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\title{$\hbL{Dynamic Behavior of Differential Pricing and}$ \\ $\hbL{Quality of Service Options for the Internet}$}
\author{$\heM{Peter C. Fishburn and Andrew M. Odlyzko}$ \\
$\he{AT\&T Labs - Research, 180 Park Avenue, Florham Park, NJ 07932}$ \\
$\heM{fish@research.att.com, amo@research.att.com}$
}
\date{}
\begin{document}
\maketitle

\noindent
{\large\bf Abstract}.
The simple model on which the Internet has operated, with 
all packets treated equally, and
charges only for access links to the network, 
has contributed to its explosive growth.
However, there is wide dissatisfaction with the delays and 
losses in current transmission.  Further, new services such as 
packet telephony require assurance of considerably better service.
These factors have stimulated the development of methods for providing
Quality of Service (QoS), and this will make the Internet
more complicated.  Differential quality will also force
differential pricing, and this will further increase the complexity of the
system.

The solution of simply putting in more capacity is widely regarded as
impractical.  However, it appears that we are about to enter a period
of rapidly declining transmission costs.  The implications of such an
environment are explored by considering models with two types of
demands for data transport, differing in sensitivity to congestion.
Three network configurations are considered: (1) with
separate networks for the two types of traffic, (2) with a single
network that provides uniformly high QoS, and (3) with a single
physical network that provides differential QoS. The best solution
depends on the assumptions made about demand and technological
progress.  However, we show that the provision of uniformly high QoS to all
traffic may well be best in the long run.  Even when
it is not the least expensive, the additional costs it imposes are
usually not large.  In a dynamic environment of rapid growth
in traffic and decreasing prices, these costs may well be worth paying
to attain the simplicity of a single network that treats all
packets equally and has a simple charging mechanism.


\newpage
\section{Introduction}
The Arpanet, which evolved into today's Internet, was a research
project that did not provide for any
payment mechanisms and treated
all packets on an equal ``best-effort'' basis.
The Internet has (with minor exceptions) inherited these properties.
Packets are basically still treated equally.
Charging usually is only for the bandwidth of the connection 
to the Internet and is independent
of the amount of data sent and received.  (See \cite{McKnightB}, especially
\cite{MacKieMV}, for a survey of the economics of the Internet.)
These features, which provide for extreme simplicity in both operation and
economics, have
contributed to the spectacular growth of the Internet.

Although there have been persistent criticisms about the lack
of Quality of Service (QoS) provision on the Internet, and
about the charging scheme, thus far they have not been sufficiently
convincing 
to modify the system.
However, there are signs that change
is imminent.  Dissatisfaction with endemic congestion on the public Internet,
which makes even Web surfing annoying, and the need to 
provide QoS for novel applications that are delay-sensitive,
such as packet telephony and videoconferencing, are leading
to demands for differential treatment of packets.  
% (See \cite{FergusonH} for techniques for providing QoS.)  
Similar
demands are coming from the corporate side.  Private line
networks use the same IP (Internet protocol) technology, 
are far larger in aggregate than the public Internet \cite{CoffmanO},
and have been providing high QoS largely through low utilization
levels \cite{Odlyzko1}.  However, with demand for bandwidth rising, corporate
network managers are also demanding tools such as prioritization
to ensure higher efficiency of network usage.  Differential service
quality will inevitably force introduction of more complicated
pricing schemes than the present one, since it will be necessary
to prevent all traffic from being sent on the highest quality
level.  The departure from the simple network operations and
charging mechanisms of the Internet would represent at least
a partial victory for the ``Bell-heads'' in the infamous
controversy over networking \cite{Steinberg}.

The ``Net-head'' approach to the problems of poor service has been to provide
greater bandwidth and keep the charging algorithm simple.  This solution
is used universally in LANs (local area networks) and has worked for
corporate and research networks in the past.  The objection to
the ``Net-head'' approach is that it is too expensive, 
at least for the public Internet,
since more than two decades of experience have shown that any
bandwidth gets saturated quickly.    

Data transport is a serious constraint on ISPs (Internet
service providers), as it accounts
for about half of the total cost of long-haul networks.  While
new optical fiber technologies led to a dramatic drop in rates
for leased lines in the 1980s and early 1990s, prices have been increasing
recently as a consequence of scarcity of supply and rapidly 
growing demand (see \cite{CoffmanO, Rendleman} for examples).
Network operators have been lowering their cost per unit
of bandwidth by moving to higher capacity lines (see Section 2 for
data and discussion of this issue)
and by signing long-term leases.
In an environment of rising prices, differential QoS and more sophisticated
pricing schemes appear essential to meet the explosive
data transport needs at affordable cost.

Do we have to give up on the simple operation and charging
mechanisms of the current Internet?  Both splitting 
of traffic into different QoS classes and complicated charging
mechanisms impose heavy costs on developers of applications
and network systems, and on network operators.  
Further factors in favor of
simple fixed-fee charging mechanisms come from customer
preferences (even those of large corporate customers),
which often lead to higher revenues for service providers
who use such pricing approaches \cite{FishburnOS}.
An early 1988 paper by Anania and Solomon (published in [AnaniaS])
already presented several arguments for 
a simple flat-rate pricing approach to broadband networks.

% In the broadband telecommunications marketplace
% we can observe a move towards simpler charging mechanisms.
% This is visible, for example,
% in the decreasing role played by distance in pricing of
% data transport.  
% Corporations are moving from
% traditional private line services to shared public networks
% such as ATM and Frame Relay.  
% All three methods involve fixed monthly fees (with the exception of SVCs,
% switched virtual circuits, where usage-sensitive pricing
% does appear inevitable and is implemented, but is unpopular,
% and is often cited as a major reason for the slow acceptance of SVCs).
% However, private line prices depend on distance (although
% even here the dependence is decreasing, see \cite{CoffmanO}, for example),
% while ATM and Frame Relay charges do not.
% The attractions 
% of simple fixed-fee charging thus appear to be great.  On the
% other hand, we do see some movement away from them,
% with an increasing number of Internet service providers
% imposing usage limits on their fixed monthly fee subscribers.
% Web hosting services already tend to have usage-sensitive 
% pricing.

Although simple flat rate pricing with uniform best-effort data transport
is attractive, it has many defects.  It provides
a single level of service quality, and does not allow users
to select what is best for their needs.  Economists in general
oppose it on the grounds that it leads to misallocation of resources.
For fuller description of the 
arguments for abandoning the traditional Internet model,
and for further references, see \cite{McKnightB}.
However, those arguments are based on experience
with an environment that is likely to change drastically.
(It is already an environment far removed from the traditional
telecommunications world studied in \cite{MitchellV}, for example,
and will diverge from it even further.)
As mentioned above, long distance data transport prices have been rising
in the last few years.
The basic fiber optic network that 
carries both voice and data traffic was designed primarily
for voice, and until a few years ago, most of the bandwidth
was devoted to voice.
In revenues, the network is still dominated by voice.
However, the bandwidth
devoted to data is already comparable to that used
for voice \cite{CoffmanO},
and data traffic is growing much more rapidly.  Thus we can
expect communications networks will grow rapidly and be increasingly
dominated by data.
Furthermore, WDM (wavelength division multiplexing)
technology allows for expanding
capacity without laying down
more fiber (at least not on long distance routes), which is
a very expensive process, especially when acquisition
of rights-of-way is included.  
Within a few years, existing fiber will provide 100 or even
1000 times the bandwidth it did a couple of years ago, at
modest additional cost.  The main determinants of network
costs will be the electronics needed to provide WDM and
switching.  However, in electronics ``Moore's Law''
reigns, with performance increasing while prices drop.
What this means is that we are likely to enter an era
in which the price of bandwidth continues dropping
dramatically for a decade or more.  The question is,
what will this mean for service providers and
consumers?

