12004 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 69, NO.

10, OCTOBER 2020

Performance Analysis of a Delay Constrained Data

Offloading Scheme in an Integrated Cloud-Fog-Edge
Computing System
Romano Fantacci , Fellow, IEEE, and Benedetta Picano , Student Member, IEEE

Abstract—The recent growth in intensive services and applica-

tions demand has triggered the functional integration of cloud
computing with edge computing capabilities. One of the main
goals is to allow a fast processing to tasks with strict real time
constraints in order to lower the task dropping probability due
to expiration of the associated deadlines. This paper deals with
the performance evaluation and optimization of a three layers
cloud-fog-edge computing infrastructure by resorting to the use of
queueing theory results. In particular, a Markov queueing system
model with reneging is proposed for the cloud subsystem, in order
to consider the premature computation requests departure due to
their deadline expiration. Furthermore, a computational resources
allocation method is proposed with the aim at maximizing the social
welfare metric, constrained to specific quality of service require-
ments. Finally, the proposed queueing theory analysis as well as
of the computational resources allocation approach is validated Fig. 1. Edge-fog-cloud Network Model.
by comparing the obtained analytical predictions with simulation
results.
health care applications or recognition assistance [2]–[5]. All
Index Terms—Mobile computing, offloading, queueing theory,
performance optimization. these new challenges have triggered the tendency to migrate
towards novel solutions, typically based on the deployment of
computational nodes, characterized by a contained processing
I. INTRODUCTION ability and storage supply, to the edges of the network, in order
URING last decades, the cloud computing [1] architecture to reduce the network response latency [6]. The novel network
D has held an undisputed dominant role in the network
computing paradigms scenario, providing massive processing
paradigms based on this approach are named edge computing
(EC) [7] and fog computing (FC), and are able to provide low
and storage capacity to the users. However, over the years, latency response and service continuity to mobile users.
the large scale diffusion of devices always getting smarter, However, in comparison to the existing public cloud based
able to exchange data information to each other and with the solutions, e.g., Microsoft Azur and Amazon AWS, the process-
surrounding environment, has led to a deep need to redesign ing capacity of EC solutions is limited [8]. In order to address
the networks architectures. More in depth, the ever increasing both the cloud and theEC/FC issues, the functional integration
presence of smart devices in our daily life has opened the doors to of these two approaches in a same computing infrastructure has
novel ubiquitous communication paradigms. In such a dynamic recently gained momentum [2]. This solution enables massive
context, the Internet of Things (IoT) is emerged to express a wide improvements in system performance and users quality of ser-
reality consisting of heterogeneous smart devices generating big vice (QoS), introducing higher levels of flexibility for rapid
volume of data traffic and applications demand. Other typical computations and high mobility patterns. In this regards, the
aspects of the IoT may be represented by the mobility of the service providers point of view has to be considered, in order
devices, which poses new challenges as concerns the seamless to provide a proper system resources exploitation. Towards this
service continuity, or real-time execution constraints required direction, the paper analyzes the performance of an integrated
for some classes of applications, such as augmented reality, cloud-fog-edge computing infrastructure depicted in Fig. 1. At
the Edge layer we have sets of EC nodes (ECNs) able to offer
computation to the underlying devices. Each set of ECNs is
Manuscript received September 9, 2019; revised February 4, 2020 and May
18, 2020; accepted June 28, 2020. Date of publication July 14, 2020; date of linked to a fog computing node (FCN) belonging to the Fog
current version October 22, 2020. The review of this article was coordinated by layer. Finally, all the FCNs are linked to the cloud [9], [10]. ECNs
Dr. K. Bian. (Corresponding author: Benedetta Picano.) and FCNs are modeled as Markov queueing systems with finite
The authors are with the University of Florence, 50121 Florence, Italy (e-mail:
romano.fantacci@unifi.it; benedetta.picano@unifi.it). capacity. The entering of a computation request into the ECN or
Digital Object Identifier 10.1109/TVT.2020.3008926 FCN is ruled by the number of requests waiting for computation
0018-9545 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information.

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 FANTACCI AND PICANO: PERFORMANCE ANALYSIS OF A DELAY CONSTRAINED DATA OFFLOADING SCHEME 12005

