Dynamic adaptive windows for high speed data networks with multiple paths and propagation delays

Computer Networks and ISDN Systems 25 (1993) 663-679 North-Holland

663

Dynamic adaptive windows for high speed data networks with multiple paths and propagation delays Debasis Mitra and Judith B . Seery AT&T Bell Laboratories, Murray Hill, NJ 07974, USA

Abstract Mitra, D . and J .B . Seery Dynamic adaptive windows for high speed data networks with multiple paths and propagation delays, Computer Networks and ISDN Systems 25 (1993) 663-679 . Recently the optimal design of windows for virtual circuits has been studied for high speed, wide area data networks in an asymptotic framework in which the delay bandwidth product is the large parameter . Based on the results of this analysis we have previously proposed and evaluated a new class of algorithms for dynamically adapting windows in single path, multi-hop networks . Here we complement our previous work by first developing a parallel theory of algorithms for dynamically adapting windows on networks having multiple paths with different propagation delays and multiple virtual circuits (VCs) on each path . A common feature of these algorithms is that the source of each VC measures the round trip response time of its packets and uses these measurements to adjust its window with the goal of satisfying certain asymptotic identities that have been proven to hold in stationary asymptotically optimal designs . These identities, which hold for all values of cross traffic intensities, serve as "design equations" for the algorithms . A major part of the work reported here is the evaluation of the performance of the new adaptive algorithms in realistic, nonstationary conditions by simulations of networks with data rates of 45 Mbps and propagation delays of up to 47 ms . One of two networks studied has 3 nodes, 2 paths and up to 16 VCs on each path ; the level of cross traffic determines whether a path has 1 or 2 bottlenecks . The simulation results generally confirm that the realizations of the adaptive algorithms give stable, efficient performance and are close to theoretical expectations . Keywords: delay-bandwidth product ; asymptotic analyses ; moderate usage ; design equation ; cross traffic; fair allocation .

1. Introduction Sliding windows for virtual circuits is a wellestablished feedback-based mechanism for exercising congestion control in data networks [4-

analysis in [13] is asymptotic with the delay-bandwidth product as the large parameter . In [16] a class of algorithms for dynamically adapting windows is proposed and extensively evaluated ; these algorithms have in common the feature that the

6,8,17,18,20,21] . The optimal design of windows is also a topic of longstanding ; but only recently attention has been given to the problem in the context of high speed, geographically dispersed networks which are characterized by large delaybandwidth products . In [13] such a study has been done in the context of the network in Fig . 1 . The

source of each virtual circuit (VC) measures the round-trip response time of its packets and uses these measurements to adjust its window to satisfy certain asymptotic identities that are proven to hold in asymptotically optimal designs . These identities, which hold for all cross traffic intensities, are called design equations .

Correspondence to: D . Mitra, AT&T Bell Laboratories Murray Hill, NJ 07974, USA . * Preliminary reports of partial results have been reported at the Workshop on Very High Speed Networks, and INFOCOM '91 .

The main contribution of this paper is to complement our previous work by, first, developing a parallel theory of algorithms for dynamic adaptive windows on networks which have multiple paths with different propagation delays and multiple VCs on each path ; the second contribution

0169-7552/93/$06 .00 © 1993 - Elsevier Science Publishers B .V. All rights reserved



664

D. Mitra, J.B . Seery / Dynamic adaptive windows for high speed data networks

A Propagation

Propagation

Delay

Delay

J

J Fig. 1 . Model of a multi-hop virtual circuit with propagation delays and sliding window control . The route of the virtual circuit is shown by the solid line . M is the number of hops and K is the window size . The dotted lines represent cross traffic from other virtual circuits.

is the extensive performance evaluation of the proposed adaptive algorithms through simulations of networks with data rates of 45 Mbps and propagation delays of up to 47 ms . The network for which the theory for adaptations is developed is shown in Figure 2 . It is a single node network with C VCs with the propagation delay and window for the jth VC denoted by i-1 and K1, respectively; the nodal processor services the VCs as well as cross traffic which has rate v . Finally, we apply the adaptive windowing algorithms developed for the networks in Figs . 1 and 2 to the network shown in Fig . 3 which has 3 nodes and 2 paths, "long" and "short" with propagation delays T L and T s , and T L /T S = 2 . Typically many VCs follow these paths and we let C L and C s denote the respective numbers (C = C L + Cs ) . This network makes an excellent test bed

since complex interactions occur and also, depending on the intensity of the cross traffic, the number of bottleneck nodes on each path may be either 1 or 2. We base our application of the prior results to this network on the viewpoint which regards the entire network as a collection of disjoint subnetworks each of which views the rest of the network as being represented by cross traffic streams. It should be noted that there is a long and extensive history of the use of such a viewpoint in the analysis of stochastic networks of both loss and queueing varieties (see, for instance, [9,23,241) . There has been a significant amount of related work done recently, notably on the DECBIT algorithm [7], the slow-start enhancement to TCP/IP [5,8] and an open-loop, rate based approaches [2,25] . Of particular importance is

Debasis Mitra received the B .Sc and Ph .D degrees in electrical engineering from London University in 1964 and 1967, respectively . He joined Bell Laboratories in 1968 and he is currently Head, Mathematics of Networks and Systems Department. During the Fall semester of 1984 he was Visiting Mckay Professor at the University of California, Berkeley, Dr . Mitra is the receipient of awards given by the Institution of Electrical Engineers, United Kingdom, the Bell System Technical Journal and of the Guillemin-Cauer Prize Paper Award of the IEEE . He is a member of the ACM, SIAM, ORSA, IFIP Working Group 7 .3, and a Fellow of the IEEE .

Judith B. Seery received the B .A. degree in mathematics from the College of St . Elizabeth, Convent Station, NJ, in 1968 and the M .S. degree in mathematics from New York University in 1972 . She joined Bell Laboratories in 1968 and is presently a member of the technical staff in the Mathematical Sciences Research Center . Her current research interests include the modelling and simulation of queueing networks representing communication and computing systems . Ms . Seery is a member of the Mathematics Association of America, the Society for Industrial and Applied Mathematics and the Association for Women in Mathematics .



D . Mitra, T.B . Seery / Dynamic adaptive windows for high speed data networks v

iv

Parameter set

V

1

665

transmission rate : 45 Mbps

J

packet size : 1 Kbytes

=> µ = 264 .

