1. Introduction
The recent advancements in mobile hardware technology have not only resulted in trendy, lightweight, and easily portable smart devices from bulky with limited functionality devices, but also paved the path for applications like augmented reality, cognitive assistance, and face recognition, to name a few. However, these applications are highly computation intensive, and this limited computing capability of portable mobile devices like iPhones, Android phones, and Google glasses are unable to support these applications. Although
mobile cloud computing enables mobile devices to access a shared pool of configurable computational and storage resources, providing ubiquitous, convenient, and on demand services, large communication latency between mobile devices and remote clouds still presents a new challenge for
low-latency applications that demand 1–100 ms end-to-end system latency [
1]. To achieve this low-latency requirement, the authors of [
2] proposed the idea of
cloudlet computing similar to fog and mobile-edge computing, where cloudlets are considered as trusted, resource-rich computer or cluster of computers. Nonetheless, cloudlets are designed to be self-managing and function with low-context-awareness. Hence, they are most suitable for deployment in adversarial environments [
3].
The authors of [
4] showed that the system performance in terms of expected latency for the requested jobs from a single-hop cloudlet server is always better than a distant cloud server for low-latency applications. Thus, the optimal placement of cloudlets over wireless access networks seemed to be an important research problem to enhance the system performance. Along with the wireless access networks, researchers started to focus on
optical and fiber-wireless (FiWi) access networks for edge-computing solutions deployment due to their wide deployment coverage, low cost per bit, very high data transmission bandwidth, efficient network virtualization and network scalability [
5]. The time-division multiplexed passive optical network (TDM-PON) has also been considered as a front/back-haul support for wireless access network, i.e., access technologies like Radio-over-Fiber (RoF), WiFi, LTE-A and millimeter-wave (mmWave) can be integrated with optical network units (ONUs) for a higher user coverage [
6].
A
static cloudlet placement framework considers that the network is static in time, i.e., it does not consider user mobility and virtual machine (VM) mobility into account, and identifies optimal cloudlet locations over an existing access network infrastructure [
7]. Nonetheless, most of the existing research on optimal cloudlet placement formulated integer/mixed-integer linear/nonlinear programming problems, which are
NP-Hard problems. Hence, authors either relied on commercial solvers or designed heuristic algorithms, e.g., we proposed a mixed-integer nonlinear programming (MINLP) based hybrid cloudlet placement cost-optimization framework over TDM-PON based FiWi networks in [
8] that relies on commercial solvers. However, both these approaches either suffer from scalability issues or from a strong bias arising from the particular dataset used for the framework evaluation. Apart from the computational aspects, these works do not provide any general insight on the behavior of the frameworks against variation in network parameters. However, these cloudlet placement frameworks provide exact cloudlet placement locations, but an existing network infrastructure is required as an input to these frameworks. Thus, when some network service provider intends to
install cloudlet servers while deploying green-field fibers, an estimation of the cloudlet deployment cost appears to be really useful, but these existing cloudlet placement frameworks fail to do so, as there is no existing network infrastructure.
To address this gap in existing literature, in this paper, we propose an analytical cost-optimization framework for the hybrid cloudlet placement based on the assumption that the underlying access network is
homogeneous, i.e., the same split ratio of each TDM-PON and the same number of users are served by each optical network unit (ONU), and the average distance between all ONUs and field, remote node (RN) and central office (CO) cloudlets are, respectively, equal. This analytical cost-optimization framework diminishes the computational complexity of the MINLP based framework proposed in [
8] to a great extent by reducing the entire volume of network to a single TDM-PON based FiWi branch and provides a quick lower bound of cloudlet deployment cost. Note that we can directly convert the integer variables of the MINLP formulation in [
8] as continuous variables and formulate a continuous convex optimization problem, which can provide a lower bound to the actual cloudlet installation cost. However, at this point, we ask the question “can we find a better lower bound for cloudlet installation cost?”, i.e., if it is possible to find a framework that can find a tighter lower bound for the cloudlet deployment cost. In this context, we feel that directly using continuous variables to denote cloudlet installation decisions is not very appropriate and hence propose an alternative method. Our primary contributions in this paper are as follows:
- (i)
We formulate a novel constrained convex optimization problem under the network homogeneity assumption, in which the objective function is linear, but the optimal solution is dictated by a quasi-convex latency constraint.