at its arrival instant, in relation to specific deadline constraints. between cloud and vehicles to host users computational offload-
Conversely, the cloud does not reject any computation request. ing is the subject of [18], in which a joint optimization of both
As a consequence, we have resorted here to a Markov queueing the computational offloading policies and resource allocation is
system model with reneging, to take into account the premature formulated, and a game theoretic approach is pursuit. Within
departure of a request due to the expiration of the associated the optimal resource allocation in EC, authors in [19] design
deadline. two solutions to optimally partitioning the edge servers resource
Summarizing, the main contributions of this paper are blocks, aiming at minimizing the average service response time
r Queueing theory analysis of the considered integrated of IoT applications having strict service delay constraints. More-
cloud-fog-edge computing infrastructure; over, computational offloading has been widely investigated
r Performance optimization on the basis of the social welfare also in [20], in which authors consider ultradense IoT networks
metric [11]–[13], constrained on a target referred to the and propose a combined game-theoretic greedy approach to
dropping probability, i.e., the probability of the occurrence minimize the overall system overhead, i.e., processing time and
of the deadline expiration; energy use, within the mobile computation offloading problem.
r Validation of the proposed analytical model and obtained Similarly, authors in [21] focus on the cloudlets offloading by
analytical predictions throughout comparisons with nu- performing virtual machine migration and power control ap-
merical results derived by performing extensive computer proaches throughout the particle swarm optimization algorithm,
simulation runs under realistic world conditions. aiming at simultaneously taking into account the computation,
The rest of the paper is organized as follows. Section II migration and transmission costs of the offloading procedure.
provides an in-depth review of the related literature. Section III Virtual machine migration and transmission power control are
deals with the proposed system analysis based on the queue- also applied in [22], in which the main goal is the minimization
ing theory models. In Section IV the problem formulation is of the average user service delay. More in details, in [22], the
addressed while Section V provides comparisons between the service delay is given by the sum of both the processing and
obtained analytical predictions with numerical results derived transmission delay terms. For this reason, the virtual machine
by resorting to extensive computer simulation runs assuming migration is proposed to lower the delay due to the processing
realistic word conditions. Finally, the conclusions are drawn in overhead, while transmission power control is proposed to re-
Section VI. duce the transmission delay. Within this context, a heuristic in
which the two approaches are integrated is proposed. Transmis-
sion power control is also adopted in [23], in which the mean
II. RELATED WORKS field game theory is applied to design an offloading framework to
Nowadays, due to the massive number of heterogeneous manage both interference and handoffs in 5G ultra-dense small
EUs devices running different and advanced applications, a cell networks. Furthermore, some other works needing mention
big challenge is to properly handle computational resources in are given by [24]–[27]. In particular, the paper [24] designs
order to host all service requests with satisfactory performance. a multi-layer edge computing framework to optimally allocate
In accordance to this, several approaches and methodologies the tasks considering the trade-off between the computing and
have been proposed in the scientific literature. In particular, communication resources. A heterogeneous architecture has
a proper computing and networking resource allocation is the been also considered in paper [25], in which a joint assignment
objective of [14], where a deep reinforcement learning based of task, computation capabilities and transmission resources is
on the Q-learning algorithm is proposed in order to minimize performed in order to minimize the latency of the whole system.
the average service time based on network conditions. The Similarly, authors in [26] address the same joint optimization
computation offloading problem is addressed also in [15]. In problem, considering the distinction between the wireless and
this case, the authors analyze the performance of a blockchain- the wired channels, connecting the edge servers with the devices
empowered mobile edge computing scheme taking into account and the cloud center with the edge servers, respectively. Also in
both the mining and data processing tasks. In [15], the authors this case, paper [26] aims at minimizing the network latency, by
propose to combine deep reinforcement learning with a genetic resorting to a min-max problem. A multi-objectives offloading
algorithm approach to speed up the convergence time of the framework is proposed also in [27], where authors propose a
deep learning strategy. A deep reinforcement learning frame- graph coloring based strategy to perform the physical resource
work is adopted also in [16], in which the user equipments blocks allocation, while the offloading decisions are performed
offload traffic on a vehicle network arranged to support users considering the computation overhead estimated by all user
computation. Furthermore, the optimal policies for dynamic equipment by the mobile edge servers, respectively.
computation offloading are provided on the basis of a semi In addition to the above methodologies, queueing theory
Markov process, considering stochastic mobility vehicle traffic analysis has recently received many attentions within the field of
models, and the maximization of the long-term utility of the the computational networks. As a consequence, several papers
computing network. Vehicular edge computing networks are dealing with different aspects of these systems are available
studied also in [17], where the workload offloading and tasks from the literature. In particular, in [28] the authors focus on
computation scheduling is analyzed, supposing network nodes the optimization of the number of processors in a real time
with high mobility levels. The paper [17] takes into account computing system, where arrivals are bursty and divided into
both the communication and computation resources by resorting two priority job classes. The two priority service requests are
to a Lagrangian relaxation problem. A collaborative approach managed by setting a fixed number of processors to serve the
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 12006 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 69, NO. 10, OCTOBER 2020