(1 .1)

mean round trip propaga-

K, Propagation Delay, t,

tion delay : 47 ms

The propagation delay is appropriate for the con-

K2 Propagation Delay, T,

tinental USA . For the optical transmission rate of 1 .7 Gbps, . p is about 10,000 . The point is that high transmission rates and non-scalable propa-

0 G Propagation y, c

gation delays give rise to large µ . Large .p is the basis of the asymptotic analysis . In the simula-

Fig . 2 . Multiple virtual circuits with various propagation delays and individual windows over a single hop . The propagation delay and window of the jth virtual circuit are Kj and rj . The cross traffic path is shown by dotted lines .

tions the unit of time is taken to be 47 ms and µ to be 264 . Our approach to adaptive windowing is quasistationary. From an asymptotic analysis of the queueing-network models of Figs . 1 and 2 we prove that for optimal steady state performance

Zhang's work [25] in which it has been shown that

the optimal static window length for VC j, Kj*, is

some adaptive windowing algorithms lead to (i)

related to R7, the mean round trip response time

large-amplitude oscillations in the queues and (ii)

for the VC, through the identities called design

unfair treatment of the virtual circuits with longer

equations (see (2 .10), (3 .11) and (3 .22)) . In (3 .22),

paths . As the simulation results show, our algo(R*/-rj -1)/K*'-bj, (1
rithms perform well in both these respects .

(1 .2)

Consider the following parameters which are

where b1 is an explicitly identified number ob-

used in the simulations . Let µ denote a represen-

tained from the optimization . The design equa-

tative service rate at the various nodes ; it has dimensions of packets/time . Let the unit of time

tion is universal in that it is oblivious of the cross

be chosen to be a representative mean round trip

rates . This obliviousness allows the window to be

propagation delay .

adapted on the basis of measurements of re-

traffic intensities as well as the nodal processing

V2 n DELAY tL/2

NODE 2 III

DELAY TL/2

LONG PATH NODE I SOURCE

DELAY is/2

DELAY ts/2 NODE 3

Fig . 3 . 3-node network with 2 paths having different propagation delays and multiple virtual circuits on each path . In the simulations 7L /-rs = 2 and also Al = N'2 = µ3 = p, . Cross traffic rates are v,, vZ, v3 .



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D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks

sponse times regardless of the (unknown) cross traffic intensities and nodal processing rates . The window adaptation is derived from the following iterative algorithm for finding the root of the equation in (1 .2): Ki,n + 1 = Kin (1

cj

-


aj (Rjn/Tj [

-

1) yKi n

- bb ]

(1 .3)

where n indexes the packets acknowledged at the source . The gain parameters ai affect responsiveness to changing exogenous conditions as well as long run variability of the window around its optimal value . The simulations reported in Section 4 study the start-up of windows from small initial values, the responsiveness of the algorithms to large, abrupt changes in the cross traffic intensities, the allocation (fair and arbitrary) of bandwidths to VCs, uniformity of behavior among VCs on the same path and with the same desired bandwidth allocation, and the effect of the gain parameters {a j } . The simulations have shown that the windows adapt rapidly to changing cross traffic rates . This is because the adaptation is done on a packet by packet basis, so when a source is transmitting quickly, it also adapts quickly . The simulations generally confirm that the realizations of the adaptive algorithms give stable, efficient performance and are close to theoretical expectations when these exist . We briefly review some of the theoretical background to the steady-state analysis of the networks in Figs. 1 and 2 for the case where the windows are fixed, i .e., static . This analysis culminates in the design equations . With static windows a closed queueing network model arises due to the constancy of the number of packets of the virtual circuit in the network which in turn is due to the assumption of "infinite data sources", i .e., the sources always have data to send [21,25] . It is known (see, for instance, [25]) that this assumption stresses the adaptive windowing algorithms . For the theoretical development the network in Figure 1 is assumed to be balanced, i.e ., A=µt-vl=µ2-v2= . . . =Am -vy, ( 1 .4 ) where, at node i, /.t i is the service rate and v i is the exogenous,cross traffic rate . For the network in Fig . 2, A = µ - v. Since the unit of time has been selected to be representative of the round

trip propagation delay, A is large for wide area, high speed networks. More precisely, for the asymptotic analysis of the networks in Figs . 1 and 2 it is assumed that A is large and Tj = O(1) (1 < j < C). Now let F be defined by the relation c K' 1 E -F (A-*co) . (1 .5) j=1 TI

Three asymptotic regimes exist : T < 1 : light usage, r = 1 : moderate usage, F > 1 : heavy usage .

(1 .6)

It has been shown [13,15] that based on considerations of throughput (T), mean round trip response time (R) and queue variability, there is an overwhelming case for operating in the moderate usage regime . Hence, the high level goal of adaptive windowing algorithms is quite simply stated : operate in the moderate usage regime . This goal is entirely plausible in the following fluid interpretation [13] of the moderate usage regime : the packets in the windows just fill the pipelines from source to destination and back.

2. Basic results on virtual circuits with identical propagation delay It will prove useful later if the main results in [13] are briefly reviewed now . It is assumed that there is a single VC and in this case it will be convenient to also assume that the propagation delay is unity, i.e ., C = 1 and r, = 1 . These results are from an asymptotic theory of the steady state behavior of the network in Fig. 1 in which the fixed, i .e . static, window conforms to the moderate usage regime . For detailed analysis in the moderate usage regime it is necessary to augment (1 .5) and (1 .6) thus, K=A -aA 1 / 2 , (2 .1) where the nodal usage parameter a does not scale with A, i .e. a = O(1); a may be either positive or negative . Also, the network is assumed to be product-form [12,20,21] . This restricts the form of service discipline at the nodes ; however, two allowed disciplines are processor-sharing (PS) and FCFS. The former is of particular interest since it



D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks

667

approximates round robin . If the service discipline is PS, then the packet service time distribu-

When the objective of design optimization is the maximization of power [3,10], P =° T/R, it has

tions are allowed to be arbitrary and if it is FCFS then the distribution is necessarily exponential . Cross traffic streams are required to be Poisson .

