapter 3).

Now we explicitly compute Eπ[q]for this policy, E_π[q] = 1

E[N]

∞

X

k=1

E ^N−1

X

l=0

q_l|N =k

P[N =k]

= 1

E[N]

∞

X

k=1

E[A1(k−1) +...+A_k−1]P[N =k]

= E[A]

2E[N](E[N²]−E[N]) =E[A](1

p−1). (7) Any policy which clears the queue independently with probability p will have mean queue length delay as (7).

Therefore, minimizing average queue length under average power in this class of policies is equivalent to maximizing psubject to the average power constraintP, i.e., solving,

maxp,

L

X

i=1

ζip(hi), (8a) subject to

E[P],

L

X

i=1

E[A]ζip(hi) αphi

≤P , 0≤ζi≤1,∀i, (8b) whereζi is the probability with which we clear the queue in channel statehi. This is a Linear Program (LP) and has a nice structure. The optimal policy is obtained as follows. Find an integerm1≥1 such that,

E[A]

pα

m1

X

i=1

p(h_L+1−i) hL+1−i

< P ≤ E[A]

pα

m1+1

X

i=1

p(h_L+1−i) hL+1−i

, (9) andp⁰ such that

E[A]

pα ^m1

X

i=1

p(h_L+1−i) hL+1−i

+ p⁰ hL+m₁

=P . (10) The optimal policy is: Clear the queue with appropriate power, needed from the linear map (3) when H_k = h_L+1−i, i = 1, ..., m₁ and clear with probability p⁰, when H_k =h_L−m1. Letp,Pm1

i=1p(h_L+1−i) +p⁰.

This algorithm is explicit, requires little computing and only the channel gains but no queue length information. Also, it does not require the statistics of the arrival process, onlyE[A].

Furthermore, it remains the optimal policy in the class of all stationary policies which use only channel state information.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 5 10 15 20 25 30

Average Power

Average Queue Length

Greedy Policy Optimal Policy

Fig. 1. Single user single channel for Poisson arrivals

We will show via examples in the next section that this policy is very close to the optimal policy obtained via MDP which uses the queue length information also and needs the statistics of the arrival and channel gain process.

B. Example

We compare our greedy policy with the optimal policy via MDP. The optimal policy is obtained via policy iteration [4].

The constraints are handled via Langrange relaxation.

We compare the greedy policy of the last section and the optimal policy for two examples. In the first example, the arrival process{Ak}is Poisson with rate 2. The channel gains take values in the set{0.3,0.5,1,2}and the respective channel gain probabilities are {0.2,0.4,0.3,0.1}. The average queue length vs the corresponding average powers of the greedy policy and the optimal policy are plotted in Fig 2. We observe that the optimal curve and the greedy curve are very close to each other. We have obtained similar results with other distribution.

C. Generalizations

If{(A_k, H_k)}is stationary, ergodic then the policy obtained in (8a)-(8b) remains optimal in the class considered. Of course now the mean queue length expression is not (7) but needs to be computed via Palm theory [1]. Then an explicit expression for mean delay is still available.

Next we consider the case when the channel states can take values in R+ and can be unbounded, e.g., Rayleigh, Rician, Nakagami distribution. For simplicity we limit ourselves toiid case. As mentioned before, we can discretize the state space to arbitrary accuracy and use the policies developed in III-A.

But we need not discretize and can handle this case directly.

For simplicity we assume thatHk has density functionp(h).

Now the optimal policy is, clear the queue ifH ≥γwhereγ satisfies:

E[A]

pα Z ∞

γ

p(h)

h dh=P , (11)

wherep=P[H1≥γ].

IV. SYSTEM WITH AN ENERGY HARVESTING SOURCE

The policy developed in Section III can be easily extended to the case where the transmitter is powered by an energy

harvesting source. Consider the system of Section III-A. Now, in addition we also have an energy harvesting source which stores its energy in a battery. LetY_k be the energy harvested during slot k and let E_k be the total energy in the energy buffer at time k. We assume{Y_k} to be stationary, ergodic.

