A Reliable Virtual Network Embedding Algorithm based on Game ...

IEEE ICC 2014 - Next-Generation Networking Symposium

A Reliable Virtual Network Embedding Algorithm based on Game Theory within Cloud’s backbone Oussama Soualah, Ilhem Fajjari† , Nadjib Aitsaadi and Abdelhamid Mellouk LiSSi, University of Paris-Est Creteil Val de Marne (UPEC): 122 rue Paul Armangot, 94400 Vitry-sur-Seine, France † VIRTUOR: 4 Residence de Galande, 92320 Chatillon, France [email protected], [email protected], [email protected], [email protected] Abstract—In this paper, we propose a new survivable virtual network mapping strategy within Cloud’s backbone enhancing the Cloud Provider’s revenue and dealing with physical failures of routers and links. In order to skirt the exponential complexity of the mapping, we propose a new reliable embedding strategy, denoted by CG-VNE, based on coordination game framework. To do so, we have formulated the problem as two interleaved coordination games. The first game addresses the virtual routers’ mapping. In fact, the actions of each virtual router player strongly depend on the mapping of its attached virtual links. Hence, the second game is launched to embed the virtual links. Note that with both games, fictitious players cooperate to reach Nash Equilibrium of which we have proven the existence and it corresponds to a social optimum. CG-VNE aims to maximise the Cloud’s provider revenue by maximising the acceptance rate of clients, as well as minimise the blackout rate of virtual networks caused by the outage of substrate routers and/or links. Based on extensive simulations, the results obtained show that CG-VNE has the best performance in terms of i) rejection rate of new clients, ii) Cloud’s revenue and iii) rate of clients impacted by physical failures.

Keywords: Cloud Computing, IaaS, Virtual Network Embedding, Reliability, Game Theory, Identical Interest Game. I. I NTRODUCTION Companies and end-users rely more and more on services (i.e., SaaS, PaaS and IaaS) offered by Cloud providers. However, the compliance of Service Level Agreements (SLA) between clients and the Cloud providers strongly depends on the availability of hardware resources. For instance, in North of America the average cost of unexpected failures in data centers is estimated to $5, 000 per minute [1]. In the context of Cloud computing, virtualization technology [2] plays a crucial role. In fact, thanks to the latter, many services (i.e., virtual machine) can coexist in the same hardware resource. Hence, hardware outage may simultaneously cripple the resources of many clients which dearly hurts the Cloud providers. In this paper, we focus on Infrastructure as a Service (IaaS) and more precisely on a Network as a Service (NaaS) in the Cloud’s backbone. Cloud Provider (CP) leases, within its backbone, virtual network resources to clients. In the context of NaaS, each client asks for Virtual Network (VN ) defined by its topology and the capacity of virtual routers and links (e.g., processing power, memory, bandwidth, etc.). In our previous work [3], we proposed a novel Preventive VN Embedding strategy denoted by PR-VNE based on Artificial Bee Colony metaheuristic. Indeed, during the embedding process of virtual routers and links, we take the reliability of the selected physical equipments into consideration. In other words, we favour the selection of equipments with high reliability rather than the old ones. However, the main drawback of PR-VNE is the absence of any curative mechanism (reactive and/or proactive). In fact, once the physical failure occurs, all the impacted VN s disappear from the Substrate Network (SN ) and 978-1-4799-2003-7/14/$31.00 ©2014 IEEE

