Dimensioning of message-switched computer communications networks with end-to-end flow control

Dimensioning of message-switched computer communications networks with end-to-end flow control An algorithm for the end-to-end flow-control window-setting problem is presented by Jackson Chan and Nicolas Georganas. A message-switched computer network with end-to-end window flow control is considered. An algorithm for the selection of the best flow-control window settings is developed using heuristics for the numerical solution of such networks. The criterion of performance is the ratio of the network average throughput to the average network delay. An example network demonstrates the results.

Flow control has been defined 1° as a system of algorithms used in a data network to prevent a single user or a user group from monopolizing the network resources to the detriment of other users. One of the flow-control schemes in use in many computer-communications networks is that of 'end-to-end' flow control. This scheme places a limit on the number of messages, in transit in the network, corresponding to each source-destination pair. This limit is commonly referred to as a 'window'. The dimensioning problem of a network with end-to-end flow control is to select the best window settings. Very few analytical models of computer networks with flow control have been reported in the literature 2, 11-13 and, to the best of our knOwledge, none have dealt with the dimensioning problem. The convolution algorithm 6 has been used for obtaining numerical solutions. This approach, however, has a computational complexity of the order of the product of all network windows, and thus is impractical for large networks. Recently, Reiser proposed certain heuristics s, 7 that reduce, by several order of magnitude, the computational requirements. An algorithm is presented, based on the Reiser heuristics, for finding the best flow-control window settings, namely those windows that maximize the network ,power,4,8. This is defined as the ratio of the average network throughput to the average network transit delay. The algorithm has been implemented in APL and an example network is given to demonstrate the results. Department of Electrical Engineering, University of Ottawa, Ottawa, Ontario KIN 6N5, Canada This work was partly supported by the Natural Sciences and Engineering Research Council of Canada, under Operating Grant NCR A-8450.

vol 2 no 4 august 1979

PROBLEM STATEMENT AND DEFINITIONS A message-switching network is considered having N switching nodes and L half-duplex channels (each fullduplex channel is decomposed into two half-duplex ones in parallel). R classes of messages are defined, each corresponding to a virtual channel that routes messages from a source node, via intermediate nodes, to a destination node. Fixed routing is considered. It is assumed that messages arrive at the source nodes in a Poisson pattern, and that messages are independent 1. End-to-end message-acknowledgement time and nodal-processing delays are considered to be negligible. The channels are modelled by simple FIFO infinite queues. Message lengths are exponentially distributed, with the same mean length for all classes. For class r, r = 1, 2 , . . . , R, an end-to-end flow-control protocol with window size E r is imposed 2, so that at any time when the class population reaches Er, the arrival process for class r is stopped. The formulation of the problem therefore guarantees that the solution for the equilibrium state probabilities will be in product form 3. The problem is to develop and study a heuristic procedure for the determination of the flow-control window settings that would optimize the network 'power' (the ratio of the network throughput to the average network delay) 4.

ANALYSIS Analytical model It was observed s that the imposition of end-to-end flow control on an open queueing network results in a closed one with multiple cyclic closed chains. The exact solution of such networks is difficult to obtain 6, so that approximate solutions obtained through computationally efficient heuristic techniques must suffice. Consider a simple cyclic closed chain with population E and throughput X (E) as in Figure 1. For any queue

i,i = 1 , 2 , . . . , M , ri

= mean service time

0 1 4 0 - 3 6 6 4 / 7 9 / 0 4 0 1 6 1 - 0 4 5 0 2 . 0 0 © 1979 IPC Business Press

161

" '""M---

A heuristic was developed to circumvent this problem 7. Define

..... Population = E

8d (r-) = Nij (E) - Nil. (r-) = Nij (E) - Nil. (E - Ur) X(E)

(8)

Here, 8/j (r-) is the increment in mean queue size for chain j at queue i when the population of chain r is augmented by one. It is intuitively appealing to assume s that

Figure l. Simple cycfic closed chain

o <. ai/(r-) <. 1 N i (E) = mean queue length As a first heuristic approximation, we rule that 7, for i = 1, 2 , . . . , L , j = # r , and/, r E R (i),

ti (E) = mean queueing time It is then clear 5 that: t i (E) = r i + r i. (mean queue length of queue i upon arrival of a message)

x (E) =

(9)

6it (r-) > > 6ij (r-)

With suitably redefined parameters s, the following system [Equations (10)-(13)] can be iteratively solved instead, starting with N i (0) = O,

M

(10)

6ir (r-) = N i (Er) - N i (Er - 1 )

ti (E) 1

(11) j~R (i)

N i (E) = X (E) t i (E)

(12)

