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Computation Offloading Strategies based on Energy Minimization under Computational Rate Constraints Sergio Barbarossa, Paolo Di Lorenzo, and Stefania Sardellitti Department of Information Eng., Electronics, and Telecommunications “Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy. e-mail: {sergio,dilorenzo}@infocom.uniroma1.it, [email protected] Abstract—The aim of this paper is to propose a computation offloading strategy to be used in mobile cloud computing in order to minimize the energy expenditure at the mobile handset, while guaranteeing a computational rate constraint imposed by the application. We start from a single mobile handset assisted by a single server and then we generalize the approach to the multiuser case. In the multiuser scenario, we derive the optimal transmit power, number of bits per symbol and CPU cycles assigned to each virtual machine in order to minimize the energy consumption at each mobile user, still satisfying a computational rate constraint dictated by the application requirements. We take explicitly into account the channel random fading and provide theoretical results proving the existence of an optimal solution of the problem. Then we corroborate our theoretical findings with simulation results, showing for which classes of application and under what channel conditions, computation offloading can provide a significant performance gain. Keywords—Mobile cloud computing, radio resource allocation, virtual machines.

I. I NTRODUCTION Computation offloading has attracted a lot of research efforts as a way to augment the capabilities of resource-constrained and energy-hungry mobile handsets by migrating computations to more resourceful servers. Offloading is useful either to enable smartphones to run more and more sophisticated applications, while meeting strict delay constraints, or to prolong the battery lifetime by limiting the energy spent at the mobile handset to run a given application, still under applicationdependent delay constraints. The idea of offloading computations is not new. A recent broad survey on computation offloading for mobile systems is [1]. A great impulse to computation offloading has come through the introduction of cloud computing. One of the key features of cloud computing is virtualization, i.e. the capability to run multiple operating systems and multiple applications over the same machine (or set of machines), while guaranteeing isolation and protection of the programs and their data. Through virtualization, the number of virtual machines can scale on demand, thus improving the system computational efficiency. Providing wireless access to the cloud has enormous potentials to augment the capabilities of mobile handsets, through what is known as mobile cloud computing [2]-[7]. One of the limitations of mobile cloud computing is the latency experienced in a wide This work has been supported by TROPIC Project, Nr. 318784.

area network (WAN). As pointed out in [8], humans are acutely sensitive to delay and jitter, and it is very difficult to control these parameters at WAN scale: As latency increases, interactive response suffers. To overcome this limitation, the authors of [8] introduced the concept of cloudlet, i.e. the possibility for the mobile handsets to access nearby static resourceful computers, linked to a distant cloud through high speed wired connections. In [8], cloudlets are imagined to be deployed much like Wi-Fi access points today. This would certainly improve accessibility with low latencies. The remaining problem would be the coverage offered by these proximity servers and the quality of service offered over such radio links. Within the current European Project named TROPIC, it has been recently proposed an alternative approach merging the small cell networks and cloud computing frameworks [20] into the new concept of femto-cloud computing. The idea is to endow small cell base stations, also known as Small Cell e-Node B (SCeNB), with enhanced cloud functionalities to enable mobile users to proximity access to cloud services. The empowered base stations are called Small Cell cloudenhanced e-Node B (SCceNB). Exploiting the deployment of LTE SCceNB’s, the femto-cloud architecture offers proximity access with easier vertical hand-over and better quality of service than Wi-Fi access. Computation offloading has been studied in several works, as e.g., [9]-[18]. The offloading decisions are usually made by analyzing parameters including bandwidths, server speeds, available memory, server loads, and the amounts of data exchanged between servers and mobile systems. The solutions include code partitioning programs [7], [9], [11], [13], [17], [14], [16], and predicting parametric variations in application behavior and execution environment [18]. Offloading typically requires code partitioning, so that part of the code runs locally, whereas the remaining part may be offloaded to a fixed server based on contextual parameters, such as battery level, delay constraints, channel state, and so on. Program partitioning may be static or dynamic. Static partition has the advantage of requiring a low overhead during execution, but it works well only if the parameters related to the offloading decisions are accurately known in advance or predicted, see, e.g. [9], [15]. In contrast, enforcing a dynamic decision mechanism makes possible to adapt to different operating conditions [16][17]. Given the variability of the wireless channel, dynamic partitioning seems more appropriate, but it has an associated higher signaling overhead, which must be taken under control.

