In this section, we present the model and the analytical results we derive thereof. Our model is built in the intersection of the three areas of related work discussed in
Section 1.2: we tackle the
pricing scheme of 5G cloud and network infrastructure providers that drives the evolution of their business topology with the methodology and tool set of
network formation games. Furthermore, in our opinion, the arrangement of future 5G business networks is exceptionally reliant of, and, in this way, will be at first dependent on the current Web topology. Thus, we expect a layered structure following the
present Internet transit and peering relations as our underlying topology model to proceed with the assessment on how the new 5G environment may adjust the initial setup. We showcase the fundamental trade-off regarding expenses of keeping up business relations, and the cost of business mediators.
2.1. Graph Model of Business Relationships
Figure 1 shows instances of our graph model of 5G providers. In this model, vertices stand for (i) network providers (likewise offering computing services) and (ii) Service Access Points (SAPs), which are reaching points of end-clients. Edges represent business connections, which, in the underlying stage can be either (i) transit relations (depicted by solid lines in the figure) or (ii) peering relations (depicted by dashed lines) between nodes. The locality of clients and delay-critical services are among the most significant driving components of multi-provider setups; accordingly, we portrayed SAPs only at Level 3 providers as those offer access service to end-clients [
25]. This is notwithstanding that in our vision all network providers are expected to become 5G cloud operators as well [
4,
6].
In the center of our model are such multi-domain services in which the end-clients of the service are clients of provider A, but they really utilize the service inside the access domain of provider B. In this situation, provider A purchases a resource slice from provider B (and perhaps from different providers interconnecting providers A and B). We model these business arrangements as paths in the graph model associating two nodes: the purchaser of the service is essentially the customer-facing provider, while the seller of the service is the provider in whose domain end-clients currently utilize the service. The delay-critical service is subsequently placed at the vendor provider’s premises near the current SAP of the end-client. The service’s descriptive characteristics, e.g., computing needs, explicitly stating VNFs to onboard that comprise the service, network QoS to the SAP, and so forth, should be satisfied and paid for by the purchaser provider. In this multi-provider arrangement, we investigate the development of business relations: we do not handle the pricing of services or that of the resource slices, this study is limited to the valuation of inter-provider organizational possibilities.
2.2. Cost of Business Relations and Price of Middlemen
Deploying a service into the resource slice that is mounted on another provider(s) infrastructure requires prior negotiations and a pre-built business link or path among the partners, similarly to the network formation models, e.g., in [
19]. As depicted in
Section 1, we assume that two opposing effects decide how these arrangements are made: either by a direct business made between the two parties, or through a chain of agents that relay between the seller and purchaser providers. In the second case, the base cost of the resource slice is supplemented by the business relay cost of the interconnecting providers. The length of the chain of mediating providers equals to the number of hops on the path between the two providers in our graph model. Alternatively, each extra immediate business connection between providers incurs an expense at the two players: setting up and keeping up business contracts have their expenses. The trade-off situation is clear: while edges increment the general managerial expenses, barring mediators from setting up resource slices in a remote operator’s infrastructure saves cost. Compared to the classical network formation games [
16,
17,
18], the utility of being strongly connected to other nodes there, is, in our case, translated to fewer middlemen to pay for mediating the business between remote nodes.
Figure 1 showcases the business connection between a chosen pair of SAPs in an underlying layered graph model. The path traverses the transit links towards the highest level of the tiered structure of providers. Nevertheless, another business interconnection can be shaped by building up business associations between any two nodes at lower tiers, e.g., between Tier 2 providers in this case, as appeared in
Figure 1. We underline the important notice that the new business connection does not change the data plane path if we assume that the newly connected providers are not neighboring ASes. Tier 1 providers will presumably provide the data plane connectivity, which might be provisioned for the relating Tier 2 providers on an alternate timescale. The new business path is, however, abbreviated as Tier 1 providers do not act as business mediators any longer.
In the following, we characterize the network formation game we use for tackling the aforementioned model. As part of the study, we infer significant characteristics that portray the equilibria of the game. Note that the fundamental contrast between our game and the current ones presented in
Section 1.2 is due to the cost terms: first, the distance measure in our situation is switched to broker costs; second, the expense of every player is decreased by the pay produced by being a mediator agent.
2.3. Game Definition
For tractability, let us use the notation of [
17]. We consider the network service providers as the players of a network formation game.
N denotes the player set
.
denotes the strategy set of player
i, which is the power set of
, in other words, the collection of possible sets of other players to create a link with.
indicates the whether
i wants a link between nodes
i and
j, so
and
. The strategy of
i is
and
. The combination of the strategies of all players provide the outcome of the game. The resulted strategy profile is denoted by
. The outcome of this one-shot game is an undirected graph
in which a given edge is built if there is consent between the two nodes, i.e.,
. That is, both players
i and
j must agree to establish a link between each other in order for it to be created.