It is instructive to consider microprocessors.  Table 1
shows the last dozen years from the history of Intel.
For each year, the microprocessor listed is the
most powerful model introduced that year, with the
price the one available at the end of that year.
(All dollar figures are in nominal dollars, and the
prices are for orders of 100 or 1000 chips at a time.)
The processing power, in mips (millions of instructions
per second) is an imperfect measure of the computing
power of processors.  Still, it illustrates how
the power of state of the art microprocessors has
been growing at an exponential rate, while their
prices have remained about constant.  At the same time,
revenues
and profits have increased.  Over the period illustrated
by Table 1, computing power has grown over 60\% per year,
with prices of the most powerful available processors
rather stable, while Intel's revenues have
grown about 30\% per year.
A similar scenario
appears to be realistic for high bandwidth communication networks
in the next decade.  What we explore are
the implications of this kind of environment for the
provision of QoS on the Internet.  If available capacity
doubles each year, or every two years, while total costs
increase much more slowly, so that the price per unit
of bandwidth decreases rapidly,
it might make sense to provide uniformly
high QoS for everybody and avoid the complexities of
the schemes that are being considered.


\begin{table*}[htb]
\caption{Intel and its microprocessors.  For each year, lists the
most powerful general purpose microprocessors sold by Intel, its
computing power, price at the end of the year (in dollars), and
Intel's revenues and profits for that year (in millions of dollars).}

\vspace*{+.1in}
\begin{center}
\begin{tabular}{llllll}
year & processor & mips & price & revenue & net profit \\ \hline
86 & 386 DX (16 MHz) & 5 & 300 & 1265 & -173 \\
87 & 386 DX (20 MHz) & 6 &  & 1907 & 248 \\
88 & 386 DX (25 MHz) & 8 &  & 2875 & 453 \\
89 & 486 DX (25 MHz) & 20 & 950 & 3127 & 391 \\
90 & 486 DX (33 MHz) & 27 & 950 & 3922 & 650 \\
91 & 486 DX (50 MHz) & 41 & 644 & 4779 & 819 \\
92 & DX2 (66 MHz) & 54 & 600 & 5844 & 1067 \\
93 & Pentium (66 MHz) & 112 & 898 & 8782 & 2295 \\
94 & Pentium (100 MHz) & 166 & 935 & 11521 & 2266 \\
95 & Pentium Pro (200 MHz) & 400 & 1325 & 16202 & 3566 \\
96 & ~ & ~ & ~ & 20847 & 5157 \\
97 & Pentium II (300 MHz) & 600 & 735 & 25070 & 8945 \\
\end{tabular}
\end{center}
\end{table*}


Existing work on QoS, surveyed in the book \cite{FergusonH},
does not contain any projections of the degree to which
the different proposals for providing QoS will lower network
utilization.  
The relation between utilization of network capacity and
perceived quality of service is a complex one.  
It is possible to have a lightly utilized network that 
delivers horrible service, but in general the lower the
utilization rate, the better the service.  Further,
many networks, such as corporate Intranets,
are already providing QoS largely through low utilization 
rates \cite{Odlyzko1}.
High-quality experimental networks such as vBNS also have very low
utilizations.  
These networks are still operated on
the ``best-effort'' basis, with no explicit guarantees (but with
sophisticated traffic engineering tools).
Congestion episodes are infrequent enough for this to be
acceptable.  In general, no matter how a network is engineered,
lowering the traffic load on it will result in better service.
The routers and switches are fast enough already that if congestion
does not cause buffers to fill up, the quality is sufficient for
all anticipated demands.

In this work, we will assume as a first approximation
that
improved QoS is associated directly with low utilization
levels.  Although schemes like
those in \cite{FergusonH} can increase
the efficiency of the network, whether it has just a single
best-effort service, or several classes of service,
it is hard to
incorporate them into an economic model until more is known
about their performance.  

To explore potential futures for QoS on the Internet with and without
differential pricing,
we will assume two types of demands in our models.
One is for transport that is delay insensitive,
such as many bulk file transfers or even email.  
The other is for transport of information that is sensitive
to delay, such as packet telephony, or even some Web
browsing.
(In effect we will thus be considering Class of Service models for the
Internet, and not the more involved QoS ones.)
We refer to delay insensitive demand, or to its users, as type $A$, and
to delay sensitive demand, or {\em its} users, as type $B$.
Within a given time period, each type has a {\em potential volume} or
potential demand, which is the total Internet transfer volume the
type would use if the transfer charge or price were essentially zero.
We denote their potential volumes by $V_A$ and $V_B$, or
simply by $V$ as a general designation.

We will vary the ratio of $V_A$ and $V_B$, but only within
a narrow range, near equality.  The justification
for this is that in current data networks, the volumes of
data sent over the congested public Internet and over
the uncongested private line networks are comparable.
If $V_A$ were much larger than $V_B$, then clearly
it would be best to send all data over an uncongested
network designed for type $A$ traffic.  On the other
hand, if $V_B$ were much larger than $V_A$, the case
for a separated network or a two-tiered network would
be much stronger.

Because real use will be price sensitive, the actual volume carried 
for a user type during the period is modelled by $P(x)V$, where $x$ is 
the price per unit of volume and $P(x)$ is the probability that a 
potential user will subscribe to the service at price $x$. 
We refer to $P$ as the {\em demand function} and assume that
$P(0)=1$ and that $P(x)$ decreases toward 0 as $x$ increases.
An approximate but revealing measure of customer satisfaction is the 
{\em demand satisfaction} expressed as the percent of potential
volume that customers subscribe to during the period,
i.e., $100P(x)$.
This should not be confused with the utilization of available network capacity
since, for example, a channel that carries priority data may have a 
high demand satisfaction yet provide very good QoS because its 
transport capacity substantially exceeds the priority demands.
Several forms will be considered for $P$ to account for the 
possibility that our conclusions may depend on assumptions about 
the demand function.

Three network configurations are examined for provision of service to 
types $A$ and $B$, as follows:
\begin{enumerate}
\item
physically separate networks are used for each of $A$ and $B$, with each 
network having its own cost, QoS, and price characteristics; 

\item
a single network is used for both $A$ and $B$, with one price for all users 
that is constructed to provide the high QoS desired by $B$;

\item
a single network is used for $A$ and $B$, but the types are logically 
separated by software that differentiates between them and allows 
different QoS and prices for the two. 
\end{enumerate}

We refer to (1) as the {\em separated} network, to (2) as the 
{\em one-price} network, and to (3) as the {\em two-tiered} network. 
We assume for (3) that the types use logically separated channels
and ignore techniques such as those in \cite{FergusonH} that can 
lead to greater efficiencies, as when
low-priority traffic is used to fill gaps in high-priority traffic.
We also ignore the large increases in utilization rates
that can be gained by
exploiting different time-of-day patterns of use (which are
discussed in detail in \cite{Odlyzko2}).  The main conclusion
of our models is that factors of two in price or utilization
do not matter much in an environment of increasing demand and
falling prices.