high priority queue, while the remaining servers are allocated on cloud with a M/M/c/K system. In [39] the mobile device resource
the basis of the waiting time on both the queues. The presence of sharing framework in a cloudlets network is proposed, to model
two classes of customers has been analyzed also in [29], in which offloading. The proposed framework consists of a M/M/c/K
a system M/M/1/K with weighted fair queueing discipline is queue system, and the optimal average service time for cloudlet
studied. The conditions and performance of a computing system is provided. This paper aims at investigating the performance of
with finite input source is analyzed in [30]. Furthermore, in [30] the integrated cloud-fog-edge computing infrastructure depicted
a M/M/C/m/m queueing system model is consider to derive the in Fig. 1 by resorting to the use of suitable Markov queueing
optimal number of processors through the fuzzy logic approach, system models for the ECNs, FCNs and cloud, respectively. In
aiming at minimizing the system maintenance cost. A novel addition to this, a performance optimization analysis is proposed
method to assess the suitable number of servers in a queueing on the basis of the social welfare metric [11]–[13], constrained
system with finite capacity is proposed in [31], on the basis on target dropping probability values. In particular, by means
of the level of customers satisfaction. In particular, the server of the proposed heuristic strategy, the suitable number of active
optimization is performed here by considering three different processors to be allocated at each ECN, FCN and cloud is derived
metrics consisting of the system cost, its acceptability and the in order to maximize the derived social welfare metric.
servers utilization rates. Authors in [32] optimize the number
of processors on the basis of a M/G/∞ system, focusing on the III. SYSTEM MODEL
maximum exploitation of the production line, by considering
a provider perspective. The study of the economic aspects of A. Reference Scenario
different versions of the computational systems have recently We refer here to the three layers computing infrastructure
emerged in many papers. An example is represented by [13] depicted in Fig. 1. The EC layer consists of a suitable number
where a dynamic control problem in an open Jackson network of ECNs, each of them located at a given Base Station (BS) of
with limited capacity is formulated, in which the aim is to deter- a high speed, high reliable, low latency fifth generation (5G)
mine the suitable admission price to maximize the long therm wireless network. Each ECN provides computation services to
social welfare system metric. Paper [33] formalizes the cloud all mobile users within its service area, i.e., within the coverage
provider maximization profit, in which both service charges and area of the related BS of the 5G network. FCNs belonging to
business costs are considered in the system optimization. the Fog layer, can be connected to a given number of ECNs
With the emergence of the new network paradigms, queueing by means of switched high speed links. FCNs provide tasks
theory has been extensively applied to provide stochastic traffic computation service to mobile users within the service area
analysis of next generation networks. Paper [34] aims at ruling of the connected ECNs on the basis of the suitable procedure
offloading considering an heterogeneous networks scenario. The described later. Finally, we have the cloud layer where a cloud
paper models both the partial and the full offloading policies, via infrastructure connected by switched high speed links with all
Wi-Fi and cellular networks, considering reneging and service the FCNs of the Fog layer can provide computing service to all
interruptions. The main aim of the paper is the optimal tradeoff mobile users within all the ECN service areas on the basis of a
between energy efficiency and system performance, and the het- suitable offloading procedure.
erogeneous offloading interfaces are represented through on/off More in detail, we focus on a reference scenario where we
Markov chain models. The offloading problem is investigated have a set of tasks from mobile users requiring computation,
also in paper [35] authors use anon/off alternating renewals and an integrated computing infrastructure composed of a set E
process which is analyzed to derive transmission delay and of ECNs, several FCNs F belonging to the set FC , and a cloud.
offloading efficiency. Then, a model with balking is proposed Each FCN offers computation support to a subset of ECNs EF
taking into account the WLAN status, the number of packets and, similarly, the remote cloud is devoted to provide support
waiting for transmission and the associated deadline. Differ- to FC . Hereafter we refer to the service area of an ECN as
ently, an alternating renewal process is also used in paper [36], the geographical area within which a task originated in that
in order to model the availability of the WiFi network for the area can be offloaded on the corresponding ECN throughout
offloading strategy. The paper proposes a complete theoretical wireless links with a negligible latency. As stated before, we
queueing analysis, and authors especially focus on a novel user assume ECNs located at the BS sites of the 5G cellular network
patience metric. The customers impatience is analyzed also which supports, in conjunction with a high speed low latency
in paper [37], where a multi-server retrial queueing system is core network, mobile users communication connections with
modeled, where customers may leave the system for balking the integrated computing infrastructure under consideration. As
or impatience. Furthermore, a more realistic customer behavior mobile users move across different cells during the time needed
is formulated by using three parameters to weight probabilities. to complete a task computation, a handover procedure can be
The offloading scheme in a hybrid cloud-fog computing system, performed between neighboring cells to guarantee a seamless
for time critical application, is proposed in [38]. Paper [38] aims connection with the integrated computing infrastructure.
at minimizing the mobile devices power consumption, consider- In this paper, due to the complexity and time consume of
ing strict restrictions on the system response time. The paper [2] a task computation migration from one computation site to a
still considers a hybrid fog-cloud network as computational neighboring one, we have considered here the communication
support in an IoT scenario. The paper addresses the healthcare plane (i.e., handover) separated by the computing plane (i.e.,
services deployment, modeling each edge node as a M/M/C tasks computation). This means that, whenever a mobile user
system, the public cloud with a M/M/∞ queue, and the private is no longer in the coverage of a given BS, it starts a handover
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 FANTACCI AND PICANO: PERFORMANCE ANALYSIS OF A DELAY CONSTRAINED DATA OFFLOADING SCHEME 12007