Cost function
c determines the player cost given the strategy profile out of the combination of strategy sets, i.e.,
. As in related work, the expense brought onto player
i when all players embrace strategy
s is comprised by the expense based on the quantity of links
that player
i sets up effectively with different nodes, and by the aggregate of the agent expenses paid to middlemen to reach every single other node. As a novel term, we represent the salary that is created by mediating business going through node
i. In our game, the total cost is defined as follows:
where
and
are the business peering cost and the middleman price introduced in
Section 2, respectively;
denotes the number of middlemen on the shortest-path between providers
i and
j in the business graph
G;
depicts the extent of services bought by
i from
j through whatever path of middlemen providers this business is realized; and
indicates whether
i is on path
, i.e., the shortest path between
j and
k. If no path exists between
i and
j, then
. Therefore, the first term of the cost formula stands for the business link creation, the second term reflects the price to pay middlemen for reaching providers indirectly, and the third term is the income that is generated by acting as middlemen for other providers’ businesses.
As in [
17], this game model reflects a setting in which building connections is expensive; however, an extensive direct network might be beneficial to build in order to limit the brokers to pay off. Likewise, the more connections a provider has, the more probable it will earn as a middleman, subsequently lowering the total cost. Naturally, providers try to limit their costs characterized in Equation (
1). If the expense of an extra business link
, the mediator cost
, and the business demand
M are all fixed, the game comes down to the question of which new connections are worth being made so as to reduce expenses.
2.4. The Effects of Link Creation
For different sorts of equilibrium, stability conditions, lower and upper limits on the price of anarchy in network formation games, we refer the reader to [
16,
17,
18]. Note that our model is different from the models found in the related work. The game variant that is the most similar to ours was introduced in [
19]; however, contrary to that model where a player’s transit traffic brings about expense, in our arrangement, the more shortest paths that traverse a node, the more income is created to that respective player. For their arrangement, the authors of [
19] demonstrated that the steady result of the game is consistently a tree, as more transit paths and the edge creation are not worth making lower distances to different nodes, once the graph is connected. In our game, the final graph
can also be a tree for high connection creation cost
: both lower distance to different nodes, and mediating more business paths decrease the expense, hence, if edge creation is generally low cost, it is useful to make a few.
As portrayed in
Section 2, we expect an underlying layered topology of providers. The objective of the work introduced in this section is to give an adequate condition under which there are no new connections made by the players. In case this condition is fulfilled, the underlying layered structure is along the stable state of our game, i.e., an equilibrium.
Assumption 1. There are business links between providers originally, and these links organize nodes in a tiered topology, denoted by , such as the one depicted in Figure 1a. Let us number the tiers from top to bottom, , and let indicate the tier that provider i belongs to. Let denote the set of providers that can be reached downwards in the tiered topology through provider i, i.e., , preferring peering links in Tier-1 to peering links in lower tiers. Now, we deduce the parametric cost saving when a new link is created.
Lemma 1. If Assumption 1 holds, the highest cost reduction a new link between two nodes, i and j, can result in is Proof. Providers
i and
j, belonging to tiers
and
respectively, would both make a cost reduction for their children in
and
by interconnecting themselves with a new link and thus lowering the second term of Equation (
1) of the children. At most,
middlemen in upper tiers are shortcut from cross paths between the two sets of children with the new link. This number might be lower if any peering links exist between parents of
i and
j, or if they have the same Tier-1 parent. Note that we assume full mesh among Tier-1 providers in
. The cost allocated to middlemen is proportional with the extent of the business which is upper bounded by
. The number of business relationships is given by
, hence the result starting from the following formula:
□
Hindered by the complexity in a general tiered topology setting, we make the following assumption on the number of children each node has, and of businesses leaf nodes make.
Assumption 2. contains a number of Tier-1 nodes connected in full mesh, and a tree subgraph under each Tier-1 node in which intermediary nodes have at least k children, and all leaf nodes are at the same depth t. Furthermore, any pair of leaf nodes exchange m amount of business; intermediary nodes do not act as service sellers or buyers.
Given the specific tiered topology of Assumption 2, we prove that the highest cost saving can be attained with new peering links in the topmost tier.
Lemma 2. Under Assumption 2, the higher tier the nodes belong to, the larger the cost saving that is attained if they create a new link.
Proof. Under Assumption 2, the size of and are lower bounded by the number of leaves of perfect k-ary trees: . The cost saving of two nodes i and j by creating a link is , where m represents the amount of business any pair of leaf nodes exchange under Assumption 2. By expressing , it is easy to see that this cost saving is higher when is larger. As and , the maximum is attained if , i.e., and , or , i.e., and . □
2.6. The Stackelberg Game of Bertrand Competitions
Finally, in this section, we describe the equilibrium point of the network formation game with the assumption of initially non-existent peering relations between low tier providers, and of missing transit links between tiers.
Assumption 3. Let us assume a 3-tier topology of providers, initially with no transit/peering links other than the Tier-1 full mesh. Furthermore, we assume that one transit link can be built by each Tier-2 (to a Tier-1) and Tier-3 (to a Tier-2) provider for no cost.
