Global Stability in the Presence of State-Dependent
Delay for Rate-Control
Priya Ranjan, Richard J. La, and Eyad H. Abed
Abstract— We characterize the stability conditions with an
arbitrary time varying state-dependent round-trip delay in the
context of single-flow and single-resource using Kelly’s optimization framework for rate control. It is shown that the stability
conditions with an arbitrary fixed delay are sufficient for the
stability of the system with a state-dependent time-varying delay.
We consider an extension of Kelly’s primal algorithms and
establish the global stability of an example of the extended model.
Keyword – Economics, control theory, mathematical programming/optimization
I. I NTRODUCTION
Kelly [7] has suggested that the problem of rate allocation
for elastic traffic can be posed as one of achieving maximum
aggregate utility of the users and proposed an optimization
framework for rate allocation in the Internet. Here the utility
of a user could either represent the true utility or preferences
of the user or a utility function that is assigned to the user by
the end user rate control algorithm, e.g., Transmission Control
Protocol (TCP). In the latter case the selection of the utility
function determines the end user algorithm, and the trade-off
between the fairness among the users and system efficiency.
Using the proposed framework he has shown that the system
optimum is achieved at the equilibrium between the end users
and resources. Based on this observation researchers have
proposed various rate-based algorithms, in conjunction with
a variety of active queue management (AQM) mechanisms,
that solve the system optimization problem or its relaxation
[7], [12].
Initially the convergence of these algorithms [9], however,
has been established only in the absence of feedback delay,
and the implications of feedback delay have been left open.
Accurate modeling of the communication delay is especially
important when the delay is non-negligible or widely varying
with uncertainty. A good example of such an environment is
multi-hop mobile wireless networks.
There has been some work on establishing the stability of
the system with delays [1], [2], [6], [13], [14]. However, these
models assume fixed delays, where the round-trip delays of
packets are fixed regardless of the rates of the users. In other
words, the queueing dynamics at the resources have been
ignored. In this paper we extend our earlier work in [14] to
model the queue dynamics in a simple single-flow, singleresource setting. We show that even with queue dynamics,
the stability conditions for the fixed delay case are sufficient
for the stability. In other words, the detailed dynamics of the
round-trip delay do not change the stability conditions in a
significant way. This is illustrated using a family of popular
utility and price functions.
This paper is organized as follows; Section II describes
the model for the simple single-flow, single-resource case and
presents the general stability conditions. Numerical examples
are provided using a family of utility and resource price
functions in Section III. We extend Kelly’s primal algorithm
in Section IV and characterize the global stability conditions
for a system with MulTCP flows.
II. S TABILITY C ONDITION : S INGLE -F LOW,
S INGLE -R ESOURCE WITH S TATE D EPENDENT D ELAY
In this section we first consider a flow traversing a single resource. The rate control problem in the optimization
framework proposed in [7] in this simple case is given by
the following simple optimization problem:
Rx
max U (x) − 0 p(y)dy
(1)
x≥0
s. t.
x≤C
where x is the rate, U (x) is the utility of the user when it
receives a rate of x, p(x) is the price per unit flow charged by
the resource when the rate is x, and C is the capacity of the
resource. The proposed end user algorithm in the absence of
delay is given by the following differential equation [9].
d
x(t) = κ (w(t) − x(t)µ(t))
dt
(2)
where w(t) is the price per unit time user is willing to pay,
µ(t) = p(x(t)), and κ, κ > 0, is a gain parameter. We will
study the stability of the system in (2) with general utility
functions that satisfy a set of assumptions to be stated shortly.
An example of a family of utility functions that satisfy such
assumptions is given in Section III.
In practice, the rate of a flow is typically limited by the
receiver buffer size. Hence, even when there is no congestion
the rate of a flow is upper bounded by some constant.
Assumption 1: The rate of the flow is bounded from above
by some arbitrary constant Brate , C ≤ Brate < ∞.
