Uwe Bauknecht
email: uwe.bauknecht@ikr.uni-stuttgart.de

Tobias Enderle
email: tobias.enderle@ikr.uni-stuttgart.de

Arthur Witt
email: arthur.witt@ikr.uni-stuttgart.de

Reduction of Delay Overfulfillment in IP-over-DWDM Transport Networks

Keywords: Delay, ILP, Network reconfiguration, Optimization, Overfulfillment, QoS

The traffic in today's transport networks is increasing dramatically due to more demanding applications like video on demand and improving access technologies like 5G. Additionally, the quality of service requirements are becoming more stringent while network operators are seeking new ways for revenue creation. We propose a multi-layer network reconfiguration approach that reduces the overfulfillment of service delay requirements. In that way it provides an incentive to customers with low-delay services to acquire a more expensive service class for their traffic. Additionally, it relieves highly utilized links for services with very strict delay requirements. We provide an ILP formulation that solves this multi-layer network problem by performing a cross-layer optimization. Further, we evaluate our approach for two nationwide backbone networks. We show that a reduction of service delay overfulfillment is possible and how that affects other network metrics.

Introduction

Since its inception, the Internet has changed significantly from a best-effort network of low-rate data exchange services into a ubiquitous and indispensable service delivery infrastructure for individuals and businesses alike. Trends like software as a service and video on demand, fueled by more potent access technologies such as FTTx and 5G, allow Internet-based business models such as cloud and content delivery services to thrive. These so-called over-the-top (OTT) services rely on the availability and quality of the underlying connection services rendered by Internet service providers (ISPs). Therefore, OTT providers require connection services that provide higher data rates and also adhere to guarantees on quality of service (QoS) metrics codified in service level agreements (SLAs). In order to ensure their fulfillment and avoid contract penalties, ISPs typically resort to overprovisioning QoS parameters on such SLA-based services. This practice translates to the ISPs' multi-layered core networks as well, where the packet forwarding layer and the optical transport layer are both subject to overdimensioning and further safety margins.

Matching the continuous growth in traffic volume of currently 26 % annually1 [START_REF]Cisco: Visual Networking Index: Forecast and Trends[END_REF] becomes increasingly expensive, especially due to improvements of core network transmission equipment approaching the limits of the deployed fiber infrastructure [START_REF] Ellis | Performance limits in optical communications due to fiber nonlinearity[END_REF]. To avoid or at least defer costly investments for commissioning new fibers, ISPs try to reduce operational costs and explore new avenues of revenue creation. Important means towards cost reduction are improving the utilization of existing equipment and reducing safety margins. Several works have been published which address this and related issues. For example, the authors of [START_REF] Lourenço | Running the Network Harder: Connection Provisioning Under Resource Crunch[END_REF] introduce an approach which increases the network utilization by exploiting the fact that some service requests can temporarily tolerate lower bandwidth. This frees resources in high load situations and allows more services to be accepted in the long run. In [START_REF] Bauknecht | Resource Efficiency and Latency in Dynamic IP-over-WSON Networks utilizing Flexrate Transponders[END_REF] we presented a latency-aware network reconfiguration approach that minimizes the amount of active network hardware. In [START_REF] Pointurier | Design of Low-Margin Optical Networks[END_REF], ways to reduce safety margins that compensate physical layer impairments and aging are discussed. Techniques like these can be employed by a software-defined networking (SDN) approach which exploits global knowledge of the network state to dynamically adjust the network's configuration such that light path and traffic routing precisely fit the QoS requirements of the services to be realized and thereby reduce overprovisioning [START_REF] Velasco | In-Operation Network Planning[END_REF].

In addition to cost reduction, a more differentiated SLA portfolio can help increase an ISP's revenue [START_REF] Walrand | Economic Models of Communication Networks[END_REF]. On the side of individual end customers with low-rate best-effort contracts, ISPs have already instated so-called "data caps", where the bandwidth is throttled by rate shaping if a customer exceeds a predetermined data volume. This way, they capitalize on heavy users by offering uncapped services at increased rates [START_REF] Sen | A Survey of Smart Data Pricing: Past Proposals, Current Plans, and Future Trends[END_REF]. For business clients who use their connections beyond the limits of their SLA contracts up to the maximum overprovisioning, rate shaping is also an option to control the data rate. Other QoS parameters are harder to limit with inexpensive measures, leaving business customers with little to no incentive to acquire contracts with more stringent guarantees, given that they are well aware of their contracts' overfulfillment. Among these parameters especially delay requirements are imperative to the aforementioned services.

Sanctioning customers by actively limiting delay overfulfillment could be realized on both, the packet layer and the optical layer. On the packet layer, incoming traffic could simply be queued in buffers for a preconfigured amount of time. However, since the buffer size necessary for delaying scales directly with the data rate and typical line cards have limited buffer memory, additional hardware would be needed given the large data rates of some business customer connections. Moreover, this also raises concerns when considering availability requirements since additional active hardware also harbors additional points of failure. On the optical layer, latency can be artificially introduced by simple fiber delay lines. Due to the low cost and deterministic delay, this approach seems superior, but also requires custom delay lines for each service and a more complicated fiber routing to match services to delays.

Rather than introducing additional hardware to artificially reduce quality, a differentiated service routing on the existing infrastructure enables reduced operational cost, provides means of sanctioning customers and allows for increased revenue services by exploiting the overprovisioning of existing hardware. Traffic of customers under sanction can be shifted to physically longer and therefore delay-inducing paths where switching equipment is otherwise underutilized while the shorter routes can be dedicated to premium services increasing revenue.

The main part of this work is organized as follows. In Section 2, we formulate the considered network architecture and reconfiguration problem. In Section 3, we describe our proposed path generation and optimization approach. Simulation setup and results are presented in Section 4. Eventually, we conclude the work in Section 5.

Problem Statement

We consider a two-layer transport network which receives aggregated client traffic demands from earlier stages of the ISP's aggregation network. We assume a controller of global knowledge on all state of the transport network. This could be realized by an SDN-controlled IP-over-DWDM architecture. Our prototypical network therefore consists of a circuit-oriented optical layer and a packetoriented electrical layer on top. The physical topology is given as the directed graph G = (V, E), where V is the set of nodes and E is the set of directed fiber links. Each node consists of an optical switch and a packet router connected to it. The optical switch acts as a contentionless, directionless and colorless reconfigurable optical add-drop multiplexer (ROADM), such that arbitrary optical circuits can be established between any fiber, which connects to another node, and any of the tunable ports in the line cards of the router. Each fiber link can provide a capacity in terms of data rate, defined by the maximum number of wavelengths multiplied by their individual data rates. We focus on the single line rate case.

Further, we are given a set of demands K that need to be routed in the network. Each demand k ∈ K is an aggregation of services. It connects a node pair (s, t) ∈ U = {(u, v) ∈ V × V : u = v} with a fixed data rate h k and a maximum allowed end-to-end delay τ k,max , also referred to as target delay. A node pair can also be connected by multiple demands.