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constrain the CP to pay the penalties due to the violation of the signed service level agreement. In this paper, we propose a novel Preventive VN embedding strategy which makes use of a reactive mechanism to repair the failures. It is named Coordination Game for Virtual Network Embedding (CG-VNE). In fact, our proposed resource allocation algorithm for CP aims to i) manage fairly the physical resources in the SN , ii) maximise the provider’s profit by accepting more new clients, iii) minimise impacted clients by physical failures and iv) re-map impacted VNs. To achieve our objective, we take the reliability of the SN (i.e., physical routers and links) into consideration during the VN embedding process. It is noteworthy that our proposal does not allocate any backup resource initially. However, it takes reactively action when a physical resource breaks down. It deals with all impacted virtual resources (i.e. nodes, links) by embedding them based on the same primary mapping process. As seen in [4], the VN embedding problem, regardless of the reliability, is a multi-objective and non-linear optimisation problem that has been proved a NP-hard problem [5]. In order to skirt the exponential complexity, our proposal CG-VNE makes use of Game Theory (GT) mathematical tool. Indeed, CG-VNE relies on the Coordination Game [6] to i) minimise the usage rate of substrate resources and ii) maximise the reliability. In other words, we formulate our reliable VN embedding problem as a cooperative game in order to motivate the players (i.e., decision makers) to collaborate together. In fact, based on the Identical Interest Game (IIG) [7], the players implicitly collaborate by sharing the same utility function in order to reach a Nash Equilibrium. The problem is formulated as two nested IIGs. The first game deals with the virtual nodes mapping and the second one addresses the virtual links’ embedding. In the former game, we consider that each virtual router is a fictitious player which is characterised by i) its initial embedding substrate node and ii) a set of actions (i.e., strategies). In our case, we define the actions set as the set of physical routers (e.g., 1-hop neighbours, etc.) in which the virtual router can move. To be able to estimate the action’s payoff of the fictitious virtual router player, we propose to run the second IIG in order to map its attached virtual links. The virtual link game is composed of fictitious players. Each one is characterised by i) a virtual link and ii) the substrate routers hosting the virtual routers connected by the aforementioned virtual link. Similarly to the virtual router game, each virtual link game player is typified by a set of actions and selects in each turn the best one in term of payoff function. It is worth pointing out that the set of actions of any virtual link player consists in a set of substrate paths connecting the player’s substrate routers and can host the virtual link. In other words, our proposal selects an action for the virtual router


player with respect to the actions performed by its attached virtual link players. We proved that our IIG is a Potential Game and hence a Nash equilibrium exists and corresponds to a social optimum. To reach it, CG-VNE makes use of Best Response algorithm [8] which ensures the convergence to a Nash equilibrium. Based on extensive simulations, we gauge the performance of CG-VNE compared with the related virtual network embedding algorithms. The results obtained show that our proposal is better in terms of i) rejection rate of new VN s, ii) rate of VN s impacted by failures and iii) Cloud provider’s revenue. The remainder of this paper is structured as follows. In the next section we will summarise the related work addressing the reliable VN mapping problem. Then in Section III, we will formulate our problem as an Identical Interest Game. In Section IV, we will describe our proposal CG-VNE. Afterwards, we will evaluate the performance of our proposal and some prominent related strategies. Finally, Section VI will conclude the paper. II. R ELATED WORK In this section, we will summarise the main virtual network embedding strategies based on a Game Theory mathematical tool. We will then enumerate the main related survivable VN mapping algorithms. To the best of our knowledge, in literature, only two research papers deal with virtual network embedding problem while making use of Game Theory. In [9] [10], the authors formulate the virtual network embedding problem as a non-cooperative game. This game is designed between a set of VN s which behave in selfish manner to get the residual bandwidth in the SN . We notice that in both papers, the authors did not take into consideration i) the reliability of SN and ii) the substrate and virtual router capacities (i.e., memory and processing power). In fact, only the bandwidth allocation is considered. Moreover, the proposal has been validated in small-size SN which is not realistic. Accordingly, the authors did not study the proposal scalability. We notice that the rejection rate of VN requests is not evaluated which is a very important metric for provider’s business. Finally, the authors did not compare their proposal with the prominent related strategies. In literature, reliable VN mapping algorithms can be classified into two groups: i) centralised and ii) distributed. However, distributed approaches [11] are not attracting the community because of the current routers and substrate network architecture which can not support multi-agent applications. Hereafter, we describe prominent centralised algorithms handling node and/or link failures. A. Link failure protection In [12], the authors proposed a Survivable Virtual Network Embedding (SVNE) algorithm dealing with substrate link failures. The proposed policy is based on a fast re-routing strategy and uses a pre-reserved quota for backup on each physical link. In fact, in order to protect the system against a single substrate link failure, SVNE dedicates a fixed percentage of bandwidth resources on each substrate link for backup. The heuristic consists of three separate stages. In the first stage, before the arrival of any VN request, the CP proactively computes a set of possible backup detours for each substrate link using a path selection algorithm. Then, during the second stage which is invoked when a VN request arrives, the authors make use of D-VINE [13] to embed VN requests. Finally,