been established that this objective is satisfied in

Proposition 2 .1. For the network in Fig . 1 in steady state, with A large and (2 .1) holding, Throughput,

actly computed and also observed to the closely approximated thus [13] :

T=A 1- A 12 (a+M/3)+o( A ),,

the moderate usage regime, the asymptotically optimal solution has been obtained and the associated constants a *, /3 * and y * have been ex-

0 .5

1

VM

7W '

(2 .2)

L

MP

+0( 1

(2 .3)

Mean nodal queue, (N1 ) =(3A'i2 + 0(1),

(2 .4)

Standard deviation of nodal queue, u(N,) = yA'/ 2 + O(1),

(2 .5)

where,

VM+l

Substituting (2 .8) in the expressions for K, T, R, (N,) and a(N1 ) in (2.1)-(2.5), give their values for the optimal design which are specified by the * symbol . While power as the performance functional is convenient, it is important to note that in the asymptotic framework any other measure which weights T and R reasonably will give the same qualitative results [13,15]. Now in (2 .3) we may substitute A' /2 by K 1 /2 and obtain (R-1)/k=M/3+0(A-'/2) .

/3(a)

( 2 .6)

M WM (( )) ,

Design equation

cylinder functions which are closely related to the normal (Gaussian) distribution and are readily computed by recursions:

(R*-1)yK* -'b,

= f0

e - v 2 / 2-a "v'dv

(m=0, 1, 2, . . .) .

Finally, y 2 (a) = 2(1 -af3)/(M+ 1) _'82 .

(2 .7)

Recall that for this section the unit of time has been selected to be the round-trip propagation delay and this accounts for the first term on the right-hand side of (2 .3). The implications of the above results are important . Contrary to conventional analysis based on the M/M/1 model, the number of queued packets in the optimal design is /3A1/2 + 0(1), not 0(1). The utilizations are much higher than are conventionally recommended . Since FA is much smaller than A, to leading order the windows are simply A (see (2 .1)). The overwhelming majority of packets in the window are not in the nodal buffers but in the process of being propagated .

(2 .9)

Furthermore, if we make use of the optimum value of /3, i .e . /3 *, then from (2.8) and (2 .9),

{W„r(a) I m = 0, 1, . . . } are the classical parabolic

W,,,(a)

y

(2 .8)

Mean round-trip response time, R=1+

1

(2 .10)

where b = M/3 * _ ~_M _. The practical importance of this equation stems from the fact that it shows how K, the window length, should be controlled from a knowledge of R which can be estimated from observations . Note that {R(K) - 1} 1K is an increasing function of K and, hence, if its value is greater (less) than b then K should be reduced (increased) . Notice too that (2.10) holds for all A, the reduced nodal processing rate, which gives it an aspect of universality . The reader will find in [16] generalizations of (2 .10) to the case of multiple VCs with identical propagation delays .

3 . Multiple virtual circuits with various propagation delays : theory In this section we first lay the theoretical groundwork leading up to the design equation for



668

D. Mitra, J.B . Seery / Dynamic adaptive windows for high speed data networks

the single node network shown in Fig . 2. Then the window adaptation algorithm is derived . The theory given here separates into two parts, the asymptotic behavior of individual virtual circuits with fixed, static windows, and design optimization . In the first part the desired fractional band-

and 13(a) is as defined in (2.6), Proposition 2.1, with M = 1 .

width allocations for the individual virtual circuits are assumed to be given, while in the second part they are obtained from the optimization . The

Little's Law states that the mean round-trip response time of the jth virtual circuit, R j =Kj /T, we obtain as a corollary to (3 .1a) and (3 .2),

asymptotic theory is largely extracted from [12]. The design optimization is carried out in terms of "network power" which is the product of the power of individual virtual circuits [1,11] . In the presence of unequal propagation delays another important consideration is fair bandwidth allocation [25]. It is shown below that the fair allocation is also the allocation which maximizes network power . Also, this allocation has a simple, intuitively appealing design equation .

The reader is invited to verify that Proposition 2 .1 when specialized to M = 1 coincides with the above when it is specialized to C = 1 . Also, since

0(a)

Rj=Tj+

+O ( A12

).

(3 .3)

A

Note also the corollary to (3 .1b) and (3 .2) which states that the total throughput over all virtual circuits, c

T

T

=A-A1/'2{a+)13(a)}A+O(1) .

j=1 (3 .4)

3.1. Asymptotics, bandwidth allocations The moderate usage scaling which conforms to (1 .5) and (1 .6), and which generalizes (2.1), is

Kj =k jA-lj 1

(j=1,2, . . ., C),

(3 .1a)

where ( 3 .1b )

to be 0(1) constants, i.e ., they do not scale with the large parameter A which, as before, is the reduced nodal processing rate, µ - v. The result summarized below is essentially contained in [12, Section VI] . Proposition 3.1. In the moderate usage regime (3 .1), for large A, the steady state throughput of virtual circuit j,

kj

1

Tj

A1/2

kj

Tj A

( C),

c

L

(3 .5)

it is required that for large A and all pairs i and j,

T

T

- 9, +0( A'1/2 ) Bj

A particular allocation of special interest, which may be called "fair," is one in which asymptotically all virtual circuits get equal share of the bandwidth, i .e ., 1

Bj =

C(1

kj/Tj ,

Cj < C) : fair allocation .

(3 .7)

It follows from (3 .2) that to achieve the bandwidth allocation requirement of (3 .6), it is necessary and sufficient that

k j =Oi ri ,

j=1

B =1,

lj + ~6(a)

where

A2

9j > 0 (j=1,2 1 . . . ,C) j=1

Recall from Fig . 2 that Tj is the propagation delay of the jth virtual circuit ; these are assumed

A

leading order, the fraction of the total throughput which is to be allocated to virtual circuit j . That is, where C

c k. - _ 1. j=1 Tj

(1 c j

Let us digress briefly to consider the desired allocation of the bandwidth to the virtual circuits . Let B 1 , 02 , . . . . Bc denote a set of fractional bandwidth allocation parameters in which Bj is, to

a

0 1

= A

C j=1

(1 cj
(3 .8)

in which case,

T =B1 A+0(A'

2) .