Demands are routed in the virtual topology which in turn is defined by optical circuits in the optical layer. An optical circuit requires one line port at the router of its source node and one at the router of its target node. Intermediate nodes are bypassed, i.e., the circuit is switched in the ROADM without conversion to the electrical domain. Depending on the path on which a demand k is routed through the network it experiences a certain delay τ k . We denote the difference between τ k,max , the maximum delay a demand is allowed to experience, and its actual delay τ k as delay overfulfillment. Further we denote as relative delay overfulfillment. We focus on core networks that span large geographical distances which in turn lead to significant propagation delays. In the core, these are the largest contributor to overall delay [START_REF] Xiao | A K self-adaptive SDN controller placement for wide area networks[END_REF], followed by queuing delay. While queuing delay in routers can generally result in large delay spikes, these occur only rarely in the core where client traffic had already been subject to prior rate shaping. Together with traffic engineering sufficient link capacity to avoid excessive congestion can be ensured, such that we assume that queuing delays can be controlled in regular operation. Since existing studies show typical queuing delays to vary significantly between different networks from a few micro- [START_REF] Choi | Analysis of Point-To-Point Packet Delay In an Operational Network[END_REF] to a few milliseconds [START_REF] Bozkurt | Dissecting Latency in the Internet's Fiber Infrastructure[END_REF], which typically remains below the propagation delay in our scenario, we choose to focus on the deterministic part of the delay by modeling a demand's delay τ k on a path by its propagation delay. In [START_REF] Enderle | Modeling Dynamic Traffic Demand Behavior in Telecommunication Networks[END_REF], we already presented a delay-aware routing approach that minimizes the amount of active hardware in the network. This approach yields short paths for most demands which results in high delay overfulfillment. Our goal is to find a routing approach that reduces the delay overfulfillment compared to a resource optimization as shown in [START_REF] Enderle | Modeling Dynamic Traffic Demand Behavior in Telecommunication Networks[END_REF]. To be more precise, we find a network configuration, i.e., a set of circuits and a corresponding routing, for a given set of demands K that minimizes the average relative delay overfulfillment for a certain subset K S ⊆ K of those demands. We study the effects on the blocking behavior for a continuous reconfiguration of the network with demands that change over time. Furthermore, we consider the effects on the amount of network hardware that is needed. We focus on router line cards because they are the lion's share of the operator's capital expenditure (CAPEX) [START_REF] Papanikolaou | Multilayer flex-grid network planning[END_REF].

∆τ k = τ k,max -τ k τ k,max (1)

Solution Approach

We solve the problem introduced in the previous section with the help of an integer linear program (ILP). The ILP is based on a link-path formulation, i.e., it selects optimal routes out of a precomputed set of candidate paths. The whole process is depicted in Fig. 1. As input the algorithm receives the network topology, the set of demands as well as the current configuration of the network. In the first step, candidate paths are generated for each of the demands. Subsequently, the demands are routed using the ILP. The result is a new network configuration. We employ a make before break migration approach meaning that if the routing of a demand changes from the previous to the current configuration, then the new path must be completely set up before the old one is torn down. Consequently, we need to ensure that enough capacity is available to realize old and new path simultaneously. The details are explained in the following sections.

Path Generation

The ILP selects optimal routes out of a set of precomputed candidate paths. The generation of these paths consists of two steps. In the first step, we find paths for the demands in the physical topology. In the second step, we create corresponding circuit realizations for each of the previously found paths. With this approach, a joint optimization of electrical and optical layer is possible.

For every demand k ∈ K that connects nodes s and t we use the first m shortest simple paths between s and t in the physical layer G = (V, E) as candidate paths. If the demand allows a maximum delay of τ k,max , then only the subset of the m paths that fulfill this delay requirement is used. We denote the set of these valid paths by L k .

We represent a candidate path l ∈ L k as a tuple of physical links, i.e., l = (e 1 , e 2 , . . .) where e i ∈ E. In order to realize such a candidate path l one or more consecutive optical circuits need to be set up. An optical circuit can span one or more physical links; therefore, if we split l into several sub-tuples we obtain a sequence of optical circuit paths that realize l. We find all possible sequences of optical circuit paths and denote them by R k,l . Sequences containing circuit paths that are longer than a predefined transparent optical reach are excluded. We denote the set of all individual circuit paths by C.

An example is given in Fig. 2 where we assume a demand from node a to c. Two simple paths, a-c and a-b-c, connect the two nodes. The path a-c can only be realized by a circuit connecting nodes a and c directly. The path a-b-c can be realized by a single circuit that bypasses node b or by two consecutive circuits, a-b and b-c.

ILP Formulation

The goal of the ILP is the joint minimization of demand blocking, delay overfulfillment and required router line cards. To this end, we define variables that encode the selected candidate paths as well as the resulting amount of circuits and hardware that is required. Specifically, we define g k,l,r ∈ {0, 1} (k ∈ K, l ∈ L k , r ∈ R k,l ) as path selector, i.e., g k,l,r is one if demand k is routed over path l using circuit realization r; b k ∈ {0, 1} (k ∈ K) as indicator for a blocked demand; w c ∈ N (c ∈ C) as the number of parallel circuits on circuit path c where C is the set of all circuit paths in the network; w c ∈ N (c ∈ C) as the number of circuits on circuit path c taking migration into account; p v,t ∈ N ((v, t) ∈ U ) as the number of ports at v for connections with t; q v ∈ N (v ∈ V ) as the number of line cards at node v; i e ∈ {0, 1} (e ∈ E) as indicator for a highly utilized link.

The objective function is

min   α k∈K b k + β v∈V q v + γ |K S | k∈KS,l∈L k ,r∈R k,l g k,l,r • ∆τ k,l + µ |E| e∈E i e   . (2)
The first term penalizes blocked demands while the second term is responsible for the minimization of active line cards. The third term penalizes the average relative delay overfulfillment of the demands in K S ⊆ K. The parameter ∆τ k,l is demand k's relative delay overfulfillment if it takes path l. The last term penalizes links that are highly utilized. This is important for the make before break migration approach because it encourages the optimizer to leave some free link capacity for later migration steps.

The following constraints complete the ILP.

l∈L k ,r∈R k,l g k,l,r = 1 -b k ∀k ∈ K (3) k∈K,l∈L k ,r∈R k,l ρ c,k,l,r • h k • g k,l,r ≤ ξ • w c ∀c ∈ C (4) k∈K,l∈L k ,r∈R k,l ρ c,k,l,r • h k • g k,l,r + k∈ K,l∈L k ,r∈R k,l ρ c,k,l,r • h k • g k,l,r - k∈K∩ K,l∈L k ,r∈R k,l ρ c,k,l,r • h k • g k,l,r • g k,l,r ≤ ξ • w c ∀c ∈ C (5) c∈C δ e,c • ξ • w c ≤ π e ∀e ∈ E (6)
1

π e c∈C δ e,c • ξ • w c ≤ ψ + i e ∀e ∈ E (7) c∈C w c • ϕ c,v,t ≤ p v,t ∀(v, t) ∈ U (8) c∈C w c • ϕ c,t,v ≤ p v,t ∀(v, t) ∈ U (9) t∈V \{v} p v,t ≤ κ • q v ∀v ∈ V (10)
In these constraints the set K contains the demands of the previous reconfiguration step. Additionally, we define the parameters ξ ∈ R as the capacity of a single optical circuit;

π e ∈ R as the capacity of link e; ψ ∈ [0, 1] as the utilization threshold for highly utilized links; κ ∈ N as the number of router ports a line card can maximally hold; ρ c,k,l,r ∈ {0, 1} as indicator showing whether circuit realization r of demand k's candidate path l uses circuit path c; δ e,c ∈ {0, 1} as indicator showing whether circuit path c traverses link e; ϕ c,u,v ∈ {0, 1} as indicator showing whether circuit path c connects nodes u and v with u as source node; g k,l,r ∈ {0, 1} as the optimal path selector of the previous configuration.

Equation [START_REF] Choi | Analysis of Point-To-Point Packet Delay In an Operational Network[END_REF] ensures that a demand is either routed on exactly one path or blocked. Constraints ( 4) and ( 5) ensure that enough circuits are installed depending on the chosen paths. In contrast to constraint (4), which only considers the demands of the current reconfiguration step, constraint (5) also takes the routing of the previous reconfiguration step into account allowing the make before break migration. ( 6) is a link capacity constraint and (7) triggers the indicator i e if the utilization of link e exceeds a certain limit ψ ∈ [0, 1]. Constraints ( 8) and ( 9) reserve a sufficient amount of router ports to accommodate the optical circuits. Finally, [START_REF] Papanikolaou | Multilayer flex-grid network planning[END_REF] ensures that enough line cards are present to hold the ports.

Evaluation

In this section we evaluate our approach. We first present the topologies and traffic demands as well as the parameters we studied. Then we discuss our results.