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Our goal in this paper is to propose a computation offloading strategy operating jointly over the communication and computation resources. The proposed optimization criterion is the minimization of the energy consumption at the mobile handset under computational rate constraints. The constraints take explicitly into account the application requirements as well as the time and energy wasted to transfer the program execution from the mobile handset to the cloud. We start from the single user case and then we extend the approach to the multiuser case. In both cases, we provide theoretical results proving the existence of an optimal solution of the resource allocation problem. II.

O FFLOADING S TRATEGY FOR THE S INGLE U SER

The proposed optimization criterion is the minimization of the energy spent at the mobile device, under the following constraints: a) power budget constraint; b) delay constraint; c) computational stability constraint. The third constraint takes into account that the net computing rate must include the time wasted to transfer the program execution from the mobile device to the server. The final net rate must exceed the rate required to run the application within the latency constraints needed to meet the user quality of experience. We consider a time slotted system with the following parameters: floc is the local computational rate (instructions/sec); fS is the server computational rate (instruction/sec); T is the duration of the time slot (it could also represent a latency constraint); B is the bandwidth allocated to transmission from MUE to SCceNB; pproc is the power spent for local processing; N is the dimension of the program state; λ is the computational rate required by the application. In principle, the degrees of freedom are: a) d ∈ {0, 1} is the decision variable, which is equal to 0 if the program is executed locally, or 1 otherwise; b) p ∈ [0, PT ] is the power spent for transmitting the program state from mobile to server; c) τ ∈ [0, T ] is the duration of the time interval used to transfer the program execution from the mobile to the server. However, these degrees of freedom can be reduced to one by exploiting their inter-relationships. First of all, the transmission delay τ is related to the transmit power p through τ=

N B log (1 + αp)

(1)

and, equivalently " 1 ! N/Bτ p= 2 −1 . α

(2)

d = u (pproc T − pτ )

(3)

where α := |h|2 /(Γ(BER)N0 ) is the (equivalent) channel coefficient, having denoted by h the channel coefficient, whereas Γ(BER) is the SNR margin necessary to meet a desired target BER, and N0 is the noise power. Furthermore, since offloading occurs only if the energy spent locally is greater than the energy required for offloading, the decision variable d is related to p and τ through

where u(·) is the unit step function. Since the transmit power is upper bounded by a budget constraint PT , the duration τ is consequently bounded as follows: N ≤ τ ≤ T. B log (1 + αPT )

(4)

Whenever the computation is offloaded, an inevitable waste of time occurs to transfer the program status from the mobile device to the server. This “communication” time translates into a computation rate loss: If within an overall time interval T , τ seconds are dedicated # to data $ transfer, the real computing rate at the server is fS 1 − Tτ . The computational rate constraint can then be formulated as follows: & % ! τ" − floc u (pproc T − pτ ) ≥ λ (5) fR := floc + fS 1 − T Now, exploiting the relations among the free variables, we can express the objective function in terms of the single variable τ , as % "& τ ! N/Bτ Es := min (pproc T, 2 −1 ! τ !α " " 2N/Bτ − 1 − pproc T := pproc T + α "" ! τ ! N/Bτ 2 −1 . (6) · u pproc T − α The optimization problem can then be formulated as follows: % "& τ ! N/Bτ min min (pproc T, 2 −1 [P.1] (7) τ α subject to N ≤τ ≤T B log (1 + αPT ) & % ! τ" C.2 floc + fS 1 − − floc u (pproc T − pτ ) ≥ λ. T After some algebraic manipulations, it is possible to show that this problem is feasible if fS ≥ λ and the (equivalent) channel coefficient exceeds a minimum value, i.e., C.1

α > αmin where

'

( λ " N 1 (1 − fS ) ! BT (1−λ/f S) − 1 αmin = max , 2 . PT pproc

(8)

(9)

This last condition states, in closed form, that offloading takes place only if the channel is sufficiently good, as expected. The interesting point is that the minimum channel value is dictated by parameters related to both radio and computational parameters. If the feasible set is non-empty, the optimal solution of problem [P.1] is given in closed form by: " N 1 ! BT (1−λ/f S) − 1 p= . (10) 2 α In practice, since the wireless channel is random, there is a nonnull probability that offloading takes place or not, depending on both channel status and computational requests. The consequence is that the real computing rate experienced

3

(11)

where DA (α) denotes the cumulative distribution function of α and αmin is given in (9), with λ∗ instead of λ. The expected value of the rate is then