(3 .9)



D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks

On substituting (3 .8) in (3 .3) and making use of the implication of (3 .1), /K1 = /kj A [1 + O(A -1/2)], we get

Bj/Tj (R1/T - 1)\/K1 _ J 1 1

1 /3(a) + O A 1/2

el/TI

669

Equation (3.12) follows from (3 .9) and (3 .3), while (3.13) follows from (3 .12) and an application of Little's Law . Note that while the number of queued packets of each virtual circuit is O(A'/2 ), the remaining packets of the window are in propagation and are O(A) in number .

(3 .10)

(j=1,2, . . .,C) .

To recapitulate, the above asymptotic identities are obtained as a consequence of the moderate usage regime, see (3.1), and the fractional bandwidth allocation in (3.6) . The converse may also be shown to be true, that is, for given fixed /3(a) and {0j}, if the identities in (3 .10) hold then the moderate usage regime is operative and the fractional bandwidth allocation is as specified in (3 .6). It is the latter viewpoint that we adopt for designing the adaptation algorithm .

3.2. Maximization of network power From (3 .2) and (3 .3) we obtain for the power of the jth virtual circuit,

T Akj 1 P=-=- 1Rj J j2 A'/ 2 T

1j kj

2/3(a) +

jA

T

( 1 +01 A

.

(3 .14)

) .

(3 .15)

Hence, network power Design equation C

(Rj

/T

j -1)~Kj -bJ

(1,j,C)

(3 .11)

P°

H Pi j=1

where

=

o1/Ti E 8'/Ti /3(a)

_ J

b

Akj II 2 J = 1 Tj 1

As in (2 .10), the monotonicity of the function [(Rj j - 1) /Kj - bj ] with respect to Kj , and knowledge of Rj gained from measurements may be used to control the window length Kj by increasing or decreasing it depending upon whether the value of the function is negative or positive, respectively . Note that in (3 .11), E b2 =p2 (a) and also that the constant a, and hence /3(a), is arbitrary ex cept for the requirement stemming from moderate usage that it be O(1). In the sequel a rationale for selecting a will be given . It will suffice here to observe that the selection contained in (2.8), which in the present context of M = 1 gives /3(a) = 1, performs consistently well . The consequences of the design in (3 .11) are to give the following statistics for the jth virtual circuit (1 < j S C) :

lj

x l- A1/2

T = O.A + O(A'/ 2 ) ; E b2

1 Rj

=T,

+ A

i/2

(

E Bi/Tt )

1/2

( 1) +O A

(3 .12)

T

(3 .13)

1

+ A

+0(

1 j

a1

T

1

Write the above expression in the following convenient form P=P (0) 1-

A'

1 (1 /2 P( ' ) +0( A

(3 .16)

where the definitions of P (J°) and P (') follow naturally . Observe that the leading term in the expansion of network power, P (0), depends on the choice of the windows only via {kj}. The first problem we formulate is the maximization of P {0} with respect to {k,} subject to the constraint in (3 .1b) which is a condition of moderate usage . This problem has a simple solution . Proposition 3 .2 . Ak i k J subject to E -J = 1 [I j=1 j Tj C

(Njl ) =A'/2 (b JBj j ) + O(1) .

E k J

/T

2/3(a)

max {k J }

P(01=



670

D . Mitra, J. B . Seery / Dynamic adaptive windows for high speed data networks

has the solution ki - 1 Ti C

stant associated with the optimal design of a single virtual circuit over a single hop . O

(1
(3 .17)

Proof. The result is a simple corollary to the fact that the product of any pair of positive, unequal numbers is increased if the numbers are made equal while their sum is left unchanged. El

Note that in the asymptotically optimal design the 0(1) coefficients {1i} may be chosen arbitrarily provided only that their weighted linear combination satisfies the following constraint,

Eli 7i/

1/CL1/Ti =a* .

i The implication of the above proposition is in the following Corollary. P ( O ), the highest order term in the asymptotic expansion of the network power P, is maximized when the allocation is fair, i.e ., the fractional bandwidth allocation parameters are all equal, Bi = 1/C (1
Let us proceed to the minimization of the next term in the expansion of P, P ( ' ), given that P (O) has been maximized . Introducing (3 .17) in P (' ) given by (3.16), we obtain PM =VC E

1/Ti

[ a+2f3(a)] .

(3 .19)

a

Not surprisingly, this problem has been encountered before when the power of a single virtual circuit was under consideration and its solution is given in (2 .8). Therefore, the solution to (3 .19) is the constant a * (for M = 1) whose close approximation is, a*=-1/2and/3*=/3(a*)=1 .

i

We substitute in (3 .10) the parameters ( O) and /3(a) as given by Propositions 3 .2 and 3 .3 for the design which asymptotically maximizes network power, and obtain, Design equation for network power maximization

(R717,-1)1/Ki* -bi (1
bi

1/T

' 1/7i

L

' 1/Ti

(3 .22)

(3 .18)

This expression is remarkable in that all the second order terms {li ) in the definition of the windows {Ki } in (3 .1) are reflected only via the coefficient a . That is, the fair allocation has reduced the C-dimensional problem of minimizing P (' ) with respect to {l) to the following one-dimensional problem: min [a+213(a)] .

(3 .21)

(3 .20)

To recapitulate we have,

This design equation plays the same central role as does its counterpart in (2 .10) in the case of a single virtual circuit and (3.11) for multiple virtual circuits . It differs from (3 .11) only in that specific selections have been made for the parameters Bi(1
Proposition 3 .3. Recall the asymptotic expansion of the network power in (3 .16). Assuming that {k i} have been chosen to conform to the fair allocation which maximizes P (O) in the moderate usage regime, the term P (' ) is minimized with respect to the remaining parameters (l i ) when a= a*, the con-

T= E T.=Ai=1

,Ai/2

2 V

E

1/Ti

C

+0(1) . (3 .23)

The following general features of the design equations in (2 .10) and (3 .22) are noteworthy . With all other quantities held fixed, (i) the parameters bi are smaller for VCs with larger prop-



D. Mitra, J.B . Seery / Dynamic adaptive windows for high speed data networks

671

agation delays, and (ii) bi is larger for VCs with

options immediately after receipt of the nth ac-

more bottlenecks in their path . In both cases the

knowledged packet . In our implementation,

square root governs the dependence . These guidelines are used to select the design parameters for the network in Fig . 3 which has multiple

Ki,n+1=K1 -1 if Kin+I Kin-1 = Ki .„

propagation delays and may also have multiple bottlenecks in the paths of individual VCs .