Network Topologies and Architecture

We selected the two backbone networks Abilene and Géant, found in the SNDlib [START_REF] Orlowski | SNDlib 1.0-Survivable Network Design Library[END_REF], to evaluate the behavior of our routing approach in different wide-area networks. We only show details for the Géant topology here since the results for Abilene are very similar.

Géant is a research network connecting various countries in Europe. As we neglect the connection to New York, we end up with 21 nodes and 68 directed links. The topology is shown in Fig. 3(a). We assume that the ports operate at a data rate of ξ = 100 Gbps with a transparent reach of 2500 km and that a router line card can hold a single port, i.e., κ = 1 [4]. Further, we assume that the number of wavelengths on a link is limited such that π e = 40 • ξ.

Traffic Demand

We synthetically generated traffic demands whose arrival and holding times follow a negative exponential distribution. We assume that the demands have wavelength granularity, i.e., the traffic value of a demand h k equals the capacity ξ of an optical circuit. We distinguish two traffic demand classes, namely delaysensitive and delay-insensitive demands. The delay-insensitive demands do not have a delay requirement meaning that they can be routed on arbitrary routes. The delay-sensitive demands do have a certain delay requirement τ k,max . In the ILP optimization they form the set K S , i.e., their delay overfulfillment is minimized. Delay-sensitive demands were generated only between those node pairs for which the shortest connecting path satisfies the delay requirement. Varying the delay requirement therefore also changes the number of node pairs with delay-sensitive traffic. Fig. 3(b) shows the delay distribution for the ten shortest paths between all node pairs. Additionally, the average shortest path delay τ sp,avg considering all node pairs is depicted. It is visible that for a delay requirement τ k,max equal to the average shortest path delay τ sp,avg , 128 node pairs can be connected with a delay-sensitive demand. 98 of them have at least one alternative path. The remaining 30 node pairs can be connected on the shortest path only. Our approach tries to avoid the blocking of demands between those node pairs by routing demands with less strict delay requirements on circumjacent links.

Reconfiguration

The reconfiguration is triggered at regular intervals where the ratio of the interval duration and the mean holding time of the traffic demands equals 0.05. In each reconfiguration step we route the demands that will arrive in the following interval while taking the present configuration of the current interval into account. We employ a make before break migration to transition between two configurations. For the candidate path generation we consider the m = 10 shortest paths between each node pair. For each path we consider all possible circuit realizations. We consider a link to be highly loaded if its utilization exceeds a value of ψ = 0.95. For the objective function of the ILP we used the parameters α = 10 000 and µ = 1000. In that way, avoiding unrouted demands is the most important optimization goal followed by the prevention of highly utilized links. By adjusting the ratio of β to γ the trade-off between the minimization of overfulfillment and active line cards can be controlled. We set β = 0.0001 and γ = 10 to focus on the maximum achievable overfulfillment reduction in this scenario.

Varied Scenario Parameters

For the evaluation of the introduced reconfiguration approach we adjusted a number of parameters relevant for the traffic generation.

The offered load describes the amount of traffic that needs to be transported through the network in relation to its capacity. Therefore, on the one hand, the offered load depends on the demand values themselves. But on the other hand it also depends on the path length of a routed demand. Demands between nodes with a physical connection can be routed single-hop whereas demands between distant nodes necessarily occupy resources on several links. We therefore introduce an offered load metric which, to a certain extent, takes these differences in the path length into account. For each demand k we compute the shortest possible path and count the number of physical links ζ k it traverses. In our model, this value ζ k is equivalent to the number of wavelengths the demand occupies when it is routed on the shortest path. We multiply the occupied wavelengths with the demand value h k and relate it to the total network capacity, i.e., the sum of all link capacities. The offered load is then given by

A = k∈K ζ k • h k e∈E π e . ( 11
)
An additional parameter is the share of delay-sensitive demands in the total number of demands φ = |K S |/|K|. Lastly, we vary the target delay τ k,max of the delay-sensitive demands. In order to compare networks with different geographical layouts we relate the target delay to the average shortest path delay τ sp,avg of the network. We introduce a target delay factor χ τ which adjusts the target delay according to τ k,max = χ τ • τ sp,avg , ∀k ∈ K S .

Using the introduced parameters we have determined demand inter-arrival and holding times. Based on these, we generated a traffic demand series of 200 demand sets for each parameter combination of offered load A, share of delaysensitive demands φ and target delay factor χ τ as depicted in Fig. 4.

Results

The presented approach on the reduction of delay overfulfillment is compared to a resource optimization. The resource optimization routes the traffic load such that the amount of active line cards is minimized while the delay requirements are satisfied. This is realized by setting γ = 0 in the ILP's objective function. Fig. 4 shows the results of our investigation as introduced for the Géant topology for an offered load ranging from 0.1 to 0.6 in steps of 0.05. Each data point represents the average of a series of 200 reconfigurations including migration. The error bars show one standard deviation. Fig. 4(a) shows the relative delay overfulfillment averaged over all delay-sensitive demands K S . As can be seen, our approach is able to reduce the relative overfulfillment by up to 60 % compared to the resource optimization for the case that the target delay factor is 2. Even for a delay factor of 0.5, for which alternative paths for delay-sensitive demands are scarce, an overfulfillment reduction is possible. The share of delaysensitive demands has only a small impact on the delay overfulfillment. Fig. 4(c) shows the blocking ratio, i.e., the ratio of unrouted demands to the total number of demands K. Blocking occurs only if all wavelengths on an physical link are occupied and no alternative route is available. As can be seen, blocking occurs for all parameter combinations if the offered load exceeds a value of 0.4. This effect is also visible with the resource optimization which suggests that an offered load of 0.4 is already close to the capacity limit of the network. For values below 0.4 we can see that the overfulfillment minimization exhibits slightly higher blocking ratios. Fig. 4(b) shows the relative increase in required line cards for the overfulfillment reducing optimization compared to the resource optimization. For a target delay factor of χ τ = 0.5 and a traffic load below 0.3 no increase in the number of line cards is visible. Also for χ τ = 1 and traffic loads up to 0.2 this number stays the same for both optimization approaches. This means that the delay overfulfillment can be reduced without the need to invest more line cards. For a target delay of twice the average shortest path delay additional line cards are necessary. However, we assume that this hardware overhead is within the margin of hardware overprovisioning which is typically found in transport networks. The amount of additional line cards depends both on the share of delay-sensitive demands and the offered load.

We have conducted the evaluation for the US network Abilene [START_REF] Orlowski | SNDlib 1.0-Survivable Network Design Library[END_REF] as well. Compared to Géant, the Abilene network has a broader delay distribution resulting in less alternative paths for delay-sensitive demands. Hence, for a delay factor of 0.5 the overfulfillment cannot be reduced significantly compared to the resource optimization. For higher target delays the results are very similar to those presented for the Géant topology. In particular, they confirm that the delay overfulfillment can be significantly reduced by our approach while the blocking behavior is only slightly affected.

Conclusions

In this paper we presented a new routing and network reconfiguration approach which addresses the overfulfillment of delay requirements in transport networks. The approach routes traffic on paths other than the shortest path to reduce the delay overfulfillment of traffic demands. In that way, customers experience service delays closer to the limits specified in their SLA providing an incentive to acquire a more expensive service class if necessary. For the ISP this provides a new opportunity for revenue creation. As a secondary goal, our approach relieves highly utilized links for those demands that have no alternative to the shortest path due to their strict delay requirements.

We presented a parameterizable ILP that jointly optimizes routing in the optical and electrical layer of the transport network. We evaluated our proposed approach for two nationwide transport networks. The results show that the delay overfulfillment can be reduced by up to 60 % without a significant increase in the blocking ratio compared to a resource optimization. The number of required line cards naturally increases because longer routes are taken. However, we assume it to remain within the margin of hardware overprovisioning by ISPs. Furthermore, the trade-off between the minimization of overfulfillment and required hardware depends on adjustable parameters which allows the ISP to adapt our approach to its business models and networking hardware.

Fig. 2 .

 2 Fig. 2. Candidate path generation for a demand from node a to node c.

Fig. 3 .