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if a substrate link failure occurs, a reactive backup detour optimisation is performed. It reroutes the affected bandwidth along candidate backup detours which were selected in the first stage. The authors formulated the problem of the second and the third stages as a linear program, they aim to respectively reduce the bandwidth consumption and the penalties caused by link failures. We notice that the authors implicitly assume that substrate nodes remain operational perpetually, which is not realistic. Moreover, the authors did not evaluate the recovery rate of impacted virtual resources. In [14], the authors proposed an algorithm named by RMap which combines VN mapping and substrate link backup. The proposed solution ensures some level of reliability against substrate link failure. In fact, RMap calculates the backup for the links that are characterised by a stress value higher than a defined threshold. Note that the link stress parameter reflects the number of virtual links embedded into the substrate edge. RMap operates as follows: i) mapping the VN , ii) computing the backup and iii) migrating virtual links when substrate link failure occurs. Based on simulations, the authors focus on the recovery rate metric of impacted virtual resources. The results obtained show that RMap approach can reach 70% of virtual link protection. We notice that authors deal only with link protection. However, substrate node failure is very critical for operating virtual networks. Besides, RMap ensures protection only for stressed links. Unfortunately, the non stressed ones are not protected. Thus, if a failure happens on one non-stressed link, all virtual networks using this substrate link will be affected. Finally, the authors do not compare RMap with already defined strategy dealing with links protection. B. Node failure protection The authors proposed in [15] an Opportunistic Redundancy Pooling (ORP) mechanism to benefit from the properties of the virtualized infrastructure. This new technique ensures reliability on critical routers thanks to backup resources. In fact, it is based on sharing redundant resources between Virtual Infrastructures (VInf). Since redundant routers are idle, those provisioned for one VInf can be reused by others. Accordingly, the number of redundant resources is minimised. The authors augment each VInf request with backup links and nodes. Then, the problem is formulated as a Mixed Integer Programming. The authors refer to Multi-Commodity Flow (MCF) problem which is formulated in [13] to jointly map routers and links. Afterwards, the mixed integer problem is solved using the open-source CBC solver. To evaluate the performance of ORP strategy, the authors compare it to a reliable approach but without sharing redundancy resources between VInfs (No-share). Besides, ORP is compared with the baseline solution in order to gauge the additional amount of resources consumed for reliability. We notice that ORP does not deal with link failures. However, assuming that links remain operational is not realistic. Moreover, the authors did not explain how the matching should be between a critical node and its associated backup, especially, when the number of redundant nodes is less than the number of critical nodes. Indeed, the authors did consider simultaneous fails. III. P ROBLEM FORMALISATION In this section, we formulate the reliable VN embedding problem. We will first describe the models of i) networks (i.e., substrate and virtual) and ii) the physical equipment (i.e., routers and links) failures. Then, we will detail the formalisation


problem, studied in this paper, based on Game Theory. We will clearly define the players, the payoff and utility functions. A. Network and physical failure models As in our previous work [3], we denote the VN by D and the SN by G. Each substrate equipment (i.e., router or link) is typified by its : i) reliability, ii) bandwidth, iii) processing power and iv) memory. Moreover, we assume that the resources previously mentioned in G are limited. Hence, G cannot host an infinite number of VN requests. Besides, the physical equipment reliability is inversely proportional to its age. In other words, the probability of physical failure increases with time. Consequently, a knowledgeable and smart VN embedding in G is necessary to maximise the acceptance rate of VN requests as well as the rate of recovered VN s impacted by physical failures. Thus, the CP’s revenue is maximised by accepting more clients and paying less penalties due to outage. The SN is formulated as an undirected graph denoted by G = (V (G) , E (G)) where V (G) and E (G) are respectively the sets of physical routers and their connected links. Each physical router, w ∈ V (G), is characterised by its i) residual processing power B (w), ii) residual memory M (w), iii) type: access or core X (w) and iv) reliability Rn (w, t) at time t. Note that if X (w) = 1, then w is an access router. Otherwise, X (w) = 0. Likewise, each physical link, e ∈ E (G), is typified by its i) residual bandwidth C (e), ii) capacity bandwidth Cˆ (e) and iii) reliability Rl (e, t) at time t. We model the Mean Time Between Failures (MTBF) as a Weibull distribution [16]. Accordingly, we can estimate the equipment’s reliability at any time. The Cumulative Distribution Function (CDF) of the MTBF is expressed as following: x b ), x ≥ 0 (III.1) F (x) = 1 − exp(− a where a and b are the parameters of the Weibull distribution. It is worth pointing out that we assume heterogeneous ages in the physical network. This means that any equipment in G has its own initial age respectively denoted by A (w) and A (e) for substrate node w and substrate link e. Consequently, we classify the equipments within the SN in three groups: i) young, ii) adult and iii) old. Note that Rn (w, t) = F (A (w) + t) and Rl (e, t) = F (A (e) + t) where t is the current simulation time. As well, a VN request is formulated as an undirected graph, denoted by D = (V (D) , E (D)) where V (D) and E (D) are respectively the sets of virtual routers and their virtual links. Each virtual router, v ∈ V (D), is characterised by its i) required processing power B (v) and memory M (v) and ii) its type X (v). In addition, each virtual link d ∈ E (D) is characterised by its required bandwidth C (d). B. Problem statement based on Game Theory As in our previous work [3], we focus on the mapping of the virtual graph D in the substrate graph G while considering the i) reliability of G and ii) virtual network embedding constraints [17], for instance: i) edge/core routers, ii) residual capacity of links/routers, iii) unsplittable substrate path, etc. In this paper, our objective is to i) minimise the rejection rate of VN requests, ii) minimise the number of embedded VN s impacted by physical failures and iii) deal with physical failures as soon as they occur. To do so, we formulate the reliable VN embedding problem as a game. In fact, Game Theory may be defined as the study of mathematical models interaction between