Therefore, throughout the rest of the paper we implicitly
assume that when the rate of the flow reaches the upper bound
Brate , the time derivative is given by
d
x(t) = min{κ(w(t) − x(t)µ(t)), 0} .
dt
Adopting the end user algorithm given in [9] we assume
0
that w(t) = x(t)·U (x(t)). The congestion signal generated at
the resource, i.e., p(x(t)), is returned to the user after a timevarying delay. We denote the round-trip delay of the packet
whose acknowledgment (ACK) arrives at the source at time t
by τ1 (t). Hence, the rate at which the ACKs or feedback signal
arrives at the source at time t is x(t − τ1 (t)). The round-trip
delay consists of two components; it has a fixed delay and a
time-varying delay that is caused by the time-varying queue
size at the resource. We denote the fixed delay by T . Since the
fixed delay can lie (i) in the forward path, i.e., in the path from
the source to the resource, (ii) in the reverse path, i.e., in the
path from the resource to the source, or (iii) in both forward
and reverse paths, there are three different cases that one can
consider.
If we assume that the fixed delay lies in the reverse path,
in the presence of general time-varying and in particular
state-dependent delay,1 the system dynamics are given by the
following set of delayed differential equations:
0
d
x(t) = κ x(t)U (x(t)) − x(t − τ1 (t))p(x(t − τ1 (t))) (3)
dt
q(t − τ1 (t))
τ1 (t) = T +
(4)
C
dq(t)
x(t) − C
if q(t) > 0
=
(5)
+
[x(t)
−
C]
if q(t) = 0
dt
where q(t) is the queue size at the resource at time t, C is
the capacity of the resource, and [a]+ = max{a, 0}. Here the
time varying delay in fact depends on the state of the system,
i.e., queue size.
If the fixed delay lies in the forward path, (3) - (5) are given
by
0
d
x(t) = κ x(t)U (x(t))
dt
−x(t − τ1 (t))p(x(t − τ1 (t) + T )))
q(t − τ1 (t) + T )
τ1 (t) = T +
C
dq(t)
x(t − T ) − C
if q(t) > 0
(6)
=
[x(t − T ) − C]+ if q(t) = 0
dt
Eq. (5) and (6) essentially state that the server is workconserving, i.e., the server is never idle while the queue is
non-empty. Although the evolution of x(t) depends on where
the fixed delay lies, in this paper we only consider the case
where the fixed delay lies in the reverse path. The other cases
can be handled in a similar manner and result is not affected
by the relative placement of fixed and queuing delays as long
as queuing delay stays bounded.
In practice, any physical buffer/memory at the resource will
be finite no matter how large it may be. Therefore, without loss
of generality we assume that the buffer size at the resource is
finite.
Assumption 2: The buffer size B is finite, i.e., 0 ≤ B < ∞.
This assumption enables us to bound the size of round-trip
delay from above, i.e., T ≤ τ1 (t) ≤ T + B
C.
1 We
say that the delay is state-dependent if it depends on the queue size
of the resource.
Rather than directly dealing with the delayed differential
equation in (3) we put it in a form that is more amenable to
analysis as follows. After normalizing time by T and replacing
t = s · T , (3) becomes
0
1 d
x(s) = κ x(s)U (x(s)) − x(s − τ (s))p(x(s − τ (s)))
T ds
0
d
ν x(s) = x(s)U (x(s)) − x(s − τ (s))p(x(s − τ (s))) (7)
ds
where ν = T1κ , and τ (s) = τ1T(t) . We normalize the queuing
delay to keep the discussion in line with our earlier results for
fixed delay cases [14].
In this paper we show that the stability properties of the
solutions of (3) can be obtained from an underlying nonlinear difference equation. This is obtained from the following
singularly perturbed system
d
x(t) = g(x(t)) − f (x(t − 1))
(8)
dt
of general nonlinear difference equation with continuous argument given by g(x(t)) = f (x(t − 1)), t ≥ 0, where
0
g(x) = x · U (x) and f (y) = y · p(y) in the context of (7).