=Kin+1

if Ki „ - 1 < Ki n + 1 < Kin + 1 if

As noted in the Introduction, the particular attraction of these asymptotic identities in the

Kin+I
where {Kin} is the solution of the idealized algo-

control of the windows is the economy of vari-

rithm which runs concurrently and is calculated

ables involved ; for instance, the identities apply for various cross traffic rates v, yet v is not

from (3 .24) . The implementation in (3 .25) has

explicitly represented . However, an implicit rep-

The role of the gain parameter will not be

resentation exists since the choice of the window

analytically analyzed here since this has been

is affected via the effect of v on the mean

done in [16] . Results there quantify the trade-off

round-trip response times .

between (i) the increased speed with which the

been designed to track the "ideal" (Kin) .

window expands and contracts in response to changing conditions, such as cross traffic intensi3 .3 . Adaptation algorithms

ties, with increased a1, and (ii) the accompanying increased fluctuations or variability, in the long

Suppose the appropriate design equation has been selected ; now consider its realization . The

run, of the adapting window size and also, but to a lesser extent, in the nodal queues .

goal of the adaptation algorithm is to find and track the root of the design equation . This suggests the following algorithm for the window of

4 . Simulations

the jth virtual circuit, The simulation results are classified into 8 Groups . Groups I-IV are for the 1-node network

Ki,n+1 = Kj, - aiBi,n

of Fig . 2 while Groups V-VIII are for the 3-node

where

(1
Bi,n = (R1,n/Ti - 1)~Kl,n -bi

(n=0, 1, . . .)

network of Fig . 3 . (Simulation results for the multiple hop, single propagation delay network of

(3 .24)

Fig . 1 are reported in [16] .) We use the notion of "paths" for both networks : the long path has TL = 1 and the short path rs = 1/2 . Each path

Here Ri n is the measured round-trip response may have multiple VCs and the numbers are time of the nth packet of the jth virtual circuit which is acknowledged at the source and Kin+1

denoted by CL and Cs ; hence the total number of VCs is C = CL + Cs . The values of (CL, Cs)

is calculated after receiving the acknowledgeconsidered in the simulations are (1, 1) and ment ; also, ai is the gain parameter . (8, 16) . Generally we let The algorithm as described in (3 .24) is not cognizant of the fact that the window length is an integer and that the control options are limited . In our simulations the control options are three : K1,n+ 1 - Ki n E { - 1, 0, 1} . That is, the respective

Ti=TL, ai=aL, bi=bt : =TS,

=as,

(1sj(CL) (CL + 1 < j c C) (4 .1)

The Groups are as follows .

options are not to replace the acknowledged

I . The 1-node network with fair allocation :

packet by a new packet from the source, to re-

Table 1 and Fig . 4 ; theoretical predictions : Table 2 .

place it and, finally, to replace it by two packets transmitted in succession . Let Kin+1 denote the implemented window size (as measured by the number of unacknowledged packets in the virtual circuit) subsequent to the exercise of the control

II . Arbitrary bandwidth allocation on the 1node network : Table 3 and Fig . 5 ; theoretical predictions : Table 4 . III . Effects of the gain parameters : Fig . 6 .



672

D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks

Table 1 Simulation results for 1-node network with fair allocation as design objective . Gain parameters : (a L , a s ) = (1/4, 1/16) for Cases 1 and 2; (a L , a s )=(1/2, 1/8) for Cases 3 and 4 (a) Case

1 2 3 4

Network&design parameters CL, Cs

bL, bs

1, 1 1, 1 8,16 8, 16

0.5773, 0.5773, 0.1581, 0.1581,

Mean queue per VC, (N l )

0.8165 0.8165 0.2236 0.2236

v

long path

short path

0 132 0 132

6 .3 4.3 0.50 0.34

6.3 4 .3 0.49 0.33

Total VC throughput T

Node utilization

251.1 122.1 252.0 122.4

0.94 0.96 0.95 0.96

(b) Case

1 2 3 4

Long path

Short path

Throughput per VC

Mean queueing delay

Mean window, (KJ )

Std . dev. (K)

Throughput per VC

Mean queueing delay

Mean window, (KJ )

Std. dev. (K)

122.8 59 .6 10.5 5.1

0.051 0.072 0.047 0.067

129.3 64.6 11 .0 5 .5

9.5 7.6 0.9 0.7

128.3 62.5 10 .5 5 .1

0.055 0.069 0.046 0.065

70.5 35 .7 5 .8 2.9

3.4 2.8 0.4 0.3

IV . Effects of cross traffic on the 1-node network . Various constant rate cross traffic : Fig. 7 ; responsiveness in the presence of large, abrupt changes in cross traffic rates : Fig . 8; fairness and uniformity among individual VCs : Fig . 9 . V . Fair and arbitrary allocations on the 3-node network. Single VC in each path : Table 5 . VI . Multiple VCs on each path : Table 6 . VII . Effects of various constant intensities of cross traffic streams in 3-node network . Single VC on each path : Table 7 . VII . Same as VII except for multiple VCs on each path: Table 8 . The simulations were performed on PANACEA [19,22], a software package for the performance analysis of queueing networks . The sources are modelled as infinite data sources and the nodal buffers are not limited . (Related work involving small nodal buffers is reported in [14] .) As mentioned in connection with (1 .1), the unit of time is 47 ms and the nodal service rates µ, are 264 . The packet service time distributions are taken to be exponential and the propagation delays are constants . For the results in Group I the parameters (b L ,b s ) for the adaptation algorithm are computed from (3 .22) and depend only on T L , -r s and

C . Notice from Table 1 that these parameters are not changed as the cross traffic rate v is substantially changed . Notice that the design goal of equal throughputs is effectively achieved in all cases. We check the agreement between selected simulation results in Table 1 and theoretical predictions . We remind the reader that the latter were derived for the static, asymptotically optimal designs . Table 2 gives values for the total throughput T from (3 .23), mean queueing delay, Ri - 7-i from (3 .12) and the mean queue (NJ,) from (3 .13). Note that in the fair allocation case these statistics do not depend on the path taken by the VC. The reader will observe that the agreement with the corresponding results in Table 1 is good . Table 2 Theoretical predictions for selected results in Table 1 Case