Equations (2) and (3) are simply Little's equations for the entire chain, and for queue i, respectively. For the class of closed multichain queueing networks under consideration, Reiser and Lavenberg 7 have shown that an arriving customer 'sees' the system with himself removed as being in equilibrium. That is:

ti (E) = ri t l + Ni (E-1) }

(is) Nir = Xr tir where N i is the mean queue size in the single chain problem. For a given network, the operations count is of the order

Extending the single chain result to the multichain case~ f o r r = 1, 2, . . . , R:

R ( Xr=

(s)

Er ) ~.~ tir

r=l

i~Q (rl

(6)

Nir = Xr tir

i~R(i)

",l I

(7)

This is an improvement over the exact solution procedure. A performance ratio P called 'power' is now defined s where: pA_X =T

where O(r)

= set of queues in chain r

E

= (E,, E 2 , . . . , ER)

(14)

and R

X = network throughput = ~

E - Ur = (El, E 2 , . . . , E r - 1 , R (i)

Er-1, Er + 1,. •., ER)

Xr

(15)

r=l

= set of chains visiting queue i

Although this mean value analysis is simple, the R dimensional recursive solution of equations (5)-(7) is computationally as complex as the original recursion using the traditional convolution algorithm approach 6. The operations count is in the order of

(16)

T = average network delay = r=|

v (r) = Q (r) - (reentrant queue from sink to source)

R

nE~ r=l

162

computer communications

MVWDIM

algorithm

The MVWDIM (mean value window dimensioning) algorithm is essentially a multidimensional pattern searching procedure 9 with function evaluations afforded by the heuristic described above. Let F = lIP be the objective function to be minimized in the procedure. Initially, the window sizes are approximated to by the vector e = (el, e2, . . . , eR), with parameter increments y = (y~,)/2 . . . . ,YR). Local eJcploration is then made by evaluating F at points on either side of the point Q (designated by the position vector e) shifted along each axis in turn by an amount equal to the increment in the current direction. At any stage, moves subsequent to the local exploration depend on the sequence of moves already made as part of the current pattern. The step size is augmented if the search successfully reduces F. Otherwise the step size is reduced and a new pattern started. The search is terminated when the step size is reduced below a prescribed value. For the initialization of the window sizes, the single chain problem is considered. If each link is modelled by a single M[M[1 queue, and if there are k hops in the chain, then P is maximized 4 (and hence Fminimized) when the window size is h. Thus a good approximation to the initial windows settings will be the vector (hi, h2, • • .,kR), where kr is the number of hops in chain r.

The network can be modelled as shown in Figure 3. In this analytical model there are two chains and nine queues. Queues 1 - 7 represent the seven channels interconnecting the nodes; queues 8 and 9 model the sources of the two chains. The effect of varying the class arrival rates $1 and 32 on the optimal window settings (that set of windows which maximizes the power P, as defined in Reference 8) is shown by results determined using the MVWDIM algorithm and summarized in Tables 1 and 2.

ChainI

6tF /...

.

.a

~

)

f

9

NUMER ICAL EXAMPLE

3 !

TO8/

Figure 3. Closed-chain model of example network The MVWDIM algorithm has been implemented in APL and run on an IBM 360/65 to study the simple network in Figure 2. This network has six switching nodes and seven half-duplex channels. Channels 1 - 5 have a capacity of 50 kbit/s each. Channels 6 and 7 have capacities 25 kbit/s each, with a FIFO queueing discipline. Two message classes are defined in the network: class 1 messages originate, with Poisson rate S~, at Edmonton and are routed to Ottawa via Winnipeg and Toronto; class 2 messages originate, with Poisson rate $2, at Montreal and are routed to Vancouver via Winnipeg and Toronto. The messages are exponentially distributed with mean lengths of 1 000 bits for both classes.

\

5'1~

( ' ~ Ninnipeg

I

~j )

Vancouver

~ Toronto

Figure 2. Network example

vol 2 no 4 august 1979

_.~ / ~ Ottawa

Table I Effect of symmetrical class Ioadings on the optimal window settings Class I Class 2 Total arrival rate arrival rate network ($1), (S2), arrival rate messagels message/s ($1 +$2) message/s

12.0

13.0

15.5 18.0 20.0 22.5 25.0 37.5 50.0 62.5 75.0

15.5 18.0 20.0 22.5 25.0 37.5 50.0 62.5 75.0

25 31 36 40 45 50 75 1 O0 125 150

Optimal window settings

Network power

5

5

5 4 4 4 3 3 3 2 2

5 4 4 4 3 3 3 2 2

| 59 173 | 79 182 184 183 190 192 194 196

From Table 1 it can be seen that, with symmetrical class Ioadings, the optimal window sizes are also symmetrical. This is readily explained by the symmetry in the routings and network parameters for the two classes. If the traffic arrival rates are increased, the maximum power also increases. Smaller window settings are needed to attain this maximum power. The heavier the traffic, the greater is the tendency for the network to be congested, and so a more stringent end-to-end control is in order. The effect of dissimilar class arrival rates on the optimal window settings is shown in Table 2. Note that, even for class arrival rates differing by a factor of up to 3 or 4 message/s, the optimal window sizes remain close to those