procedure but its offloaded task continues to receive computation M/M/K/H queue system where task computation requests are
service from the original ECN, even if the mobile user is not served in their order of arrival (FCFS scheduling) and with each
directly linked to the original ECN anymore. This is so, because of them requesting an exponential service time with mean value
a mobile user can be linked with the remote ECN by means of TE . Moreover, in this case, we have that the at most H task
any BS to which the mobile user is connected that acts as a relay computation requests can be in the ECN at a time. This term
node. is dependent on K in relation to a specific QoS requirement.
The arrival processes of the task computation requests at the Therefore, H is the highest value for which the resulting proba-
ECNs are assumed as independent and identically distributed bility PEOU T (K, H) that the task computation request entering
Poisson processes with same mean arrival rate Λ. the system while there are H − 1 requests, waiting for service
At a first instance, a mobile user within a given EC service completion, does not complete its service before expiration of
area submits a task computation request to the related ECN. its deadline, is less than a target value, PEOU T,tg .
This new arrived request is accepted if, on the basis of the Hence, given K and PEOU T , the maximum number H of task
task computation requests already in the ECN, it results, with computation requests accepted by each ECN is derived under the
suitable probability, that it can complete computation before assumption (worst case) that each task computation request ac-
expiration of its time deadline. Otherwise, the BS associated to cepted by ECN completes service within its deadline. It follows
the ECN redirects the task computation request to the linked that the time needed to a task computation request arrived while
FCN. Here again, the task computation request is accepted if, in the ECN there are H − 1 requests tocomplete  its service,
on the basis of the number of task computation requests already i.e., T , can be defined as the sum of k = K H
+ 1 independent
in the FCN, it is possible to guarantee with a suitable statistical exponentially distributed random variables with mean values
uncertainty that its computation is completed within its time α = TKE , and an independent exponentially distributed random
deadline. Differently, it is redirect through the core network to variable with mean values β = TE .
the cloud. Each new arrived task computation request at the cloud Therefore, the pdf of T can be obtained throughout the convo-
is accepted, hence, in this case, we can have task computations lution of an Erlang distribution generating the random variable
reneging due to the expiration of the tasks deadlines. corresponding to the sum of the k independent exponential
In performing our analysis we assume the task computation random variables previously introduced, and the exponential
time at the ECNs, FCNs and cloud of the integrated computing distribution with mean values β. After some algebraic manipu-
infrastructure exponentially distributed with appropriate mean lations, the corresponding pdf results to be
value (related to the different computation capabilities of each 
αk βe−αt t
site). We will validate the goodness of this assumption by fT (t) = (t − τ )(k−1) e(α−β)τ dτ. (1)
comparing the obtained analytical predictions with simulation (k − 1)! 0
results derived by assuming realistic world task computation Hence, let E be the random variable exponentially distributed
time distributions. In addition to this, we consider that each task with mean value μ1D referred to the time deadline of the task com-
computation request has associated a deadline which expires putation request entering the ECN system in the H-th position,
after a time exponentially distributed [35], [40], [41], with mean we have that PEOU T (K, H) is
value dependent on the specific computation site (EN, FN or
cloud), in order to take into account the impact of the resulting PEOU T = P {T > E} = 1 − P {T ≤ E}
different communication delays.  ∞  ∞ 
−μD σ
As a consequence, on the basis of our assumption, we have =1− μD e dσ fT (τ )dτ
0 τ
resorted to a M/M/K/H queueing system to model each ECN  ∞
behavior, where K denotes the maximum number of task com- =1− e−μD τ fT (τ )dτ. (2)
putations accepted by each ECN and H the number of available 0
CPUs at each ECN. Similarly, the FCNs have been modeled Consequently, parameters K, H have to be defined in order
as independent M/M/F/A queueing system. Finally, the cloud, to have
being in this case the rejection of task computation request
not allowed, has been modeled as a M/M/S queueing system. PEOU T (K, H) ≤ PEOU T,tg (3)
It is important to note that parameters K, H, F, A, S have to
Once K, H have been defined to satisfy (3), any new task
be derived on the basis of a suitable optimization approach in
computation request arrived while the ECN is in state H is
relation to specific QoS requirements in terms of task reneging
redirect to the linked FN for a successive consideration. By
probability at the cloud site less than a target value. It is important
referring to the state diagram of the M/M/K/H queue system
to note that at both the ECNs and FCNs, the task computation
under consideration, shown in Fig. 2, we have
completion is assumed guaranteed, since any new arrived task

computation request is admitted if this is assured with a suitable Λ, 0 ≤ n < H
probability, as detailed later. λn = (4)
0, otherwise
and
B. Edge Subsystem Analysis 
On the basis of our previous assumptions, we have that nμs , 1 ≤ n < K
μn = (5)
each ECN belonging to E can be modeled as an independent Kμs , K ≤ n ≤ H,
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 12008 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 69, NO. 10, OCTOBER 2020

Fig. 2. ECN Markov chain.

