Copyright © IFAC Theory and Application of Digital Control New Delhi. India 1982
DIGITAL COMPUTER CONTROL OF COMMUNICATION NETWORKS J.
Warrior and K. S. P. Kumar
Department of Electr£cal Engineering, University of Minnesota, 123 Church Street Southeast, Minneapolis, MN 55455, USA Abstract. This paper uses the theory of hierarchical control to arrive at routing algorithms for messages in computer communication networks, some aspects of implementation are also considered. Keywords. Hierarchical control; computer control; communication computers applications. messages are delivered correctly to their destination in a time t < tmax.
INTRODUCTION The need to multiplex expensive transmission facilities and switching equipment to efficiently handle bursty data channels has long been recognized by communications engineers. This need becomes a prime consideration when the data channels are part of a computercommuni~ation network. Packet transmission networks have evolved over the past ten years as an attempt to solve this problem. User data is divided into small, fixed sized units called packets, each carrying destination information. These packets are passed from one node in the network to another until they arrive at their destination, where they are reassembled and delivered.
d.
It appears that the control theoretic approach provides a framework in which these problems can be posed and answered. MODEL FORMULATION Consider a packetswitched network that has "N" nodes indexed by the set {1,2, ••• ,N}. Let (i,j), i,j £ N denote a directed link from node i to node j, and define L to be the set of all such links that actually exist in the network. Define
All packet transmission schemes are designed to share a resource that is inadequate to meet the simultaneous peak demand of all subscribers but can adequately support average traffic flows in the network. The essence of the design problem is to obtain a balance between high average utilization and acceptable levels of congestion under peak loads. An early attempt to solve this problem in a decentralized control theoretic framework is by Meditch and Mandojana (1979).
Priority SchedulingAllow users to pay more (or less) for a larger (or smaller) fraction of the available bandwidth and thus receive better (or worse) service.
b.
FairnessGuaranteeing that messages receive a level of service commensurate with their priority class, and for each priority class minimizing the variance within that class.
c.
ReliabilityGuaranteeing that, regardless of priority class, all
P(i)
{j/N(i,j)
£
L};
Q(i)
{t/N(t,i)
£
L}.
P(i) represents the set of network nodes physically reachable from node i. Q(i) represents the set of nodes which can physically reach node i. Let t = t,t+l,t+2, ••• be discrete (uniformly spaced) instants in time of length T and define:
Interest is now focusing .. not only on the basic "system" problems associated with routing (i.e., maintaining stable flows in the network), but on "user" problems as well: a.
Fault ToleranceGuaranteeing that every message is delivered correctly even in the presence of system failures. This implies that the routing procedure is both loopfree and deadlockfree, Sauri, Wong and Field (1980).
/(t) l.
number of packets at node "i" at time "t" whose destination is node "k" where i,k £ Nand k#i. rate (packets/sec) at which packets whose destination is node "k" are transmitted over link (i ,j ) during time interval (t,t+l), where k,i,j £ N, k*i and (i,j) £ L.
w~( t) l.
85
number of packets that arrive at node "i" from outside the network during time interval
J. Warrior and K. S. P. Kumar
86
(t,t+l) and whose destination is node "k" where k1oi, and i,k e: N. Clearly, conservation requires that:
x~(t+l)
=
l.
x~(t)

l.
+
T E
,te:Q(,t)
w.
T E U~j(t) je:p( i) l. i10j
l.
k
u,t'( t) l.
Vk
where t=t,t+l, ••• ; i,k e: Nand k#i. Clearly this equation is valid only for i,k e: N for which there is at least one directed path from the first node to the second. The cases with k=i correspond to packets that have arrived at their destination and these are removed from the network. As in Meditch (1977), we will permit x(t), u(t) and w(t) to be real valued in order to apply discrete optimal control theoretic results. wik(t) models the arrival packets to the network from external sources over (t,t+l). It is possible to describe w(t) as a random process or as a sequence of known form (e.g., polynomial) but with unknown coefficients Meditch (1977). Segall (1977) considers the modeling of Wik(t) as a compound Poisson process. State and control constraints are required to complete the model. It is obvious that these must be constrained to be nonnegative. In addition, capacity constraints on each link (i,j) e: L, and restrictions on the buffer capacities available at the nodes result in further constraints. Thus, we have the channel capacity constraints: k
~ u ij ( t) .. Uij
k#i where Uij is the capacity (packets/sec) of the channel connecting (i,j) and the statevariable constraint (either due to limited buffer storage available at a node or the implementation of priority or buffer management schemes)
=
1 2 il i+l N)T (w.w .••• w . w .•• • w. l. l.
l.
