E Primeros auxilios.
SECCIÓN 2.ª NORMAS ESPECÍFICAS PARA DETERMINADOS TIPOS DE ANDAMIOS
Bourreau et al. [2] have developed a mathematical model to compare the effects of competition between ISPs under a NNN and NN regime. While the ISPs mechanically benefit from NNN in a monopoly, it is less clear that switching to an NNN regime would benefit competing ISPs. The model also considers investments by ISPs and CPs, multi-homing CPs and heterogeneous CPs and IUs. The setting is quite similar to that of Krämer et al. However, this model also includes competing ISPs. The notation used in the article is presented in the following table.
i ϵ {A, B} h ϵ [0, ∞)
Index of the two horizontally-differentiated competing ISPs Continuum of CPs with mass 1
λ a R v d t
Constant number of visits per user, which is the same for all web- sites
Per click advertising revenue IUs utility of connecting to as ISP
IUs preference for product variety supplied by the CPs
Parameter for the IUs preference for the speed of the connection The standard Hotelling unit transporting cost
wi xi ̅
̃
Congestion on ISP i’s network
Number of end-users subscribing to ISP i
Marginal CP which is indifferent between connecting to ISP I and not connecting
Marginal CP which is indifferent towards priority lane and non- priority lane
pi μi
Subscript for IUs to connect to ISP i’s network ISP i’s network capacity investments
Table 7: Indexes, parameters and variables of the mathematical model by Bourreau et al. [2]
With a click trough rate on ISP i for CP h of (1-hwi), the advertising revenue for CP h is
( )
( 30 )
Under NN (N), the profit of CP h is
{
(
)
(
)
(
)
( 31 )
Under NN, all CPs are active at ISP I are treated equally and face the same average level of congestion wi
N
. However, under the NNN (D), a CP may choose to pay a fixed fee to ISP i to benefit from a priority (P) lane where the congestion is lower. The profit for CP h under NNN is given by
{
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
( 32 )
In the NNN regime the CP that connects to ISP choose either to pay for access to the priority lane or to use the non-priority (BE) lane for free.
The two ISPs are located at the extremities of a linear city of length one, with ISP A located at point 0 and ISP B located at point 1. The investment costs C(μi) is increasing and convex in μi (C’ > 0 and C’’ > 0). Under NN the profit function of ISP i is
In the NNN regime, the ISPs can also charge a fixed fee, fi, to the CPs that opt for the priority lane. The ISP profit function in the NNN regime is
( ̅
̃)
(
)
( 34 )
Due to capacity constraints the IUs might suffer from congestion. Congestion is measured by the waiting time for IUs when they request content from CPs. The M/M/1 queue model is used to determinate the average level of congestion as a function of network capacity and traffic. Under the NN regime, the average level of congestion for ISP i is
̅
( 35 )
Under the NNN regime, each ISP sorts CPs into two traffic lanes, the priority lane and the non-priority lane. The congestion for the priority lane (P) operated by ISP i is given by
( ̅
̃
)
( 36 )
The congestion for the non-priority lane (NP) is given by
̅
( 37 )
The formula implies that congestion is always higher in the non-priority lane. If bi = 1 - ̃ is the share of CPs that buy priority from ISP i, note that the average congestion under the NNN regime satisfies
( )
̅
( 38 )
So if the volume and capacity are the same under the NN and NNN regime, the average congestion will be the same (wi
N = wi
D
). This is a well-known property of the M/M/1 queuing model.
There is a unit mass of IUs uniformly distributed along the unit interval. Each IU subscribes only to one ISP (single-homes). Under NN, a user located at xj on the unit interval and who subscribes to ISP A, obtains utility
̅
( 39 )
A similar expression can be obtained for ISP B. It is assumed that R is sufficiently high so that the market is covered in equilibrium in both regimes. Under the NNN regime, the end user located at xj obtains utility
̅
( 40 )
It is also assumed that the IUs value content sufficiently compared to the disutility they suffer from congestion. In particular, in a symmetric equilibrium, it must be that
( 41 )
To see why, put ( 38 ) into ( 40 )
(
) ̅
( 42 )
For consumers to value the CPs, the bracket has to be positive. Therefore, , which implies what is stated above as in an equilibrium solution xi
D = 1/2.
In the NN regime, the model is solved in a two stage game with the following set- up
1. The two ISPs choose their capacities μA N and μ
B
N, and set the subscription fees to the end users pA
N and pB
N .
2. The CPs choose which ISP(s) to connect to (if any), and the end users choose which ISP to subscribe to.
The model is solved backwards to find the symmetric subgame perfect equilibrium (SPE).
In the NNN regime, each ISP offers a priority lane and a non-priority lane to CPs. The CPs that opt for priority at ISP i pay a fixed fee fi, whereas the non-priority lane is offered for free. The two-stage game is modified accordingly:
1. The two ISPs choose their capacities μA D and μ
B D
, set their subscription fees to the end users, pA
D and pB
D
, as well as the fees for their priority lanes fA and fB. 2. The CPs choose which ISP(s) to connect to (if any) and whether to pay for
priority, and the IUs choose which ISP to subscribe to.