The advantage of (3) over (1) is that the unified network can
take advantage of economies of scale.  We will not consider
the added costs of providing for logical separation of the two
traffic types on a two-tiered network.

As is shown in \cite{Odlyzko1}, current data networks are
an inefficient amalgam of the separated network and the one-price network.
They do resemble a separated network,
with the public Internet operating in a congested mode with
relatively high utilization rate (although lower than that of
the switched voice network), while corporate networks have
very low utilization rates.  However, this is not the
separated network of our model, since all corporate data,
whether it is sensitive to delay or not, travels over 
underutilized networks, while all public Internet traffic
goes over congested links.  Thus we have two separate
one-price networks.

The economic models used to determine prices for the three configurations 
we study are based on providers' costs and revenues. 
Costs include ongoing operational costs, depreciation and other overhead 
charges, and a reasonable rate of return or profit that might be 
limited by competition or regulatory constraints. 
We assume for each period that total cost is a function of actual
volume carried, as described in the next section.

Per-period revenue equals price times actual volume, i.e., 
\begin{center}
Revenue = $xP(x)V.$
\end{center}
We then compute the actual price charged as the smallest $x$ at which
revenue equals cost.
In doing this, we are not trying to maximize profit because it
is already built into costs; we seek only to determine a reasonable
price based on equality between costs and revenues.
If no value of $x$ satisfies the 
Revenue = Cost equation, then revenue is insufficient to cover cost 
at any price, and we refer to the configuration as {\em infeasible}.
Although our models are based on 
equilibrium between revenue and cost rather than optimization schemes 
{\em per se},
we will compare prices, demand satisfactions, and revenues of the   
three network configurations to assess their performances with respect 
to each other. 

We regard our models as informative but very rough approximations 
to an extremely complex environment and uncertain future.
Explanations of aspects of cost, including economies of 
scale, effects on cost of enhanced QoS, and how costs may change 
over time in a competitive marketplace with rapidly increasing volume
are described in the next section.
Section 3 specifies the models more completely for the three network 
configurations in a static one-period scenario and describes 
solution procedures. Section 4 then extends the models to the 
dynamic scenario of a succession of periods in which potential volumes, 
costs, and implied prices change from period to period. 

For computational simplicity in the dynamic analysis, we will assume that 
potential volume doubles from period to period. This assumption is 
made palatable by not fixing period lengths in advance. 
For example, two-year periods 
might be assume.
See \cite{CoffmanO} for history and projections of growth
patterns in data traffic.  While voice traffic has been
growing at around 10\% per year, Internet traffic (measured
in bytes) has been just about doubling each year in the 1990s,
with the exception of 1995 and 1996, when it grew by factors
of about 10 in each of those two years.

We have already mentioned that different market demand functions will 
be considered. A further accommodation for an uncertain future will 
be made by considering two very different patterns for changes in costs 
over time. 
The first is a {\em conventional pattern} in which costs change only 
because of the potential volume doubling from period to period. 
The second, which we refer to as the {\em dynamic pattern}, reflects 
not only the doubling assumption but also cost reductions driven by 
competition and technological advances.
Dynamic-pattern revenues increase from period to period
(except in one extreme scenario where they remain constant), but at a much slower 
rate than conventional-pattern costs. 
Both patterns are specified more completely in the next section.

As we will see in Section 4, the implications of our dynamic models
depend on our different demand functions and cost patterns, 
but some trends emerge. For example, in comparisons between the    
separated network and two-tiered network, the prices for both $A$ (ordinary service) and $B$
(premium service) tend to be slightly higher for the separated network ,
whereas demand satisfactions are comparable. 
An anticipated finding is that dynamic-pattern costs drive prices 
substantially below those for conventional-pattern costs in all three 
networks. 
Another result that was not anticipated at the outset, is that 
the one-price network with its uniformly high QoS is competitive 
with the others under several assumptions. In regard to revenues 
(= costs), which are aggregated for $A$ and $B$ in the separated network, 
the highest revenues occur for either the separated network or the 
one-price network, whereas the lowest revenues occur for either the 
one-price network or the two-tiered network. The differences in the 
revenue picture are caused more by the different cost patterns than 
by the different demand functions. A more complete picture of 
these matters is given at the end of Section 4.  The main conclusions,
though, are that 
differences between the different networks are not
great.

How can the one-price network be superior to the separated one?
We show this with an example that simplifies our model by
ignoring effects of price on demand.  Suppose that type $A$ and
type $B$ traffic are the same when measured in bytes, but that 
type $B$ transmission requires much less congested networks,
with capacity 4 times as large as that for type $A$.
Suppose also that the cost of a network of bandwidth $x$ is
${x}^{1/2}$.  (Section 2 discusses cost formulas in detail.)
Then the cost of the separated network is 3 ($= 1 + {4}^{1/2}$), 
whereas that of the the one-price network is ${8}^{1/2} = 2.8284$,
as the capacity has to be 8 times that of just the $A$ network.
Thus in this scenario, providing uniformly
high QoS to everybody saves 6\% of the cost.  A much larger
saving comes from having a single network, which makes life
simpler for customers.  On the other hand, there are also
costs.  For example, if there is no way to charge different
prices for $A$ and $B$ traffic on a single network (as we
will be assuming throughout the paper), then type $A$ users
will pay 1.4142 (half of total cost) instead of 1.0
for their own separate network, whereas type $B$ users
would see their charges drop from 2 to 1.4142.  Thus
different types of networks have varying impacts on
social welfare.  However, we argue that in the long
run such costs might be bearable in the interests of
simplicity.  The reason is that rapidly decreasing costs
of data transport mean everyone is as well off within
one or two time periods as they would be with any other
network solution.

The temporal aspects of technological change have a large
impact on the marketplace.  For example, for a long time,
Intel microprocessors were slower, usually by at least
a factor of 2, than comparably priced RISC chips.
However, Intel was usually able to provide comparable
price/performance ratio within two or three years.
This, combined with the advantages of compatibility
(i.e., lower costs to customers in upgrading) allowed
Intel to increase its dominance in the processor business.
Similar effects might favor simple schemes (such as the
one-price network) over more efficient and socially optimal
ones in data networks.

A summary of our study is provided in Section 5. 

\section{Economies of scale and other cost factors}
Forecasting prices of telecommunications services has historically
been a risky enterprise.  As an example, we cite the paper
\cite{Irvin}.  Written in 1992 and published in 1993, it
develops two models for leased line prices in the United
States.  Both models fit the data up to 1992 extremely well,
and both models predicted a drop in prices of about 50\%
by 1998.  Instead, prices have increased by approximately 50\%
since 1992, so they are about three times as high as predicted
by Irvin's model.  However, we feel that this was an anomaly,
caused by unexpectedly high demand for data network bandwidth
and little new growth in supply.  At some point in the future,
prices are likely to resume their decline.