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rational decision makers (i.e., players) [18]. A game Γ (N , S, U) is defined by three parameters: 1) N = {N1 , · · · , Np }: Set of players. 2) S is the action profile for all players N . It is equal to S = Ni ∈N SNi . Note that SNi is the action set of player Ni . 3) U is the vector of game’s utility function. U is equal to UN1 , · · · , UNp . Note that UN1 is the utility function of player Ni defined as, UN1 : S → R. We formulate our problem as a coordination game [6] where all fictitious players {Ni } collaborate together to reach a Nash Equilibrium. It is worth noting that the collaboration between {Ni } is motivated by an Identical Interest Game (IIG) [7]. In fact, the decision makers {Ni } share the same utility function. Formally, ∀Ni , Nj ∈ N : UNi = UNj . This can be justified by the fact that all players N must play on behalf of the Cloud Provider in aim to increase its revenue. We recall that our problem consists in mapping virtual routers and virtual links. Besides, we notice that the mapping of virtual links strongly depend on virtual routers embedding. In fact, these virtual routers correspond to the sources and destinations of virtual links. Based on this observation, we propose to model the reliable VN embedding problem as two nested (i.e., interleaved) games. The first one Γ1 deals with the virtual routers. During each round of the virtual router game, we run the second game Γ2 dealing with virtual links embedding. Hereafter, we define the parameters for both games. In the first game Γ1 N 1 , S 1 , U 1 , we define a set of fictitious players N 1 corresponding to the set the virtual routers V (D). We assume that each virtual router player has an initial mapping 1 in the substrate graph G. The actions set SN 1 of each fictitious i 1 player Ni is composed of the K-hop neighbourhood of the current substrate router hosting the player Ni1 and a set of substrate routers randomly selected in V (G). It is worth pointing 1 out that all substrate routers sNi1 ∈ SN 1 i) have sufficient i 1 resources (i.e., B Ni ≤ B sNi1 and M Ni1 ≤ M sNi1 ) and are thus able tohost the player and ii) have the same type 1 (i.e., X Ni1 = X sNi1 ). The payoff function UN 1 of each i 1 player Ni depends on the i) remaining resources (i.e., memory and processing power) in the substrate node wNi1 hosting the player, ii) its reliability and iii) the payoff earned by its attached virtual links. Formally,

2 1 n w 1 1 R − 1 U + = (w , T ) · ∇ UN exp 1 Ni Ni d d∈ξ N 1 Ni1 i i (III.2) where Ud2 is the payoff function of the virtual link d, ξ Ni1 is the set of attached virtual links to the virtual router player Ni1 , Rn (wNi1 , TNi1 ) is the reliability of the substrate router w hosting the virtual router player at time TNi1 . Note that TNi1 corresponds is the to the departure time of the virtual network D , ∇w Ni1 residual resources of substrate router w after the mapping of virtual player Ni1 . The utility function 1is equal to: U 1 = N 1 ∈N 1 UN 1 (III.3) i i The second game Γ2 N 2 , S 2 , U 2 deals with the virtual link embedding in a social way. In fact, we define a new set of fictitious players N 2 . Each player Ni2 corresponds to one virtual link d ∈ E (D) and the two substrate routers hosting its virtual router extremities. Players implicitly collaborate together by sharing the same utility function. Each virtual link player Ni2 is


2 characterised by the actions set SN 2 . The latter is composed of a i set of substrate paths connecting the two physical routers hosting 2 the virtual router extremities. Note that each path sNi2 ∈ SN 2 i 2 2 can host the virtual link player Ni (i.e., C Ni ≤ C sNi2 ). 2 2 The payoff function UN at an 2 of a virtual link player Ni i 2 2 action (i.e., substrate path) sNi ∈ SN 2 depends on i) the i residual bandwidth of sNi2 and ii) the reliability offered by sNi2 . Formally,