- (ii)
We solve this modified constrained optimization problem by using KKT conditions on the Lagrangian functions and derive closed-form expression for the cloudlet deployment cost.
- (iii)
We use these closed form expressions to illustrate that this optimal solution yields a tight lower bound for the framework proposed in [
8] by making a performance comparison over an urban deployment area against 1 ms, 10 ms, and 100 ms target latency values with an optical access network split ratio of
,
.
- (iv)
We also perform a detailed parametric analysis to observe the changes in behavior of cost optimization framework against different network parameters like split ratio of TDM-PON, population density and quality-of-service (QoS) latency target.
The rest of this paper is organized as follows.
Section 2 provides a brief review of some recent related works.
Section 3 briefly discusses the TDM-PON based FiWi hybrid cloudlet placement network architecture.
Section 4 formulates the constrained optimization problem for optimal cloudlet placement.
Section 5 justifies the validity of the proposed analytical framework.
Section 6 presents a detailed parametric analysis and explains the impacts of several network parameters on cloudlet cost-optimization frameworks. Finally,
Section 7 concludes the paper by summarizing our primary findings.
2. Related Work
In this section, we explore some very recently published works on different aspects of cloudlet networks. The authors of [
9,
10,
11] proposed computation offloading frameworks for energy saving in edge devices, assuming that cloudlets are already deployed over access networks. However, usually the user distribution under a typical wireless network deployment is very complicated. In addition, in crowded areas, cloudlets can be accessed by a large number of mobile users, whereas in sparsely populated areas only a few users will intend to access cloudlets. Thus, optimal placement of cloudlets over wireless access networks and placement of optimal number of VMs in cloudlets appeared to be essential research problems to several researchers because these aspects improve the cloudlet resource utilization significantly [
12]. Note that, before the genesis of cloudlets and other edge-computing paradigms, researchers explored various VM migration techniques to optimize the power consumption in
Network Function Virtualization (NFV) environments under a dynamic traffic scenario and became aware of its potential benefits [
13,
14]. The authors of [
15] proposed a linear programming solution for computation offloading by considering the QoS requirements of mobile users while maximizing the revenue of service providers. The authors of [
7,
16] proposed an edge-cloud network design framework that first determines where to install cloudlet facilities among available sites, and then assign sets of access points to cloudlets that supports VM orchestration as well satisfies service-level agreements. The authors of [
17,
18,
19,
20,
21] proposed heuristic algorithms for
static network planning to optimally place cloudlets over the existing wireless access network and followed by
dynamic job request allocation to the cloudlets. The authors of [
22,
23] focused on minimizing the energy consumption of the mobile devices while computation offloading to the cloudlets over wireless channels. Several researchers like [
24,
25,
26,
27,
28,
29] also took interest to design efficient dynamic resource allocation algorithms.
On the other hand, the authors of [
30,
31] identified that
optical access networks can be useful in the next few decades to support edge computing technologies. In [
32], the authors studied the performance of centralized and decentralized bandwidth allocation algorithms in a long-reach optical access network to investigate the feasibility of computation offloading to edge-computing servers and develop an analytical framework to validate against simulated results. The authors of [
1] presented the implementation of a cloudlet framework for human–machine interactive applications with control server at the CO of a fiber-based access network. The authors of [
33] presented the idea of cloud and cloudlet empowered FiWi-heterogeneous network architecture for LTE-A and designed a cloudlet-aware resource management algorithm that aims to reduce the offload latency and prolong mobile-devices’ battery life. The authors of [
34] designed a joint optimization algorithm of multiple jobs scheduling and investigated lightpath provisioning to minimize average completion time in fog computing micro data-centers elastic optical networks. Recently, in [
35,
36], we provided a high-level overview of latency-aware cloudlet placement frameworks over TDM-PON based FiWi environment. In [
37], we proposed an MINLP based cost optimization framework for cloudlet placement in the field locations.