There is no simple formula for costs of communication networks.
It is almost always true that larger transfer volume or bandwidth
purchases are less expensive per unit of volume or bandwidth than smaller ones,
but even that is
not always the case.
For example, in April 1998, UUNet \cite{UUNet} was citing the
following prices for dedicated Internet connections (not including
the cost of local connections to the nearest UUNet site):
\begin{center}
\begin{tabular}{ll}
speed & price per month \\  [+.1in]
56 Kbps & \$595 \\
1.5 Mbps (T1) & \$1,795 \\
45 Mbps (T3) & \$54,000 \\
\end{tabular}
\end{center}
In this case, a 24-fold increase in bandwidth from a 56 Kbps line
to a T1 incurs only a 3-fold increase in price, but the 28-fold
increase in speed from a T1 to a T3 raises the cost by a factor 
of 30.  This pricing may reflect scarcity of high-capacity lines,
and possibly of handling the traffic from a T3 connection on
a network that consists largely of T3 links.  Similar linear
pricing in bandwidth applies to speeds between T3 and OC3.
(Sprint charges for these three speeds are \$897,
\$2,062 and \$20,620, respectively, according to data
at [Boardwatch], but these figures may not be strictly
comparable to UUNet's because of special conditions and
features.)

A better view of transmission costs might be offered by
examining leased line prices.  In April 1998, the tariffed monthly rates for
an approximately 300 air mile private line,
with about 5 miles of local connections that are leased from
a local phone company were about as follows:
\begin{center}
\begin{tabular}{ll}
speed & price per month \\ [+.1in]
9.6 Kbps & \$1,150 \\
56 Kbps & \$1,300 \\
128 Kbps & \$3,000 \\
256 Kbps & \$3,800 \\
512 Kbps & \$5,100 \\
1.5 Mbps (T1) & \$7,000 \\
7.7 Mbps & \$37,000 \\
45 Mbps (T3) & \$66,000 \\
\end{tabular}
\end{center}
(In practice, long-term leases and bulk purchase discounts
might reduce these costs by up to 50\%, see \cite{Leida}, for example.
It is worth noting that the local access connections account
for about 60\% of the cost of a 9.6 or 56 Kbps line and about 17\% of a T1
or a T3.)  The exact figures depend on distance \cite{Leida},
but this dependence has decreased greatly over time \cite{CoffmanO}.

Using the leased line prices cited above, 
we can see that a moderately good fit for the cost of carrying a 
given volume in one time period at the most common
speeds between 56 Kbps and 45 Mbps is obtained by making the cost
proportional to the volume, raised to a power in the range of 0.5 to 0.7
that we denote by $s$ and refer to as the {\em economy-of-scale parameter}.
(In the general economics literature, $1/s$ is known as the
{\em elasticity of scale,} and we are assuming it is constant.)
Economies of scale can arise from reduced requirements
for the multiplexing equipment needed to provide low speed links
on a high-capacity network as well as lower costs of sales,
administration, maintenance, and related operational costs.
It is reasonable to suppose that the same $s$ value will apply in
the future for
greater volumes.  Although later examples
assume a value of $s=2/3,$ we write our cost formulas for
general $s$.
(For comparison, \cite{Harms} uses a value of $s=1/2$.)
Today, $s=2/3$ applies only through T3 speeds, and charges for
OC3 (155 Mbps) private lines are reportedly often higher than
for equivalent capacity in T3 lines.  However, as traffic
grows, and new technologies are deployed, it is not unreasonable
to expect that our cost formula will apply at higher bandwidths as well.

In particular, we will assume that the cost for demand type $A$ in a 
period with potential volume $V_A$ and demand probability $P(x)$ at 
price $x$ is given by
$$\mbox{Cost for} ~~A = [P(x) V_A]^s ~.$$
This applies to the separated network, where costs are scaled in units
determined for the separated $A$ case.
Using the same scale, we assume that the cost for demand type $B$ under 
similar conditions is
$$
\mbox{Cost for}~ B = [ \psi P(x) V_B]^{s} ~,
$$
where $\psi$, which we refer to as the {\em premium factor}, is a parameter
that exceeds 1 to account for higher cost and enhanced QoS for type $B$ users.
Reasonable values for $\psi$ might lie in the range of 2 to 4,
judging by the comparison of different networks in \cite{Odlyzko1}.
For example, if $\psi =2$, then the $B$ part of the separated network 
is arranged to carry the same volume as the $A$ part at twice the capacity.
Single-period costs for one-price and two-tiered networks have related
forms that are described in next section.

The preceding costs apply to an initial period, which can be taken 
to be the present or some other base period.
The conventional pattern for costs, in which costs change from period to period
only as a function of the doubling of potential volume, implies that costs $t$
periods in the future from the base period will be
$$[P(x) 2^t V_A]^{s} = [P(x) V_A]^{s} 2^{ts} \quad {\rm for} \quad A$$
and
$$
[ \psi P(x) 2^t V_B]^{s} = [\psi P(x) V_B]^{s} 2^{ts} \quad {\rm for} \quad B
$$
in the separated network.

However, competition and technological advances along with rising demand
may lead to substantially lower costs than those given by the
conventional pattern.
Among other things, developments in WDM mean that fiber capacity is 
not a limiting factor.
Instead, the electronics that connect end users to the fiber are becoming
the main obstacle, and improvements in optical and silicon technology are 
likely to induce rapid decreases in the price/performance ratio.
Although prices of connections of a fixed speed might not drop dramatically,
the bulk of the data transport capacity that is purchased is likely to cost
far less per unit of volume than at present.  (That is the pattern seen
in prices of microprocessors and DRAMs.)
We model such effects in our dynamic pattern for costs by dividing the 
conventional next period cost by $\sqrt{2}$, a factor that accumulates 
exponentially over time.
For example, the present cost of $[P(x)V_A]^{s}$ for $A$ in the separated
network translates into the dynamic-pattern cost of
$$[P(x) 2^t V_A]^{s} 2^{-t/2} = [P(x) V_A]^{s} 2^{t(s-1/2)}$$
$t$ periods in the future, which is substantially less than the figure 
of $[P(x)V_A]^{s} 2^{ts}$ for the
conventional pattern.
We regard $\sqrt{2}$ as a fairly drastic dynamic factor, representing
an extreme case for unit cost reduction.
For example, if $s=1/2$ then total cost remains the same as potential 
volume doubles.

Because period lengths are flexible, we allow for varying rates of decrease
in unit cost as time progresses.
If period length is one year and $s=2/3$, the conventional pattern presumes a yearly
decrease of about 20\% in unit cost, and the dynamic pattern presumes a yearly
decrease of about 44\% in unit cost.
If period length is two years and $s=2/3$, the yearly unit cost 
decreases are 10\% for
the conventional pattern and 22\% for the dynamic pattern.



\section{One-period static analysis}
This section discusses our models for a fixed period in which
$A$ has potential volume $V_A$, $B$ has potential volume $V_B$, and both have
demand function $P$.
As before, $\psi$ is the premium factor for higher QoS
and $s < 1$ is the economy-of-scale parameter.
The example later in this section takes $V_A =V_B$, $\psi =3$ and $s=2/3$.
The next section considers other arrangements for $V_A$, $V_B$, $\psi$ and $s$.