 3 Fig.3. Logical fiber topology (a) and propagation delay distributions of the 10 shortest paths (b) for the European research network Géant[START_REF] Orlowski | SNDlib 1.0-Survivable Network Design Library[END_REF] without the node in New York. The path length index in (b) increases from left to right, i.e., the red curve on the left corresponds to the shortest path. The average delay of the shortest path at τsp,avg = 7.7 ms is depicted with the red vertical line.

Fig. 4 .

 4 Fig. 4. Results of the introduced reconfiguration approach for wavelength demands with Poisson arrivals and departures. Shown metrics are the average relative delay overfulfillment (a), the relative increase in required line cards compared to a delayaware resource optimization (b) and the blocking ratio (c). The error bars represent one standard deviation.

  Overview of the solution approach.

	Input	Optimization	Output
	Traffic Demands
	Network Topology Configuration Current Network	Candidate Path/Circuit Generator	ILP Solver	New Network Configuration
	Fig. 1.

Compound Annual Growth Rate

to 2022

Acknowledgements

This work has been performed in the framework of the CELTIC EUREKA project SENDATE-TANDEM (Project ID C2015/3-2), and it is partly funded by the German BMBF (Project ID 16KIS0458). The authors alone are responsible for the content of the paper.


Frank Slyne
email: fslyne@tcd.ie

Bruno Cornaglia

Marco Boselli

Marco Ruffini

3-stage Hierarchical Quality of Service for Multi-tenant Passive Optical Networks

Keywords: Quality of Service, Two Rate Three Color Marker, Scheduler, Policer, Passive Optical Networks, Virtualization, Slicing

Passive Optical Networks (PONs) are an economically efficient means of providing high bandwidth services to end users because end-points can share the cost and the benefit of fibre optic connection. There is a growing interest in the use of PONs as a bearer for services such as Long Term Evolution (LTE), however work remains to be done to ensure stable, low latency and low packet loss for full duplex Transmission. In this paper we are mostly concerned with the characteristics of the downstream channel so as to further the case for PON sharing. We review typical downstream schedulers used in the industry and propose a three-stage downstream scheduler that assures quality of service, accurately, across Virtual Network Operators (VNOs) in multi-tenant PONs environment. We benchmark our scheduler against other algorithms, obtaining almost ideal scheduling performance.

Introduction

Up until recently, residential broadband has been the dominant service delivered over PONs. However, because PONs are an economically efficient means of providing high capacity bandwidth across wide rural and urban geographical areas [START_REF] Ruffini | Protection strategies for long-reach PON[END_REF], they are becoming increasingly attractive for the delivery of non-retail services such as LTE [START_REF] Alvarez | Experimental Demonstration of SDN-controlled Variable-rate Fronthaul for Converged LTE-over-PON[END_REF], where dedicated fibre capacity would not be economically feasible. PON sharing across multiple VNOs is also becoming increasingly important, in order to foster competition without replicating costly hardware infrastructure. Thus, in a multi-tenant scenario each VNO should have the ability to assign its portion of assured and best effort traffic to its customers, without affecting or being affected by other VNOs. Given that LTE, when implemented as Cloud Radio Access Networks (C-RAN) has a low tolerance to excessive packet jitter and latency, the VNO needs granular slicing and resource control in order to share PON infrastructure with non-critical services such as residential broadband.

Achieving separation between VNOs in a PON environment is complicated by the fact that upstream and downstream transmissions function in different manners. Unfortunately, granular resource control is only defined for upstream traffic in the PON standards, with downstream left to the devices of operators and whatever capabilities their vendor supplied equipment provide. By its nature, upstream transmission in a PON is a TDM channel where there is an inherent control over how much and when each Traffic Container (TCont) of each Optical Network Terminations (ONT) can transmit, down to the granularity of each 32-bit word [START_REF] Rec | 10-Gigabit-capable passive optical network (XG-PON) systems[END_REF]. In this context a TCont can be thought of as a dedicated queue at the ONT to which a specific class of service can be assigned. The scheduling of the upstream transmission is regulated by the Dynamic Bandwidth Allocation (DBA) mechanism [START_REF]Broadband optical access systems based on Passive Optical Networks (PON)[END_REF], which allocates capacity to the ONTs located at the end user side [START_REF] Skubic | A comparison of dynamic bandwidth allocation for EPON, GPON, and next-generation TDM PON[END_REF]. One solution to provide full control to the VNOs over their upstream capacity allocation, by virtualizing and slicing the DBA mechanism, was recently proposed in [START_REF] Elrasad | Virtual dynamic bandwidth allocation enabling true PON multi-tenancy[END_REF], prototyped in [START_REF] Slyne | Experimental Demonstration of DPDK Optimized VNF Implementation of Virtual DBA in a Multi-Tenant PON[END_REF] and standardized in [START_REF]BBF technical report tr-402, functional model for PON abstraction interface[END_REF].

Of most pertinence in this paper, is the downstream channel of the PON which functions in the manner of a statistically multiplexed packetized channel or shared bus. The packetization of data followed by statistical multiplexing through a shared transmission channel is an essential means of increasing the utilization of the channel, when compared to time or frequency division multiplexing means over the same channel. This technological trend is similar to how synchronous SDH and cell-based ATM networks in the core and metro networks are being replaced by multi-gigabit Ethernet [START_REF] Winzer | Beyond 100G ethernet[END_REF] networks running over Dense Wave Division Multiplexing (DWDM) fiber [START_REF] Anderson | Optical transceivers for 100 gigabit Ethernet and its transport [100 gigabit ethernet transport[END_REF], at times over an Optical Transmission Network (OTN) sublayer [START_REF] Roese | Optical transport network evolving with 100 gigabit Ethernet evolving with 100 gigabit ethernet [100 gigabit ethernet transport[END_REF]. However, a drawback for real-time services that use a statistical multiplexed channel is the lack of a guarantee over the data information rate, packet loss and latency, which go together to define QoS. As central office infrastructure is moving towards virtualization and slicing [START_REF] Ruffini | Moving the Network to the Cloud: the Cloud Central Office Revolution and its Implications for the Optical Layer[END_REF], the ability to provide assurance over QoS is becoming increasingly important to provide slice independence across multiple tenants, as though they were being served by dedicated infrastructure. This is typically achieved through appropriate queue management, to separate traffic with different QoS requirements and policing the rate of incoming traffic. However, such methodology is not defined in PON standards, where decisions on how to implement downstream QoS are left to vendors and operators. Typically, the options vary between over-dimensioning capacity on each PON, executing vendor specific scheduling and policing, or shaping traffic prior to entering the Optical Line Termination (OLT) [START_REF] Alvarez | Experimental Demonstration of SDN-controlled Variable-rate Fronthaul for Converged LTE-over-PON[END_REF].