Thus,

Ek+1= (Ek−Pk) +Yk (12) wherePkis the energy used in slotk. There is no other energy source and hence Pk ≤Ek. Also, as before we assume that the energy is spent only in transmission and energy buffer is infinite. Various generalizations to energy spent on other activities and battery leakage can be taken into account as in [13] and due to space limitations we ignore them in this paper.

Now the problem is to minimize (4a) but instead of (4b), the constraint at timekis P_k ≤E_k.

The solution proposed in Section III-A can be used for this system if we takeP =E[Y₁]− where >0 can be taken arbitrarily small. Now, of course if the policy dictates that in a slot we need to clear the queue, but we do not have enough energy to transmit all the data, we use Ek (in slot k) and transmitαEkHk bits. With this policy,Ek is lower bounded by a random walk that tends to ∞ a.s. Thus, Ek → ∞ a.s.

and eventually the power policy becomes the policy of Section III-A, providing stationary mean delays as in that policy.

In the next section we develop an algorithm for a multiuser system without energy harvesting. We can extend the devel- oped algorithm to the energy harvesting case in the same way as done here for the single user case.

This algorithm can be further extended to the system where the transmitter is powered by an energy harvesting source and power grid which will be the more common scenario. But due to lack of space we exclude it from this paper.

V. MULTIUSER SYSTEM

In this section, we consider the discrete time system of Section III-A with M users (could represent the BS of a cellular system). In a given time slot, only one user can transmit. Our objective is to minimize,

lim sup

n→∞

1 nE

M

X

j=1 n−1

X

k=0

W_jq_k^j, (13a) subject to

n→∞lim 1 nE

M

X

j=1 n−1

X

k=0

P_k^j≤P . (13b) Existence of an optimal policy for this problem has been proved in [13]. However, there is no closed form solution and it is computationally very complex to obtain the optimal policy.

Thus, inspired by the single user algorithm in Section III, we propose a greedy suboptimal policy for the above multiuser problem which uses only the channel state information.

Letζjhbe the probability that we will clear the jth queue in a slot if the system state vector in that slot is h. Then if P

hζjh > 0 for each j, the system queue length process

5 5.2 5.4 5.6 5.8 6

20 40 60 80 100 120 140

Average Power

Sum of Average Queue Lengths

TOCA Policy

Opportunistic Greedy Policy EECA Policy

Fig. 2. Five users for Poisson arrivals

{qk}, qk = (q_k¹, ..., q_k^M), has a stationary distribution and if E[(A^j)^a] < ∞, a ≥ 1, then the jth queue has a⁰th finite stationary moment for a⁰ < a. In the following we assume E[A^j] < ∞, for all j. Let p^j = P

hζjhp(h). Then the jth user has stationary mean queue length E[A^j](_p¹j −1). Thus we consider the optimization problem: find ζjh that

min

M

X

j=1

W_jE[A^j]

p^j (14a)

subject to,

M

X

j=1

X

h∈H

P(j, h)ζ_jh≤P , (14b)

M

X

j=1

ζjh≤1, ∀h, ζjh≥0,∀j, ∀h, (14c)

whereP(j, h) = ^E[A_α ^j]p(h)

jp^jh_j .

In the above optimization problem, objective function and constraints (14b) are non-convex. The nonlinear constraints (14b) can be converted into linear constraints by defining new unknows Pj, j= 1, ..., M,P

jPj =P in the optimization problem. Now (14b) can be replaced by

X

h∈H

P(j, h)ζjh≤Pj, Pj ≥0, j= 1, ..., M. (15) X

j

P_j≤P . (16)

Now the constraints become linear. The objective function (14a) is sum of ratios of linear functions. This problem is non-convex and can have local minima. But it has been exten- sively studied in recent years ([11], [20]) and now efficient algorithms are available that provide a global optimal. We use these in our examples below to obtain a globally optimal solution.