Bertrand competitions [
26] describe interactions among suppliers that set prices and their customers that choose quantities to purchase from them at the prices set. The Bertrand competition model assumes that (i) there are at least two suppliers producing a homogeneous (undifferentiated) product and cannot cooperate in any way, (ii) suppliers compete by setting prices simultaneously and consumers want to buy everything from a supplier with a lower price (since the product is homogeneous and there are no consumer search costs), (iii) if suppliers charge the same price, consumers’ demand is split evenly between them, (iv) all suppliers have the same constant unit cost of the product or service, so that marginal and average costs are the same, and (v) each supplier has sufficient capacity to serve all customers.
Bertrand proved that, if suppliers chose prices strategically, then the competitive outcome would occur with the equilibrium price equal to marginal cost. In our specific case, the product to sell is the middleman service; therefore, we suppose that there is no maximum capacity imposed on the suppliers, and they all offer the same service.
We argue therefore that the game of setting individual middlemen prices in Tier-1 and Tier-2 are Bertrand competitions, as lower tier providers seek to build their transit link to one with the lowest price. Consequently, the Nash Equilibrium of the Bertrand game in Tier-1 results in a homogeneous middlemen price for all Tier-1 providers, covering the marginal cost of the full mesh linkage. Considering the average transit business they take care of, the following statement displays the value of .
Lemma 3. In equilibrium, Tier-1 providers all set their middlemen prices to , where is the number of Tier-1 providers, γ is the fraction of businesses reaching Tier-1 and m is the grand sum of business matrix M.
Proof. As in a Bertrand duopoly competition, the only equilibrium price for the competing providers is at the marginal cost, since any provider setting a higher price would lose its customers. The total income is provided by the fraction of businesses flowing through Tier-1 providers, i.e., not via Tier-2 peering links, denoted by . Supposing a uniform distribution of businesses flowing through the Tier-1 providers, and an equal share of peering costs, then applying covers the cost of full mesh peering at each Tier-1 provider. □
As Lemma 3 shows, the Bertrand game equilibrium is partly defined by the fraction of business () that will reach Tier-1 providers in the first place. This fraction is, in turn, dependent on the middlemen price that Tier-1 providers set, i.e., because, for certain Tier-2 providers that exchange relatively large amount of business, creating a direct link might be beneficial compared to paying the Tier-1 middlemen. In order to grasp this condition, we introduce the empirical distribution of such business amounts.
Definition 1. Let us denote the empirical distribution of the amount of business between Tier-2 provider pairs by . Similarly, let us denote the distribution of those between Tier-3 providers by .
Both Tier-2 and Tier-3 providers have the option of creating peering links in case it is less costly than dealing with middlemen. As middlemen prices grow, more and more provider pairs decide so. Therefore, the fraction of business from Tier-2 to Tier-1, and from Tier-3 to Tier-2, decreases with the rise of middlemen prices: demand is monotone decreasing in
and
, respectively. Furthermore, peering links are created in the decreasing order of the amount of business between the two endpoints, as the middlemen price grows. This phenomenon creates a Stackelberg game [
21] nature of the middlemen price setup between Tier-1 (being the leaders) and Tier-2 (being the followers). The Stackelberg competition suits well the situation as the leaders, i.e., Tier-1 providers move first by setting their middleman prices homogeneously, and then the followers, i.e., Tier-2 providers move sequentially, deciding about the link creation in function of Tier-1 prices. Let us see how we can deduce the equilibrium price in Tier-2.
Lemma 4. In equilibrium, the Tier-2 providers’ middlemen price iswhere δ is the fraction of business reaching Tier-2, and denotes the number of Tier-2 providers. Proof. Similarly to Lemma 3, in equilibrium, all Tier-2 providers apply the same middlemen price , and their total income precisely covers their cost. The overall Tier-2 income is given by the Tier-3 providers, with their business not traversing through their own peerings, i.e., . The middlemen fee and the creation of peerings constitute the total cost of Tier-2 providers, i.e., and , respectively, where denotes the amount of business over which the peering is cheaper than via Tier-1 middlemen. This latter condition gives . Substituting the value of with from Lemma 3 yields the formula for . □
Finally, we can draw the amount of business that Tier-3 providers will carry through Tier-2 providers. The following statement expresses the necessary formulas for substituting and in Lemmas 3 and 4.
Lemma 5. In equilibrium, the fraction of business flowing through Tier-2 and Tier-1, respectively, are: , and .
Proof. Similarly to the proof of Lemma 4, let us denote by the amount of business over which the peering is cheaper than through Tier-2 middlemen. Analogously to , . The statement then follows. □
The equilibrium prices can be determined numerically if are given. Moreover, if Tier-3 providers select their Tier-2 transit partner randomly, and the ratio between the number of Tier-2 and Tier-3 providers is sufficiently small, the central limit theorem might be applicable in order to determine based on . Hindered by the analytic complexity of the problem, we now turn to a comprehensive numerical analysis of the business network formation game.