Under certain natural invertibility conditions on g(·), it leads
to a much studied equation [15]
ν
x(t) = F̂ (x(t − 1)) , t ≥ 0
(9)
where F̂ (·) = g −1 (f (·)). For the solution of (9) to be
continuous for t ≥ −1, along with the continuity of F
and initial condition φ(·), a so-called consistency condition
limt→−0 φ(t) = F (φ(−1)) is required [5], [15]. It turns out
that a great deal about the asymptotic stability of (8) can
be learned from the asymptotic behavior of the following
difference equation, with Z+ denoting the set of non-negative
integers [4]:
xn+1 = F̂ (xn ), n ∈ Z+
(10)
Unfortunately, the concave utility functions and resource price
functions assumed in [7], [9] and the time varying nature of
round-trip delay do not satisfy the assumptions in [4], and
we cannot directly apply their results. However, the general
approach used in the paper for establishing the stability can
be extended to study the convergence of (3).
In the following subsections we first establish general convergence results for one dimensional case described here. In
Section III we illustrate how our results in subsection II-A can
be applied to study the convergence of the system described
in this section to the optimum with utility and resource price
functions given in the section.
A. Convergence Results for State-Dependent Delay
In this subsection we establish the conditions for convergence of the system in (3) regardless of the communication
delay T , gain parameter κ, and the nature of queuing delay.
Consider the following substitution:
0
y(t) = x(t)U (x(t)) := g(x(t))
and f (x(t)) := x(t)p(x(t)).
(11)
We first make the following assumptions on the functions g(x)
and f (x).
Assumption 3: (i) The function g(x) is strictly decreasing
0
with g (x) < 0 for all x > 0, (ii) the function f (x) is strictly
increasing for all x > 0, (iii) both g(x) and f (x) are Lipschitz
continuous on [ , ∞), where is an arbitrarily small positive
constant, and (iv) the derivative of time delay function τ (t)
is bounded in magnitude, i.e., there exists some constant B ∈
IR+ such that |τ̇ (t)| ≤ B.
Assumption (i) implies that users are rather inelastic, i.e., users
do not react sharply to the changes in resource prices. From
(11) one can see that a sufficient condition for (ii) is that
the resource price functions are strictly increasing in the rates
traversing the resources. The utility functions and resource
price functions used in Section III satisfy the above assumptions.
Assumption 3 allows us the following change of coordinate:
x(t) = g −1 (y(t)) ⇒ ẋ(t) =
ẏ(t)
g
0
(g −1 (y(t)))
(12)
0
ν ẏ(t) = g (g −1 (y(t)))(y(t) − f (g −1 (y(t − τ (t)))))
where the inverse g −1 (·) exists from Assumption 3. Let
0
κ(y(t)) := −g (g −1 (y(t))). Clearly, κ(y(t)) > 0 under
Assumption 3. Using this substitution in (12) we get the
following form.
ν ẏ(t) = κ(y(t)) f (g −1 (y(t − τ (t)))) − y(t)
(13)
We study (13) and show that there is a close correspondence
between invariance and global stability properties of the discrete time map
yn+1 = f (g −1 (yn )) := F (yn )
(14)
and those of (13). In particular, we will prove that if y n+1 =
F (yn ) has a stable fixed point then (13) will have a uniformly
constant solution for all possible time delays T ≥ 0 if
the initial function’s range is contained in the immediate
basin of attraction of this fixed point. The proofs are based
on the invariance property of the underlying map F (·) and
the monotonicity of function g(·). Note that the map F (y)
is strictly decreasing because g −1 (y) is strictly decreasing
and f (x) is strictly increasing under Assumption 3 and a
composition of a strictly increasing function and a strictly
decreasing function is a strictly decreasing function.
Assumption 4: Suppose that I ⊂ {x : x ≥ } is a closed
invariant interval under F , i.e., F (I) ⊂ I. In particular, let
I = [a, b] be compact.
Let us first define the notation that will be used throughout
this section:
B
dmax = 1 +
CT
Mt =
sup
y(s)
mt =
t−dmax ≤s≤t−1
inf
t−dmax ≤s≤t−1
y(s)
We denote the set of continuous functions over [−d max , 0]
with the range of D by C([−dmax , 0], D). Let X :=
C([−dmax , 0], IR+ ), and XI := {φ ∈ X : φ(s) ∈ I ∀ s ∈
[−dmax , 0]}. Under this assumption, we have invariance for
the solution of (13) for all time t ≥ 0 and for all ν > 0.