In this paper, we first review the current practices used by operators for the scheduling of downstream traffic in a PON, typically divided into one-stage and the two-stage scheduler architectures. We will show that while these methods can be effective in single-operator scenarios, they fail to provide the necessary support for VNOs sharing the same infrastructure. We thus propose a novel three-stage scheduler that includes a QoS stage for providing additional separation between VNOs. In order to make the case for the scheduler, we build the architecture for the one-, two-and three-stage schedulers in a simulator, which are then exercised using a common set of scenarios. The performance of the three simulators are then compared using typical metrics such as packet loss. Throughout the experiment, we are careful to distinguish between the packet handling capabilities for both High Priority (HP) and Low Priority (LP) traffic. Examples of HP traffic includes Signal data and traffic related to services that deteriorate significantly due to packet loss, such as Voice over IP (VoIP) [START_REF] Collins | Carrier grade voice over IP[END_REF]. Examples of LP traffic includes streaming Video and raw data, both of which can be buffered and replayed in the event of packet loss. Typically, downstream PON QoS is implemented through queue management functions defined by Internet Engineering Task Force (IETF) Request For Comment (RFC)s. These functions include Strict Priority (SP), Weighted Round Robin (WRR), Weighted Fair Queuing (WFQ) and two rate three color marker (trTCM) which operate on network packet flows and which we distinguish as follows. A SP queue function has N input ports (numbered from 0 to N-1) and 1 output port. It will serve packets from port 0, if they are present, ahead of a lower priority (1 to N-1) port. Likewise, a SP queue will serve packets from port 1, if they are present, ahead of a lower priority (2 to N-1) port. While a SP can guarantee packets from higher priority ports to be delivered to the output port, there is no rate-limiting function that prevents the output port from being overloaded by lower priority packets. A WRR queue function has N input ports (numbered from 0 to N-1) and 1 output port. Packets are served from the input ports according to a weighted distribution function, typically denoted by φ (phi), and averaged over a unit of time. The downside of the WRR queue function is that it only distinguishes between packet counts and not packet sizes. A small number of very large packets served to low weighted port can skew the queue output in its favor. This is rectified in the WFQ. The weighted distribution function φ (phi) is defined in a similar way to that of the WRR queue function, except that it operates according to bytes offered to each port, as opposed to packets. The trTCM is an example of a Policer that can implement rate-limiting functionality. trTCM accepts flows of data on typically a single physical input port, which are then sent to a single output port. These flows can be passed through (logically tagged Green), dropped (Red) or remarked and passed (Yellow) depending on predefined data rate rules. Examples of the specific operation of the trTCM are given below. The mechanism by which these queue management functions can be composed into typical downstream QoS schedulers are shown in Figure 1. Firstly, we see a single-stage QoS mechanism in Figure 1 (a), where QoS is only implemented at the ONT level, with WRR scheduling traffic within the same priority class, and SP operating across the classes to provide, for example, strict higher priority to voice and signaling with respect to video and other data types. Secondly, Figure 1 (b) shows an alternative two stages vendor solution. The first stage operates at the ONT level and is composed of 4 queues, a SP scheduler and trTCM policer. The SP scheduler handles traffic in the decreasing priority of signaling, voice and the weighted fair queuing balancing of video and data. The SP assigns a color to the traffic based on the source. Both signaling and voice are HP and thus marked as green. Video and data are instead LP and marked as yellow. Because the color of the traffic is already configured by SP, trTCM is configured in color-blind mode, according to RFC 4115 [START_REF] Aboul-Magd | A differentiated service two-rate, three-color marker with efficient handling of in-profile traffic[END_REF], and passes the colored traffic directly to the egress Committed Information Rate (CIR) and Excess Information Rate (EIR) ports respectively. A weighed fair queuing is applied at the ONT level between video and data, while strict priority is assigned to voice and signaling. trTCM policers enforce proportionality between CIR and EIR by marking the packets according to RFC2698 (trTCM) color definition [START_REF] Heinanen | A single rate three color marker[END_REF].

II. DOWNSTREAM SCHEDULING

Packets that are marked as red are dropped, green packets are passed through, yellow packets are remarked packets that may be dropped at a later stage or passed if there is sufficient available capacity. At the OLT port level, a weighted round robin mechanism is applied to schedule packets across all ONTs, respecting aggregate levels of CIR and EIR. The second stage is the PON port module, composed of two WRR queues operating on the CIR and EIR queues. The weighting of the WRR queues is based on the total service profile, CIR and EIR respectively, that each ONT should carry as a proportion of the total capacity allowed in the PON port. For example, the weighting given to the packets from the i th ONT

𝑤 = 𝐶𝐼𝑅 ∑ 𝐶𝐼𝑅 , 𝑤 = 𝐸𝐼𝑅 ∑ 𝐸𝐼𝑅
Where 𝑤 and 𝑤 are the fractions of configured CIR and EIR for that ONT over the total CIR and EIR respectively for all (k) ONTs. The total weightings for each WWR then sums to 1.

𝑤 = 1, 𝑤 = 1

It should be noticed that QoS scheduling becomes especially important in multitenant scenarios, where different VNOs can have different QoS contracts with their own customers and with the infrastructure provider. However, current scheduling mechanisms such as those described above, do not take into consideration the existence of multiple VNOs, and thus there is no QoS differentiation at the VNO level. In this paper we attempt to tackle this issue by proposing a novel three-stage scheduler that adds VNO-oriented scheduling layer and by comparing its performance improvement against the baseline mechanisms shown in Figure 1.

Finally, we can define the behavior of the ideal scheduler, against which current and proposed architectures can be compared. To do this, we calculate 𝑋 and 𝑋 the expected HP and LP traffic egressing from the PON related to the i th ONT, given offered traffic 𝑍 and 𝑍 . The high priority traffic marked as green should not exceeded the CIR allocated to that ONT. In the event, that HP traffic exceeds the CIR, the excess is recolored as yellow.

𝐺 = 𝑚𝑖𝑛(𝑍 , 𝐶𝐼𝑅 ), 𝑌 = 𝑍 -𝐺

Any excess CIR may be used to allow LP traffic to be colored as green. The excess LP traffic is marked yellow.

𝐺 = 𝐶𝐼𝑅 -𝐺 , 𝑌 = 𝑍 -𝐺

The expected HP traffic egressing from the PON related to the i th ONT is a sum of HP traffic marked as green and HP traffic marked as yellow contending for the total EIR EIRT. 𝑤 and 𝑤 are as defined above.

𝑋 = 𝐺 + 𝑤 . 𝐸𝐼𝑅 . 𝑌 𝑌 + 𝑌

Similarly, the expected LP traffic egressing from the PON related to the i th ONT is a sum of LP traffic marked as green and LP traffic marked as yellow contending for the total EIR 𝐸𝐼𝑅 .

𝑋 = 𝐺 + 𝑤 . 𝐸𝐼𝑅 . 𝑌 𝑌 + 𝑌

In the next section we explain the architecture of our proposed three-stage QoS scheduler and the implementation of the simulation environment. This allows us to compare the results of our scheduler with the one-stage and two-stage schedulers described above, showing the advantages brought by our three-stage design for multitenant operations. The architecture of our Hierarchical three-stage scheduler is shown in Figure 2, and uses the same array of functions (SP, WRR, WFQ and trTCM) as the one-and twostage schedulers. We note that in recent years the IETF has developed algorithms and techniques for Queue management and schedulers, such as Controlled Delay (CoDel) and Active Queue Management (AQM) respectively that deal with anomalous treatment of TCP/IP transport flows due to excess buffering of packets. This anomalous behavior is a result of a phenomenon known as Bufferbloat [START_REF] Gettys | Bufferbloat: Dark buffers in the internet[END_REF], that is, the use of excessively large buffers in routers, switches and modems, that disrupt how normal functioning of TCP/IP flow control mechanism. IETF has combined CoDel and AQM into a single RFC (8290) known as Flow Queue CoDel Packet Scheduler and Active Queue Management Algorithm (FQ-CoDel) [START_REF] Hoeiland-Joergensen | The Flow Queue CoDel Packet Scheduler and Active Queue Management Algorithm[END_REF]. With respect to the reference twostage scheduler shown Figure 1 (b), we have added a further WRR followed by a trTCM function that operate at the individual VNO level. The scope of this additional stage is to further partition the overall capacity, so that any excess capacity from a VNO is preferentially re-distributed across the ONTs belonging to the same VNO. Hierarchical Scheduler is located as a single functional block at the head end of the PON, that is at the OLT. It functions at 3 levels, that is, at the level of the ONT, at the level of the Operator and at the level of the PON, where WRR, SP and WFQ assure proportionality between contracted bandwidths. Downstream from the ONT and Operator schedulers, trTCM policers enforce proportionality between CIR and EIR by marking the packets according to RFC2698 (trTCM) color definition [START_REF] Heinanen | A single rate three color marker[END_REF]. Packets that are marked as Red are dropped, Green Packets are passed through, Yellow packets are remarked packets that may be dropped at a later stage or passed if there is enough bandwidth. In an ideal scenario, but not implemented here, a feedback path from the PON functional block, similar to Call Admission Control (CAC), directs specific ONTs to block new flow requests, where the CIR is higher than a specific percentage (typically 50%) of the total capacity of the PON or indeed should the CIR of an operator exceed a specific threshold agreed with the PON infrastructure provider.