Let $x,y,$ and $z$ denote the prices for type $A$ in the separated network,
for type $B$ in the separated network, and for both types in the one-price
network, respectively.
The costs for these networks are as follows:
\begin{eqnarray*}
\mbox{separated: $A$ cost} & = & [P(x)V_A]^s \\
\mbox{$B$ cost} & = & [ \psi P(y) V_B]^s \\
\mbox{Total} & = & [P(x)V_A]^s + [\psi P(y) V_B ]^s \\
\mbox{one-price: Cost} & = & \{ \psi [P(z) V_A + P(z) V_B]\}^s \\
& = & [ \psi P(z)]^s (V_A + V_B)^s ~.
\end{eqnarray*}
For one-price, $\psi$ applies to both $A$ and $B$ because this network
offers the premium service to both types.

The Revenue $=$ Cost equations for the preceding networks are
\begin{eqnarray*}
xP(x) V_A & = & [P(x) V_A]^s \\
yP(y) V_B & = & [ \psi P(y) V_B ]^s
\end{eqnarray*}
and
$$zP(z) (V_A + V_B) = [ \psi P(z) (V_A + V_B)]^s ~.$$
In the first equation, $xP(x)$ for the forms we use for $P$ increases
to a maximum and then decreases for larger $x$,
whereas $P(x)^s$ on the right side decreases from 1 at $x=0$ and approaches 0 as $x$ gets large.
If the single-peaked curve for $xP(x) V_A$ lies beneath the decreasing
curve for $[P(x)V_A]^s$, i.e., if $xP(x) V_A < [P(x) V_A]^s$ for all $x \ge 0$,
then the $A$ part of the separated network is infeasible.
Otherwise, there will typically be two $x$ values, say $x_1 < x_2$, where the curves
cross.
We take $x_1$ as our price solution to $xP(x) V_A = [P(x) V_A]^s$ because
it gives a lower price, higher revenue, and greater utilization than $x_2$.
Similar remarks apply to the other Revenue $=$ Cost equations.

We introduce a new parameter for the two-tiered network.
It is the ratio $\la \ge 1$ of the higher to the lower price in this
network, i.e.,
$\la = y/x$ when $y$ is the premium price and $x$ is the ordinary price.
When $\lambda$ is not made explicit, the two-tiered Revenue $=$ Cost 
equation is
$$xP(x) V_A + yP(y) V_B =  [P(x) V_A + \psi P(y) V_B ]^s ~.$$
Unlike the one-price case, $\psi$ applies here only to the premium 
service because of the two-tiered structure.
In keeping with the rationale of a two-tiered network, we regard 
this network as feasible only if the
preceding equation holds for some $(x,y)$ with $y \ge x$.
We note also that costs could be
increased slightly for the two-tiered network because of
the additional costs of network operators, as well
as those of users, who have to adjust to a more complicated pricing
scheme.
However, we do not believe that this matters very much since the 
models are approximate in the first place.

A feasible two-tiered network offers more freedom of choice than the others
because it typically has a continuum of $(x,y)$ solutions to the 
Revenue$~=~$Cost equation in which $y$ increases as $x$ decreases in 
moving away from the equal-prices solution where $x=y$.
We have found that two-tiered revenue is often greatest when $x$ and $y$ 
are close together, but note also that
$x=y$ defeats the purpose of a two-tiered network.
We shall therefore regard $\lambda = y/x$ as a control variable subject to policy decision.
Reasonable values for $\lambda$ range from about 2 to 4, so that the premium service costs about two to four times as much as the ordinary service per
unit volume.
Our use of $\la$ also eases the computational burden of solving the
Revenue$~=~$Cost equation since, when $\la$ is given, we need only solve for $x$ and then obtain $y$ from $y = \la x$.

When $\la x$ is substituted for $y$ in the preceding two-tiered 
equation, it becomes
$$x [P(x) V_A + \la P( \la x )V_B ] = [P(x) V_A + \psi P(\la x) V_B]^s ~.$$
As for the other networks, the solution is taken as the smallest
$x$ that satisfies the equation when it is feasible.

We consider three forms for the demand function $P$ in the example that
follows.
They are
\begin{eqnarray*}
P_1 (x) & = & e^{-{x}^2} \quad {\rm for} \quad x \ge 0 ~, \\
P_2 (x) & = & \frac{e^{-x}}{1+ x} \quad {\rm for} \quad x \ge 0 ~, \\
P_3 (x) & = & \frac{1}{1+x^4} \quad {\rm for} \quad x \ge 0 ~.
\end{eqnarray*}
Figure 1 illustrates the differences between the three.
$P_1$ and $P_3$ begin high for small $x$,
decrease rapidly as $x$ gets into a mid-range,
and have very narrow tails.
$P_2$ begins its descent immediately, levels off sooner than $P_1$ and $P_3$ and
has a fat tail.
When prices are low, $P_2$ is much more sensitive than the others 
to small price changes.
This is the most important difference between them because most 
of the solutions we have seen for our networks have prices well below 1.
\begin{figure}[htb]
\centerline{\psfig{file=P1.ps,width=4in}}
\caption{Three demand functions}
\end{figure}

We now turn to an example with parameter values $\psi =3$, $s= 2/3$, 
and $\la \in \{2,4\}$.
The example has six scenarios in the 2-by-3 cross classification 
$\{$low potential volume, high potential volume$\} \times \{P_1, P_2, P_3 \}$.
With $V_A = V_B$, we set
the low potential volume for each of $A$ and $B$ at $V_1 =32$,
and set the high potential volume at $V_2 = 64 V_1$.

We consider the separated and one-price networks first.
The Revenue$~=~$Cost equations for $A$ separate, $B$ separate,
and the one-price network are, for $V=V_1$,
\begin{eqnarray*}
xP(x) V_1 & = & [ P (x) V_1]^{2/3} \\
xP(x) V_1 & = & [3 P (x) V_1]^{2/3} \\
xP(x) (2V_1) & = & [3 P(x) (2V_1) ]^{2/3}
\end{eqnarray*}
respectively.
These simplify to
$$
\left.
\begin{array}{ll}
P_1 : & x^3 e^{-x^2} \\
P_2 : & x^3 e^{-x} / (1+x) \\
P_3 : & x^3 /(1+x^4)
\end{array}
\right\} = \left\{
\begin{array}{ll}
1/32 & A ~\mbox{separate} \\
9/32 & B~ \mbox{separate} \\
9/64 & \mbox{one-price.}
\end{array}
\right.
$$
The right-hand sides of these equations are multiplied by 1/64 to
obtain the corresponding equations for $V_2$.