where μs = 1/Ts and the system state represented by the num- analysis, is the maximum value for which the following con-
ber n is the task computation requests in the system. straint is satisfied
Therefore, pn ,i.e., the probability of having n task computa-
PF OU T (F, A) ≤ PF OU T,tg , (10)
tion requests in the ECN, for 1 ≤ n < K, results to be
where now PF OU T (F, A) is
Λn
pn = p0,E , (6) PF OU T = P {U > E} = 1 − P {U ≤ E}
n!μns
 ∞
while for K < n ≤ H is =1− e−μD τ fU (τ )dτ, (11)
0
Λn in which
pn = p0,E . (7)
K n−K K!(μ s)
n 
π θ ξe−πt t
fU (t) = (t − τ )(θ−1) e(π−ξ)τ dτ, (12)
The term p0,E , i.e., the probability of having no task computation (θ − 1)! 0
requests in the ECN system, in (9) and (7) can be derived by
and, as in the edge subsystem case, we have π = TFF , ξ = TF ,
imposing the validation of the state probability normalization A
 and θ = F + 1. In this case we have that any new task com-
condition p0,E + ∞ n=1 n = 1 according to [10] obtaining
p
putation request arrived while the FCN is in state A is routed to
K−1 H −1 the cloud. It is important to highlight that being each FCN linked
Λn Λn with a number Y of ECNs, the task computation requests arrival
p0,E = + . (8)
n=0
n!μns K n−K K!μns process is Poisson with mean rate Γ, equals to YΛPB , resulting
n=K
from the superposition of Y independent Poisson processes with
Moreover, its is easy to note that the task computation request equal mean rate ΛPB [10]. The state diagram of the M/M/F/A
blocking probability PB equals to the probability of having the queue system under consideration is shown in Fig. 4. The system
ECN system in state H, i.e., PB = pH . state is again represented by the number n of task computation
Then, the mean time spent in the system by any task com- requests in the FN and the queue system parameters are given
putation request results, through application of the Little’s for- by
mula [10] 
H Γ, if ∃ε ∈ EF s.t. n = H
λf = (13)
x=0 xpx 0, otherwise
Tp,E = , (9)
Λ(1 − pH ) 
f (μF + μD ), 1 ≤ f < F
where Λ(1 − pH ) is the mean task computation requests arrival μf = (14)
F μF + f μD , l ≥ A.
rate at each ECN system.
Hence, the state probability pf of having f requests in the
C. Fog Subsystem Analysis system, 1 ≤ f < F , is

According to our assumptions, we have here again that the Γf

pf = p0,F N , (15)
FCNs can be modeled as independent M/M/F/A queue systems f !(μF + μD )f
where task computation requests are served in their order of and, for F ≤ f < A
arrival (FCFS scheduling) and each task computation request
requires an exponential service time with mean value TF . As Γf
pf = p0,F N f
.
for the ECNs, we have that at most A task computation requests (F − 1)!(μF + μD )(F −1) y=F (yμD
+ F μF )
can be in each FCN at a time with this term dependent on F in (16)
relation to the specific QoS requirement previously introduced, As before, the term p0,F N , i.e., the probability of having no task
i.e., PF OU T (F, A) ≤ PF OU T,tg . Also in this case, given the F computation requests in the considered FCN, in (15) and (16) can
and PF OU T values, the number A of task computation requests be derived by application of the state probabilities normalization
accepted by each FCN, according to the considered worst case condition as in (17), shown at the bottom of this page, with the

⎡ ⎤−1
F −1 A
Γf Γf
p0,F N = ⎣ + ⎦ (17)
f !(μD + μF )f (F − 1)!(μD + μF )F −1 f
y=F (yμD + F μF )
f =0 f =F

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 FANTACCI AND PICANO: PERFORMANCE ANALYSIS OF A DELAY CONSTRAINED DATA OFFLOADING SCHEME 12009

Fig. 3. FCN Markov chain.

Fig. 4. Cloud Markov chain.

task computation requests blocking probability PB , in this case, validating of the state probability normalization condition for
equals to the probability to have the considered FCN in state A, this case. Similarly to the previous analysis, the mean time spent
i.e., pA . by each task computation request in the cloud system is
As before, the mean time spent by each task computation ∞
xpx
request in a FCN through the Little’s formula is Tp,C = x=0 . (24)
∞ Φ
xpx Finally, in this case the reneging probability, i.e., the prob-
Tp,F N = x=0 . (18)
Γ(1 − pA ) ability that a task computation request in the cloud does not
where Γ(1 − pA ) is the mean task computation requests arrival complete its service due to its deadline expiration, PD , can be
rate. obtained according to [42], as
PD = Tp,C μD . (25)
D. Cloud Subsystem Analysis
In the cloud case, we have no limitations on the accepted IV. PROBLEM FORMULATION
tasks computation requests. Hence, it follows that the cloud can The main aim of the paper is to pursuit an optimization
be modeled as a M/M/S queue system, with FCFS selection procedure in order derive the proper number of active processors
policy and S available CPUs. In this case, we have that the allocated to each ECN, FCN and cloud, as well as the maximum
cloud may be linked to a number of F FNs. So that, according to number of computation requests accepted by each ECN and
our assumptions, we have that the task requests arrival process FCN, respectively. The optimization procedure is intended here
is Poisson with mean rate Φ equal to FΛPA . The associated as the maximization of the social welfare [11]–[13] function,
system state diagram is provided in Fig. 4 where the system through which both the users and provider points of view can
state is considered as the number of task computation requests be simultaneously taken into account, reaching a good trade-off
m in the cloud. In particular, we have between the parts involved in the proposed network infrastruc-