To the best of our knowledge, for the very first time, we proposed a cost-optimization framework for hybrid cloudlet placement in [
8], where cloudlets are allowed to be installed in the field, RN and at CO locations subject to the capacity and latency constraints. In this work, we formulated an MINLP based cost-minimization problem and implemented using commercial solvers that use spatial branch-and-bound algorithms to find a global optimal solution. We applied this framework over stochastically generated 5 km × 5 km urban, suburban, and rural areas with population densities 4000, 2500, 1500 persons/km
, respectively, with QoS latency targets 1 ms, 10 ms, and 100 ms to find optimal cloudlet placement locations. Through this exercise, we realized that, although this framework is capable of providing us the exact cloudlet placement locations, but it does not provide any general insight about cost-optimal cloudlet deployment strategies, depending on the underlying network scenario. Moreover, the solution algorithm does not scale very well with large data sets. These fundamental shortcomings motivated us to develop the analytical cost-optimization framework proposed in this paper. This analytical framework is based on network homogeneity assumption and performs an average analysis with a few network parameters e.g., split-ratio, population density, available bandwidth, and QoS latency requirements. Therefore, it can provide us a first-hand estimation of the deployment cost and several useful insights on the cost-optimal deployment strategies for any underlying network scenario, without any scalability issues.
3. Hybrid Cloudlet Placement Architecture
A static cloudlet placement framework considers the network status as static in time, i.e., it neither takes user mobility nor VM mobility into account and identifies optimal cloudlet locations over an existing access network infrastructure. This is essential to a cost-optimal network design framework for cloudlet placement and assignment of ONUs to cloudlets [
16]. In this paper, we consider the same hybrid cloudlet placement framework over the TDM-PON based FiWi access network proposed in [
8], as shown in
Figure 1. Recall that we considered tree-and-branch network topology of TDM-PON with split-ratio
, where
. Each ONU has an integrated wireless access point to serve multiple edge devices by wireless connection. The cloudlets can be suitably installed either in the field, at RN or at CO locations. Based on the assumption of
network homogeneity, instead of analyzing the entire volume of the network, we can reduce our analysis to just a single TDM-PON based FiWi branch.
Each CO has multiple (e.g.,
) optical line terminals consisting of line cards and optical transceivers. We recall from [
8] that ONUs can be connected to the
field cloudlets via point-to-point fiber links (brown links). In that case, a new set of optical transceivers are installed both at the ONU and corresponding field cloudlet.
We consider the scope of installing cloudlets at RN locations along with the passive splitters. Each RN cloudlet uses one or multiple new time-shared wavelengths for communication with ONUs in both uplink and downlink. In this case, a new set of optical transceivers are also required to be installed both at the ONU and RN cloudlet.
The
CO cloudlets are installed at CO and hence are furthest from the ONUs. However, ONUs can use the spare bandwidth of the default uplink and downlink channels for communication with the CO cloudlets and hence no additional transceivers are required to be installed at ONUs or COs. However, in this work, we restrict our attention only to RN and CO cloudlets, as we showed in our previous work in [
8] that these are the most cost-effective schemes for most practical cloudlet deployment.
4. Optimal Cloudlet Placement Problem
In this section, we present the system model and constrained convex optimization problem formulation to overcome the scalability issues observed with the MINLP formulation in [
8]. Cloud servers are usually over-provisioned and possess huge computational and storage resources, hence can be assumed as M/M/∞ queuing system [
21]. On the other hand, the cloudlets contain a finite number of processors with
application virtualization that uses a VM to provide an execution environment for the offloaded job requests [
38]. Due to this reason, cloudlets can simultaneously perform different tasks and hence we consider that the cloudlet hardware is parallel processing enabled in this work [
39]. From Google cluster-usage traces, it can be shown that job request arrival and their service times follow
exponential distributions, and hence can be considered as
Poisson processes [
40]. Note that an
queue provides a lower-bound on processing latency as long as a single cloudlet has the aggregated processing rate of all the processors. Therefore, to compute the
average processing latency, we model the cloudlets as
queueing systems [
20].
We consider a single TDM-PON based FiWi branch where cloudlets can be installed at RN and CO locations, as shown in
Figure 1.
Table 1 briefly outlines the definitions of the required network parameters. In this work, we consider the maximum job request arrival rate as
(jobs/s), where
p denotes the maximum number of users each ONU can serve and
N denotes the split-ratio of each TDM-PON under network homogeneity assumption. The total transmission latency between a cloudlet and ONU is the sum of to-and-fro data transmission latency and average polling cycle latency, i.e.,
, where
denotes the average length and
denotes the average polling cycle latency between cloudlet at
and ONUs, and
denotes the velocity of light in optical fiber. We assume that the job request packets from ONUs and job response packets from cloudlets are highest priority packets in the network, and are processed within one polling cycle of the TDM-PON standard considered.