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III. O FFLOADING S TRATEGY IN THE M ULTIUSER C ASE In the multiuser case, the optimization can be cast as the minimization of the sum of the energies spent by the mobile handsets to run their applications, under individual stability constraints and a maximum server capacity, as follows: * K "+ ) τk ! Nk /Bτk min 2 min (pkproc T, −1 [P.2] (13) αk τ ,f k=1

subject to

Nk ≤ τk ≤ T, k = 1, . . . , K B log (1 + αk PT ) & % ! τk " k k C.4 floc − floc + fk 1 − T , "τk ! Nk /Bτk k · u pproc T − −1 ≥ λk (14) 2 αk k = 1, . . . , K K ) fk ≤ fS . C.5 k=1

2x2 MIMO

(12)

where we made explicit the dependence of P from λ (through αmin ). Imposing this average rate to be equal to the target rate λ, we end up with a nonlinear equation in λ∗ . This equation admits a unique solution. As a numerical example, Fig. 1 (top) shows λ∗ as a function of the distance between mobile user and base station, for different antenna configurations. The simulation parameters are: fS = 1010 , floc = 107 , pproc = 0.1, PT = 0.1, N = 5 × 103 , B = 106 , T = 10−2 , λ = 108 . As expected, at short distances, λ∗ tends to coincide with λ, because the channel attenuation is negligible and offloading occurs most of the times. However, as the distance exceeds a certain value, λ∗ starts increasing considerably to compensate for the missing opportunities to offload. In parallel, Fig. 1 (bottom) shows the average energy spent for offloading as a function of the distance between mobile user and base station, for different antenna configurations. As expected, the energy increases at larger distance until reaching a constant value dictated by the energy required to run the application locally. It is interesting to see how MIMO transceivers yield a larger saving and then, ultimately, widen the area over which offloading is beneficial.

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P := Prob{α > αmin } = 1 − DA (αmin )

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over a fading channel is going to depend on the channel statistics and typically will be lower than the rate achievable under ideal channel conditions. However, building on previous expressions, we can derive the equivalent computing rate λ∗ , to be asked by the mobile device in order to ensure the desired rate λ, in the presence of fading, provided that the fading statistics are known. More specifically, the probability of offloading is simply

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Fig. 1. (Top) Equivalent rate λ∗ and (bottom) Average Energy spent for offloading, vs. distance between mobile user and access point, for different communication strategies.

where τ = [τ1 , . . . , τK ] and f = [f1 , . . . , fK ]. Let us first check the problem feasibility, assuming fS > λk ∀ k. Denoting with τk∗ = τ ∗ (αk ) the unique solution of the following equation associated to user k " ∗ τ∗ ! pkproc T = k 2Nk /(Bτk ) − 1 , (15) αk user k is enabled to offload its computations if the following necessary conditions are both verified F.3 F.4

Nk ≤T B log (1 + αk PT ) ∗ τ (αk ) ≤ T (1 − λk /fS ).

(16) (17)

After some algebra, conditions F.3 and F.4 can be reduced to the following necessary single condition on the channel coefficients: F.5

αk > αkm

(18)

4

with

, -⎤ Nk " Nk λk BT (1−λk /fS ) ) 2 − 1 (1 − fS ⎢ 2 BT − 1 ⎥ ⎥. , αkm = max ⎢ ⎣ ⎦ k PT pproc ⎡!

(19)

In case condition F.5 is not verified, user k performs its computations locally. Let us denote with S the set of users satisfying F.5 and define |S| as the cardinality of S. From condition C.4, in case of offloading we get fk ≥ fk,min with λk + . Then, to ensure that fk,min = * Nk 1− BT log (1 + αk PT ) the feasibility set is non-empty, the further global necessary condition must be verified by all users in S: ) F.6 fk,min < fS . (20)

Algorithm 1 : Multi-user minimum energy offloading strategy 1. Find F , i.e. the collection of all the subsets S¯i satisfying F.5, F.6; 2. If F = ∅ all the users run computations locally; else for each subset S¯i : a) compute the solution (τ ∗S¯i , f ∗S¯i ) of problem [P.4]; b) evaluate the optimal energy spent as " ) ) τ∗ ! ∗ k k 2Nk /(Bτk ) − 1 ; Es (S¯i , τ ∗S¯i , f ∗S¯i ) = Eloc + αk ¯ ¯ k̸∈Si

3.

k∈Si

Find the optimal subset S¯∗ and the optimal pair (τ ∗S¯i , f ∗S¯i ) such that Es (S¯∗ , τ ∗S¯∗ , f ∗S¯∗ ) = arg min Es (S¯i ) {S¯i }L i=1

k∈S

Hence, if conditions F.5 and F.6 are verified for a given subset S¯ ⊆ S, then the feasible set in (14) is non-empty and problem [P.2] can be formulated as " 4 4 τk ! Nk /(Bτk ) k minτ ¯,f ¯ 2 −1 [P.3] ¯ Eloc + ¯ k̸ ∈ S k∈ S S S αk s.t.