Since the functions involved in (13) are Lipschitz continuous
by assumption, solutions do exist for all t ≥ 0 and are unique
for any initial function φ ∈ XI . Furthermore, the invariance
property of the solutions, which is stated below (Theorem 1),
ensures that they stay positive and bounded by the initial set
they start in, which is assumed to be invariant under map F .
Theorem 1: (Invariance) If φ ∈ XI , the corresponding
solution y(t) = y(t; φ) satisfies y(t) ∈ I for all t ≥ 0. This
means that the set I is invariant under (13).
Proof: We prove the theorem by contradiction. Suppose
that the claim is not true and there exists φ ∈ X I such that
y(t; φ) 6∈ I for some t ≥ 0. Let t0 be the first time when
solution y(t; φ) leaves I, i.e.,
t0 = inf{t ≥ 0 | for all [t, t + δ), δ > 0,
there exists t1 , t < t1 < t + δ,
such that y(t1 ; φ) 6∈ I} .
(15)
First, assume that y(t0 ) = b. In this case, for every (t0 , t0 +δ),
δ > 0, we can find a point t2 , t0 < t2 < t0 + min(δ, 1), such
that y(t2 ) > b and ẏ(t2 ) > 0. However, since y(t2 − τ (t)) ≤
Mt ≤ b from (15) and due to the fact that 1 ≤ τ (t) ≤ dmax ,
we have
κ(y(t2 )) f (g −1 (y(t2 − τ (t)))) − y(t2 )
<0
ẏ(t2 ) =
ν
from (13) and Assumption 4 that I is invariant under F ,
i.e., F (y(t2 − τ (t))) = f (g −1 (y(t2 − τ (t)))) ≤ b. This
contradicts the earlier assumption that ẏ(t 2 ) > 0.
Similarly, suppose that y(t0 ) = a and the trajectory exits
from the left end of the interval. Then, for every interval
(t0 , t0 + δ), δ > 0, we can find t2 , t0 < t2 < t0 + min(δ, 1),
such that 0 < y(t2 ) < a and ẏ(t2 ) < 0. However, because
y(t2 − τ (t)) ≥ mt ≥ a, we have
κ(y(t2 ))(f (g −1 (y(t2 − τ (t)))) − y(t2 ))
>0
ν
from (13) and Assumption 4. This, however, contradicts the
assumption ẏ(t2 ) < 0. Hence, the theorem follows.
ẏ(t2 ) =
Next theorem considers the case when the map F has an
attracting fixed point y ∗ with immediate basin of attraction
J0 : F n y0 → y ∗ for any y0 ∈ J0 , which is assumed to be a
closed set invariant under F . Let XJ0 = C([−dmax , 0], J0 ).
Then, the following theorem holds.
Theorem 2: (Stability) For any ν > 0 and φ ∈ XJ0 , we
have limt→∞ yφ (t) = y ∗ .
Before proving the theorem we will first state a lemma
which is the key to the proof of Theorem 2.
Lemma 1: Suppose that a closed interval J is mapped by
F to itself. If neither of the endpoints of the interval F (J)
is a fixed point, then for every φ ∈ XJ = C([−dmax , 0], J)
there exists a finite t0 = t0 (φ, ν, κ) such that yφ (t) ∈ F (J)
for all t ≥ t0 .
Proof: From Theorem 1 it is clear that yφ ∈ J for all
t ≥ 0. The claim here is that after some time t0 it will belong
to F (J) ⊂ J.
We consider two cases. First, assume that φ(0) ∈ F (J).
Then it can be shown that yφ (t) ∈ F (J) for all t ≥ t0
by contradiction. Suppose that this is not true, and let t 0
be the first time when yφ (t) leaves the interval F (J). In
particular, assume that it leaves the interval through the right
end, i.e., every time interval (t0 , t0 ), where t0 > t0 , contains
a point t1 such that yφ (t1 ) > sup F (J). The interval (t0 , t0 )
also contains a point t2 such that yφ (t2 ) > sup F (J) and
ẏφ (t2 ) > 0. However, as yφ (t) ∈ J for all t ∈ [t0 − dmax , t0 ],
we have
κ(yφ (t2 ))(f (g −1 (yφ (t2 − τ (t)))) − yφ (t2 ))
<0
ν
from (13). This contradicts the earlier assumption that
ẏφ (t2 ) > 0. The other case where yφ (t) leaves the interval
from the left end can be handled similarly.