Total VC throughput, T Mean queueing delay, R i - T Mean queue per VC, (N l )

1

2

3

4

254 .1

125.0

253.5

124 .6

0 .050

0.071

0.048

0.067

6.6

4.7

0.52

0.37



D. Mitra, J .B . Seery / Dynamic adaptive windows for high speed data networks

673

Table 3 Simulation results for 1-node network with design objective of T1 / Ts = 1.5 and 0 .67 in Cases I and 2, respectively . Gain parameters: (a L , a s )=(1/2, 1/8) (a) Case

Mean queue per VC, (NjI )

Network & design parameters

1 2

C L , Cs

b L , bs

v

long path

short path

8, 16 8, 16

0.1846, 0 .2132 0.1336, 0 .2315

0 0

0.65 0.36

0 .43 0.53

Total VC throughput

Node utilization

252.0 252.0

0.96 0.95

(b) Case

Long path

1 2

Short path

Throughput per VC

Mean queueing delay

Mean window, (K J )

Std, dev. (K1 )

Throughput per VC

Mean queueing delay

Mean window, (K)

Std. dev. (K1 )

13.3 7.9

0.049 0.046

14.0 8.3

1.0 0.5

9.1 11 .8

0.048 0 .045

5 .0 6 .4

0.0 0.5

Figure 4 shows further results for Case 3 of Table 1 . The long run throughputs and mean windows of each of C (= 24) VCs are exhibited .

T-

15 r

= E Da

10

O Tc cr

5

r ~ p,

0 8 9 16 VIRTUAL CIRCUITS

I 24

Proceeding to Group II we consider two cases in which the design objective is not the fair allocation . Table 3 presents simulation results for two cases in which the design objectives are TL /Ts = 1 .5 and 0 .67, respectively, where T L and Ts are the throughputs of the individual VCs on

(a) 15

00

Z N

10

a

the long and short paths, respectively . The parameters of the design equation b L and b s are computed from (3 .11) in which we have set /3(a) = 1. Observe that the realized values of TL /Ts are 1 .46 and 0 .67 . Table 4 gives theoretical predictions computed from (3 .12) and (3 .13) and comparison with Table 4 .3 shows that the agree-

W

-



I

I I L_ '

I-

8 9

1s -

I--LONG PATH--I F-

SHORT PATH

VIRTUAL CIRCUITS (b)

150

This data shows that the behavior among VCs is uniform. Figure 4c exhibits the time-dependent behavior of the cumulative windows over the two paths . The absence of large amplitude oscillatory behavior should be noted . Notice that the frequency spectrums of the random fluctuations in the windows depend on the length of the paththe longer path has higher frequency components .

SHORT PATH

U)

0 0 Z

ment is rather good .

3 J 200

400

800

TIME (C)

Fig. 4. 1-node network in Fig. 2 in which VCs (1-8) in long path have r L = 1, b L = 0.1581, a L = 1/2, and VCs (9-24) in short path have r s = 1/2, b s = 0.2236, a s = 1/8. Design goal is fair allocation. (This is Case 3 of Table 1 .) The throughput and mean window of individual VCs are shown in (a) and (b). The total window for each path is shown in (c) .

Table 4 Theoretical predictions for selected results in Table 3 Case

1 2

Mean queue per VC, (NjI ) long path

short path

Mean queueing delay, R1 - rj

0.69 0.38

0.46 0.58

0 .049 0 .047



674

D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks 150

150

SHORT PATH a=1/4 N a) U N

a CD

8 9

16

24

VIRTUAL CIRCUITS

0O z

(a)

3

Z LONG PATH a1=1/2

15 0

o 3 Z W

(a)

0

a

150

2

200

N 8 9

16 SHORT PATH I-- LONG PATH---I 1VIRTUAL CIRCUITS 1

U)

24

U

m a CD 3 0

(b)

0 z

150 Co

8 Z 3

(b)

jl

r

WR1f'' RIr9TU1~'llfll 4. A

sad/ As

LONG PATH

0 200

SHORT PATH 400 TIME

r IN

I

200

3

TIME (c) 600

800

(C)

Fig. 5 . 1-node network in Fig . 2 in which VCs (1-8) in long path have T L = 1 .0, b L = 0.1846, aL = 1/2, and VCs (9-24) in short path have T s = 1/2, b s = 0.2132, a s = 1/8 . Design goal : TL =1 .5 x Ts. (This is Case 1 of Table 3 .) The long term average throughput and window of individual VCs are shown in (a) and (b). The total window for each path is shown in (c).

Figure 5 from this Group exhibits further results for Case 1 of Table 3 . The qualitative conclusions drawn from this figure are the same as from Fig . 4. Figure 6 from Group III is presented to show that the short-term transient behavior is substantially affected by the choice of the gain parame-

0

200 TIME (d)

Fig. 6. Effects of gain parameters in 1-node network of Fig . 2. The network and algorithm parameters are otherwise the same as in Fig . 4.

ters (a) . The parameters of the four vignettes in this figure differ only in the gain parameters and the other parameters are from Case 3 of Table 1 . It will be observed that only for the selection of (a L , a s ) = (1/2, 1/8) do the time responses neither overshoot nor undershoot and the performance obtained from this selection is clearly the best . For a L /a s < 4 the cumulative window on the short path overshoots, while for a L/a s > 4 the cumulative window on the long path overshoots . The fact that (a L , a s) = (1/2, 1/8) per-

Fig. 7. 1-node netwrok in Fig . 2 in which VCs (1-8) on long path have T L = 1, bL = 0.1581, aL = 1/2, and VCs (9-24) in short path have T s = 1/2, b 5 = 0.2236, a s = 1/8 . The top pair of cumulative windows for each path is for no cross traffic, and the bottom pair is for cross traffic with rate v = 132 .