2 + exp 1 + Rp (sNi2 , TNi2 ) exp 1 + C sNi2 UN 2 = i − length sNi2 − length DijkstraNi2 × C Ni2 (III.4) where i) Rp (sNi2 , TNi2 ) is the reliability offered by the substrate path sNi2 at TNi2 when the virtual network D leaves the

Hereafter, we enumerate two interesting properties of a potential game: • Every finite (i.e., cardinal of actions |S| is finite) potential game has at least one pure strategy to reach a Nash Equilibrium [19]. • All Nash Equilibria are either local or global maximisers of the utility function U [19]. So we can conclude that NE exists in our reliable VN problem. Moreover, it corresponds to a social optimum which is our main objective. In the next section, we will detail our proposal CG-VNE that reaches the Nash Equilibrium of both games Γ1 N 1 , S 1 , U 1 and Γ2 N 2 , S 2 , U 2 . IV. P ROPOSAL : CG-VNE In this section, we describe our algorithm CG-VNE that ensures the to NE convergence of the above games Γ1 N 1 , S 1 , U 1 and Γ2 N 2 , S 2 , U 2 . We recall that in the previous section, we formulated our reliable VN embedding problem as two interleaved games Γ1 and Γ2 . Besides, we proved the existence of (NE) which corresponds to a social optimum. To reach a NE, we make use of Best Response (BR) [8] algorithm. In fact, BR ensures the convergence to a steady state (i.e., NE) which matches in our case with a social optimum. Each player Ni runs Best Response at each turn by selecting the best action in its action set SNi . To reach a NE, we run the generic Algorithm 1 based on BR scheme. It operates as follows. First, for each player Ni , it determines its action set SNi . Then, the turn of each player Ni is determined. Afterwards, each player Ni , during its turn, selects the best action in SNi which maximises its payoff function UNi (i.e., run the BR algorithm). Once all the players N = {Ni } played, the utility function U is evaluated. If the gap between the previous value and the new one is less than a predefined threshold Γ , we can conclude that NE is found. Otherwise, the same process is repeated from the turn determination stage. It is worth noting that in our case, if a NE is reached, a social optimum is obtained.

physical backbone network G, ii) length sNi2 corresponds to the number of hops of sNi2 and iii) DijkstraNi2 is the shortest path hosting the virtual link player Ni2 . We recall that the capacity C sNi2 of the substrate path sNi2 is equal to the minimum residual bandwidth of its substrate links. Besides, the reliability Rp (sNi2 , TNi2 ) of the substrate path sNi2 is equal to the product of reliabilities of all its substrate links and nodes. The utility function is equalto: 2 U 2 = N 2 ∈N 2 UN 2 (III.5) i i It is quite straightforward to see that the reliability of VN 1 2 is maximised thanks to the payoff functions UN 1 and UN 2 . In i i fact, the latter functions are proportional to the reliability of selected substrate resources. Moreover, the acceptance rate of new VN requests is considered in both payoff functions by 1 including the amount of residual resources. Similarly, UN 1 and i 2 UN 2 are proportional to the residual resources in the SN . By i doing so, we avoid the congestion of substrate resources and we maximise the load balancing of their usage rate. The main objective of any game strategy is to reach a Nash Equilibrium (NE). NE is a set of actions, one for each player Ni , such that no player has incentive to unilaterally change its action. In other words, NE is a steady state where each player cannot improve its benefit anymore by individually deviating its choice, assuming that other players keep constant their strategies. Formally, a strategy S ∗ = (s∗N1 , . . . , s∗Np ) ∈ S is a NE if: ∀ Ni ∈ N , ∀ sNi ∈ SNi : ∗ ∗ UNi s∗Ni , S−N ≥ UNi sNi , S−N i i ∗ = S ∗ \s∗Ni . where S−N i However, the existence of NE is not guaranteed. Hereafter, we will prove that our model contains at least one Nash Equilibrium. As exposed above, our reliable VN embedding is formulated as an Identical Interest Game (IIG). It is worth noting that an IIG is a special case of a Potential Game [19]. The proof of this lemma is straightforward. In fact, the potential function can be defined equally to the common utility function shared by all players. A Potential Game is interesting if we can define a function that can express the player deviation. Formally, a Potential Game is characterised by a Potential function φ satisfying:

φ:

R SN 1 × · · · × S N p → 1 ∈ N , ∀ s , s2Ni ∈ SNi : where ∀ N i Ni 1 2 1 2 φ sNi , s−Ni − φ sNi , s−Ni = UNi s1Ni , s1−Ni − UNi s2Ni , s2−Ni

Algorithm 1: Calculation-of-NE 1 Inputs: Γ (N , S, U ) ∗ ∗ ∗ ∗ 2 Output: s = (sN , · · · , sN ) is a NE, s ∈ S 1 p 3 Determine the action set SNi of each player Ni ∈ N 0 4 Calculates the initial payoff UN of each player Ni i 0 0 5 U ← Ni ∈N UNi 6 repeat 7 {Nj } ← TurnPlayers (N ) 8 for j ← 1 to |N | do 9 s∗Nj ← {s∗Nj ∈ SNj : UNj (s∗Nj ) = maxsNj ∈SNj UNj (sNj ) } ∗ 10 UN ← UNj (s∗Nj ) j ∗ 11 U ∗ ← Nj ∈N UN j ∗ 0 12 tmp ← U − U 13 U0 ← U∗ 14 until tmp ≤ Γ ;

(III.6)

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We recall that the first game Γ1 N 1 , S 1 , U 1 deals with the mapping of virtual routers. However, the virtual router’s payoff 1 UN 1 (see equation III.2) depends on the payoff of its attached i virtual links. To map the virtual link, we resolve the game Γ2 N 2 , S 2 , U 2 . As we can observe, in that sense both games


Γ1 and Γ2 are interleaved. The pseudo-code of our proposal is illustrated in Algorithm 2. Algorithm 2: CG-VNE Inputs: G = (V (G) , E (G)), D = (V (D) , E (D)) 2 Output: Mapping of D in G 3 Determine the initial embedding of D 1 1 4 Build the player set of Γ : N ← V (D) 1 1 1 5 Determine the action set S 1 of each player Ni ∈ N Ni 1 0 1 6 U0 ← Ni ∈N UNi (Ni ) 7 repeat 8 {Nj1 } ← TurnPlayers (N 1 ) 9 for j ← 1 to |N | do 1 1 10 for each Nj,k ∈ SN 1 do j 11 Build the player set of second game Γ2 : N 2 ← Nj1 ’s attached virtual links 2 12 Determine the action set SN 2 of each player k 2 2 Nk ∈ N Calculation-of-NE Γ2 N 2 , S 2 , U 2 13 1

14

15 16 17 18

1∗ 1 Select the best candidate Nj,k ∈ SN 1 which j 1 maximises the payoff function UN 1 (The maximum j 1 is equal to Ubest (Nj1 )) 1 1 Ubest ← N 1 ∈N Ubest (Nj1 ) 1 j 1 tmp ← U0 − Ubest 1 U01 ← Ubest until tmp ≤ Γ1 ;

1 Note that the candidates of each virtual router player SN are i formed by the H-Neighbourhood of the initial substrate router hosting the virtual player Ni and a subset of substrate candidates randomly selected in order to avoid local optimum. By doing so, we favour the exploration of both the neighbourhood and the distant solutions. Concerning the candidates of virtual link paths, we generate L first K-Shortest path between the source and the destination. Finally, in both games Γ1 and Γ2 , the players’ turn (i.e., TurnPlayers procedure) follows a discrete uniform distribution. CG-VNE handles reliability based on two steps. The first one is preventive and executed during the selection of candidates S 1 and S 2 . In fact, payoff and utility functions of the two nested games Γ1 and Γ2 promote the choice of resources having higher reliability (i.e., lower failure probability). Accordingly, our proposal prevents the breakdowns by avoiding non reliable substrate resources (i.e., routers and links). The second step is a reactive mechanism. Once a failure occurs, CG-VNE performs a recovery process. Indeed, our algorithm re-embed the impacted virtual resources (i.e., routers, links). We make use of the same games Γ1 and Γ2 to embed the impacted virtual resources. The reactive mechanism is motivated by the optimisation of substrate resource consumption. In fact, we alleviate the network to dedicate backup resources that might not be used. To ensure the recovery, we design a central substrate controller [20] which is used to save checkpoints of all virtual resources. V. P ERFORMANCE E VALUATION

In this section, we will assess the effectiveness of our proposal CG-VNE based on extensive simulations. We will start by describing our discrete event simulator while considering the reliability of the SN and equipments’ ages. Then, we will define