Φ, if ∃η ∈ FC s.t. f = A ture.
λm = (19) Furthermore, assuming ΔEf f as the whole infrastructure rate
0, otherwise
of satisfied requests, the social welfare metric is defined as

m(μC + μD ), 1 ≤ m < S F(ΔEf f , b, c, d, K  , H  , F  , A , S  )
μm = (20)
SμC + mμD , m ≥ S.
= U (ΔEf f,E + ΔEf f,F + ΔEf f,C )
Consequently, the state probability pm , that is the probability of − V (Tp,E + Tp,F N + Tp,C )
having m tasks in the cloud subsystem, for 1 ≤ m < S, is
K F S
Φm
pm = p0,C , (21) − rbz − ucw − sdj ,
m!(μC + μD )m z=1 w=1 j=1
while for m ≥ S, we have (22), shown at the bottom of this page.
(26)
In (21) and (22) the term p0,C , i.e., the probability of having no
task computation requests in the cloud system, is given as in where ΔEf f,E = Λ(1 − pH ), ΔEf f,F N = Γ(1 − pa ), which
(23), shown at the bottom of this page, by imposing again the represent the mean values of the computation requests arrival

Φm
pm = p0,C m (22)
(S − 1)!(μD + μC )S−1 g=S [gμD + SμC ]

S−1 ∞ −1
Φm Φm
p0,C = + m (23)
m=0
m!(μD + μC )m (S − 1)!(μD + μC )S−1 g=S [gμD + SμC ]
m=S

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 12010 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 69, NO. 10, OCTOBER 2020

rates accepted by each ECN and FCN for which we have as- where
sumed guaranteed the service completion. Likewise, ΔEf f,C = r Constraint (29) represents a QoS target constraint referred
Φ(1 − PD ) is the mean rate of computation requests completing to each ECN, FCN and cloud site on the probability that a
service without deadline expiration at the cloud site. In partic- computation request does not complete its service due to
ular, ΔEf f represents the sum of the single accepted requests its deadline expiration;
flows deriving from the single subsystem, although Λ, Γ, and Φ r Constraints (29)–(33) represent the architectural restricts
are the arrivals rate of the edge node, fog node, and the cloud sub- of the integrated cloud-fog-edge computation infrastruc-
systems, respectively. Therefore, the mean number of computa- ture;
tion requests accepted by the three computation subsystems are r Constraint (34) refers to the stability condition for the cloud
given by ΔEf f,E = Λ(1 − pH ), ΔEf f,F N = Γ(1 − pa ), and subsystem, interpreted in its most strict form.
ΔEf f,C = Γ(1 − PD ) which define ΔEf f as the sum of these
three quantities, i.e., ΔEf f = ΔEf f,E + ΔEf f,F N + ΔEf f,C .
It is important to highlight that (26) represents the gap between A. Proposed Heuristic
the revenue due to the users service accomplishments and the
penalty associated to the time spent in the computation sub- Due to the intrinsic difficulty of problem (28)–(29), this paper
systems, minus the provider costs associated to different CPUs. proposes an approximated heuristic to determine the suitable
Therefore, by considering this metric, we indirectly take into ac- number of CPUs to be allocated at each computation site, i.e.,
count the service provider and the users interests, since a higher ECNs, FCNs, and cloud, in addition to the maximum number of
number of users accomplishments implies greater service gains computation requests accepted at each ECN and FCN in order
and, from the other side, an efficient resource exploitation, i.e., to guarantee a specific QoS in relation to the probability that
the CPUs activation policy, triggers a greater number satisfied a computation request does not complete its service at a given
users. Furthermore, K  , H  , F  , A , S  represents the system pa- site due to its deadline expiration. The proposed social welfare
rameters values for the integrated cloud-fog-edge computation maximization procedure is iterative and, for each computation
infrastructure, that have to satisfy the following architectural site, acts as follow
contsraint: 1) Let S, F , K, H, and A = 1 be the maximum architectural
capacity of the whole system. Start with S  = 1, F  = 1,
K  ≤ K, H  ≤ H, F  ≤ F, A ≤ A, S  ≤ S. (27) K  = 1, H  = 1, and A = 1, φ = 0, ξ = 0, θ = 0, π = 0,
In addition to this, in (26) we have that ψ = 0;
r U is the gain associated to the client service accomplish- 2) Compute PD . If PD ≤ PD,target then terminate, other-
ment while V is a penalty associated to each unit of time wise evaluate
spent by a computation request in an ECN, FCN or cloud; a) If K  = K, i.e., it cannot be incremented, set
r Assuming the service provider operating cost associated to φ = 1 and jump to b), otherwise evaluate A1 =
a CPU dependent on its location, i.e., ECN, FCN or cloud, F(ΔEf f , b, c, d, K  , H  , F  , A , S  ) with S  , K  =
r represents the service provider cost for CPU available at K  + 1, H  , F  , A ;
a ECN, u is the cost related to a CPU located at a FCN, b) If H  = H, i.e., it cannot be incremented, set
while s is the cost associated to an active CPU at the cloud ξ = 1 and jump to c), otherwise evaluate A2 =
site; F(ΔEf f , b, c, d, K  , H  , F  , A , S  ) with S  , K  ,
r b, c and d are binary vectors whose elements bi , with H  = H  + 1, F  , A ;
1 ≤ i ≤ K, ci with 1 ≤ i ≤ F , and di 1 ≤ i ≤ S equal to c) If F  = F , i.e., it cannot be incremented, set
1 if at the ECN, FCN or cloud site, respectively, the i-th θ = 1 and jump to d), otherwise evaluate A3 =
CPU is active, or 0 otherwise. F(ΔEf f , b, c, d, K  , H  , F  , A , S  ) with S  , K  , H  ,
Hence, being our goal that of maximizing the social welfare F  = F  + 1, A ;
metric defined in (26), we can formulate our problem as d) If A = A, i.e., it cannot be incremented, set
π = 1 and jump to e), otherwise evaluate A4 =
max F(ΔEf f , b, c, d, K  , H  , F  , A , S  ), (28) F(ΔEf f , b, c, d, K  , H  , F  , A , S  ) with S  , K  , H  ,
b,c,d,K  ,H  ,F  ,A ,S 
F  , and A = A + 1;
s.t. e) If S  = S, i.e., it cannot be incremented, set
PDeadline ≤ PD,traget , (29) ψ = 1 and jump to 3), otherwise evaluate A5 =
F(ΔEf f , b, c, d, K  , H  , F  , A , S  ) with S  = S  +