Three-Stage Scheduler Description and Architecture

Simulator

In order to compare the QoS performance improvement of the three-stage scheduler in PON multi-tenant scenarios we have developed a QoS discrete event simulator based on the Python SimPy [START_REF] Matloff | Introduction to discrete-event simulation and the simpy language[END_REF] package. We use several fundamental QoS modules (trTCM, WRR, WFQ and SP) to construct the downstream PON schedulers operating in the OLT. The trTCM object class marks packets with the three colors: green, yellow and red, depending both on any original marking and on the CIR and the EIR parameters. It should be noticed that the decision on whether incoming data is above or below the CIR and EIR thresholds is computed based on average values across a given burst data size. In the simulation we used burst sizes values for the CIR and the EIR typically configured in Cisco and Juniper routers of 64 Kbytes and 128 Kbytes, respectively. The WRR module accepts packets from several inputs streams and assures fairness based upon a predefined distribution based on number of packets in the input buffer for each stream. WFQ behaves similarly to WRR but uses number of bytes in the input buffer for each stream as a means of determining the fairness. Finally, SP selects packets from different queues in ascending order of priority and being strict it will only select packets from a lower priority queue once the higher priority ones are empty.

The ONT module is made up of 4 queues each with a buffer size of 128 Kbytes, a SP scheduler and trTCM policer. The SP scheduler handles traffic in the decreasing priority of signaling, voice and the weighted fair queuing balancing of video and data. The SP assigns a color to the traffic based on the source. Both signaling and voice are HP and thus marked as green. Video and data are instead LP and marked as yellow. Because the color of the traffic is already configured by SP, trTCM is configured in colorblind mode, according to RFC 4115 [START_REF] Aboul-Magd | A differentiated service two-rate, three-color marker with efficient handling of in-profile traffic[END_REF], and passes the colored traffic to the egress CIR and EIR ports respectively. This is constructed similarly to the ONT QoS module, however there is a pair (that is for both CIR and EIR) of WRR schedulers for each VNO, whose weightings are calculated based on the total capacity assigned to the specific VNO. Thereafter a trTCM policer colors the aggregate traffic from the SP scheduler as passed (green), remarked (yellow) or dropped (red). The PON port QoS module is composed of two WRR queues whose weighting are based on the aggregate service profiles, CIR and EIR respectively, that each other operator carries. The PON QoS module does not have a policer, but instead must aggregate both CIR and EIR onto the common downstream physical link. To assess the performance of the Hierarchical QoS (H-QoS), we also emulate one and two stage QoS architectures. The one stage architecture is composed of just the PON module from the three-stage architecture, which has been extended to have 4 WRR queues, terminating the Signaling, Voice, Video and Data links. The weighting of the WRR queues is based on the total service profile CIR and EIR respectively that each other ONT carries as a proportion of the total for the PON.

The two-stage architecture Figure 1(b) is composed of the ONT and PON modules from the three-stage architecture Figure 2, omitting the Operator QoS modules. The ONTs connect directly to the PON. The weighting of the PON WRR queues is based on the total service profile CIR and EIR respectively that each other ONT carries as a proportion of the total for the PON.

Simulation Scenarios and Results

In our simulation, we assume that there are 2 VNOs which share a common PON downstream capacity of 2.488 Gbps [START_REF] Rec | 10-Gigabit-capable passive optical network (XG-PON) systems[END_REF], of which 1.760 Gigabits per second (Gbps) is allocated to CIR. We define a service profile as a fixed mix of committed and excess information rates that may be assigned by any VNO to one of its ONTs. We denote this mix as a tuple (CIR, EIR). In our simulations, we define two service profiles, Profile-1 (10,100) and Profile2 (100,1000), measured in Megabits per second (Mbps). In order to gauge the ability of the QoS mechanisms described to schedule capacity appropriately across VNOs we introduce an imbalance between their user base. Operator A has 24 ONTs, of which half are Profile-1 and half are Profile-2.

Operator B has 4 ONTs with Profile-1 and 4 with Profile-2. The net result is that Operator A carries three quarters of the total traffic and Operator B carries one quarter. We are especially interested in what happens to bandwidth which is offered to an ONT that is either in excess or in deficit of its service profile's CIR and EIR, which we call the Nominal information rates. For this purpose, traffic offered at each ONT is transmitted at either +20% or -20% of the nominal CIR or EIR for the configured service profile. For instance, ONTs with service profile Profile-1 may be offered [START_REF]BBF technical report tr-402, functional model for PON abstraction interface[END_REF]80), [START_REF]BBF technical report tr-402, functional model for PON abstraction interface[END_REF]120), [START_REF] Ruffini | Moving the Network to the Cloud: the Cloud Central Office Revolution and its Implications for the Optical Layer[END_REF]80) or [START_REF] Ruffini | Moving the Network to the Cloud: the Cloud Central Office Revolution and its Implications for the Optical Layer[END_REF]120) Mbps. Finally, in addition to providing a comparison between our three-stage QoS mechanism and the two other baseline algorithms described in section 1, we also provide the output of an ideal scheduler as described previously, under the same conditions, that is, whose output is computed mathematically rather than simulated. This allows use to assess the three QoS architectures against an ideal scheduler.

The results of our simulations are reported in Figure 3 showing the throughput of the different schedulers for HP and LP traffic. The x axis represents the different ONTs, which are identified by the triplet (x:y:z), where x identifies the VNO (A or B), y is the (CIR/EIR) profile (i.e., Profile-1 or Profile-2, as described in the previous section) and z is the ONT identifier. To simplify the plot, we only show the results for a subset of ONTs, which is representative of the performance provided by the three schedulers. In the Figure 4(a) plot, we see that both the proposed three-stage (in orange), the baseline two-stage (in gray) and the baseline one-stage (in yellow) algorithms follow the ideal output (shown in blue) quite closely. This is to be expected since the purpose of CIR is to strictly cap the capacity at the agreed levels, which can be easily enforced already at the ONT level. As we see in Figure 3, where the three-stage mechanism provides considerable improvements over the two-stage and the one-stage is in the handling of LP traffic. Since the EIR traffic is only scheduled if there is spare capacity, our additional VNO-level QoS can reassign such capacity appropriately within each VNO.

The three-stage algorithm provides a much closer output to the ideal scheduler both across ONTs and across VNOs, showing a maximum deviation of +/-5%, compared to the two-stage scheme that deviates between +/-25%. This accuracy is highly desirable in a multi-tenant Fixed Access Network Sharing (FANS) scenario [START_REF] Cornaglia | Fixed access network sharing[END_REF], where each VNO pays the infrastructure provider for assuring a given level of aggregate CIR and EIR. The one-stage shows a +/-26% deviation, however this on the basis that video, and data share the same WRR queue. If this were the not case, then the video queue would always be served at the expense of the data (least priority) queue in this scenario. Figure 4 shows the capacity variance for HP traffic (top graph) and for LP traffic (bottom graph) with respect to the ideal case for the three different scheduling algorithms (labeled three-stage, two-stage and one-stage). The different bars group the results for all ONTs that offer, respectively, 8, 12, 80 and 120 Mb/s of HP traffic (the first two rates are for ONTs in Profile-1 with a 10Mb/s CIR and the second two rates for ONTs in Profile-2 with 100 Mb/s of CIR). We see that the proposed three-stage architecture provides the lowest variance for all traffic levels, especially for the allocation of capacity to LP traffic. Finally, the one-stage architecture performs poorest with a deviation from the desired output between +80% and +200%.

