Simplification of Network Dynamics
in Large Systems
Xiaojun Lin and Ness B. Shroff
∗
Abstract
In this paper we show that significant simplicity can be exploited for pricing-based control of large net-
works. We first consider a general loss network with Poisson arrivals and arbitrary holding time distributions.
In dynamic pricing schemes, the network provider can charge different prices to the user according to the
current utilization level of the network and also other factors. We show that, when the system becomes large,
the performance (in terms of expected revenue) of an appropriately chosen static pricing scheme, whose price
is independent of the current network utilization, will approach that of the optimal dynamic pricing scheme.
Further, we show that under certain conditions, this static price is independent of the route that the flows take.
This indicates that we can use the static scheme, which has a much simpler structure than the optimal dynamic
scheme, to control large communication networks. We then extend the result to the case of dynamic routing,
and show that the performance of an appropriately chosen static pricing scheme with bifurcation probability
determined by average parameters can also approach that of the optimal dynamic routing scheme when the
system is large. Finally, we study the control of elastic flows and show that there exist schemes with static
parameters whose performance can approach that of the optimal dynamic resource allocation scheme (in the
large system limit). We also identify the applications of our results for QoS routing and rate control for
real-time streaming.
1 Introduction
In this work, we use pricing as the mechanism of controlling a network to achieve certain performance objec-
tives. The performance objectives can be modeled by some revenue- or utility-functions. Such a framework has
received significant interest in the literature (e.g., see [1, 2, 3, 4, 5] and the references therein) wherein price
∗
X. Lin and N. B. Shroff are with School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907,
U.S.A. E-mail: {linx, shroff}@ecn.purdue.edu. This work has been partially supported by the National Science Foundation through the
NSF award ANI-0099137, and the Indiana 21st Century Research and Technology Award 1220000634.
1
provides a good control signal because it carries monetary incentives. The network can use the current price of a
resource as a feedback signal to coerce the users into modifying their actions (e.g., changing the rate or route).
In [6], Paschalidis and Tsitsiklis have shown that the performance (in terms of expected revenue or welfare)
of an appropriately chosen static pricing scheme approaches the performance of the optimal dynamic pricing
schemes when the number of users and network capacity becomes very large. Note that a dynamic pricing
scheme, is one where the network provider can charge different prices to the user according to the varying levels
of congestion in the network, while a static pricing scheme is one where the price only depends on the average
levels of congestion in the network (and is hence invariant to the instantaneous levels of congestion). The result
is obtained under the assumption of Poisson flow arrivals, exponential flow holding times, and a single resource
(single node). This elegant result is an example of the type of simplicity that one can obtain when the system
becomes large. In this paper, we find that simple static network control can also approach the optimal dynamic
network control under more general assumptions and a variety of other network problems.
For simplicity of exposition, we structure the paper as follows:
We first extend the result of [6] to a general loss network with arbitrary holding time distributions. Note that
while the assumption of Poisson arrivals for flows in the network is usually considered reasonable, the assump-
tion of exponential holding time distribution is not. For example, much of the traffic generated on the Internet
is expected to occur from large file transfers which do not conform to exponential modeling. By weakening
the exponential service time assumption we can extend our results to more realistic systems. We show that a
static pricing scheme is still asymptotically optimal, and that the correct static price depends on the service time
distribution only through its mean. A nice observation that stems from this result is that under certain conditions,
the static price depends only on the price elasticity of the user, and not on the specific route or distance. This
indicates, for example, that the flat pricing scheme used in the domestic long distance telephone service in the
US may be a sufficiently good pricing mechanism.
We then investigate whether more sophisticated schemes can improve network performance (e.g., schemes that
have prior knowledge of the duration of individual flows, schemes that predict the future congestion levels, etc.).
We find that the performance gains using such schemes become increasingly marginal as the system size grows.
We then weaken the assumptions of fixed routing and fixed bandwidth flows. In our dynamic routing model,
flows can choose among several alternative routes based on the current network congestion level. In our elastic
flow model, users are allowed to modify their rates when facing different prices, similar to the way in which TCP
and some elastic multimedia traffic react to changing network conditions. In these more general models, when
the system is large, we show that the invariance result still holds, i.e., there still exists a static pricing scheme
whose performance can approach that of the optimal dynamic scheme.