We now use (5) to rewrite (1)
+ C. z. (t) l.l.
where N
E Lij~j (t)
z. (t)
l.
min .. x. (t) .. x~x
~i
l.
l.
max u.
l.
It is interesting to note that Bi & Lij are connectivity (routing) tables for the nodes 'i' and 'j', padded with zero matrices of appropriate dimensions. The Ai'S & Ci'S are identity matrices. ~(t) is a vector describing the effects of the control variables applied at other nodes in the network and is thus an interaction vector. The overall optimization problem for the network can now be written in terms of the formulation for the sUbsystems: t
+
l.
where
U.
l.
~.
1/2 11 u i ( k) 11 ~}
(8 )
subject to:
2
+ w.(k) l.
(4 )
11
l.
o l.
(1/2 11 ~i ( k) 11 Hi
1/2 11 ~i (k) 11 ~ . )
(1/211 ~i(k)
Ef
o +
(6 )
j=l
i=l R=t
tj
(3) to give
A.x.(t) + B.u.(t) + w.(t) l.l. l.l. l.
N
k:~
jH
l.
We have chosen a quadratic performance index since this leads to mathematically simpler solutions. In addition the quadratic performance index very naturally introduces the concept of fairness into the problem by penalizing excessively large control vectors. We are able to handle priority classes in this framework, either on the basis of source and destination of the message (by altering the elements of the H&R matrices), or on a classbyclass basis by introducing corresponding state variables into the problem formulation. When used with the constraints on the state and controls this allows us considerable flexibility in determining the nature of the priority scheme implemented.
Minimize J
where Xi is the maximal buffer capacity (in packets at node i. We now define a performance index, Ji, for each node as follows:
l.
z . (k) = l.
N E L. j~j(k) j=l l.
l.
min x.
.
l.
x.( k)
..
'l.
min u.
.
l.
u. (k)
..
l.
l.
max
X.
max u.
i=1,2, ... ,N
Digital Computer Control of Communication Networks PROBLEM SOLUTION In this section we will outline the use of the interaction prediction method Singh and Titli (1978) for the solution of the problem stated in equation (8,9) of the previous section. We will focus our attention on the dynamic model, and only indicate the handling of the constraint equations. Our approach will be to assume that an optimum flow assignment has been calculated and attempt to develop a control algorithm that will maintain traffic flows in the network as close as possible to this optimum (based on average or ensemble values). This steadystate optimization is carried out relatively infrequently in comparison to the routing algorithm. Its purpose is to use traffic statistics to adjust to "longterm" trends in traffic patterns. The control algorithm will be run off line and the results will be used to update the parameters for the routing algorithm. The development of the interaction prediction method proceeds as follows. We have the Lagrangian for the system of equations (8)  (9) as t Ef i=l k=t N
+
(1/211 ~i(k) 11
~.
~.
)
~
T
N
+ A.(k)(~.(k) ~
E L.j~j(k)) j=l ~
~
+ P:(k)[x . (k+l) + A.x . (k) ~
~
lower level problems are solved iteratively by time decomposition in order to incorporate the state and control constraints in the simplest possible manner. This also allows us to handle other constraints on the state and control vectors at this level by introducing appropriate Lagrange multipliers. We will summarize the various steps that are involved in the above modified interaction prediction principle of optimization. The total time period of the process is divided into m consecutive coordination intervals starting at coordination instants k=0,tl,t2,'" ,tml'
The nodes solve their problems assuming no interactions in order to determine an initial reference control at the first coordination instant k=O.
The coordinator uses the solutions to determine Ai*(' ) & ~*(. ) which define the interactions which would occur if this control was applied for k=O to tf. These are supplied to the nodes.
~
o
1/2 11 ~i (k) 11
87
~~
The nodes use the values for Ai* & ~* to solve their control problems by timedeposition to produce an improved set of controls for k=O to tf for each "i", i=l, ••• ,N.
The improved controls are actually applied to the controllers at each node over the first coordination interval from k=O to k=tl
This can be rewritten as
t Ef i=l R=t N
+
The subsystems update their estimates of the system state by using an appropriate measurement scheme. These are used by the coordinator to produce the new ~*( • ) & ~*( • ) for k=tl to tf. The whole procedure now repeats from steps 2 through 5.
