ÐS (6ÐÐ # S Wg u ()Ð Ð H 'ÐB Щ F uh wi Hj $4Ð # G H $жР# I H '8B Ð )5 as the governing equation for the occupancy rate of the destination buffer. By making ...
Some Recent Techniques in Delay-Differential Equations and the Converging Flows Problem G. Pecelli and B. G. Kim Computer Science Department University of Massachusetts Lowell Lowell, MA 01854 kim, giam @cs.uml.edu tel: (978) 934-3639; fax: (978) 934-3551
Abstract. This paper provides a rigorous analysis of the stability properties of a protocol for congestion and fairness management. The protocol was analyzed earlier [Roche 94, 95] both by simulation and with less precise mathematical tools. The techniques introduced - by now well known in the field of Delay-Differential Equations - are fairly general, and can be applied in a number of more complex contexts.
Key Words.
Network Congestion, Stability of Delay-Differential Equations, Hopf Bifurcation.
Acknowledgment. The research was partially supported by N.S.F. grant NCR-9628058.
1
Introduction.
Several problem of congestion control arise in the management of ATM networks [Jain 95]. In this context, Roche and others [Roche 94, 95] have studied the problem of distributing bandwith among several sources with identical or similar characteristics in such a way that all sources will eventually send at the same rate. This paper extends the analytical approach using various results from the field of DelayDifferential Equations. Discrete simulation studies provide confirmation of the theoretical predictions. We will assume the reader has access to the standard literature for Delay-Differential Equations ([Bellman 63] and [Hale 93]) and has a reasonable facility with the standard theory for Ordinary Differential Equations.
2
The Problem.
The description is taken, with slight modifications, from [Roche 95]. We consider a destination and queues with buffer sizes , arrival two sources, modelled as queuing systems. These are FIFO rates , queue lenghts and service rates . The subscript will denote the destination, while the subscripts and will denote the sources. We further assume that the sources are connected to the destination via a high-speed connection with no bandwidth limitation. We let denote the percentage of each buffer being used. The destination receives this load information and stores it in the form of a vector . It then determines the amount of bandwidth to allocate to each source, notifying each sorce of the service rate it should use. At reception, each source resets its service rate
"! $# &% ('
)
"
1
according to the value specified. If we assume a propagation delay of that
+,
*
in both directions, we can assume
- /. 10 * # 2. ) 10 * #435. # +
The control algorithm is constructed according to the following criteria:
i) One determines the amount of bandwidth available at the destination at time , say ii) One estimates the fraction .
7;% 2<
iii) One chooses
7
3
6 .
of available bandwidth to be allocated to source . Note that
7 !8:9
)= >. 7= 6 , 35. # .
7 is: A B C !80EDFH 9 0 - 0EDF 0 % E 0 F D 0 A BGC A B C
[Roche 94, 95] makes the case that a good choice for
+?
7= @.
DI . For bandwidth availability, the same source makes the case for +?J 6 >.LKNMO0 $P Q59 P #4MSR # '# where P is the departure rate from the destination, and K and M are appropriately chosen positive numbers, with M the asymptotic buffer occupancy rate. Then, for 35. # , we have T @. = /0 " >. = /0 )= 10 * /. /0 6 /0 * 7= U0 * . = /0 6 U0 * A ! B C /0 0E* DF Q V90 - U0 0E* DF 0 &% 10 * E 0 F D 0 " A BGC A BG C U0 &* Q E 0 S D 0 . = /0 6 U0 * A B C A BGC 0EDF 0 W! U0 &* H9 A B C 0EDF 0 % 10 &* Q for some
Taking a difference, we have
+YX
! T! 10 % T% @.
! Z0 %&/0 6 U0 * > [ \] ^SNDF !8U0 &* /0 %10 &* + The assumption that the source buffer sizes are equal ( -! . % ) lets us simplify +_X to
+?`
5 T59 6 U0 * a[b\] ^SND U0 &* c. where < -! . % , /< W! /0 % and /< d! Z0 ;% .
Using
+?J , we can also derive P5 TP @. 6 U0 W * 1 0 " P . KNMe0 P U0 & * QV9 P 1 0 &* 10 P $#
and, finally
+?f
P5 $TP 59gK P U0 W* c.
KhMi0j "P /0 P 10 &* Q #
as the governing equation for the occupancy rate of the destination buffer.
=+ `
By making different assumptions on the functions that are explicitly time-dependent in and we can study the problem at varying levels of complexity. We identify several distinct problems. 2 / 12
+?f ,
F< lkm
6 >0 * i< 6 I
A) , non-negative without loss of generality, and . This case arises when the source arrival rates differ by a constant, and the decisions on departure rates are made strictly on the percentage use of the source buffers. The implicit assumption is that the receiver service rate is constant. The relevant Delay-Differential Equation is
+?n
5 T o9 6 [b\] ^SpD /0 &* " U. # ! % which has q.¿¹ X;+ ¹¹ . The ”period” of the oscillations is also, approximately, }/* . The next simulation uses D. + } ¹ f , Figure 4.4 : k = 0.85,
just after (theoretical) transition into instability. The radius of oscillation is large (the predicted value is ), closer to the value expected for , and the system has not stabilized after 4 seconds. The three plots correspond to the same entities as in Fig. 4.4.