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our performance metrics. In fact, these metrics aim to compare our proposal with related prominent survivable embedding virtual network strategies: i) SVNE [12], ii) RMap [14] and iii) our previous proposal PR-VNE [3]. Finally, we will gauge the effectiveness of our proposal. A. Simulation Environment We implemented a discrete event reliable VN mapping simulator. To this aim, we make use of GT-ITM [21] tool to generate random SN and VN topologies. Note that we model the VN requests’ arrival by a Poisson Process with rate λA . We model VN lifetime by exponential distribution with mean μL , too. As stated in our previous work [17] [3], we set the SN size to 100. In this case, the ratio of access and of core routers are respectively fixed to 20% and 80%. Moreover, we set VN size basing on discrete uniform distribution, using values given between [3, 10]. We keep the same proportion of access virtual nodes like SN 20% for each VN . It is worth mentioning that in both cases (VN and SN ), each pair of routers is randomly connected with a probability of 0.5. The arrival rate λA and the average lifetime μL of VN s are respectively set to 4 requests per 100 time unit and 1000 time units. We calibrate the capacity of substrate nodes and links (i.e., B (w), M (w) and Cˆ (e)) according to a continuous uniform distribution taking values in [50, 100]. Furthermore, we set asked virtual resources (i.e., B (v), M (v) and Cˆ (d)) based on a continuous uniform distribution, using values between [10, 20]. Regarding SN reliability, the parameters a and b of the Weibull distribution modeling the MTBF are respectively set to 25 × 104 and 1.5. It is noteworthy that a and b are calibrated in order to obtain a lifetime for each physical equipment (≈ 6 × 105 ) roughly equal to 10 times the simulation duration (≈ 6 × 104 time units). Moreover, the proportional of young equipments in SN is set to 30%. We study the performance with respect to the adult proportion (hence old proportion) in the SN . Besides, physical equipment’s initial age follows a uniform distribution within the bounds of its group (i.e., young, adult and old). In our simulator, we implemented CG-VNE alongside the related strategies: i) PR-VNE ii) SVNE and iii) RMap. We set the number of VN requests to 2000. We respectively fix the parameters H, L, Γ1 , Γ2 of CG-VNE to 1, 15, 0.1 and 0.001. It is worth pointing out that each performance value of pseudo-random strategies is equal to the average of 15 simulations. Furthermore, simulation results are always presented with confidence intervals corresponding to a confidence level of 95%. Note that tiny confidence intervals are not shown in the following figures. B. Performance Metrics Hereafter, we define performance metrics used to gauge CG-VNE. 1) Q: is the reject rate of VN requests during the simulation. In other terms, the rate of VN s that have not been mapped in the SN . 2) F: is the rate of deployed VN s that has been impacted by physical failures during the simulation. In fact, a VN is impacted if at least one of its components (i.e., virtual router and/or link) is impacted by the physical failure. 3) G: evaluates the SN provider’s profitability by calculating the total benefit G1 of all accepted VN requests minus the paid penalties G2 to the clients induced from the physical failures during all the simulation lifetime (G = G1 − G2 ). G1 is defined as in [13]. It depends on the amount of requested


Cumulative failure rate of VN

Reject rate of virtual networks

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PR-VNE SVNE RMAP CG-VNE

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0.01 0.35

0.4 0.45 0.5 Rate of adult physical equipments

Fig. 1.

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Reject rate - Q

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Fig. 2.

resources and the lifetime of the virtual network in the backbone network SN . We define the penalty G2 as the reimbursement to the clients. The cost is proportional to the residual time of a VN heightened by a penalty δ. Formally, for each VN denoted by D: T) (V.7) × G1 (D) × δ G2 (D) = (T −Δ T where ΔT is the hosting duration in the SN before the physical failure and δ is the penalty rate. In our simulations, we set the penalty to 50%.

Cumulative Gain (Penality = 50%)