2
In networks of today and in the future, the capacity will be very large, and the network will be able to support
a large number of users. The work reported in this paper demonstrates under general assumptions and different
network problem settings that, when a network is large, significant simplicity can be exploited for pricing based
network control. Our result also shows the importance of average information when the system is large, since
the parameters of the static schemes are determined by average conditions rather than instantaneous conditions.
These results will help us develop more efficient and realistic algorithms for controlling large networks. We have
identified the applications of our results in QoS routing and rate control for real-time streaming.
Our work also has similarities to the work in [7, 8], and the reference therein. However, in their work, the
price is fixed, and the focus is on how to admit and route each flow. Our work (as well as [6]) explicitly models
the users’ price-elasticity, and consider the optimality of the pricing schemes. Our model of elastic flows is also
similar to the optimization flow control model in [3, 9, 4, 5]. However, their models assume that the number of
users in the system is fixed. Hence their optimization is done for a snapshot in time, while we explicitly consider
the dynamics of the network by taking into account the flow arrivals and departures.
2 Pricing in a General Multi-class Loss Network
2.1 Model
The basic model that we consider in this section is that of a multi-class loss network with Poisson arrivals and
arbitrary service time distributions. There are L links in the network. Each link l ∈ {1, ..., L} has capacity R
l
.
There are I classes of users. We assume that flows generated by users from each class have a fixed route through
the network. The routes are characterized by a matrix {C
l
i
, i = 1, ..., I, l = 1, ..., L}, where C
l
i
= 1 if the route
of class i traverses link l, C
l
i
= 0 otherwise. Let ~n = {n
1
, n
2
, ..., n
I
} denote the state of the system, where n
i
is the number of flows of class i currently in the network. We assume that each flow of class i requires a fixed
amount of bandwidth r
i
. The fixed routing and fixed bandwidth assumption will be weakened in Sections 3 and 4,
respectively.
Flows of class i arrive to the network according to a Poisson process with rate λ
i
(u
i
). The rate λ
i
(u
i
) is a
function of the price u
i
charged to users of class i. Here u
i
is defined as the price per unit time of connection. We
assume that λ
i
(u
i
) is a non-increasing function of u
i
. Therefore λ
i
(u
i
) represents the price-elasticity of class i.
We also assume that for each class i, there is a “maximal price” u
max,i
such that λ
i
(u
i
) = 0 when u
i
≥ u
max,i
.
Therefore by setting a high enough price u
i
the network can prevent users of class i from entering the network.
Once admitted, a flow of class i will hold r
i
amount of resource in the network and pay a cost of u
i
per unit time,
until it completes service, where u
i
is the price set by the network at the time of the flow arrival. The service
3
times are i.i.d. with mean 1/µ
i
. The service time distribution is general.
The bandwidth requirement determines the set of feasible states Ω = {~n :
P
i
n
i
r
i
C
l
i
≤ R
l
∀l}. A flow
will be blocked if the system becomes infeasible after accommodating it. Other than this feasibility constraint,
the network provider can charge a different price to each flow, and by doing so, the network provider strives to
maximize the revenue collected from the users. The way price is determined can range from the simplest static
pricing schemes to more complicated dynamic pricing schemes. In a dynamic pricing scheme, the price at time
t can depend on many factors at the moment t, such as the current congestion level of the network, etc. On the
other hand, in a static pricing scheme, the price is fixed over all time t, and does not depend on these factors.
Intuitively, the more factors a pricing scheme can be based on, the more information it can exploit, and hence the
higher the performance (i.e., revenue) it can achieve.
The dynamic pricing scheme we study in this section is more sophisticated than the one in [6]. Firstly, we
allow the network provider to exploit the knowledge of the immediate past history of states up to length d. Note
that when the exponential holding time assumption is removed, the system is no longer Markovian. There will
typically be correlations between the past and the future given the current state. In order to achieve a higher
revenue, we can potentially take advantage of this correlation, i.e., we can use the past to predict the future, and
use such prediction to determine the price.
Secondly we allow the network provider to exploit prior knowledge of the parameters of the incoming flows.
In particular, the network knows the holding time of the incoming flows, and can charge a different price accord-
ingly. In order to achieve higher revenue, the network can thus use pricing to control the composition of flows
entering the network, for example, short flows may be favored under certain network conditions, while long flows
are favored under others. We assume that the price-elasticity of flows is independent of these parameters.