K ≤ K, (30) 1, K  , H  , F  , and A ;
H  ≤ H, (31) 3) If (ψ ∧ ξ ∧ θ ∧ π ∧ φ) is equal to 1 terminate,
otherwise select the resource allocation among

F ≤ F, (32) A1 , A2 , A3 , A4 , and A5 which minimizes
A ≤ A, (33) F(ΔEf f , b, c, d, K  , H  , F  , A , S  );
4) Perform allocation in accordance with the previous step;
Φ < SμC , (34) 5) Repeat from step 2).

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 FANTACCI AND PICANO: PERFORMANCE ANALYSIS OF A DELAY CONSTRAINED DATA OFFLOADING SCHEME 12011

TABLE I
SYSTEM PARAMETER

V. NUMERICAL RESULTS
In order to validate the effectiveness of the proposed analyt-
ical approach, this section provides comparisons between the
obtained analytical results (AR) based on the assumed Markov
queueing system models with numerical results (SR), derived by
resorting to extensive computer simulation runs by considering
Fig. 5. Computation requests blocking probability at each ECN as function
actual arrival and service time distributions under the assumption of Λ.
of same mean values [10]. Furthermore, where clearly visible,
the confidence bars have been inserted in the corresponding
plots. In performing our analysis, conducted on the basis of
specifications reported in Table I, weassume the cloud system
equipped with a maximum number of S = 16 CPUs, each FCN
with a maximum number F of CPUs equal to 13 maximum, and
each ECN with a maximum of K = 10 CPUs. Furthermore, we
refer to an edge layer composed of 6 ECNs and a fog layer with
3 FCNs.
According to [43], in order to test the validity of the proposed
theoretical model, we have analyzed here, under the assumption
of equal mean values, the case of computation requests arrivals
modeled as a Normal distribution, with service time following a
hyperexponential distribution as in [10]. Furthermore, we have
assumed that applications require an equal mean computation
time at ecah computation site. in particular, in the cloud case,
it results equal to 0.4 s, with a μC = 2.5 s−1 . Likewise, the
mean requests computation time on each FCN is 0.2 s, with
μF = 5 s−1 , while the mean requests computation time on each Fig. 6. Computation requests blocking probability at each FCN as function
of Λ.
ECN has been assumed equals to 0.1 s, with μE = 10 s−1 .
The system performance is measured in terms of computation
requests completion failure probability at each ECN, FCN and
cloud site, PB limited to ECNs and FCNs, social welfare metric V = 4, U = 15 per time unit, r = 20, u = 16, and s = 12, are
referred to the integrated cloud-fog-edge computing infrastruc- provided in Fig. 7 as a function of the parameter Λ. The values
ture under consideration. of the systems parameters, i.e., number of CPUs allocated at
All the numerical results presented here are referred to the each ECN, FCN and cloud, maximum number of computing
mean values obtained from 2000 independent runs of simula- requests accepted by each ECN and FCN, that maximize (26),
tions. Fig. 5 depicts the behavior of PB at an ECN as a function are shown in Figs. 8 and 9, respectively, as a function of Λ.
of Λ for H and K values derived in order to satisfy the constraint Finally, in Figs. 10–12 is shown the resulting computation
of a resulting PEOU T (K, H) less than 3.00 × 10−3 . requests completion failure probability at each ECN, FCN and
Likewise, Fig. 6 shows the behavior of PB at each FCN as a cloud site, respectively, as a function of Λ. The assumed compu-
function of Λ with the system parameters A and F derived in tation requests completion failure probability target value (i.e.,
order to satisfied the constraint PF OU T (F, A) ≤ 3.00 × 10−3 . 3.00 × 10−3 ) for all the computation sites is given in Fig. 10. In
It is important to stress that in all previous figures a very all these figures comparisons with the simulation results derived
good agreement between analytical predictions based on the under proper assumptions for the computing requests arrival
considered Markov queueing system models with the simula- process and service time are also given in order to highlight the
tion results derived under real world arrival and service time goodness of our Markov approach. Furthermore, it is important
distributions [10]. to note that the trends of H and A in Fig. 8, in relation to Figs. 9
The results related to the proposed heuristic based on the and 7, reveal that the size of H and A increases when the cost of
maximization of the social welfare metric, defined in (26) with the activation of new processors is higher in comparison to the