Now assume that φ(0) ∈
/ F (J). In particular, let φ(0) >
sup F (J). We claim that yφ (t) is decreasing for all t ∈ [0, t0 ],
where t0 ≤ ∞ is the first time such that yφ (t0 ) = sup F (J).
We first argue that t0 < ∞ by contradiction. Suppose that
yφ (t) > sup F (J) for all t ≥ 0. From (13), for all t ≥ 0, we
have
ẏφ (t2 ) =
ẏφ (t) =
κ(yφ (t))(f (g −1 (yφ (t − τ (t)))) − yφ (t))
<0
ν
because f (g −1 (yφ (t − τ (t)))) ≤ sup F (J). Then, there exists
a limit y = limt→∞ yφ (t) ≥ sup F (J) due to BolzanoWeierstrass theorem [3] which says that every strictly decreasing sequence which is bounded from below has a limit. As y
is not a fixed point of map F , κ(y)(y − f (g −1 (y))) := δ > 0.
This tells us from (13) that
ẏφ (t) =
κ(yφ (t))(f (g
−1
(yφ (t − τ (t)))) − yφ (t))
δ
<−
ν
2ν
for all sufficiently large t. This implies that y φ (t) → −∞ as
t → ∞, which is a contradiction, because this means that y φ (t)
crosses sup F (J) for some finite t. Hence, t0 < ∞. Now, we
can apply the argument used in the first case, i.e., φ(0) ∈
F (J), to the system with y(t0 ) = sup F (J) and y(t) ∈ J
for all t ∈ [t0 − dmax , t0 ] to prove that y(t) ∈ F (J) for all
t ≥ t0 . The other case where φ(0) < inf F (J) can be handled
similarly.
Now we provide the proof for Theorem 2 using Lemma 1.
Proof: (Theorem 2) For any φ ∈ XJ0 , define m =
inf{φ(s), s ∈ [−dmax , 0]} and M = sup{φ(s), s ∈
0
[−dmax , 0]}. Clearly, [m, M ] ⊂ J0 . Let J be the smallest
closed invariant interval containing [m, M ] which is a subset
of J0 . The existence of such an interval is guaranteed from
the assumption that J0 is invariant under F . Then, from
0
the existence of a stable fixed point of the map F , J ⊃
0
0
0
F (J ) ⊃ F (F (J )) ⊃ . . . and ∩i≥0 F i (J ) = {y ∗ }. Using
the invariance result and Lemma 1 repeatedly one can find
a sequence of ti , i ≥ 1, such that for all t ≥ ti , yφ (t) ∈
0
F i (J ). The convergence now follows from the assumption
0
that F i (J ) → {y ∗ }.
The above theorem tells us that if the initial function lies
in XJ0 , then the rate x(t) converges to the solution of (1)
regardless of the value of T , κ, or exact dynamical nature
of queuing delay. Hence, it establishes a strong convergence
result in the presence of a time-varying communication delay
without any conditions on the value of T or κ. This contrasts
the results obtained in [2], [13].
III. N UMERICAL E XAMPLES
In this section we present numerical examples to verify the
analytical stability criterion obtained in the presence of timevarying state-dependent queing delay.
We first describe the utility and price functions that we use
for the numerical examples. The utility functions are assumed
to be of the form
1 1
Ua (x) = − a , a > 0 .
(16)
ax
In particular, a = 1 has been found useful for modeling
the utility function of TCP algorithms [10]. With the utility
functions of the form in (16) one can easily show that the
price elasticity of demand decreases with a as follows. Given
a price per unit flow p, the optimal rate x∗ (p) of the user that
1
maximizes the net utility Ua (x) − p · x is given by p− 1+a . The
price elasticity of demand, which measures how responsive the
demand is to a change in price, is defined to be the percent
change in demand divided by the percent change in price [16].