Table 2 shows approximate solution values in terms of price $x$,
demand satisfaction $S$ and
revenue $R$.
The one-price network price in each row is midway between the prices 
for $A$ and $B$ in the separated network, $P_2$ induces slightly higher
prices than $P_1$ and $P_3$, and prices drop dramatically with high volume.
The ratios of premium service to ordinary service prices for the separated
network lie between 2 and 3.5.
\begin{table*}[htb]
\caption{Prices, demand satisfactions, and revenues for separated and one-price networks}

\begin{center}
\begin{tabular}{clrrlrcrlrclrcrl}
~ & ~ & \multicolumn{3}{c}{~} & & \multicolumn{3}{c}{~} && \multicolumn{2}{c}{Separated} & & \\
~ & ~ & \multicolumn{3}{c}{~} & & \multicolumn{3}{c}{~} && \multicolumn{2}{c}{network} && \multicolumn{3}{c}{One-price} \\
~ & ~ & \multicolumn{3}{c}{$A$ separate} & ~~~ & \multicolumn{3}{c}{$B$ separate} & ~~~~~~ & \multicolumn{2}{c}{totals} & ~~~ & \multicolumn{3}{c}{networks} \\
~ \\
~ & ~ & $x$ & $S$ & $R$ && $x$ & $S$ & $R$ && $S$ & $R$ && $x$ & $S$ & $R$ \\ \cline{3-5} \cline{7-9} \cline{11-12} \cline{14-16}
$V_1$ & $P_1$ & .33 & 90 & 9.4 && .82 & 51 & 13.4 && 71 & 22.8 && .58 & 71 & 26.6 \\
(Low) & $P_2$ & .40 & 48 & 6.1 && 1.4 & 10 & 4.6 && 29 & 10.7 && .85 & 23 & 12.6 \\ 
~ & $P_3$ & .32 & 99 & 10.0 && .71 & 80 & 18.1 && 90 & 28.1 && .53 & 93 & 31.6 \\ \hline
$V_2$ & $P_1$ & .079 & 99 & 160 && .17 & 97 & 331 && 98 & 491 && .13 & 98 & 527 \\
(High) & $P_2$ & .084 & 85 & 146 && .18 & 70 & 266 && 78 & 412 && .14 & 76 & 444 \\
& $P_3$ & .079 & 100 & 162 && .16 & 100 & 336 && 100 & 498 && .13 & 100 & 536
\end{tabular}
\end{center}
\end{table*}

In all cases, demand satisfaction is substantially higher for $P_1$ 
and $P_3$ than $P_2$,
and aggregated demand satisfaction for the separated network is about the same
as one-price satisfaction.

Revenues are obviously higher for the high volume cases, but the high-to-low
ratios are smaller than the 64-fold increase in volume because of
economies of scale.
Moreover, because
$P_3$ implies greater propensity to subscribe
than $P_2$, and $P_1$ implies greater propensity to subscribe than $P_2$ for
all prices in the table, revenues run highest for $P_3$,
next highest for $P_1$ and lowest for $P_2$.
There is significantly less difference proportionately between
revenues at high volume than at low volume.

\begin{figure}[htb]
\centerline{\psfig{file=Pprice.ps,width=4in}}
\caption{Evolution of prices for $A$ users in a separate network
(line with squares),
for $B$ users in a separate network (line with crosses), and in a
one-price network (line with circles), for the scenario of Table 7.}
\end{figure}



We now bring the two-tiered network into the picture with $x$
the cheaper two-tiered
price and $y= \lambda x$
the premium service price.
The Revenue$~=~$Cost equation noted earlier for the two-tiered network reduces to
$$\frac{xe^{-x^2} + \la x e^{- ( \la x )^2}}{[e^{-x^2} + 3e^{- (\la x )^2}]^{2/3}}
= \frac{1}{(32)^{1/3}} \quad {\rm for} \quad P= P_1, \quad V = V_1
$$
\begin{eqnarray*}
\lefteqn{\frac{xe^{-x} / (1+x) + \la xe^{- \la x} / (1+ \la x)}{[e^{-x} / (1+x) + 3e^{- \la x} / (1+ \la x)]^{2/3}}} \\
& = & \frac{1}{(32)^{1/3}} \quad {\rm for} \quad
P= P_2 , \quad V= V_1  \\
& ~ & \\
\lefteqn{\frac{x/(1+x^4) + \la x / (1+ (\la x)^4)}{[1/(1+x^4) + 3/(1+(\la x)^4 )]^{2/3}}} \\
 & = &
\frac{1}{(32)^{1/3}} \quad{\rm for} \quad P = P_3 , \quad V=V_1 ~.
\end{eqnarray*}
The right sides of these are multiplied by $1/ (64)^{1/3} = 1/4$ for $V_2$.

Table 3 shows the two-tiered results to the right of the one-price results.
Comparisons between $\la =2$ and $\la =4$ for the two-tiered case reveal little difference in demand satisfaction or revenue.
In each row, the two prices for $\la =4$ (e.g., .18 and $4 \times .18 = .72$)
surround the prices for $\la =2$ (e.g., .28 and $2 \times .28 = .56$).
Without exception, the one-price network price is greater than the average
of the two two-tiered prices
for a given category $(V_i , P_j , \la )$, and can be greater than the larger of these two
as for $\la =2$ in rows 1 and 2.
Finally, the one-price network has uniformly higher revenue and uniformly
lower demand satisfaction than the two-tiered network. 
\begin{table*}[htb]
\caption{Prices, demand satisfactions, and revenues for one-price and two-tiered networks}

\begin{center}
\begin{tabular}{cllllrlllrlll}
~ & ~ & \multicolumn{3}{c}{One-price} & ~~~~~~ & \multicolumn{7}{c}{Two-tiered network} \\
~ & ~ & \multicolumn{3}{c}{network} && \multicolumn{3}{c}{$\la =2$} && \multicolumn{3}{c}{$\la =4$} \\ \cline{7-9} \cline{11-13}
~ & ~ & $x$ & $S$ & $R$ && $x$ & $S$ & $R$ && $x$ & $S$ & $R$ \\ \cline{3-5} \cline{7-9} \cline{11-13}
$V_1$ & $P_1$ & .58 & 71 & 26.6 && .28 & 83 & 21.8 && .18 & 78 & 19.2 \\
(Low) & $P_2$ & .85 & 23 & 12.6 && .36 & 40 & 12.5 && .28 & 37 & 10.9 \\
~ & $P_3$ & .53 & 93 & 31.6 && .27 & 96 & 24.4 && .17 & 92 & 23.2 \\ \hline
$V_2$ & $P_1$ & .13 & 98 & 527 && .065 & 99 & 397 && .040 & 99 & 400 \\
(High) & $P_2$ & .14 & 76 & 444 && .071 & 82 & 349 && .044 & 82 & 339 \\ 
~ & $P_3$ & .13 & 100 & 536 && .066 & 100 & 407 && .040 & 100 & 406 \\
\end{tabular}
\end{center}
\end{table*}

\section{Dynamic Analysis}
We present results of our dynamic analysis primarily for the separated and one-price networks to keep matters fairly simple.
The results for the two-tiered network in the dynamic case are similar to those in the preceding section in comparison to the other networks,
and their trends over time are similar to the trends described in this section.
For example, for an array of parameters, $\la =2$ and $\la =4$ have very
similar demand satisfactions and revenues although their prices, $x$ and
$\la x$, obviously differ.
The cheaper two-tier price at $\la =4$ is about 60\% of the cheaper price at
$\la =2$, so the premium price at $\la =4$ is about 20\%
higher than the premium price at $\la =2$.
Revenue comparisons show a general pattern in which a two-tiered network
either has the lowest revenue or the middle revenue of the three networks.