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 12012 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 69, NO. 10, OCTOBER 2020

Fig. 7. Social Welfare metric as function of Λ.

Fig. 10. Computation requests completion failure probability at each ECN as
function of Λ.

Fig. 8. Maximum number of computation requests accepted by each ECN and

FCN resulting by our heuristic as function of Λ.
Fig. 11. Computation requests completion failure probability at each FCN as
function of Λ.

Fig. 9. Number CPUs to be allocated at each ECN, FCN and cloud resulting
by our heuristic as function of Λ. Fig. 12. Computation requests completion failure probability at the cloud site
as function of Λ.

increment of the size of H or A. It is important to stress that,

the ever-increasing trend of the social welfare function shown the aim at broadening our analysis, in Fig. 11, we have also
in Fig. 7, demonstrates that the system parameters provided in considered the case of a Weibull distribution for the service
Figs. 8 and 9 allow a good exploitation of system resources times modeling [10]. Also in this case, the closeness between the
in reference to the set QoS target, on the basis of the service AR and the results derived from simulation is clearly evident,
provision cost which characterized each subsystem. Finally, with confirming the validity of the proposed approach.

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 FANTACCI AND PICANO: PERFORMANCE ANALYSIS OF A DELAY CONSTRAINED DATA OFFLOADING SCHEME 12013

VI. CONCLUSION [16] Y. Liu, H. Yu, S. Xie, and Y. Zhang, “Deep reinforcement learning
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Romano Fantacci (Fellow, IEEE) received the M.S.

degree in electrical engineering from the University
of Florence, Italy, and the Ph.D. degree in computer
networks from the University of Florence, Italy. He is
a Full Professor of Computer Networks at the Univer-
sity of Florence, Florence, Italy, where he heads the
Wireless Networks Research Lab. He is the Founding
Director of the Information Communication Technol-
ogy Inc. (TiCOM), with Leonardo Spa and of the
Wireless Communications Research Centre (LiRS)
with Telecom Italia. His current research interests
encompass several fields of wireless engineering and computer communication
networking including, in particular, performance evaluation and optimization
of wireless networks, emerging generations of wireless standards, cognitive
wireless communications and networks, satellite communications and systems.
Dr. Fantacci was elected Fellow of the IEEE in 2005 for contributions to wireless
communication networks. He received several awards for his research, including
the IEE Benefactor Premium, the 2002 IEEE Distinguished Contributions to
Satellite Communications Award, the 2015 IEEE WTC Recognition Award, the
IEEE sister society AEIT Young Research Award and the IARIA Best Paper
Award, the IEEE IWCMC’16 Best Paper Award and the IEEE Globecom’16
Best Paper Award. He served as Area Editor for IEEE TRANSACTIONS WIRELESS
COMMUNICATIONS, Associate Editor for IEEE TRANSACTIONS ON COMMUNICA-
TIONS, IEEE TRANSACTIONS WIRELESS COMMUNICATIONS. Regional Editor for
IET COMMUNICATIONS and Associate Editor for several non-IEEE technical
journals. He guest edited special issues for IEEE Journals and Magazines
and served as symposium chair of several IEEE conferences, including VTC,
WCNC, PIRMC, ICC and Globecom. Dr. Fantacci currently serves on the
Board of Governors of the IEEE sister society AEIT, as Area Editor for IEEE
IOT JOURNAL and as member of the Steering Committee of IEEE WIRELESS
COMMUNICATIONS LETTERS.

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