Nk ≤ τk ≤ T, ∀ k ∈ S¯ B log (1 + αk PT ) ! τk " fk 1 − ≥ λk , ∀ k ∈ S¯ T

k ≥ λk , ∀ k ̸∈ S¯ floc 4 k∈S¯ fk ≤ fS ¯ |S|

k where τ S¯, f S¯ ∈ R+ and Eloc = pkproc T . Let us now denote by F the collection of all, let us say L, subsets S¯i of S satisfying conditions F.5 and F.6, for i = 1, . . . , L, with L ≤ 2|S| − 1. Our goal is to find the optimal∗ users subset S¯∗ ⊆ F and the |S¯ | optimal vectors τ ∗S¯, f ∗S¯ ∈ R+ which solve problem [P.2]. To this end, we perform a search on all the possible subsets of F by solving for each of them the optimization problem k [P.3]. Assuming that the inequalities floc ≥ λk are verified ¯ ∀ k ̸∈ S, [P.3] can be rewritten for each S¯ ⊆ F as " ) τk ! minτ ¯,f ¯ 2Nk /(Bτk ) − 1 [P.4] S S α ¯ k k∈S

s.t. τk,min ≤ τk ≤ T, ∀ k ∈ S¯ ! τk " fk 1 − ≥ λk , ∀ k ∈ S¯ T 4 k∈S¯ fk ≤ fS

(21)

Nk . It is easy to see that B log (1 + αk PT ) problem [P.4] is convex. First, the objective function is sum with τk,min =

of convex functions of τ S¯. Then, the second constraint is a log-concave function of (τk , fk ) for each k,#thus guaranteing $ all its superlevel sets {(fk , τk ) ∈ R2+ | fk 1 − τTk ≥ λk } are convex. Thus, it follows that the set in (21) is feasible, when conditions F.5 and F.6 hold true, and it is also convex as non-empty intersections of convex sets. In summary, the multiuser offloading strategy is reported in Algorithm 1. Numerical Example - Optimal Allocation of communication and computation resources : Some numerical examples are useful to assess the performance of the proposed algorithm. In our numerical simulations we set pkproc = 10 mW, ∀k, T = 20 msec, fS = 107 cycles/s, PT = 10 mWatt, B = 200 MHz. Fig. 2 shows the effect of the stability constraint on the optimization strategy. It can be noted that, when λk is large, the optimal computational rate fk∗ tends to be large as well, while the optimal transmission delay τk∗ decreases. This behavior can be explained by observing that the stability constraint is always active. Moreover note that user 6 is excluded from computation offloading since its channel coefficient does not meet the conditions F.5, F.6. Finally Fig. 3 (a) shows the ∗ optimal offloading energy per user Eof f,k , associated to the optimal variables shown in Fig. 2. The corresponding channel coefficient ak and program states Nk are reported in subplots 3 (b) and (c), respectively. From Fig. 3, we can observe the joint effect on the optimal energy consumption of both channel conditions and class of running application. In particular note that user 5 entails the maximum energy consumption since it must transmit a large program state, with a stringent stability constraint, over a wireless channel inducing a strong fading. IV.

C ONCLUSIONS

In this paper, we have proposed a computation offloading framework for minimizing the energy consumption at the mobile handset, incorporating a computational stability constraint dictated by the application. The approach has been also extended to the multiuser case, where a single SCceNB is serving multiple mobile handsets. The proposed strategy

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Fig. 3. a) Optimal per user energy consumption corresponding to the channel profile in b) and to the program states in c).

[14] [15]

solves, jointly, an admission control (evaluating the set of users admitted to offloading), and a combined radio and computing resource allocation for each mobile user. The resource allocation is dynamic, depending on channel conditions. If some users are not admitted, the computation resources are distributed among the admitted ones. Simulation results show how MIMO transceivers contribute to augmenting the computation capabilities of each user. Furthermore, we have also shown how the application features, in terms of size of program state and latency constraints, play a key role in the offloading decisions. In summary, an effective offloading mechanism can be setup only by taking into account, jointly, communication and computation aspects, together with user quality of experience. R EFERENCES [1]

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