Exploiting Computation Replication for Mobile Edge Computing: A Fundamental Computation-Communication Tradeoff Study

Authors

• 2 publications
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I Introduction

The explosive growth of Internet of Things is driving the emergence of new mobile applications that demand intensive computation and stringent latency, such as intelligent navigation, online gaming, virtual reality (VR), and augmented reality (AR)[3]. The limited computation and energy resources at mobile devices pose a great challenge for supporting these new applications[4]. Mobile edge computing (MEC) is envisioned as a promising network architecture to address this challenge by providing cloud-computing services at the edge nodes (ENs) of mobile networks, such as wireless access points and base stations[5, 6]. By offloading computation-intensive tasks from mobile users to their nearby server-enabled ENs for processing, MEC systems have great potential to prolong the battery lifetime of mobile devices and reduce the overall task execution latency.

analysis lies at the degree of freedom (DoF) analysis of the so-called circular cooperative interference-multicast channels. We obtain the optimal per-receiver DoF for the uplink channel, and an order-optimal per-receiver DoF for the downlink channel. Based on these DoF regions, we then develop an order-optimal achievable communication latency pair at any integer computation load. In particular, the NULT is exactly optimal and the NDLT is within a multiplicative gap of 2 to the optimum. We show that the NDLT is an

inversely proportional function of the computation load in the interval , which presents the computation-communication tradeoff. We also reveal that the decrease of NDLT is at the expense of increasing the NULT linearly, which forms another NULT-NDLT tradeoff. Part of this result is submitted to IEEE ISIT 2019[2].

The rest of this paper is organized as follows. Section II presents the problem formulation and definitions. The computation-communication tradeoffs are presented in Section III for binary offloading and Section IV for partial offloading. The conclusions are drawn in Section V.

Notations: denotes the set of complex numbers. denotes the set of positive integers. denotes the transpose. denotes the set of indexes . denotes the largest integer no greater than while denotes the minimum integer no smaller than . denotes the cardinality of set . denotes the set of integers . denotes the set of integers .

denotes the vector

. denotes the set .

Ii Problem Formulation

Ii-a MEC Network Model

We consider an MEC network consisting of single-antenna ENs and single-antenna users, as shown in Fig. 1. Each EN is equipped with a computing server and they all communicate with all users via a shared wireless channel. Denote by the set of ENs and the set of users. The communication link between each EN and each user experiences both channel fading and an additive white Gaussian noise. Let denote the uplink (downlink) channel fading from user (EN ) to EN (user ). It is assumed to be independent and identically distributed (i.i.d.) as some continuous distribution.

Definition 1.

For a given task assignment scheme , the computation load , , is defined as the total number of task input bits computed at all the ENs, normalized by the total number of task input bits from all the users, i.e.,

 r≜∑i∈M∑j∈N∑Φ:i∈Φ|Wj,Φ|NL. (1)

Similar to [27], the computation load can be interpreted as the average number of ENs to compute each task (for binary offloading) or each input bit (for partial offloading) and hence is a measure of computation repetition.

Each user employs an encoding function to map its task inputs and channel coefficients to a length- codeword , where is the transmitted symbol at time . Each codeword has an average power constraint of , i.e., . Then, the received signal of each EN at time is given by

 Yi(t)=∑j∈Nhij(t)Xj(t)+Zi(t),    ∀i∈M, (2)

where is the noise at EN . Each EN uses a decoding function to map received signals and channel coefficients

to the estimate

. The error probability is given by

 Pue=maxi∈M P⎛⎝⋃j∈N,Φ⊇{i}{^Wj,Φ≠Wj,Φ}⎞⎠. (3)

 Pde=maxj∈N P(^˜Wj≠˜Wj). (4)

Ii-C Performance Metric

Definition 2.

 τu(r) ≜limPu→∞limL→∞EH[Tu]L/logPu, (5) τd(r) ≜limPd→∞lim˜L→∞EH[Td]˜L/logPd. (6)

Further, the minimum NULT and NDLT are defined, respectively, as

 τu∗(r) ≜inf{τu(r):∀τu(r) is % achievable at the computation load r}, (7) τd∗(r) ≜inf{τd(r):∀τd(r) is % achievable at the computation load r}. (8)

Note that (or ) is the reference time to transmit the input (or output) data of (or ) bits for one task in a Gaussian point-to-point baseline system in the high SNR regime. Thus, an NULT (or NDLT) of or indicates that the time required to upload (or download) the tasks of all users is or times of this reference time period.

Definition 3.