0.3

C. Evaluation Results In Fig. 1 we compare the reject rate Q of VN requests obtained by both of our proposal CG-VNE and the related strategies. We notice that our proposal outperforms the related strategies only when the proportion of adult equipments is larger than 45% hence the old equipment is less than 25%. We recall that the young proportion is fixed to 30%. The main rational behind this results consists in the strategy of resource allocation during the recovering of failures. In fact, CG-VNE maintains a VN in the Cloud’s backbone even a physical failure occurs if it is able to re-allocate the impacted virtual resources. We recall that it was not the case for PR-VNE. If a physical failure impacts at least one equipment within a virtual network, the whole resources dedicated for it are released. In other words, PR-VNE does not react during physical outages and all resources dedicated to the impacted VN s are released. It is worth pointing out that CG-VNE rejects less than RMap and SVNE even if they do not react to outages. In Fig. 2, we evaluate the impact rate F of unrepaired VN s caused by physical failures. It is straightforward to see that CG-VNE outperforms the rest of strategies thanks to the recovery mechanism and the prevention performed during the mapping (see payoff functions). For example, when the rate of adult equipments is equal to 30% which corresponds to a portion of old equals to 40%, CG-VNE reduces by approximately 85% the rate of VN s crashed due to outages. We can remark that the reduction of impacted VN s is more noticeable with respect to the adult population rate. The main reason is the reduction of failures in the SN . In fact, when the adult population increases, that means SN is younger. In Fig. 3 we evaluate the CP’s revenue G obtained at the end of simulation. We notice that G strongly depends on the acceptance rate of VN s (see Fig. 1) and the penalties caused by the physical crashes (see Fig. 2). Furthermore, it should be

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Fig. 3.

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reminded that the total amount paid by penalties is less than the benefit earned by accepting new clients. However, in this paper we have not evaluated the impact of Cloud provider’s bad reputation on the arrival rate of Poisson process. We assume that even if PR-VNE impacts more clients, the Cloud provider’s reputation does not degrade which is not realistic. It is straightforward to see that even the impact of CP’s bad reputation on the arrival rate of clients is not considered, our proposal CG-VNE has approximately the same gain. We believe that the consideration of frequent crashes of Cloud provider will increase the gain of our proposal. It is worth noting that our proposal improves the reputation, hence attracts more new clients. In Fig. 4, we illustrate the rate of virtual routers impacted by physical outages. We notice that the plot of our proposal CG-VNE is not shown since it is equal to 0 (i.e., 0 cannot be illustrated in log scale of y-axis). In fact, CG-VNE succeeds to recover all virtual nodes impacted by substrate router outages. In Fig. 5, we illustrate the rate of virtual links impacted by physical failures (router and/or links). We notice that our proposal CG-VNE deeply reduces the rate of impacted virtual links compared to the related strategies. For instance, when the rate of adult equipments is equal to 30% (i.e., the oldest configuration), CG-VNE reduces by approximately 88% the rate of impacted virtual links. We can claim that the virtual games Γ1 and Γ2 are very efficient. At the last but not the least, we compare our proposal by disabling the reliability objective. We assume that SN remains working without any outages.

Cumulative failure rate of Virtual router


tive algorithm named CG-VNE. To the best of our knowledge, we are the first to deal with the reactivity in VN environment. Our proposal is based on Game Theory. We designed two nested IIGs to address reliable VN embedding complexity. This model aims to maximise the Cloud provider’s revenue and minimise the blackout request rate caused by SN failures. Extensive simulations show that the performance of CG-VNE are better compared with the prominent related strategies in terms of i) rejection rate, ii) CP’s revenue and iii) rate of VNs impacted by substrate outages. ACKNOWLEDGMENTS This work is supported by ”La Direction Générale de la Compétitivité, de l’Industrie et des Services, France” within the CELTIC + TILAS projcet. R EFERENCES


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To do so, we evaluate the reject rate of VN requests and the revenue obtained by our proposal CG-VNE and the most related strategies found in literature. We make use of the same scenario used in our previous work [17]. The results obtained are illustrated in Table I. Based on extensive simulations, CG-VNE outperforms all the related virtual network embedding strategies , except D-VINE [13]. In fact, CG-VNE rejects 1.01±0.13% of VN requests and D-VINE only 0.65%. In fact, the latter stagey handles link mapping in splittable way which is not the case of our proposal. Accordingly, D-VINE is more likely to accept more clients. Despite of this, CG-VNE achieves a good level of performance which are very close to splittable based approach D-VINE’s one. VI. C ONCLUSION In this paper, we studied the reliable VN embedding problem that is NP-hard problem. Accordingly, we proposed a new reacTABLE I C OMPARISON WITHOUT SUBSTRATE RELIABILITY CONSTRAINT Algorithm CG-VNE PR-VNE [3] D-VINE [13] VNE-AC [17] VNE-Greedy [22] VNE-Cluster [22] VNE-Least [22] VNE-Subdividing [22]

Reject Rate (%) 1.01 ± 0.13 8.11 ± 0.84 0.65 4.56 ± 0.40 12.95 69 73.90 67.45

Revenue (× 10000) 49.54 ± 0.16 44.30 ± 0.32 49.85 47.70 ± 0.26 42.51 13.47 10.05 10.85

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