In our case the price elasticity of demand is given by
1
−1
p dx∗ (p)
p
−1 − 1+a
−1
=
= − 1 ·
p
. (17)
x∗ (p) dp
1
+
a
1
+a
1+a
p
Therefore, one can see that the price elasticity of demand
decreases with a,2 i.e., the larger a is, the less responsive the
demand is.
The class of resource price functions that we adopt is of the
form:
y b
,
(18)
p(y) = c ·
C
where b > 0, c is some positive constant, and C is the
capacity of the resource. However, C can be replaced with
any positive constant, e.g., virtual capacity in AVQ, so that
the price function can be dynamically adjusted based on
the current load using the virtual capacity as the control
variable [11]. Throughout this paper we assume that c = 1.
This kind of marking function arises if the resource is modeled
as an M/M/1 queue with a service rate of C packets per unit
time and a packet receives a mark with a congestion indication
signal if it arrives at the queue to find at least b packets in the
queue.
It is shown in [14] that if the delay is fixed, then the
necessary and sufficient condition for the stability is given
by a > b + 1. We take the example in our earlier work [14]
2 When comparing the price elasticity, typically the absolute value of (17)
is used.
10
where the price parameter b = 3, capacity C = 5, propagation
time T = 50, and utility parameter a is varied to show its
effect on stability.
9
8
7
6
rate−>
10
9
5
4
8
3
7
2
rate−>
6
1
5
0
4
0
100
200
300
400
3
500
time−>
600
700
800
900
1000
600
700
800
900
1000
(a)
2
75
1
0
0
100
200
300
400
500
time−>
600
700
800
900
1000
70
queing delay−>
(a)
10
9
65
60
8
7
55
rate−>
6
5
50
4
3
0
100
200
300
400
500
time−>
(b)
2
Fig. 2.
delay
1
0
0
100
200
300
400
500
time−>
600
700
800
900
Plot of rate (a = 1.5) and queueing delay. (a) rate, (b) queueing
1000
(b)
given by
Fig. 1.
Plot of rates. (a) a = 7, (b) a = 4.1
Fig. 1 shows the plots of the rate when a > b + 1. One can
see that under the stability condition a > b + 1 the rate of the
user converges to the solution (1) even with state-dependent
delays. Clearly, when a is considerably larger than b, the
system settles down faster, and as a approaches b + 1 from
above the settling time increases as witnessed in Fig. 1(b).
We also study the period-doubling type of instability when
the stability condition a > b+1 is violated in Fig. 2. Similarly
as in our earlier results, oscillations appear for a large enough
delay if the stability condition a > b + 1 is not satisfied.
Oscillations can be shown to appear via a period-doubling
bifurcation. We also plot the time periodic behavior of queuing
delay shown in Fig. 2(b). The plot shows that the user goes
through periods of underestimating and overestimating the
congestion level at the resource without any sign of settling
down. This behavior of queing delay is markedly different
from that of the case where the stability condition a > b + 1
holds and the round-trip delay settles to the fixed propagation
delay.
IV. A G ENERAL M ODEL FOR M ARKET-D RIVEN
E ND -U SER A LGORITHM
One can extend the end-user algorithm given by (3) in
several directions. Here we consider one of such extensions
dx
= κ1 (x(t), x(t − τ (t))) (g1 (x(t)) · f2 (x(t − τ (t)))
dt
− f1 (x(t − τ (t))) · g2 (x(t)))
(19)
where x(t) ∈ [0, +∞) := IR+ denotes the amount of resource
consumed at time t. The various components in (19) represent
the following:
1) κ1 (x(t), x(t − τ (t))) - a gain function that may depend
on both current and delayed state value x(·).
2) g1 (x(t)) - an adaptation term of the sender. This term,
in general, may be augmented by the past history/state
given by f2 (x(t − τ (t))).
3) f1 (x(t − τ (t))) - a feedback term that represents the
feedback information from the resources. The effect of
this feedback information may be weighted by some
function of current state x(t). We denote this multiplicative term by g2 (x(t)).