º+ "f ¹
D. =+ ¹
Source & Buffer Occupancy Rates 180 160 140 120 100 80 60 40 20 0 0
0.5
1
1.5 2 Tine in Sec.
Figure 4.5 : k = 0.896, 7 / 12
2.5
3
;% /.L¹ X;+ ¹¹ .
3.5
4
The theoretical development assumes that input and output are sufficiently balanced not to cause overflow. The simulation is likely to overflow when the input rate is close to the output rate. This tendency to overflow can be countered by increasing slightly, or by modifying the arrival rates and , maintaining their difference constant. In particular, reducing to (90Kb/sec less input into the system, lowering the point of transition to instability, but decreasing the possibility of system saturation) produces a simulation with a radius of oscillation and apparent asymptotic stability of the oscillations (over 8 simulation seconds). This can be seen in Fig. 4.6, where we show only the time between 6 and 9.5 seconds, to show that the system appears to have attained stable oscillation:
P
;%
;%
¹ X;+ ¹ ·¸
H!
º¸ + `
Source & Buffer Occupancy Rates 120 100 80 60 40 20 0 6
7
8
9
Time in Sec.
:% 2.L¹ X;+ ¹ . % /.L¹ X;+ ¹¹ ·¸ . We end with a simulation with D. + ¹ , ; Figure 4.6 : k = 0.896,
Source & Buffer Occupancy Rates 180 160 140 120 100 80 60 40 20 0 -20 0
0.5
1
1.5
2
2.5
3
3.5
4
Time in Sec.
: % >.¿¹ X;+ ¹¹ . In this case, the predicted radius of oscillation is ºÀ+ ¹ XJJ}"n . This indicates that the oscillations will Figure 4.7 : k = 0.92,
attempt to increase even further in amplitude, with substantial losses due to buffer overflow. In all cases the period of the oscillations is , as predicted by theory. In a simulation with , the system stabilizes after about 5 seconds, with an oscillation radius - and also substantial losses due to buffer overflow, as we can see in the next and last figure. Because of difficulties in seeing all three rates, we limit ourselves to showing the Differential Occupancy Rate alone. Asymptotic stability of the oscillations appears reached by about 6 seconds with a radius of, , compared to the 1.29 predicted. Substantial losses would be expected.
º}Z*
º|=+ ¹
%&.¹ X;+ ¹ ·|
-% 20 &! ºÁ+ `
8 / 12
Buffer Occupancy Rate 180 160 140 120 100 80 60 40 20 0 -20 -40
6
Time in Sec.
Figure 4.8 : k = 0.92,
5
10
8
;% /.L¹ X;+ ¹ .
Conclusions.
We have a provided a rigorous analysis of a mathematical model of congestion and fairness management. The result obtained allow us to predict the exact parameter configurations that correspond to transitions to instability, along with the exact oscillation frequencies. We can also predict the amplitude of the oscillations, at least in regions ”near” stability. The results predicted by the theory are also shown to have reasonable agreement with simulation ones. The very fast increase in the oscillation radius (and the simulation results) as the destabilizing parameter increases should be a good indication that this kind of system will be very sensitive to being destabilized.
6
Appendix.
+ n
+ }
Proof of Proposition 3.2. This follows from linearizing and around the constant solutions, computing the characteristic equations and applying Lemma 3.1. We first show some details for . We first replace
59 u% b[ \] ^ 5 T 59 6 6 9 9
with
=+ n
v ! $ xw in + n , and simplify: 6 [b\] ^SpD U0 &* " . 5+ [b\] ^SpD U0 &* "
Next compute the expansion to order 3 (the higher order terms will be used later, in the non-linear analysis) - Maple [Char 91a, b] was used for the purpose:
J +?
9 6 b[ \] ^F ) 6 9 b[ \] ^F ) .
9 6 % 0% 6 6 % 0 6 0
% ) 0 6 % 0 % U 6 % 0 >J %65 Â J/65Ã
The linearized equation is thus:
% % TH9 FD 6 2 H0 6 ²0 W* Ä. 9 / 12
. =+
% % ) )Â
+
Rescaling time so that the retardation is normalized to , we obtain
J +?J
% % To9 * SD 6 d 6 0 U0 @.