D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks

675

forms well is not surprising in the light of results ~P=132

in [161. Figure 7, the first from Group IV, exhibits time-dependent behavior for Cases 3 and 4 from Table 1 . The cross traffic is simulated as Poisson streams. The numerical data shows that the standard deviation of an individual VC's window changes slightly between v = 0 and v = 132, while

WINDOW -

--SHORT PATH

LONG PATH

(a) 150 v=0 100 w

in contrast the frequency spectrum changes substantially. Figure 8 is for the same system as in Fig. 7 except that here the cross traffic rate is switched at time = 350. Notice the asymmetric behavior of the window in (a) and (b), i.e., the window contracts more quickly than it expands . The underlying theoretical reasons for this fortuitous behavior is given in [161 . The queues shown in this figure are for the VCs only and exclude queued packets from the cross traffic . Notice in (a) the presence of a spike in the queue just after the instant that the cross traffic is switched on . It is followed by a precipitous contraction of the window in about 1 unit of time (i .e. round-trip delay) . Figure 9 shows the behavior of 4 individual VCs for the same network and scenario as Fig . 8. The main purpose of exhibiting this figure is to show that the windows operate in a narrow band and also that there is no significant disparity between VCs .

Y

a a

50

00

200

400

600

800

TIME (b)

Fig . 8. Non-stationary cross traffic on 1-node network in Fig . 2 in which VCs (1-8) on long path have z L = 1, b L = 0.1581, aL = 1/2, and VCs (9-24) on short path have T s = 1/2, b s = 0.2236, a s = 1/8 . Cross traffic is switched on and off between v = 0 and v = 132 in (a) and (b) at time 350 . The cumulative window over each path is shown . The queue shown is that of the VCs only .

Table 5 from Group V gives simulation results for the 3-node network in Fig . 3 . In all the cases considered in this table there is no cross traffic and there is a single VC on each of the 2 paths . The 3 cases considered differ only in having different objectives for the value of TL /Ts . The design parameters b L and b s are obtained from (3 .11) by substituting the appropriate bandwidth

Table 5 Simulation results for 3-node network with no cross traffic and (C L , CS) =(1, 1). Design objectives : TL /T5 = 1 .0, 1 .5 and 0.67 in Cases 1, 2 and 3, respectively . Gain parameters : (aL , a s)=(1/4, 1/16) (a) Case

1 2 3

Design parameters bL, bs

0.5773, 0.8165 0.6546, 0.7559 0.5, 0.8660

Node 1

Node 2

Mean queue per VC, (Nil ) long path

short path

5 .4 6.4 4.3

5 .3 4.6 5 .8

Node 3

Utilization

Mean queue per VC, (NJ )

Utilization

Mean queue per VC, (Nj3)

Utilization

0.93 0.93 0.93

0 .8 1 .1 0 .6

0.46 0.53 0.39

0.9 0.6 1 .2

0.47 0.40 0.54

(b) Case

1 2 3

Long path

Short path

Throughput

Mean queueing delay

Mean window, (K;)

122.0 141 .6 101 .9

0.051 0 .054 0.048

128.5 149 .7 107.0

Std . dev. (Kj ) 9 .0 11.8 6 .1

Throughput

Mean queueing delay

Mean window . (K j )

Std . dev . (K1)

124.0 105 .0 143 .9

0.049 0.050 0.049

68.2 57.8 79.0

3 .0 2.5 3 .2



676

D. Mitra, J.B. Seery / Dynamic adaptive windows for high speed data networks

Table 6 Simulation results for 3-node network with no cross traffic and multiple VCs on each path : (C L , Cs ) =(8, 16) . Design objectives : TL /Ts =1 .0, 1 .5 and 0 .67 in Cases 1, 2 and 3, respectively . Gain parameters : (a L , as )=(1/2, 1/8) (a) Case

1 2 3

Design Parameters b L, b s

Node 1

Node 2

Mean queue per VC, (NJi )

0.1581, 0.2236 0.1846, 0.2132 0.1336, 0.2315

Utilization

long path

short path

0.45 0.57 0.34

0.38 0.35 0.40

Mean queue per VC,

Node 3 Utilization

(Niz ) 0.93 0.93 0.93

0.06 0.09 0.05

Mean queue per VC,

Utilization

(Ni3) 0.34 0.42 0.28

0.09 0.07 0.11

0.59 0.52 0.65

(b) Case

1 2 3

Long path

Short path

Throughput per VC

Mean queueing delay

Mean window per VC

Std. devf of window

Throughput per VC

Mean queueing delay

Mean window per VC

Std . devf of window

11 .2 13 .7 9.1

0.046 0.048 0.043

11 .8 14 .4 9 .6

0.9 1 .2 0.7

9.8 8.6 10.8

0.048 0.049 0.047

5.4 4.8 5 .9

0.5 0.4 0.3

Table 7 Simulation results for 3-node network with various cross traffic and fair allocation as design objective . Single VC on each path, i .e . (CL , CS) =(1, 1) . Gain parameters: (a L, a s )=(1/4, 1/16), except for Case 5 where (a L , a s )=(1/16, 1/16) . In (a), numbers are circled to draw attention to the existence of bottlenecks ; the boxed numbers indicate marginal bottlenecks . (a) Case

Parameters

Node 1

Design b L , bs

Cross traffic v 1 , v 2 , v3

1

0 .5773, 0 .8165

66, 0, 0

2

0.75, 0.92

3

Node 2

Total VC queue

Node 3

Utilization

Total VC queue

9 .5

0.94

0.51

0.34

0.54

0.35

0, 66, 0

12 .4

0.94

1 .7

0.72

0.86

0.47

0.7, 1 .05

0, 0, 66

12.8

0.94

0.92

0.48

1 .6

0.71

4

0.5773, 0 .8165

132, 0, 0

7.9

0.96

0.29

0.22

0.30

0 .23

5

0.96, 0 .57

0, 132,0

7.0

0.90

6.5

0.80

0 .45

6

0.39, 1 .24

0, 0, 132

6.6

0.88

0.80

Utilization

0.44

Total VC queue

Utilization

5.5

(b) Case

1 2 3 4 5 6

Long path

Short path

Throughput

Mean queueing delay

Mean window

89.8 124.7 126.7 59.4 118.0 116.1

0.059 0.065 0.062 0.072 0.085 0.035

95 .6 133 .3 134 .8 64 .4 128 .3 120 .7

Std . dev . of window 8.3 11 .0 11 .4 7.8 6.8 8 .7

Throughput

Mean queueing delay

Mean window

Std. dev . of window

92.7 123.0 121 .2 61 .8 119.6 116.8

0.057 0.056 0.063 0.068 0.036 0.076

51 .7 68 .4 68 .4 35 .3 64.2 67.6

3 .0 3 .6 3 .6 2 .8 3 .6 5.4



D. Mitra, J$ . Seery / Dynamic adaptive windows for high speed data networks

a YU

20

the design parameters b L and b s is unchanged .