Figure 1

 1 Figure 1 Architecture of single-stage (a) and two-stage (b) reference schedulers

Figure 2

 2 Figure 2 Architecture of the proposed three-stage QoS scheduler

Figure 3

 3 Figure 3 Comparison of (a) High Priority (HP) and (b) Low Priority (LP) Traffic

Figure 4

 4 Figure 4 Variance of (a) HP data and (b) LP data against expected ideal behavior

Acknowledgement

Financial support from Science Foundation Ireland (SFI) grants 14/IA/2527 (OSHARE) and 13/RC/2077 (CONNECT) is gratefully acknowledged

Conclusions

In this paper, we have presented a three-stage scheduling architecture that can implemented at the head-end of the PON or the OLT. This has several benefits. Firstly, the OLT has full knowledge of the traffic that is being offered to the PON and provides real-time insight to network planners as to how the network is complying with contracted QoS metrics. Indeed, the three-stage scheduler is in the best position to react to changing network demands. Secondly, while the data-path of the three-stage scheduler may be implemented in high-speed Application Specific Integrated Circuit (ASIC), the parameters of the WRR, WFQ and trTCM can be dynamically tuned at the discretion of a centralized Software Defined Network (SDN) control plane or orchestrator. Thirdly, the centralization of the scheduler simplifies the functionality of the ONT, allowing for easier functional upgrades, and virtualization of the PON.

We have assumed that the CIR for each operator is equal to the sum of the CIR of all its' customers. However, in our future work, we will analyze where the CIR is less than the sum of the CIR of all its customers (over-subscription) as well as greater than the sum of the CIR of all its customers (under-subscription).


A N Stassinakis
email: a-stasinakis@phys.uoa.gr

H E Nistazakis

G K Varotsos

G S Tombras
email: gtombras@phys.uoa.gr

Performance of underwater wireless optical link under weak turbulence and pointing errors using heterodyne QAM technique

Keywords: Underwater Optical Communications, BER, Gamma distribution, Turbulence, Pointing Errors

Underwater optical wireless communications (UOWC) have gained attention due to the advantages they offer that can greatly alter the performance of a link. But such systems are affected by absorption and non predictable factors that may decrease the availability and the reliability of the system. In this work, the bit error rate (BER) of an underwater optical link with heterodyne detection modulated with M-QAM technique in presence of turbulence and pointing errors will be investigated. This investigation will lead to a closed form mathematical expression for the BER as a function of link parameters, that is very important in order to deploy an effective link. Various numerical results will also be presented for the reliability of the system.

Introduction

Optical wireless communications have attracted research and commercial interest due to the plenty advantages they offer that fulfill the requirements of modern networks [START_REF] Ghassemlooy | Terrestrial Free-Space Optical Communications[END_REF]- [START_REF] Alzenad | FSO-Based Vertical Backhaul/Fronthaul Framework for 5G+ Wireless Networks[END_REF]. Such advantages are the high bit rate they can achieve in low operational and installation cost. Such systems are point to point links that use a laser beam in order to transmit a signal between a receiver and a transmitter. So, the use of narrow beamwidth laser for transmitting information creates a license free communication link. Such systems were mostly used for terrestrial and satellite communications (FSO/sat-FSO) [START_REF] Ghassemlooy | Terrestrial Free-Space Optical Communications[END_REF], [START_REF] Henniger | An Introduction to Free-Space Optical Communications[END_REF], [START_REF] Gopal | Performance analysis of ground to satellite FSO system with DAPPM scheme in weak atmospheric turbulence[END_REF], but during the last few years, optical wireless communications have also been applied for underwater communications in order to solve the problem of low bit rate in relative short distances[5]- [START_REF] Boucouvalas | Underwater Optical Wireless Communications with Optical Amplification and Spatial Diversity[END_REF]. On the other hand, the laser beam that propagates along the water is affected by various factors that decrease the performance of the system. The main factor that decreases the possibilities of UOWC systems is the strong power absorption of the laser beam. Another important and non predictable factor that affects underwater OWC systems is the presence of turbulence that creates power fluctuations at the receiver due to the local variations of the refractive index, a phenomenon called scintillation[5], [START_REF] Fu | Performance of heterodyne differential phase-shift-keying underwater wireless optical communication systems in gamma-gamma-distributed turbulence[END_REF]- [START_REF] Zhu | Free-space optical communication through atmospheric turbulence channels[END_REF]. In order to investigate how the link is affected by turbulence, various statistical distributions are used depending on the turbulence strength [START_REF] Majumdar | Free-Space Laser Communication Performance in the Atmospheric Channel[END_REF], [START_REF] Nistazakis | Performance Analysis of Free-Space Optical Communication Systems Over Atmospheric Turbulence Channels[END_REF]. In this work, that weak turbulence is assumed, gamma statistical distribution is suitable to model the channel [START_REF] Epple | Simplified Channel Model for Simulation of Free-Space Optical Communications[END_REF]. Another factor that deteriorates the performance of an underwater OWC system is the misalignment between the transmitter and the receiver due to vibrations that come from various sources mostly from internal mechanisms of the vessel. The statistical distribution of Beckman's model is used in order to model the fluctuations of pointing errors [START_REF] Ninos | Block error rate performance of OOK free-space optical links over gamma-gamma turbulence channels with generalised non-zero boresight pointing errors[END_REF]- [START_REF] Boluda-Ruiz | Novel approximation of misalignment fading modeled by Beckmann distribution on freespace optical links[END_REF]. The modulation technique that will be used in this OWC system, is the quadrature amplitude modulation (QAM), that is an extensively used modulation scheme in modern communication systems [START_REF] Bekkali | Transmission Analysis of OFDM-Based Wireless Services Over Turbulent Radio-on-FSO Links Modeled by Gamma-Gamma Distribution[END_REF]- [START_REF] Nistazakis | K modeled turbulence and nonlinear clipping for QAM OFDM with FSO and fiber serially linked[END_REF]. The demodulation of QAM system will be deployed using heterodyne detection, a technique that presents many advantages as high sensitivity and better frequency selectivity in case of wavelength division multiplexing [START_REF] Fu | Performance of heterodyne differential phase-shift-keying underwater wireless optical communication systems in gamma-gamma-distributed turbulence[END_REF]. In order to investigate how turbulence and pointing errors affect the performance of the system described, a closed form mathematical expression for the bit error rate (BER), will be extracted. BER is an important metric for the reliability of a communication system [START_REF] Nistazakis | Performance Analysis of Free-Space Optical Communication Systems Over Atmospheric Turbulence Channels[END_REF], [START_REF] Goran | Average BER performance of FSO SIM-QAM systems in the presence of atmospheric turbulence and pointing errors[END_REF]. In section II, the underwater channel will be analyzed and in section III the BER of the system will be calculated. Finally in section IV the corresponding numerical results will be presented.

2

Underwater Channel Modeling main unpredictable factor that affects UOWC communications is the oceanic turbulence that is the chaotic change of pressure and flow velocity of the water. This phenomenon is responsible for variation of several characteristics of water through which the laser beam of an UOWC system propagates, creating lack of homogeneity of the refraction index. So the power that the receiver intercepts presents fluctuations, a phenomenon called scintillation. Such fluctuations may lead to deterioration of the performance of the system depending on its impact. A parameter that measures the strength of this effect is turbulence structure constant, C n 2 , that can be calculated using the following equation [5], [START_REF] Fu | Performance of heterodyne differential phase-shift-keying underwater wireless optical communication systems in gamma-gamma-distributed turbulence[END_REF]:

  2 2 7/6 11/6 00 2 16 Re (s, , L) E(s, , L) | E(s, , L) | ( ) L n n C k L ds d                         (1)
where ω is the magnitude of the spatial frequency,

  2 1 2 3 ( , , ) ik exp 0.5 (L s) / i s kL             , k
n v ww                                (2)
where ε is the rate of dissipation of kinetic energy per unit mass of the fluid, w provides the ratio of temperature to salinity contributions to the refractive index spec-trum, v is the kinematic viscosity and In order to investigate how the turbulence affects the performance of the system, various statistical models are used to model the channel depending on the turbulence strength [START_REF] Peppas | Semiconductor Optical Amplifiers for Underwater Optical Wireless Communications[END_REF]- [START_REF] Majumdar | Free-Space Laser Communication Performance in the Atmospheric Channel[END_REF]. A suitable model for weak turbulence is the gamma distribution and the pdf as a function of irradiance, I t , is given by the following equation, [START_REF] Epple | Simplified Channel Model for Simulation of Free-Space Optical Communications[END_REF]:

1 (I ) I exp( I ) () t I t t t f           
(3) where Γ(.), signifies the Gamma function and parameter ζ can be estimated as [START_REF] Epple | Simplified Channel Model for Simulation of Free-Space Optical Communications[END_REF]:

2 1 p    (4
) with σ 2 p being the power scintillation index which is the normalized variance of the received optical power, I, and can be calculated using the following equation [START_REF] Majumdar | Free-Space Laser Communication Performance in the Atmospheric Channel[END_REF], [START_REF] Epple | Simplified Channel Model for Simulation of Free-Space Optical Communications[END_REF], [START_REF] Bekkali | Transmission Analysis of OFDM-Based Wireless Services Over Turbulent Radio-on-FSO Links Modeled by Gamma-Gamma Distribution[END_REF]: , k=2π/λ is the optical wave number, λ is the operational wavelength of the link, D is the aperture diameter of the receiver and L is the distance between the transmitter and the receiver. Parameter δ 2 , is the Rytov variance given as, [START_REF] Majumdar | Free-Space Laser Communication Performance in the Atmospheric Channel[END_REF]: Due to the narrow beamwidth of the laser beam that UOWC systems use, there must be a precision in the alignment between the receiver and the transmitter. But several phenomena like mechanical vibrations of the system or rip currents in the water may cause pointing errors that lead to power fluctuations at the receiver. In order to investigate this pointing errors, Beckmman's model is suitable and accurate to model the channel and its pdf as a function of Irradiance, I p , is given by the following equation [START_REF] Ninos | Block error rate performance of OOK free-space optical links over gamma-gamma turbulence channels with generalised non-zero boresight pointing errors[END_REF]-[20]:

with g being the ratio between the equivalent beam radius at the receiver, w zeq , and the pointing error displacement standard deviation, σ s , and is given by the following equation [START_REF] Zhu | Free-space optical communication through atmospheric turbulence channels[END_REF]- [START_REF] Epple | Simplified Channel Model for Simulation of Free-Space Optical Communications[END_REF]:

2 zeq s w g   ( 10
)
The parameter w zeq can be calculated using [START_REF] Ninos | Block error rate performance of OOK free-space optical links over gamma-gamma turbulence channels with generalised non-zero boresight pointing errors[END_REF]- [START_REF] Yang | Free-space optical communication with nonzero boresight pointing errors[END_REF]:

2 2 (u) 2 exp( u ) z zeq w erf w u    ( 11
)
where w z is the beam waist, /2 z u r w

  and   2 0 (u) A erf 
with w z represents the radius of the Gaussian beam on the receiver plane.

In order to investigate both turbulence and pointing errors effect, a combined pdf will be extracted using the following mathematical property [START_REF] Sandalidis | Optical Wireless Communications With Heterodyne Detection Over Turbulence Channels With Pointing Errors[END_REF]:

| (I) ( | ) (I ) dI tt I I I t I t t f f I I f   (12)
where I=I t I p . Substituting equations ( 3) and ( 9) in ( 12), the combined pdf for weak turbulence strength and pointing errors will be given as [START_REF] Ninos | Block error rate performance of OOK free-space optical links over gamma-gamma turbulence channels with generalised non-zero boresight pointing errors[END_REF]:

2 2 2,0 1,2 2 0 0 () () 1, 1 I g gI f I G A A g           (13)
The combined pdf can be expressed as a function of the instantaneous SNR, μ, and the average SNR, ξ, using the following equation for the case of heterodyne detection systems, [START_REF] Fu | Performance of heterodyne differential phase-shift-keying underwater wireless optical communication systems in gamma-gamma-distributed turbulence[END_REF], [START_REF] Sandalidis | Optical Wireless Communications With Heterodyne Detection Over Turbulence Channels With Pointing Errors[END_REF]:

I    (14)
So applying random variable transformation in equation ( 13) using equation ( 14), the combined pdf as a function of SNR will be given by the following equation:

2 2 2,0 1,2 2 0 0 () () 1, 1 g g fG A A g                (15)

Bit Error Rate Calculation

A very important metric concerning the performance of a link is the bit error rate that represents the probability of receiving an error bit at the receiver. The modulation technique that is assumed to be applied in the underwater link will be M-QAM with heterodyne detection.

The BER for the received M-QAM signal is given by [START_REF] Bekkali | Transmission Analysis of OFDM-Based Wireless Services Over Turbulent Radio-on-FSO Links Modeled by Gamma-Gamma Distribution[END_REF], [START_REF] Nistazakis | QAM and PSK OFDM RoFSO Over$M$-Turbulence Induced Fading Channels[END_REF]:

  1/2 2 21 3 () log (M) 4(M 1) M P erfc            (16)
Using equation ( 16), the average BER can be obtained by averaging over the combined distribution of equation ( 15) :

  1/2 2 20 2 2,0 1,2 2 0 0 21 () log (M) A ( ) 3 4(M 1) 1, 1 av Mg P g erfc G A g                           (17)
The above integral can be solved by expressing the erfc(.) function using Meijer G function and after some mathematical manipulations [START_REF] Adamchik | The Algorithm for Calculating Integrals of Hypergeometric Type Function and its Realization in Reduce System[END_REF] the BER of an underwater optical link in a channel with weak turbulence and pointing errors using M-QAM modulation will be given as:

      1/2 2 20 2 2,2 3,3 2 0 8 1 1 () 3 log (M) A ( ) 1 0, , 41 2 3 1, 1, 1 av Mg P g G g                           (18)

Numerical Results

In this section the results of the average BER, Pav, will be presented for an underwater optical communication system using M-QAM modulation technique according to equation [START_REF] Ninos | Block error rate performance of OOK free-space optical links over gamma-gamma turbulence channels with generalised non-zero boresight pointing errors[END_REF]. We have chosen three to values for the modulation index M=4 and 16 and three values for the ratio of temperature to salinity contributions to the refractive index, w=[-0.05, -1, -3], the rate of dissipation of kinetic energy per unit mass ε=[10 -1 , 10 -2 , 10 -4 ]m 2 /s 3 and the viscosity, v=[0.01, 0.002, 10 -5 ]. All values were chosen for low values of the scintillation index m 2 [5]. The wavelength of the laser beam is assumed to be λ=532nm, the aperture diameter of the receiver D=1cm and the distance L=80m. Concerning the pointing errors, we assume normalized spatial jitters with equal to σ x /r=0.05 and σ y /R=0.1. In Fig. 1, the BER of the underwater OWC system for 4-QAM is presented as a function of links parameters. It is clear that as parameters w, ε and v are not stable, the BER of the system may change even for 2 orders of magnitude. More precisely, as the absolute value w increases, the performance of the system decreases, as the BER becomes higher. Furthermore, the reliability of the system increases when parameter ε, decreases. Viscosity also affects the BER of the systems as the systems performs better for lower values of v. In fig. 2, the corresponding results of BER are presented in case of 16-QAM. It is clear that the system's performance deteriorates as the BER increases and as in figure 1, the system's BER strongly depends on link's and channel's parameters.

Conclusions

In this work we derived a closed form mathematical expression for the BER of an underwater optical link modulated with M-QAM with heterodyne detection that is affected by weak turbulence and pointing errors. According to the results presented, UOWC systems are strongly depended on oceanic turbulence and pointing errors, so the choice of link parameters must be careful in order to deploy a communication system with high performance.

Fig. 1 .

 1 Fig. 1. Average BER of UOWC for 4-QAM

Fig. 2 .

 2 Fig. 2. Average BER of UOWC for 4-QAM

  is the wave number, L is the dis-

	( ) 0.388 10	12	1/3	11/3	 1 2.35	2/3 1/2 1/6	
	2		2	2 exp( 1.863 10 ) exp( 1.9 10 ) 4
	3 2 w exp( 9.41 10 )	
	tance between the transmitter and the receiver and Φ n (ω) is given as [5],[8]:

Acknowledgment: This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No: 777596