A communication latency pair at a computation load is said to be achievable if there exists a feasible task offloading policy . The optimal communication latency region is the closure of the set of all achievable communication latency pairs at all possible computation load ’s, i.e.,

 T≜closure{(τu(r),τd(r)):∀(τu(r),τd(r)) is achievable, ∀r∈[1,M]}. (9)

In this section, we present the analysis of the optimal communication latency pair at any given computation load, including both achievable scheme and converse, for binary offloading.

Iii-a Main Results

Theorem 1.

(Achievable result). An achievable communication latency pair at an integer computation load , for binary task offloading in the MEC network with ENs and users, is given by

 τua(r) =min{1+NrM,N}, (10) τda(r) =min{1+NM,Nr}, (11)

when . If is not an integer, one can always find two integers and so that is the closest integer to and the above results still hold by adding more users and deactivating ENs.

We prove the achievability of Theorem 1 in Section III-B.

Theorem 2.

(Converse). The optimal communication latency pair at any given computation load , for binary task offloading in the MEC network with ENs and users, is lower bounded by

 τu∗(r) ≥min{1+NrM,N}, (12) τd∗(r) ≥Nmin{M,N}. (13)

Based on Theorem 1 and Theorem 2, we can obtain an inner bound denoted as and an outer bound denoted as , respectively, of the optimal communication latency region by collecting the latency pairs at all the considered computation loads ’s. Fig. 2 shows the bounds in the MEC networks with .

Corollary 1.

(Optimality). At an integer computation load , the achievable NULT in (10) is optimal, and the achievable NDLT in (11) is within a multiplicative gap of to its minimum.

The proof for Theorem 2 and Corollary 1 is given in Section III-C.

Now, we demonstrate how the computation load affects the achievable communication latency . By discussing the function terms in (10) and (11), we have the monotonicity of the achievable computation-communication function :

• The NULT increases strictly with the computation load for , and then keeps a constant for .

• The NDLT keeps a constant for , and then is inversely proportional to the computation load for .

Remark 1.

The achievable computation-communication function has two corner points and , corresponding to and , respectively. They are explained as follows:

• For input data uploading, before increases to , the NULT is increasing since more traffic is introduced in the uplink. When grows to more than , there is no need to increase the NULT since all tasks can be uploaded within time slots by using TDMA.

• For output data downloading, before increases to , the potential transmission cooperation gain brought by computation replication cannot exceed the existing interference alignment gain without computation replication and thus the NDLT keeps fixed. When grows to more than , interference neutralization can be exploited which outperforms interference alignment, and thus the NDLT begins to decrease with .

It can be easily proved that for all . Hence, we have the following remark to characterize the envelope of the inner bound of the optimal communication latency region, present the tradeoff between computation load and communication latency, and illustrate the interaction between the NULT and NDLT.

Remark 2.

The envelope of the inner bound of the optimal communication latency region for binary offloading can be divided into three sections, each corresponding to a distinct interval of the computation load :

1. Constant-NDLT section: , , when ;

2. NULT-NDLT tradeoff section: , , when ;

3. Constant-NULT section: , , when .

In particular, in the NULT-NDLT tradeoff section, as the computation load increases, the NDLT decreases in an inversely proportional way, at the expense of increasing the NULT linearly.

It is seen from Fig. 2(b) that the envelope of the inner bound is composed of three sections corresponding to three different intervals of the computation load, and the middle section at (dotted line) presents the NULT-NDLT tradeoff, in an inversely proportional form.

Consider that the system parameters and satisfy such that holds for , where is the given integer computation load and is an integer in . In the proposed task assignment method, we let each task be executed at exactly different ENs and let each EN execute distinct tasks with even load. Note that if is not an integer, we can inject () tasks and let () ENs being idle and use the remaining ENs for task offloading, such that is the integer closest to , denoted as . In this way, we still have for , and can use the new and to replace and to obtain the corresponding analytical results.

To ensure even task assignment on each EN, we perform circular assignment. Specifically, the set of tasks assigned to EN is given by

 Ti={Wj+1:j∈[(i−1)n:(in−1)](modN)}. (14)

Given the above task assignment in (14), the uplink channel formed by uploading the tasks to their corresponding ENs is referred to as the circular interference-multicast channel with multicast group size . This channel is different from the X-multicast channel with multicast group size defined in [30, 31], where any subset of receivers can form a multicast group, resulting in multicast groups, and each transmitter needs to communicate with all the multicast groups. In our considered circular interference-multicast channel, there are only multicast groups which are performed circularly by the receivers and each transmitter only needs to communicate with one multicast group. The optimal per-receiver DoF of this uplink channel is given as follows.