The
decision variables are defined as follows:
the fraction of total incoming workload assigned to the RN cloudlet,
the fraction of total incoming workload assigned to the CO cloudlet,
(binary variable) takes the value of 1 if an RN cloudlet is installed,
(binary variable) takes the value of 1 if a CO cloudlet is installed, and
total service rate of all the processors installed at the RN and CO cloudlets (jobs/s). A summary of all the decision variables are tabulated in
Table 2 for convenience. All these variables are allowed to be only non-negative i.e.,
. Thus, the objective function to minimize the overall cloudlet installation expenditures is given below:
where the binary variables
and
follow the boundary constraints
and
. This ensures that a cloudlet is installed at RN and CO only when
and
. In objective function (
1), the first component indicates
the total cost of installing processors because
denotes the average number of jobs a single processor can simultaneously process and
denotes the cost of installing a single processor. As
and
denote the cost of new infrastructure installation for cloudlets, the second and third components indicate the
cloudlet installation costs at RN and at CO locations, respectively. However, due to the presence of
and
, the problem still remains a mixed-integer programming problem and cannot be treated as a convex optimization problem. Therefore, to avoid this issue, we reformulate the problem. Instead of using
and
in the objective function, we use only a constant
that indicates cloudlet installation cost in general and the modified objective function is:
To ensure that all ONUs are served either by RN or CO cloudlet, we consider the constraint below:
Again, to ensure the non-negativity of the decision variables, we also consider the following constraints:
The
latency constraint that ensures that the overall system latency does not exceed
is given below:
Here, each of and are multiplied to compute the weighted average of the processing latency of the cloudlet, the propagation latency, and the total transmission latency, i.e., the latency to offload total bits for the job request from ONU to cloudlet, and the latency to receive total bits post-processing by ONU from cloudlet, respectively for RN and CO cloudlets. Note that and denote processing latency at RN and CO cloudlets, respectively. The terms and denote propagation latencies between ONUs and cloudlets at RN and CO locations, respectively. In addition to this, and denote the average number of bits an ONU sends to cloudlet and the average number of bits an ONU receives from cloudlet. These parameters are used to compute the total uplink and downlink transmission latency by the expressions and , where denotes the number of wavelengths ONUs share to communicate to their corresponding RN cloudlet, and denote the maximum available bandwidth (both in uplink and downlink) between ONUs and cloudlets at RN and CO, respectively, and and denote the background load in the uplink and downlink, respectively, of the considered TDM-PON.
Now, using the constraint (
3), moving denominator terms to numerator and rearranging all terms, we rewrite constraint (
5) as below:
where
Therefore, the reformulated optimization problem can be written as a
standard optimization problem as follows:
Proposition 1. The function is non-convex/concave but quasi-convex in nature.
Figure 2 shows the contours of
as a function of the variables
and
against different values of split-ratio
N and
= 1 ms. By carefully observing the contour plots, we can understand the quasi-convex nature of the curves. A more rigorous proof is given in the
Appendix A.
Theorem 1. To find the global optimal solution of the constrained optimization problem with objective function (2) and constraints (3)–(5), the first-order KKT conditions are necessary and sufficient.
Proof. This theorem can be proven in a straightforward manner. We observe that the objective function (
2) and the constraint (
4) are linear and the Proposition 1 shows that the constraint (
5) is quasi-concave. Therefore, if there exists at least one Lagrange multiplier that satisfy the KKT conditions, then any local optimum is the global optimum for this problem. Hence, the first-order KKT (necessary) conditions are the sufficient conditions for optimality [
41]. ☐
Therefore, we write the Lagrangian function for the objective function (
2) with constraints (
3)–(
5) as follows:
and derive the first-order KKT conditions for optimality from the Lagrangian function (
A1) as follows:
Theorem 2. The cloudlet cost optimization framework installs all cloudlets either at RN locations when , , or at CO locations when , . However, when both and are satisfied together, then the cloudlet installation location is chosen based on and .
Proof. From the KKT conditions (
10) and (
11), we observe that if
,
, then
or,
, and
. This implies that
and
. To keep the processing time finite, we must have
, i.e.,
and hence
. However, the latency constraint can be tight, which implies that
and
. Hence, we find the optimal solutions as below:
Therefore, we calculate the optimal cloudlet installation cost as follows, where all cloudlets are installed at RN locations (
,
):
Again, note that if
,
, then
and
, which implies that
and
. We have
and
, same as before. In addition,
and
. Thus, the optimal solutions are computed as follows:
These values yield the following optimal cloudlet installation cost, where all cloudlets are installed at CO locations (
,
):
When both and are satisfied together, we can freely install cloudlets at either of RN and CO locations, but both of these do not provide the cost optimal solution simultaneously, if (we consider ). Therefore, in this case, we should compare both and first, and then choose to install cloudlets at RN locations if , or CO locations if .