15

We find from Table 6 that the realizations in the 3 cases give T L/Ts = 1 .14, 1 .59, 0 .84 . A clue to

a 3 0 0

Z

677

10

.

5

as

understanding this shift to values higher than in Table 5 is that now the utilization of Node 3 is higher and not completely ignorable . However, importantly, once again no node operates beyond the moderate usage regime .

. .

3 200

400

800

TIME

Fig. 9. Individual windows for VC No . 1, 8, 16, 24 for scenario shown in Fig . 8a .

Table 7 from Group VII gives simulation results for the 3-node network for various levels of cross traffic and with a single VC on each path . The cross traffic has the effect of creating various combinations of bottlenecks : in Cases 1-4 only

allocation parameters {O }, and 8(a) = 1 . The rationale for following this procedure is that nodes 2 and 3 of the network, see Fig . 3, are lightly loaded and consequently the network may be reduced to the 1-node network of Fig . 2 . In Table 5, the throughputs that are realized in the simulations are such that T L /Ts = 0.98, 1 .34, 0.71 which should be compared to the objectives of 1 .0, 1 .5 and 0 .67, respectively . Importantly, the usage is

Node 1 is definitely a bottleneck (although in Cases 2 and 3, Nodes 2 and 3 have non-negligible utilizations), in Case 5 Nodes 1 and 2 are bottlenecks, and in Case 3 Nodes 1 and 3 are bottlenecks . The design objective are fair allocation while maintaining moderate usage in the bottlenecks . The parameters in the adaptation algo-

consistently in the moderate regime as indicated by the utilization and queue length . Table 6 from Group VI differs from the framework of Table 5 only in that there are multiple VCs on each path . The procedure for selecting

rithm, b L and b s , were selected by experimenting, except for Cases 1 and 4 in which case there is a clear reduction to the 1-node network and in these cases the parameters were obtained from (3.11). The realizations give T L / Ts = 0 .97, 1 .04,

Table 8 Simulation results for 3-node network with various cross traffic and fair allocation as design objective . Multiple VCs on each path : (C L , CS) =(8, 16) . Gain parameters : (a L , a s ) = (1/2, 1/8) . (a) Case

1 2 3 4 5 6

Parameters

Node 1

Node 2

Node 3

Design bL, bs

Cross traffic v2, v3 V ,

Total VC queue

Utilization

Total VC queue

Utilization

Total VC queue

Utilization

0 .1581, 0 .2236 0 .1581, 0 .2236 0 .135, 0 .23 0 .1581, 0 .2236 0 .1581, 0 .2236 0 .1438, 0 .2889

33, 0, 0 0, 33, 0 0, 0, 33 66, 0, 0 0, 66, 0 0, 0, 66

9 .3 9 .5 8 .5 8 .7 9 .4 9 .0

0.94 0.93 0.92 0.94 0.93 0.92

0.40 0.64 0.40 0.32 0.72 0.48

0.29 0.46 0.31 0.24 0.57 0.32

1 .12 1 .44 2.24 0.80 1 .44 3 .84

0.52 0.60 0.73 0.45 0.61 0.86

(b) Case

1 2 3 4 5 6

Long path

Short path

Throughput per VC

Mean queueing delay

Mean window per VC

Std. dev . of window

Throughput per VC

Mean queueing delay

Mean window per VC

Std. dev . of window

9 .5 10.9 10.2 7.9 10 .5 10.5

0.050 0.047 0.041 0.054 0.047 0.043

10 .0 11 .5 10.6 8 .4 11 .0 11 .0

0.8 0 .8 0 .8 0 .8 0 .8 0 .8

8.6 9.9 10.0 7.4 10.1 10.0

0.051 0 .048 0.049 0 .055 0 .047 0.061

4.8 5 .5 5 .5 4.1 5 .6 5.6

0.4 0.5 0.5 0.3 0.5 0.5



678

D. Mitra, J. B. Seery / Dynamic adaptive windows for high speed data networks

1 .05, 0.96, 0.99, 0 .99, which should be compared to 1, the objective of fair allocation . Note that the aforementioned design objectives may be judged to have been achieved . Table 8 from Group VIII gives simulation results for the 3-node network for various levels of cross traffic and multiple VCs on each path . It differs from the conditions of Table 7 in the last respect and also in having a different set of cross traffic intensities . Notice that the parameters of the adaptation algorithm, (b L , bs ) in Cases 1, 2, 4 and 5 coincide with those of Case 1 in Table 6 and are computed from (3 .22). Cases 3 and 6 differ in that, in addition to the common bottleneck in Node 1, Node 3 also has nonnegligible congestion. The realizations give TL/Ts = 1 .10, 1 .10, 1 .02, 1 .07, 1 .04, 1 .05 which, as before, should be compared to 1 . Note that here too no node operates beyond the moderate usage regime . Hence, the design objectives may again be judged to have been achieved .

5. Conclusions This paper begins by considering a single node network with multiple propagation delays . In an extension of prior asymptotic analysis, in which the large parameter is the delay bandwidth product, we obtain asymptotic identities which hold in the important moderate usage regime . These asymptotic identities have the requisite properties to serve as design equations of the algorithms for dynamically adapting windows of VCs . Such adaptations are necessary in the presence of unpredictable changes in intensities of the exogenous cross traffic . The simulation results show that the algorithms perform stably, efficiently and in correspondence to theoretical expectations . The paper has also presented extensive simulation results on the performance of the aforementioned adaptive algorithms in a complex network in which the VCs may have multiple bottlenecks in their paths, as well as various propagation delays . We have based our application of the adaptive algorithm to this network on a particular view of the network as a collection of several disjoint subnetworks, each of which views the rest of the network as being represented by its own cross traffic stream . The simulation results show that this approach is a viable one . However, fur-

ther theoretical and simulation effort needs to be undertaken to refine and validate the approach .

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