We compare our results with Neely’s [16] Tradeoff Optimal Control Algorithm (TOCA) and Energy Efficient Control Algorithm (EECA). We simulated for five users. The arrival

and channel statistics of the five users are same and inde- pendent of each other. The arrival process {Ak} is Pois- son with rate 2. The channel gains take values in the set {0.01,0.25,0.64,1} and the respective channel gain probabil- ities are{0.15,0.25,0.4,0.2}. The sum average queue length vs the corresponding average powers of our opportunistic greedy policy and the TOCA and EECA policies of [16] are plotted in Fig. 2. We see that our greedy policy works much better than TOCA and EECA.

VI. RATE GUARANTEE

In this section, we consider data queues which always have packets to transmit. Our objective is to maximize the average sum rate of the users subject to an average power constraint and minimum rate guarantee to individual users. We will consider the linear rate-power function. We consider M >1.

Let the average service rate demanded by userjberj. Rest of the earlier multiuser system model remains same. Rate-power are related by (3). Thus, we want to

max lim sup

n→∞

1 n

n

X

k=1 M

X

j=1

WjE[R_k^j], (17a) subject to,

lim inf

n→∞

1 n

n

X

k=1

E[R^j_k]≥r_j, ∀j, (17b)

lim sup

n→∞

1 n

M

X

j=1

E[P_k^j]≤P . (17c) The optimal policy has the property that a user transmits only when it has its best channel. Of course, we have minimum rate requirements (17b) and want to maximize (17a). Thus, the overall policy can be given as follows. Let ζjh be the probability that user j will clear its queue when the overall channel state vector is h. Let P(j, h)be the power used by userj in state h.

To maximize (17a), a user from the set M(h) = {j:hj=h^j_L

jandWjαjh^j_L

j = max

l:hl=h^l_Ll

Wlαlhl} should be picked for transmission in channel state h. To be explicit we pick any element ofM(h)with same probability (for maximizing (17a) it doesnot matter which is picked). Let p(h^j_i)>0 for all i= 1, ..., h_Lj, for allj (drop the h^j_i from the channel states if its probability is zero). If

WjαjhL_j ≥max

l6=j Wlαlh^l₁ (18) for a user j then user j is picked with a positive probability by the above rule. Otherwise, pick user j with probability 1 only in state h= (h_l =h^l₁, l 6=j andh_j =h^j_L

j). Let A be the set of such users. Now,

p_j =X

h

ζ_jh>0 ∀j. (19) We also take P(j, h) = 0 or P(j) > 0 for an appropriate constant P(j) to be specified later. Now we have fixed ζjh

4 4.5 5 5.5 6 6.5

8 10 12 14 16 18 20 22

Average Sum Rate

Average Power

Fig. 3. Five users average sum rate vs average power consumption

for each j and h. Thus, p_j = P

hζjh is fixed and we get P(j)from the following relations

αjh^j_L

jp_jP(j) =rj, (20) for jA. Also computeP⁰(j)from (20) forjA^c. If

X

jA

P(j) +X

jA^c

P⁰(j)≤P , (21) we allocate the rest of the power P − (P

jAP(j) + P

jA^cP⁰(j))equally among the users in the set, B = {j:Wjαjh^j_L

j = max

l αlh^l_L

l},

with equal power and then appropriately increase P⁰(j) for jB toP(j)while for the usersjA−B, P(j) =P⁰(j). This reallocation should be such that P

jP(j) =P .If (21) is not satisfied then we do not have sufficient power to satisfy the QoS of all the users.

There are multiple optimal policies. The policy provided has the additional property that its peak power is low among such optimal policies.

We computed this policy for five users. We take weighting factor vector w = (W1, W2, W3, W4, W5) = (1,2,4,3,2), alpha vector α = (α1, α2, α3, α4, α5) = (2,1,0.5,1,2), and the best channel gains for the five users as (1,0.5,2,1,1.5) with respective probabilities of getting best channels as (0.2,0.15,0.5,0.35,0.4)The respective average rate guarantee vector r = (r1, r2, r3, r4, r5) is (1.5,0.5,1.25,2,4). For our policy we only need best channel statistics of all the users.