Lemma 1.

The optimal per-receiver DoF of the circular interference-multicast channel with transmitters and receivers satisfying and multicast group size is given by

 DoFur=max{NrNr+M,rM}, r∈[M]. (15)
Proof.

First, we use partial interference alignment scheme to achieve a DoF of for each receiver, which is similar to the achievability proof of the DoF of -user interference channels in [23]. Then, we compare it to the DoF of achieved by TDMA. The detailed achievable scheme and proof of optimality are given in Appendix -A. ∎

The per-receiver rate of this channel in the high SNR regime can be approximated as . The traffic load for each EN to receive its assigned tasks is bits, then the uploading time can be approximately given by . Let and , by Definition 2, the NULT for each EN at computation load can be given by

 τua(r)=NrMDoFur=min{NrM+1,N},  r∈[M]. (16)

Lemma 2.

An achievable per-receiver DoF of the circular cooperative interference channel with transmitters and receivers satisfying and transmitter cooperation group size is given by

 DoFdr=max{MN+M,rN},  r∈[M], (17)

and it is within a multiplicative gap of 2 to the optimal DoF.

Proof.

When , we use partial interference alignment scheme to achieve a DoF of for each receiver. The achievable scheme is similar to that for the user X channel [34]. We then compare it to a DoF of achieved via TDMA. When , we prove that the achievable per-receiver DoF is , where we first use interference neutralization to achieve a DoF of for each receiver, and then compare it with the per-receiver DoF of achieved by only using interference alignment. Summarizing these two cases, we have (17). Please refer to Appendix -B for the detailed achievable scheme and optimality proof. ∎

The per-receiver channel rate in the high SNR regime can be approximated as . The traffic load for each user to download its task output data is bits, then the downloading time can be approximately given by . Let and , by Definition 2, the NDLT for each user at computation load is given by

 τda(r)=1DoFdr=min{NM+1,Nr},  r∈[M]. (18)

By (16) and (18), we thus have the achievable communication latency pair at an integer computation load for binary offloading.

Iii-C Proof of Converse

Iii-C1 Lower bound and optimality of NULT

We prove the lower bound of the NULT at any given computation load , i.e., . First, we use genie-aided arguments to derive a lower bound on the NULT of any given feasible task assignment policy with computation load . Then, we optimize the lower bound over all feasible task assignment policies to obtain the minimum NULT for a given computation load .

Given a computation load . Consider an arbitrary task assignment policy where the number of tasks assigned to each EN is denoted as , , and satisfies

 ∑i∈Mai=Nr, (19) ai∈[0:N],  i∈M. (20)

Note that we only need consider case since means no task is assigned to EN and we can remove EN from the EN set , which will not change the results. Consider the following three disjoint subsets of task input data (or message):

 Wr ={Wj,Sj:j∈N,i∈Sj}, (21) Wt ={Wj,Sj:j=to,i∉Sj}, (22) ¯¯¯¯¯¯W ={Wj,Sj:j≠to and i∉Sj}, (23)

where denotes the input message of task that is assigned to all ENs in subset , and denotes one of the users that do not offload their tasks to EN , i.e., . It is seen that the set indicates the messages that EN need decode, i.e., ; The set is a nonempty set with cardinality when EN is not assigned all tasks (or ), since user exists in this case; Otherwise, we have for . We will show that set has the maximum number of messages that can be decoded by EN .

Let a genie provide the messages to all ENs, and additionally provide messages to ENs in . The received signal of EN can be represented as

 ^yi =M∑j=1,≠toHijxj+Hitoxto+^zi, (24)

where , , are diagonal matrices representing the channel coefficients from user to EN , signal transmitted by user , noise received at EN , over the block length , respectively. Note that we reduce the noise at EN from to by a fixed amount such that its received signal can be replaced by . The ENs in have messages , which do not include the message of user . Using these genie-aided information, each EN can compute the transmitted signals and subtract them from the received signal. Thus, the received signal of EN can be rewritten as

 ¯yk=yk−N∑j=1,≠toHkjxj=Hktoxto+zk. (25)

Since the message is intended for some ENs in , denoted as , the ENs in can decode it. By Fano’s inequality and (25), we have

 H(Wt|yk,¯¯¯¯¯¯W,Wr)≤Tuϵ,  k∈Rt. (26)

Consider EN , it can decode messages intended for it. By Fano’s inequality, we have