It is interesting to note that when
, then
holds but
does not have any unique optimal value. In this situation, we have
and the following optimal values:
Nonetheless, even in this situation, it is best to install cloudlets only at one location because installing cloudlets at two locations will lead only to a higher cloudlet installation cost as follows:
Clearly, the above cases show that optimal cloudlet placement framework should install cloudlets only at either of RN and CO locations, because and , always. ☐
6. Results and Discussion
In this section, we perform a parametric analysis and discuss some of the key insights obtained about the behaviour of the analytical cost optimization framework against variations in values of different network parameters.
In
Figure 5, we present the workload distribution among the RN and CO cloudlets against TDM-PON split-ratio over a circular area with diameter = 4 km, population density = 4000 people/km
, and
= 1 ms. With smaller split-ratio values, e.g., 1:4 and 1:8, the total workload can be processed by CO cloudlets, hence
and
. Thus, all cloudlets are installed at CO locations and none at RN locations. However, for a higher split-ratio of 1:16, there appears a bandwidth crunch for the ONUs to access the CO cloudlets. Therefore, under this condition, the cost optimization framework chooses to install cloudlet at RN locations over CO locations and the same is observed from the plot as well.
In
Figure 6, we show the variation of normalized cloudlet deployment cost/100 users against TDM-PON split-ratio
, where
with
= 1 ms. We vary the population density from 1000 to 4000 people/km
in steps of 1000. From these plots, we observe that the normalized cloudlet deployment cost decreases with increase in split-ratio from 1:4 to 1:8, because a lower number of TDM-PON branches are required to handle the workload and all cloudlets are installed at same CO locations. However, there is a slight increase in the normalized cost as the split ratio is further increased to 1:16. This happens because the framework chooses to install all cloudlets at RN locations, where the cost of cloudlet installation is higher than that of CO locations. We further observe that the normalized cost is higher for lower population density with smaller split-ratio, e.g., with split-ratio 1:4, normalized cost for 1000 people/km
is higher than that of 4000 people/km
. However, this behaviour is reversed with a higher split-ratio 1:16.
Figure 7 shows the variation of normalized cloudlet deployment cost/100 users against population density of the considered circular cloudlet deployment area. We vary the TDM-PON split-ratio
where
and
= 1 ms. We observe that the overall cloudlet installation cost increases with increasing population density, but the normalized cost/100 users slowly decreases for a particular split-ratio. Whenever a new TDM-PON is added, we observe a sudden jump in the normalized cost. In general, the normalized cost is minimum with split-ratio 1:8 and maximum with split-ratio 1:4.
Next,
Figure 8 shows the variation of normalized cloudlet deployment cost/100 users with population density 4000 people/km
against TDM-PON split-ratio
where
while varying
. These plots follow similar trends as we observed in
Figure 6. The normalized cloudlet deployment cost first decreases as split-ratio increases from 1:4 to 1:8, and then the normalized cost slightly increases again for the split-ratio 1:16. Note that the normalized costs for a stringent QoS requirement of
= 1 ms are highest and gradually continue to become lower as the QoS requirement is relaxed to
= 100 ms. This behaviour is observed due to the fact that a higher amount of computational resources (i.e., processors) are required to meet a stringent
= 1 ms than a lenient
= 100 ms.
Finally, in
Figure 9, we show a comparison of normalized cloudlet deployment cost/100 users against
= 1 ms with
homogeneous and
non-homogeneous network deployment. We choose the same circular area with a diameter of 4 km and population density 4000 people/km
. To simulate non-homogeneity, we choose a bunch of TDM-PONs and randomly vary their split-ratios such that their mean is equal to the corresponding homogeneous split-ratio
where
. For example, for a homogeneous split-ratio of 1:8, we uniformly choose split-ratios of the TDM-PONs from the interval
such that their mean is 8, and so on. We observe from our simulated results that the performance of the non-homogeneous framework is not worse than 5% of that with the homogeneous framework.