+ P
The first claim follows from Lemma 3.1. The rest follows from the observation that plays the same role for the linearized form of the first equation in that plays for . The linearized system is with the constant solution translated to the origin corresponding to
+ }
6
+?}
+ n
£ 5 T Z0 K PU0 * 59 FD P% 0 % 10 &* c. ¡¢ " P > "P P T 5 j 9 K P / 0 . + & * P ¢¤ In order to use Lemma 3.1, we first observe that replacing $P /0 * by a function Å >0 Z* , leads to a new system in # Å where all the retardations are /* , and with the same stability properties. The only difference is that each solution of the system with P and initial condition N- # $P N- on 0 &* # ' induces a solution of the new system with initial condition N # Å N . P Ni9 * on 0 &* # ' . This capacity to ”shift delays” in a network so that only the total delay is significant, was already observed by [Browning 94] and was used extensively in [Pecelli 99]. We will continue using in the sequel even in the new context - no confusion should arise.
P
A simple computation shows that the characteristic equation decouples
% % ( Æ 9 * SD 5 P "P 0 ( -Æ 9 / * P K /. #
and the rest of the proposition follows, since we need both factors to have the required types of zeros. Proof of Corollary 3.4.
We rewite
+ n
to terms of order 3 as
% % % % % T 59 * FD 6 H 6 0 /0 U 0 D/ 6 U0 H9 J2 -0/6 6 % ;D 10 % /. # where time has been rescaled so that the total retardation is . * DF 6 % 0 % do not The Hopf bifurcation simply follows from the fact that the derivatives of H6 vanish, except for the derivative with respect to at . . The supercriticality follows from Theorem 3.5, since we now have % % % . 0
FD Q }n
Çp0
% } 9 X 9 6 % `
6 # and this expression is always negative. We can observe that, if . , % % . 0 -f N
% D 9 X Proof of Corollary 3.5.
10 / 12
+ } $P
First, we rescale time in so the round-trip retardation is , move the constant solution to the by , and use only the terms up to third order (which is all we need origin, change the phase of to compute in the generic case), to obtain:
J +?f
TN/0 ¡¢¢ % ¢ 0 * DOKÈ 5 P P% 0 £ ¢¢¢ * D % K P% 0 % 9 > o ÂP ¢¢ ¢¢ ¢¢ ¤
It is clear that
+?}
and
/* K P 10 59 * DS P% 0 % U0 5"P 5"P % P 10 U0 Z0 * D % P% % 0 % 10 % % >0 o % P U J2 % 0 P% D * Â P % $P U0 50 9 U0 Â . 2 5 ÂP . TP o9 / * P K $P U0
J + f
cannot have bounded solutions unless
#
/* K y
" . P
The existence of the bifurcation follows from Lemma 3.1 and the non-vanishing of the relevant derivatives. The character of the bifurcation follows from a direct computation that provides
.
% Q n
Ç0 } % 9 `
6 % F
D 0 # } p
% 9 X 6 %
the same expression obtained in Corollary 3.4. Since the second equation of the system will always give rise to damped oscillations (by assumption), it is intuitively reasonable that only the first equation will have any effect on the character of the bifurcation.
7
Bibliography
[Bellman 63 ] Bellman, Richard and Kenneth L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. [Browning 94 ] Browning, Douglas W., Flow Control in High-Speed Communication Networks, IEEE Transactions on Communications, Vol. 42, No. 7, 1994, 2480-89. [Char 91a ] Char, B. W., K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan and S. W. Watt, MAPLE V Language Reference Manual, Springer Verlag, New York, 1991. [Char 91b ] Char, B. W., K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan and S. W. Watt, MAPLE V Library Reference Manual, Springer Verlag, New York, 1991. [Chow 77 ] Chow, S. N, and J. Mallet-Paret, Integral Averaging and Bifurcation, Journal of Differential Equations, 26, 1977, 112-159. [Hale 93 ] Hale, Jack K., S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, 1993. [Jain 95 ] Jain, R., Congestion Control and Traffic Management in ATM Networks: Recent Advances and a Survey, to appear in Computer Networks and ISDN Systems, 1995. [Roche 94 ] Roche, C., and S. Fdida, A dynamic resource management mechanism for LAN interconnection across high-speed networks, IEEE Infocom ’94, June 1994. [Roche 95 ] Roche, C., and Plotkin, N. T., The converging flows problem: an analytical study, IEEE Infocom ’95, March 1995. 11 / 12
[Pecelli 93 ] Pecelli, G., Delay-Differential Equations: Symbolic Computation and the Hopf Bifurcation, Univ. of Mass. Lowell Technical Report, 1993. [Pecelli 97 ] Pecelli, G., Prey-Predator Systems with Delay, Mathematical and Computer Modeling, Vol.25, No. 10, pp. 77-98, 1997. [Pecelli 99 ] Pecelli, G., and B. G. Kim, Network Models with Delays and Fixed Points, University of Massachusetts - Lowell Technical Report, 1999.
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