We plot the average sum rate vs average power consumption in Fig. 3. We observe that the average sum rate increases with the average power if the average power is greater than minimum average power required to meet all users’ respective rate guarantee constraints.

VII. MINIMIZING MEAN DELAYS AND RATE GUARANTEES

In this section we combine the real time scenario of Section V and data user scenario of Section VI. LetUrbe the set of real time users and Ud be the set of data users. Each user j Ud has minimum rate requirement of rj and we want to minimize

lim sup

n→∞

1 n

X

jU_r n

X

k=1

E[W_jq^j_k],

subject to average power constraint, lim sup

n→∞

1 n

n

X

k=1

X

jU_d∪Ur

E[P_k^j]≤P .

We will limit ourselves to the policies considered so far where the channel and power allocation in a slot is done based on the channel state information only. The notation remains as above and we retain the iidframe work.

We first consider the data users. We use the policy of Section VI except that now we will only ensure the minimum rate to each data user. Thus once ζ_jh, jU_d is fixed as in Section VI, we only allocate powers P(j), jA and P⁰(j), jA^c and do not use any extra power on the data users. Let P1 = P

jUdP(j). We assume P1 ≤P; otherwise the QoS of data users themselves cannot be met. Let Pˆ = P −P1. The rest of the powerPˆ will be used on the real time users of Section V (allocating power to data users first allows us to use as much power on the real time users as possible). We make one more change to the allocation policy of Section VI.

In Section VI we tried to increase ζ_jh in order to reduce the peak transmit power. This of course would not improve the performance of the system (and average power consumed to satisfy their QoS) as stated in the problem. Now of course we should reduce ζ_jh for jU_d as much as we can (in principle we can bring it as close to zero as we wish). This increases the peak power used on data users but provides more time resource to the real time users. For real time users it can be used to reduce their mean waiting time. Thus if we ensure that P

jU_dζjh ≤ δ ≤ 1 for all h (where δ > 0 and in fact we should make sure that we have equality in this condition for each h) then we solve (14a) withP in (14b) replaced by Pˆ and1in (14c) replaced by 1−δ.

VIII. CONCLUSIONS

In this paper, we have presented near-optimal policies for a single user with a fading channel which minimizes average queue length under average power constraint. Later, we have also extended these results to the multiuser case. We have also shown that these policies can be used for a system with an energy harvesting source. Next we have developed an efficient algorithm to guarantee minimum rates to data users. Finally, we combine the above algorithms to obtain efficient algorithms that guarantee minimum rates to data users and minimum average delay for delay sensitive users.

Our policies have very low computational complexity, are shown to perform better than other low complexity algorithms and do not need queue length information and arrival statistics except arrival rate. In future we would like to extend our algorithms to the more realistic case of finite buffer queues [21].

REFERENCES

[1] S. Asmussen, “Applied probability and queues”, 2nd ed, Springer-Verlag, 2003.

[2] F. Berggren and R. Jantti, “Asymptotically Fair Transmission Scheduling over Fading Channels”, IEEE Transactions on Wireless Communica- tions, Vol. 3, pp. 326-336, No. 1, Jan 2004.

[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, May 2002.

[4] Dimitri P. Bertsekas, “Dynamic Programming and Optimal Control”, Fourth Edition, Vol. 2, 2012.

[5] W. Chen, U. Mitra and J.Neely, “Energy-efficient scheduling with individual delay constraint over a fading channel”, WiOpt, pp. 1-10, Apr 2007. First edition, 1996.

[6] A. Fu, E. Modiano and J. Tsitsiklis, “Optimal Energy Allocation for Delay-Constrained Data Transmission over a Time-Varying channel”, Proc. IEEE INFOCOM, Vol. 2, pp. 1095-1105, Mar 2003.

[7] A. Fu, E. Modiano and J. Tsitsiklis, “Optimal Transmission Scheduling over a Fading Channel with Energy and Deadline Constraints”,IEEE Transactions on Wireless Communications, Vol. 5, No. 3, pp. 630-641, March 2006.

