tipos penales vulnerados

In this section, we first introduce the system model and notations, and then define the problems precisely.

4.3.1 System model

Given an edge-cloud G = (V∪C,E) where V is the set of AP nodes, C is a set of cloudlets co-located with some of the AP nodes withC⊂V, and Eis the set of links

78 QoS-Aware Virtual Network Service Deployment in the Edge-Cloud

between AP nodes. Each cloudletcj ∈ Chas computing capacitycapjwith 1≤ j≤m andm = |C|. Assuming that time is divided into equal time slots. The amounts of available resources in each cloudlet in the beginning of each time slot vary, due to request admissions and departures. Letcapj(t)be the available computing capacity of cloudletcj at time slot twith 1≤t ≤T and 1≤ j≤ m, where Tis the monitoring period in terms of numbers of time slots.

Computing resource in cloudlets is used to instantiate a certain number of net- work functions in VMs, referred to as Virtualized Network Functions (VNFs), to implement offloading requests from mobile users. We thus assume that there is a set

F of VNFs. Denote by fi a type of VNF in F, where 1 ≤ i ≤ N and N = |F |. If the implementation of an offloading request with a VNF f_i ∈ F demands the basic resource unit of fi, we term this implementation as an instance of network function

fi for the request, or an VNF instance of fi; otherwise (if the implementation of the request needs x (≥ 1) times the basic resource unit of fi), we term that the request implementation takes x instances of fi. We further assume that each cloudlet has instantiated some instances of each type of VNF fi in each cloudlet already, and let

nij(t)be the number of instantiated VNF instances of fi in cloudletcj at time slott. Each mobile device can offload its tasks to its requested VNF instances in cloudlets inC, via the nearby AP of its user. LetS(t)be the set of user task offloading requests at time slot t. Each user request rk ∈ S(t)is represented by a tuple (lock,Fk,λk,dk), whereloc_k is the location of the user request that is usually an AP inV,F_k is the net- work service that rk requests, λk ≥0 is the packet rate ofrk, anddk is its end-to-end delay requirement. Specifically,the user of request r_k accesses the edge-cloud G via an AP, known as its location lock, and requests to offload his task to an instance of

F_k in some cloudlet of the edge-cloud. As his task usually needs to process a certain amount of data while being offloaded, his requestrk usually has a packet rate ofλk. Note that the value ofλk can be derived from historical information of similar type of user requests. We thus here assume that the packet rate of each request is given as a priori.

4.3.2 End-to-end delay of offloading requests

The end-to-end delay experienced by each admitted requestrk includes the queueing delay incurred while waiting for an available instance of its VNF, processing delay by its assigned VNF instance, instantiation delay of creating a new VNF when necessary, and network latency from its locationlockto its assigned cloudletcj. We here describe these four types of delays as follows.

queueing delay and processing delay: Each offloaded packet ofrk with packet

rateλ_kwill be queued in a cloudlet waiting for an instance ofF_kprior to its processing

by the instance, which will incur both queueing and processing delays when each packet passes through the VNF. To differentiate user requests with different delay requirements, the requests in each cloudlet ck are partitioned into N groups with each group consisting of requests for the same type of network function service. We thus assume that there is anM/M/nqueue at each cloudlet for each type of network function fi ∈ F. Each group of requests will be processed by the VNF instances of network function fi ∈ F, with 1 ≤ i ≤ N. The average queueing delay of the

M/M/nqueue for function fi at cloudletcj thus is

τ_ij(λ) = 1

nij(t)·µij−λ, (4.1)

whereλis the sum of packet rates of the requests assigned to cloudletcj that require VNF fi, and µij is the data processing rate of a VNF instance of fi in cloudlet cj. Given the data processing rate of VNF fi, its the processing delay is µ1ij.

Instantiation delay: Without loss of generality, we assume that the instantiation

delay of an VNF instance is a given constantdins_i for VNF fi.

Network latency: Assuming that data traffic in network G is transferred via a

shortest path between each pair of source and destination nodes, the network latency of requestrk thus is the accumulative delay incurred in the edges of a shortest path

plock,cj from its source location lock to its assigned cloudlet cj. Let d(e) be the delay of linkein network G. Denote by dnet_k the network latency experienced byrk via the

80 QoS-Aware Virtual Network Service Deployment in the Edge-Cloud

shortest path plock,cj, which can be calculated by:

dnet_k =

∑

d(e). (4.2)

The end-to-end delay Dk experienced by request rk for a VNF instance of fi in cloudlet cj depends on whether it has been assigned to an existing or a newly- created instance of fi. Specifically, if it is assigned to an existing instance of fi, there will be no instantiation delay for it. Otherwise, a new VNF instance for it needs to be instantiated, and no queueing delay is incurred since it is the first one to use it. Therefore, Dk can be calculated by

Dk =      τ_ij(λ) + _µ1 ij +d net k , ifnij(t)>0 dins_i + 1 µij +d net k , otherwise. (4.3)

The end-to-end delay requirement of each offloading requestrk thus is

Dk ≤dk. (4.4)

4.3.3 The admission cost

For each request r_k ∈ S(t) with a specified network function Fi

k (we include the superscript i to indicate that the requested VNF function Fk of request rk is fi), its implementation can either make use of existing VNF instances of f_i in cloudlet c_j if it joins in other already admitted requests that are making use of the VNF instances of fi. Specifically, let Rij be the set of offloaded requests with network function fi in cloudlet cj when rk is being considered for admission. If the admission of rk to Rij will not violate the delay constraint of any request in Rij∪ {rk}, the admission cost by admittingrkin cloudletcjthen is the cost sum of its data packet transmission cost (between its location via its nearby AP) and cloudletcjand its processing costc(F_ki)at

cj. Otherwise, we allocate the demanded resources forrk by creating more instances for Fi

k in cj, if there are sufficient available computing resources in cloudlet cj. The admission costw(rk)of rk per packet thus is the cost sum of its packet transmission

cost, the creation of new instances forrk, and its processing cost in cloudlet cj.

4.3.4 Problem definition

Given an edge-cloudG(V∪C,E), a set of user requestsS(t)at time slott with each request r_k having a given packet rate λ_k and an end-to-end delay requirement d_k,

with the assumption that the set of VNFs by the requests in S(t) is F, and some instances of each VNF inF have already been instantiated in the setC of cloudlets,

the operational cost minimization probleminGis to find a schedule of request admissions such that as many requests as possible can be admitted during a given time horizon

T, while the cumulative operational (implementation) cost of admitted requests is minimized, subject to the computing resource capacity constraint of each cloudlet in

C, and the end-to-end delay requirementdk of each admitted requestrk.

Theorem 4.1 The operational cost minimization problem in G(V∪C,E)is NP-hard.

Proof We consider an extreme case where there are only two cloudlets with iden-

tical computational capacity in the network. We assume that each request in S(t)

has a different network function service. We can ignore the delay requirement of each request. Our task is to assign the requests in S(t) to the two cloudlets to see whether all of the requests can be admitted. Clearly, for each request rk ∈ S(t), we need to create a VM for implementing its network function that is associated with computing resource demandck, subject to the computing capacity constraints on the two cloudlets.

We reduce the well-known summation problem to the mentioned assignment problem in polynomial time as follows. Given n positive integers a1,a2, . . . ,an, the summation problem is to partition thenintegers into two subsets such that the sum of integers in each subset is equal, which is NP-hard. As the special case of the minimum operational cost problem is equivalent to the summation problem, the operational cost minimization problem thus is NP-hard, too.

82 QoS-Aware Virtual Network Service Deployment in the Edge-Cloud