For convenience of exposition, we restrict ourselves to the case when the range of the service time can be
partitioned into a series of disjoint segments, and the price is the same for flows that are from the same class and
whose service times fall into the same segment. In particular, let {a
k
}, k = 1, 2, ... be an increasing series of
positive numbers, i.e., 0 < a
1
< a
2
< ... and let a
0
= 0. We assume that at any time t, for all flows of class i
whose service times T
i
fall into segment [a
k−1
, a
k
), we charge the same price u
ik
(t), i.e. we do not care about
the exact value of T
i
as long as T
i
∈ [a
k−1
, a
k
).
The dynamic pricing scheme can thus be written as u
i
(t, T
i
) = u
ik
(t) = g
ik
(~n(s), s ∈ [t − d, t]), for T
i
∈
[a
k−1
, a
k
), where ~n(s), s ∈ [t − d, t] reflects the immediate past history of length d, T
i
is the holding time of the
incoming flow of class i, and g
ik
are functions from Ω
[−d,0]
to the set of real numbers R. By incorporating the
past history in the functions g
ik
, we can study the effect of prediction on the performance of the dynamic pricing
scheme without specifying the details of how to predict. Let ~g = {g
ik
, i = 1, ..., I, k = 1, 2, ...}.
4
The system under such a dynamic pricing scheme can be shown to be stationary and ergodic under very general
conditions. For example, when the arrival rates λ
i
(u) are bounded above by some constant λ
0
, one can construct
a so-called “regenerative event” (due to the Poisson nature of the arrivals), which is the event that the system is
empty in the time interval [t−d, t]. One can show that such an event is a stationary event and occurs with positive
probability. This ensures that any stochastic process that is only a function of the system state is asymptotically
stationary and the stationary version is ergodic. See Appendix for the details.
We are now ready to define the performance objective function. For each class i, let
˜
T
ik
= E {T
i
|T
i
∈ [a
k−1
, a
k
)}
be the mean service time for flows of class i whose service time T
i
falls into segment [a
k−1
, a
k
). The expectation
is taken with respect to the service time distribution of class i. Let p
ik
= P{T
i
∈ [a
k−1
, a
k
)} be the probability
that the service time T
i
of an incoming flow of class i falls into segment [a
k−1
, a
k
). We can decompose the origi-
nal arrivals of each class into a spectrum of substreams. Substream k of class i has service time in [a
k−1
, a
k
). Its
arrival is thus Poisson with rate λ
i
(u)p
ik
, since we assume that the price-elasticity of flows is independent of T
i
.
For any dynamic pricing scheme ~g, the expected revenue achieved per unit time is given by
lim
ζ→∞
I
X
i=1
1
ζ
E
"
Z
ζ
0
∞
X
k=1
λ
i
(u
ik
(t))u
ik
(t)
˜
T
ik
p
ik
dt
#
=
I
X
i=1
∞
X
k=1
E
h
λ
i
(u
ik
(t))u
ik
(t)
˜
T
ik
p
ik
i
,
where the expectation is taken with respect to the steady state distribution. The limit on the left hand side as the
time ζ → ∞ exists and equals to the right hand side due to stationarity and ergodicity. Note that the right hand
side is independent of t (from stationarity).
Therefore, the performance of the optimal dynamic policy is
J
∗
, max
~g
I
X
i=1
∞
X
k=1
E
h
λ
i
(u
ik
(t))u
ik
(t)
˜
T
ik
p
ik
i
.
When the exponential holding time assumption is removed, we can no longer use the MDP approach as in [6]
to find the optimal dynamic pricing scheme. We will instead study the behaviour of the dynamic pricing scheme
and its relationship with the static pricing scheme when the system is large. In particular, we will establish an
upper bound for the performance of dynamic pricing schemes and show that the performance of an appropriately
chosen static pricing scheme can approach this upper bound as the system is large. We will then conclude that,
when the system is large, the performance of an appropriately chosen static pricing scheme can approach that
of the optimal dynamic pricing scheme. Further, we show that the performance gains of schemes that use such
sophisticated mechanisms as prediction and charging based on prior knowledge of the holding times are minimal
when the system is large.
5