Clearly, the Kelly’s primal algorithm is a special case of (7)
0
with κ1 (x(t), x(t − τ (t))) = κ, g1 (x(t)) = x(t) · U (x(t)),
f1 (x(t − τ (t))) = x(t − τ (t))p(x(t − τ (t))), and f2 (x(t −
τ (t))) = g2 (x(t)) = 1.
Another example of a similar mechanism is MulTCP [8],
which is a variant of the original TCP and is designed to
yield smoother operation by mimicing the aggregate behavior
of multiple TCP flows. Due to its attempt to follow the
behavior of multiple TCP flows, it can be better modeled by a
differential equation. Suppose that a set I of users share a set
L of resources. Under the basic model the end user algorithm
of a user i is given by
mi
d
xi (t) = 2 −
dt
Ti
mi
xi (t)2
+
2
Ti
2mi
pi (t)
(20)
where pi (t) is the totalPprice per unit flow along the route
of user i, i.e., pi (t) = j∈ri µj (t), ri (⊂ L) is the route of
user i, i.e., the set of resources user i’s packets traverse, T i is
its round trip time, and mi is a parameter used to determine
the smoothness of the algorithm. In order to keep x i (t) nonnegative, we take the maximum between (20) and zero when
xi (t) = 0. For (20) Kelly has established the global stability
for all values of Ti .
Theorem 3: [8] The Function
√
X 2mi
x i Ti
√
U(x) =
arctan
Ti
2mi
i∈I
P
xj
XZ
j:l∈rj
−
pl (y)dy
l∈L
0
is a Lyapunov function for the system of differential equations.
The unique value x maximizing U(x) is a stable point of the
system, to which all trajectories converge.
Here we consider a single-resource case with a delay. Let M
be the number of flows using the single resource. Suppose that
mi = 1, i.e., each flow behaves like a single TCP flow, and
all flows are identical with the same delay T . If we denote
by x(t) the total rate of users, then from the symmetry of
individual flows, xi (t) = x(t)/M , and from (20) after taking
the delay T into account, we have
M x(t − T )
d
xi (t) =
(1 − p(x(t − T )))
dt
T 2 x(t)
x(t)x(t − T )
−
p(x(t − T ))
2M
M x(t)x(t − T )(1 − p(x(t − T )))
=
T2
1
p(x(t − T ))T 2
×
. (21)
−
x2 (t) (1 − p(x(t − T )))2M 2
Notice the striking similarity between (21) and our more
general model given by (19). It is clear that (21) fits our
)))
, g1 (x(t)) = x21(t) ,
model with κ = M x(t)x(t−T )(1−p(x(t−T
T2
f1 (x(t − T )) =
T )) = 1.
p(x(t−T ))T 2
(1−p(x(t−T )))2M 2 ,
and g2 (x(t)) = f2 (x(t −
A. Global Stability Criterion for MulTCP
In this subsection, using our results in Section II-A we
derive the global stability of the system with M identical
MulTCP flows with mi = 1 described above. From (21) and
(12) - (14) one can see that the underlying discrete time map
is given by
1
x2n+1
⇒ xn+1
p(xn )T 2
(1 − p(xn ))2M 2
s
√
2M
1
=
−1 .
T
p(xn )
=
(22)
(23)
By differentiating (23) and evaluating it at the equilibrium we
can compute the eigenvlaue λ:
0
M
p (x)
(24)
λ = −√
3p
2T (1 − p(x)) 2 p(x) x=x∗
where x∗ is the solution to (1). For stability we need this eigenvalue to lie within the unit circle. This yields the following
local stability condition:
0
M
p (x)
(25)
√
<1
3p
2T (1 − p(x)) 2 p(x) √
0
2T
p (x)
(26)
⇒
<
3p
M
(1 − p(x)) 2 p(x)
In fact, using Sharkovsky’s classification theorem, we can conclude that stability is global because the map is monotonically
decreasing and has at most one fixed point with period two.
Note that unlike in Section III, due to the fact that the utility
function of the end users depends on the delay T [8], the
global stability depends on T as well.
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