For the separated and one-price networks, we begin our dynamic process at
period $t=0$ with low potential volumes, and run each network through 11
periods.
In our initial runs, which are partly shown in Tables 4 through 7, we took
$V_A = V_B$, with a value of 4 at $t=0$ and $2^t (4)$ for $t \ge 1$.
These tables also use $\psi =3$ and $s= 2/3$.
An infeasible situation is shown by asterisks.

Tables 4 through 7 consider the ``low'' and ``high'' demand functions
$P_2$ and $P_3$ (see Figure 1) along with the conventional cost pattern
$C_I$ and the extreme dynamic pattern $C_{II}$ of rapidly decreasing
unit cost.
The tables pertain to $(P_2, C_I)$, $(P_2, C_{II} )$, $(P_3 , C_I)$ and $(P_3, C_{II} )$,
respectively.

We considered changes in $\psi$, $s$, and $V_A$ and $V_B$ to see how much they affect the nature of the results shown in the tables.
The specific changes include $\psi = 2$, $s=1/2$, $(V_A, V_B) = (4,16)$ 
and $(V_A, V_B) = (16,4)$ for the initial period.
We comment on these briefly after noting aspects of Tables 4--7.


\begin{table*}[htb]
\caption{$P_2$, $C_I$}

$$
\begin{array}{crrccrcccrccrccc}
~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & \multicolumn{2}{c}{\mbox{Separated}} & ~ & \multicolumn{3}{c}{\mbox{One-price}} \\
~ & ~ & \multicolumn{3}{c}{\mbox{$A$ separate}} & ~ & 
\multicolumn{3}{c}{\mbox{$B$ separate}} & ~ & \multicolumn{2}{c}{\mbox{totals}} & ~ & \multicolumn{3}{c}{\mbox{network}} \\
t & ~ & \multicolumn{1}{c}{x} & S & R & ~ & x & S & R & ~ & S & R & ~ & \multicolumn{1}{c}{x} & S & R \\ \cline{1-1} \cline{3-5} \cline{7-9} \cline{11-12} \cline{14-16}
0 & ~ & 1.3 & 13 & .63 & ~ & \ast & \ast & \ast & ~ & 6& .63 & ~ & \ast & \ast & \ast \\
1 & ~ & .79 & 25 & 1.60 & ~ & \ast & \ast & \ast & ~ & 13 & 1.60 & ~ & \ast & \ast & \ast \\
2 & ~ & .55 & 37 & 3.28 & ~ & \ast & \ast & \ast & ~ & 19 & 3.28 & ~ & 1.4 & 10 & 4.61 \\
3 & ~ & .40 & 48 & 6.15 & ~ & 1.4 & 10 & 4.61 & ~ & 29 & 10.8 & ~ & .85 & 23 & 12.6 \\
4 & ~ & .30 & 59 & 11.0 & ~ & .85 & 23 & 12.6 & ~ & 40 & 23.6 & ~ & .59 & 35 & 26.3 \\
5 & ~ & .23 & 65 & 19.0 & ~ & .59 & 35 & 26.3 & ~ & 50 & 45.3 & ~ & .43 & 46 & 49.9 \\
6 & ~ & .18 & 71 & 32.3 & ~ & .43 & 46 & 49.9 & ~ & 59 & 82.2 & ~ & .32 & 55 & 89.8 \\
7 & ~ & .14 & 77 & 53.8 & ~ & .32 & 55 & 89.8 & ~ & 66 & 144 & ~ & .24 & 63 & 156 \\
8 & ~ & .11 & 81 & 88.9 && .24 & 63 & 156 && 72 & 245 && .19 & 70 & 266 \\
9 && .084 & 85 & 146 && .19 & 70 & 266 && 78 & 412 && .14 & 76 & 444 \\
10 && .066 & 88 & 237 && .14 & 76 & 444 && 82 & 682 && .11 & 81 & 732 \\
\end{array}
$$
\end{table*}

\begin{table*}[htb]
\caption{$P_2$, $C_{II}$}

$$
\begin{array}{crrccrcccrccrccc}
~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & \multicolumn{2}{c}{\mbox{Separated}} & ~ & \multicolumn{3}{c}{\mbox{One-price}} \\
~ & ~ & \multicolumn{3}{c}{\mbox{$A$ separate}} & ~ &
\multicolumn{3}{c}{\mbox{$B$ separate}} & ~ & \multicolumn{2}{c}{\mbox{totals}}
& ~ & \multicolumn{3}{c}{\mbox{network}} \\
t & ~ & \multicolumn{1}{c}{x} & S & R & ~ & x & S & R & ~ & S & R & ~ & \multicolumn{1}{c}{x} & S & R \\ \cline{1-1} \cline{3-5} \cline{7-9} \cline{11-12} \cline{14-16}
0 && 1.3 & 14 & .63 && \ast & \ast & \ast && 6 & .63 && \ast & \ast & \ast \\
1 && .47 & 42 & 1.6 && \ast & \ast & \ast && 21 & 1.6 && 1.1 & 17 & 2.9 \\
2 && .23 & 65 & 2.4 && .59 & 35 & 3.3 && 50 & 5.7 && .43 & 46 & 6.2 \\
3 && .12 & 79 & 3.1 && .28 & 60 & 5.3 && 69 & 8.3 && .21 & 67 & 9.0 \\
4 && .066 & 88 & 3.7 && .14 & 76 & 6.9 && 82 & 10.6 && .11 & 81 & 11 \\
5 && .036 & 93 & 4.3 && .077 & 86 & 8.5 && 90 & 12. 8 && .061 & 89 & 14 \\
6 & & .020 & 96 & 4.9 && .043 & 92 & 10 && 94 & 15.0 && .034 & 94 & 16 \\
7 && .012 & 98 & 6.0 && .024 & 95 & 12 && 97 & 17.7 && .019 & 96 & 19 \\
8 && .007 & 99 & 7.1 && .014 & 97 & 14 && 98 & 21.0 && .011 & 98 & 22 \\
9 && .004 & 99 & 8.1 && .008 & 98 & 16 && 99 & 24.3 && .006 & 99 & 24 \\
10 && .002 & 100 & 8.2 && .005 & 99 & 20 && 99 & 28.4 && .004 & 99 & 33
\end{array}
$$
\end{table*}