[8] A. J. Goldsmith, P. Varaiya, “Capacity of Fading Channels with Channel Side Information”,IEEE Transaction on Information Theory, Vol. 43, No. 6, pp. 1986-1992, Nov 1997.

[9] M. Goyal, A. Kumar and V. Sharma, “Power Constrained and Delay optimal policies for Scheduling Transmission over a Fading Channel,”

IEEE Infocom, Vol. 1, pp. 311-320, Apr 2003.

[10] A. Gut, Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer, 2009.

[11] H. Jiao, Q. Feng, P. Shen and Y. Guo, “Global Optimization for Sum of Linear Ratios Problem Using New Pruning Technique”,Mathematical Problems in Engineering, Article ID 646205, Vol. 2008.

[12] V. Joseph, V. Sharma, U. Mukerji and M. Kashyap, ”Joint Power Control, Scheduling and Routing for Multihop Energy Harvesting Sensor Networks“, ICUMT, Russia, 2009.

[13] A. Lalitha, S. Mondal, Satya Kumar. V and Vinod Sharma, “Power- Optimal Scheduling for a Green Base Station with Delay Constraints”, in Nineteenth Annual National Conference on Communications (NCC), India, pp. 1-5, 2013.

[14] J. Lorincz, T. Garma and G. Petrovic, “Measurements and Modelling of Base Station Power Consumption under Real Traffic Loads”,MDPI AG Sensor, Vol.12, pp. 4281-4310, Apr 2012.

[15] M. J. Neely, “Intelligent Packet Dropping for Optimal Energy- Delay Tradeoffs in Wireless Downlinks”, IEEE Transactions on Automatic Control, Vol. 54, No. 3, pp. 565-579, Mar 2009.

[16] M. J. Neely, “Optimal Energy and Delay Tradeoffs for Multiuser Wireless Downlinks”,IEEE Transactions on Information Theory, Vol.

53, No. 9, pp. 3095-3113, Sep 2007.

[17] N. Salodkar, A. Bhorkar, A. Karandikar and V. S. Borkar, “An On-Line Learning Algorithm for Energy Efficient Delay Constrained Scheduling over a Fading Channel”,IEEE Journal on Selected Areas in Communi- cations, Vol. 26, No. 4, pp. 732-742 May 2008.

[18] Satya Kumar V, A. Lalitha, V. Sharma, “Power and Delay Optimal Poli- cies for Wireless Systems”,National Conference on Communications (NCC), India, pp. 1-4, Feb 2012.

[19] Satya Kumar V, V. Sharma, “Low Complexity Power and Scheduling Policies for Wireless Networks”, Information Theory and Applications Workshop (ITA), Sandiego, 2014.

[20] S. Schaible and J. Shi, “Fractional Programming: the sum-of-ratios case”,AMS Classification, Vol. 18, No. 2, Oct 2003.

[21] V. Sharma and J. T. Virtams, ”A Finite BUffer Queue with Priorities Performance Evaluation“, Vol. 47, pp. 1-22, 2002.

[22] V. Sharma, ”Some Limit Theorems for Regenerative Queues, Queueing Systems“, Vol. 30, pp. 341-363, 1998.

[23] V. Sharma, Utpal Mukherji, Vinay Joseph and Shrey Gupta, “Optimal Energy Management Policies for Energy Harvesting Sensor Nodes”, IEEE Transactions on Wireless Communications, Vol.9, No.4, pp. 1326- 1336, Apr 2010.

[24] B. S. Vineeth and Utpal Mukherji, “Tradeoff of Average Power and Av- erage Delay for Point-to-Point Link with Fading”,National Conference on Communications (NCC), India, pp. 1-5, Feb 2013.

[25] H. Wang and N. Mandayam, “A Simple Packet transmission Scheme for Wireless Data over Fading Channels”,IEEE Transactions on Commu- nications, Vol.52, No.7, pp. 1055-1059, Jul 2004.