1 63

Table 2 Effect of dissimilar class Ioadings on optimal window settings

Class 1 arrival rate ($1), message/s 12.0 10.0 8.4 7.0 5.0 18.0 15.0 12.0 9.0

Class2 Total NetworkSl/S2 Optimal Network arrival arrival rate window power rate ($2), (S 1 +$2), settings message/s message/s 13.0 15.0 16.6 18.0 20.0 18.0 21.0 24.0 27.0

25 25 25 25 25 36 36 36 36

1.08 1.50 1.98 2.57 4.00 1.00 1.40 2.00 3.00

5 5 5 5 5 4 5 5 5

5 5 4 4 4 4 4 3 3

159 157 153 147 138 179 177 172 161

for symmetrical loading. This is especially appealing as instantaneous window sizing is almost impractical, and so the window settings should be as insensitive to traffic fluctuations as possible. The degradation in the maximum power increases as the class arrivals differ more and more. It is therefore necessary to operate the network with similar loading for the classes. In probing the global optimality of the window sizes selected, the power P was obtained for the full range of arrival rates with different window sizes. The results are shown in Figure 4. For window sizes greater than or equal to (5, 5), and wit[, increasing applied traffic, the power initially builds up rapidly to a maximum value, but then degrades just as fast to some steady-state value, to remain there unaffected by further increase in loading. It is evident that window sizes that exceed 5 are inferior as they always

E=(2,2)

200

180 E=(5,5) 160

140 ¢~ 120 "1-

-F ~

-I-

+

+

E = Oo,lo) + ~-I-

I00 z

80 60

E=(25,25)

40

E =(50,50) 20 I

0

I

20

I

I

I

I

I

I

40 60 80 I00 120 140 Class traffic arrival rote (S~),pocket/s

I

160

II 180

Figure 4. Network power against class traffic arrival rate (Sl = S2)

164

give smaller power than for E = (5, 5) at almost any traffic loading. If the throughput and delay requirements are not too stringent, such that a window size less than 5 can be used, the network will be operating with a power that is monotonically increasing to a steady-state value as the arrival rates build up.

CONCLUSIONS The MVWDIM algorithm for resolving the end-to-end flow control window setting problem has been presented. MVWDIM essentially comprises the well-known multidimensional search technique called Pattern searching. The required function evaluations are afforded by the heuristics of the mean-value analytical model recently developed by Reiser. The computational requirement of MVWDIM is modest compared with a similar search built around the exact analytical model (e.g. using the traditional convolution algorithm approach). The algorithm has been implemented in APL and used in dimensioning the end-to-end windows of a simple network example. Results obtained from this example can be extended to provide insights to similar analysis of larger networks.

REFERENCES 1 Kleinrock, L 'Queueing Systems - Vol 2' Wiley, USA (1976) 2 Pennotti, M and Schwartz, M 'Congestion control in store and forward tandem links' IEEE Trans. Commun. Vol COM-23 No 12 (December 1975) pp 1434-1443 3 Baskett F, et al 'Open, closed, and mixed networks of queues with different classes of customers' J. A CM Vol 22 No 2 (April 1975) pp 248-260 4 Kleinrock, L 'On flow control in computer networks', Prec. ICCToronto, Canada, (June 1978) pp 27.2.1.27.2.5 5 Reiser, M 'A queueing network analysis of computer communication networks with window flow control' private communication 6 Reiser, M and Kobayashi, H 'Queueing networks with mu.ltiple closed chains: theory and computational algorithms' IBM Res. Dev. Vol 19 (1975) pp 285-294 7 Reiser, M and Lavenberg, S S 'Mean-value analysis of closed multichain queueing networks' IBM Res. Rep. RC 7023 USA (1978) 8 Giessler A, et al 'Deadlock-free packet networks' Comput. Networks (to appear) 9 Wilde, D I and Beightler, C S 'Foundations of optimization' Prentice-Hall, USA (1967) 10 Rudin, H 'Flow control: session chairman's remarks' Prec. 3rd International Conference on Computer Communications Toronto, Canada, (August 1976) pp 463-466 11 Georganas, N D 'Numerical solution of queueing networks with multiple semi-closed chains' Prec. lEE, Vol 126 No 3 (March 1979) pp 229-231 12 Wang, J W and Unsay, M S 'Analysis of flow control in switched data networks' Prec. 1 9 7 7 / l i p Congress Toronto, Canada, (August 1977) pp 315-320 13 Chatterjee A, et al 'Analysis of packet-switched computer-communication networks with end-to-end congestion control and random-routing' IEEE Trans. Commun. Vol COM-25 No 12 (December 1977) pp 1485-1489

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