\begin{table*}[htb]
\caption{$P_3$, $C_{I}$}

$$
\begin{array}{crrccrcccrccrccc}
~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & \multicolumn{2}{c}{\mbox{Separated}} & ~ & \multicolumn{3}{c}{\mbox{One-price}} \\
~ & ~ & \multicolumn{3}{c}{\mbox{$A$ separate}} & ~ &
\multicolumn{3}{c}{\mbox{$B$ separate}} & ~ & \multicolumn{2}{c}{\mbox{totals}}
& ~ & \multicolumn{3}{c}{\mbox{network}} \\
t & ~ & \multicolumn{1}{c}{x} & S & R & ~ & x & S & R & ~ & S & R & ~ & \multicolumn{1}{c}{x} & S & R \\ \cline{1-1} \cline{3-5} \cline{7-9} \cline{11-12} \cline{14-16}
0 && .67 & 83 & 2.2 && \ast & \ast & \ast && 42 & 2.2 && \ast & \ast & \ast \\
1 && .51 & 94 & 3.8 && \ast & \ast & \ast && 47 & 3.7 && 1.2 & 32 & 6.2 \\
2 && .40 & 98 & 6.3 && 1.2 & 32 & 6.2 && 65 & 12.5 && .71 & 80 & 18.1 \\
3 && .32 & 99 & 10.0 && .71 & 80 & 18.1 && 90 & 28.1 && .53 & 93 & 31.6 \\
4 && .25 & 100 & 16.0 && .53 & 93 & 31.6 && 96 & 47.6 && .42 & 97 & 51.8 \\
5 && .20 & 100 & 25.4 && .42 & 97 & 51.8 && 99 & 77.2 && .33 & 99 & 83.2 \\
6 && .16 & 100 & 40.4 && .33 & 99 & 83.2 && 100 & 124 && .26 & 100 & 133 \\
7 && .13 & 100 & 64.5 && .26 & 100 & 133 && 100 & 198 && .21 & 100 & 212 \\
8 && .10 & 100 & 102 && .21 & 100 & 212 && 100 & 314 && .16 & 100 & 336 \\
9 && .079 & 100 & 162 && .16 & 100 & 336 && 100 & 497 && .13 & 100 & 536 \\
10 && .063 & 100 & 258 && .13 & 100 & 536 && 100 & 794 && .10 & 100 & 852
\end{array}
$$
\end{table*}

\begin{table*}[htb]
\caption{$P_3$, $C_{II}$}

$$
\begin{array}{crrccrcccrccrccc}
~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & \multicolumn{2}{c}{\mbox{Separated}} & ~ & \multicolumn{3}{c}{\mbox{One-price}} \\
~ & ~ & \multicolumn{3}{c}{\mbox{$A$ separate}} & ~ &
\multicolumn{3}{c}{\mbox{$B$ separate}} & ~ & \multicolumn{2}{c}{\mbox{totals}}
& ~ & \multicolumn{3}{c}{\mbox{network}} \\
t & ~ & \multicolumn{1}{c}{x} & S & R & ~ & x & S & R & ~ & S & R & ~ & \multicolumn{1}{c}{x} & S & R \\ \cline{1-1} \cline{3-5} \cline{7-9} \cline{11-12} \cline{14-16}
0 && .67 & 83 & 2.2 && \ast & \ast & \ast && 42 & 2.2 && \ast & \ast & \ast \\
1 && .36 & 98 & 2.8 && .84 & 66 & 4.5 && 82 & 7.3 && .61 & 88 & 8.6 \\
2 && .20 & 100 & 3.2 && .42 & 99 & 6.5 && 99 & 9.7 && .33 & 99 & 10.4 \\
3 &&.11 & 100 & 3.6 && .23 & 100 & 7.4 && 100 & 11.0 && .18 & 100 & 11.8 \\
4 && .063 & 100 & 4.0 && .13 & 100 & 8.4 && 100 & 12.4 && .10 & 100 & 13.3 \\
5 && .036 & 100 & 4.6 && .073 & 100 & 9.3 && 100 & 14.0 && .058 & 100 & 14.8 \\
6 && .020 & 100 & 5.1 && .041 & 100 & 10.5 && 100 & 15.6 && .033 & 100 & 16.9 \\7 && .012 & 100 & 6.1 && .023 & 100 & 11.8 && 100 & 17.9 && .019 & 100 & 19.5 \\
8 && .007 & 100 & 7.2 && .013 & 100 & 13.3 && 100 & 20.5 && .011 & 100 &22.5 \\
9 && .004 & 100 & 8.2 && .008 & 100 & 16.3 && 100 & 24.6 && .006 & 100 & 24.6 \\
10 && .002 & 100 & 8.2 && .005 & 100 & 20.5 && 100 & 28.7 && .004 & 100 & 32.8
\end{array}
$$
\end{table*}


\paragraph{Revenue.}
Except for very low potential volume, a provider who offers
either the $A$ service or the $B$ service for comparable potential
volumes in the separated network makes more money from the premium $B$ service.
A provider who offers one of the two main network
configurations shown in the tables earns more with the one-price
network, but the difference between the two is not great in any case.

\paragraph{Separated versus one-price prices.}
The one-price network price always falls between the $A$-separate and $B$-separate prices.
It tends to be about midway between the separated network prices when $C_I$
applies, and is closer to the $A$-separate price when $C_{II}$ applies.

\paragraph{Demand satisfaction.}
As time passes, demand satisfactions approach 100\%.
The approach is much more rapid for $C_{II}$.
In either case, the forces that drive down unit cost make the service
affordable to virtually every potential user.

\vspace{0.1in}
The main trends noted above and in the preceding section do not
change substantially when other values of the parameters are used.
In most cases, we are near a full-utilization scenario of 100\%
by $t=10$, so there is no significant difference among 
$P_1, P_2$, and $P_3$ for larger $t$.
Revenues at such a time are a bit higher for the one-price
network, but the difference is not great.  There is clearly
an advantage in price for priority users with the one-price
network, which penalizes ordinary users by about 30\% or higher
prices than in the separated network.  However, the lower
$A$-separate price is approximately equal to the single
price for the one-price network one or two periods hence,
so in a dynamic world the ordinary users do not fare too badly
and might even become attracted to the QoS provided by a
one-price network.

Fig. 2 shows the prices from Table 7 for the separate and
one-price networks.  It shows graphically how quickly the
the prices on the one-price network get reduced to the
levels of the separate network for $A$ users.

\section{Summary}
Our purpose has been to compare three network configurations for 
data transmission over the Internet when user demands are divided 
into delay-sensitive and delay-insensitive demands.
Prices for the demand types were based on transfer volume and 
determined by equality between network costs and revenues.
Dynamic uncertainties were accounted for by considering alternative futures
for demands and costs,
including economies of scale for costs and possible effects of
competition and technological advances.

The three network configurations investigated were a separated network for
the demand types, a single one-price network that provides high QoS to 
all users, and a two-tiered network that logically distinguishes between types.
Dynamic analysis showed that network comparisons can be sensitive to 
demand and cost scenarios, no network is obviously superior 
to the others, and as $t$ gets large the trends are pretty
well fixed.
In terms of prices, the premium-service one-price network benefits
delay-sensitive users but penalizes delay-insensitive users, and the two-tiered
network usually gives a modest advantage over the separated network to 
both types.
The largest revenues occur either for the separated network or the one-price
network.
Demand satisfaction percentages for the three are comparable, with no network 
uniformly superior to the others.
Potential user participation approaches 100\% as time passes, and this 
happens quickly
when unit costs and prices decrease rapidly.
Even the delay-sensitive users see their prices and demand
satisfactions approach what they could obtain on a separate
network within one or two time periods.

\paragraph{Acknowledgements:}

We thank Dave Belanger, Chuck Kalmanek, Tony Lauck, and Clem McCalla
for helpful comments.



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