 H(Wr|^yi,¯¯¯¯¯¯W)≤|Wr|Tuϵ. (27)

Using genie-aided messages and decoded messages , EN can compute the transmitted signals , and subtract them from the received signal. We thus have

 ¯yi=^yi−N∑j=1,≠toHijxj=Hitoxto+^zi. (28)

By reducing noise and multiplying the constructed signal at EN by , we have

 ¯yki=HktoH−1ito¯yi=Hktoxto+^z′k, (29)

where represents the reduced noise. It is seen that is a degraded version of at EN in , so EN must be able to decode the messages that ENs in can decode. Thus, we have

 H(Wt|^yi,¯¯¯¯¯¯W,Wr)≤H(Wt|yk,¯¯¯¯¯¯W,Wr)≤Tuϵ,  i∈Rt. (30)

All the above changes including genie-aided information, receiver cooperation, and noise reducing can only improve capacity. Therefore, we have the following chain of inequalities,

 (|Wr|+|Wt|)L =H(Wr,Wt) (31) (a)=H(Wr,Wt|¯¯¯¯¯¯W) (32) (b)=I(Wr,Wt:^yi|¯¯¯¯¯¯W)+H(Wr,Wt|^yi,¯¯¯¯¯¯W) (33) (34) (d)≤I(Wr,Wt:^yi|¯¯¯¯¯¯W)+|Wr|Tuϵ+Tuϵ (35) (e)≤I(x1,x2,⋯,xai,xto:^yi|¯W)+(|Wr|+1)Tuϵ (36) (f)≤TulogPu+(|Wr|+1)Tuϵ, (37)

where (a) is due to the independence of messages, (b) and (c) follow from the chain rule, (d) uses Fano’s inequalities (

27) and (30), (e) is the data processing inequality, and (f) uses the DoF bound of the MAC channel. By dividing on , and taking and , we have .

Thus, for any given feasible task assignment , the NULT satisfies for , i.e., the minimum NULT of the task assignment policy is lower bounded by

 τu∗(r,a)≥maxi∈Mmin{ai+1,N}=min{maxi∈Mai+1,N}. (38)

Hence, the minimum NULT of all feasible task assignment is given by

. It can be lower bounded by the optimal solution of the following linear programming problem,

 P1: mina min{maxi∈Mai+1,N} \mathnormals.t. (???),(???)

By relaxing the integer constraint into a real-value constraint , the optimal solution is still a lower bound of the minimum NULT . Since the objective is equivalent to minimizing the term , the optimal solution can be obtained easily as , . Hence, the minimum NULT is lower bounded by

 τu∗(r)≥min{NrM+1,N}. (39)

The proof of the lower bound of NULT is thus completed. Comparing (39) with (10) in Theorem 1, we see that they are the same. Thus, the achievable NULT in (10) is optimal.

Iii-C2 Lower bound and gap of NDLT

Let denote the signal transmitted by each EN , and the signal received at each user , over the block length . Consider the computed results decoded by users, we have the following chain of inequalities,

 N˜L =H(˜W1,⋯,˜WN)=I(˜W1,⋯,˜WN:y1,⋯,yN)+H(˜W1,⋯,˜WN|y1,⋯,yN) (40) (g)≤I(˜W1,⋯,˜WN:y1,⋯,yN)+∑j∈NH(˜Wj|yj) (41) (h)≤I(x1,x2,⋯,xM:y1,⋯,yN)+NTdϵ (42) (i)≤min{M,N}TdlogPd+NTdϵ, (43)

where follows from , follows from the data processing inequality and Fano’s inequality, and uses the capacity bound of the MISO broadcast channel with a -antenna transmitter and single-antenna receivers. By dividing on , and taking and , we have

 τd≥Nmin{M,N}. (44)

Hence, the minimum NDLT is lower bounded by . It can be easily proved that the multiplicative gap between the achievable NDLT in Theorem 1 and this lower bound is within for , i.e., . We complete the proof of the lower bound and gap of the NDLT for binary offloading.

In this section, we present the analysis of the optimal communication latency pair at any given computation load, including achievable scheme and converse, for partial offloading.

Iv-a Main Results

Theorem 3.

(Achievable result). An achievable communication latency pair at an integer computation load , for partial task offloading in the MEC network with ENs and users, is given by

 τua(r) =N−1Mr+1, (45) τda(r) =max{N−rM+1,1}. (46)

For general , the achievable communication latency pair is given by the lower convex envelope of the above points .