1/2(
E
o
11 u i ( k) 11
~: ) ~
TNT + A. (k)z.(k)  E Aj(k)L .u.(k) j ~:t ~ ~ j=l + P:(k)[x.(k+l) + A. x . (k) ~
~
~
~
+ B.;u.;(k) + C. z.(k) + w.(k)] ~
~
~
~
~
Now, the Langrangian is additively separable for given ~ and~. We will use Ai(k) and zi(k) as our coordination vector. The optimization now proceeds in two levels. The upper level (coordinator) minimizes the Lagrangian with respect to Ai and~. The minimization at the lower levels (nodes) is carried out with respect to ~, ~ and ~ subject to the constraints on ~ and~. The
IMPLEMENTATION The interaction prediction approach appears to be fairly simple and efficient in implementation. Central to the proposed scheme is the concept of clustering of nodes in a network. This is a means of classifYing network nodes into related groups on the basis of some meaningful measure of similarity (geographical location, the magnitude of internode traffic, or combinations of these and other factors) Sauri, Wong and Field (1980). Typical cluster sizes will be from five to fifteen nodes. One node from the cluster will be designated as the cluster controller, and it acts as a secondlevel coordinator for the cluster. Note that since
88
J. Warrior and K. S. P. Kumar
the secondlevel algorithm is simple and requires no extra hardware, we are able to handle node failures or reconfiguration of the network by simply designating some other node as a controller. Cluster controllers are also responsible for handling intercluster traffic. This routing problem is handled in exactly the same manner as intracluster routing except that it is expected from the nature of the traffic and clustering that this routing algorithm will operate with considerably slower time scales. We thus have a natural level of hierarchies, each using the same single routing algorithm. These concepts can clearly be extended if necessary. Another important advantage of clustering is that it results in what is effectively a relatively small, closelyco~pled, computer communication network. This results in reduced numerical demands on memory and processing capability at each node and in an algorithm that is more immediately sensitive to the state of the local network since the diameter of the cluster is small. Assuming that a cluster controller has been chosen, we can now complete the description of the implementation of our proposed routing algorithm. Available to each node is the state of the ~ueues and the connectivity matrices (B,L). This allows each node to calculate its optimal control policy over subsequent time intervals, provided it obtains an interaction vector from the second level controller. Once the control policy has been determined, each node sends its current queue length statistics up to the secondlevel controller. The secondlevel controller calculates the interaction vectors by a simple summation and sends the updated interaction vectors out to the local controllers in each cluster. This iterative process is carried out until a satisfactory optimum is reached. Since the resulting control policy is open loop, it is necessary to reinitiate the coordination dialogue at periodic intervals, Gupta (1980). Finally, we will briefly comment on how the particular choice of model and performance index allow us to implement or relate features of the model to some of the approaches to the incorporation of "user" features that are currently being discussed in the literature. The approach described in this paper uses our assumed nominal operating point for the state of the network, and attempts to regulate its behavior so as to stay as close to this point as possible. In some sense this approach corresponds to attempts to use isarithmetic control schemes that keep the number of packets in the network a constant throughout its operation. Our approach is considerably more flexible because 1) the choice of the number of packets in the network is updated at regular intervals (by resolving the steady state flow assignment problem), and 2) the scheme does not require the overhead normally
associated with isarithmetic control in maintaining a fixed number of packets. We have already mentioned in an earlier section the possibility of priority classification of packets according to sourcedestination node pairs by adjusting weighting factors and constraints for the statevectors. Twolevel control schemes that use hard limits on the number of messages in each class at a node can also be incorporated through these and the constraints on the control variables. ay introducing time varying parameters for the system matrices, it is also possible to incorporate linearly increasing priorities for messages in a class. Finally, we can guarantee a fair and equitable level of service to each set of sourcedestination pairs by adjusting the values of ~min, Ui max insuring that each packet gets a minimal (or maximal) fraction of the available bandwidth. Considerable flexibility is thus built into the model to handle such "useroriented" refinements. CONCLUSIONS This paper has outlined an efficient algorithmic procedure for routing messages in a data communication network. The method uses ideas from Hierarchical control theory and a modified interaction prediction principle. REFERENCES Gupta, M. M. (1980). Hierarchical Dynamic Optimization for Linear Discrete Systems. Journal of Cybernetics, 10, 4175. Meditch, J. S. (1977). On the State Space Approach in Modeling Data Communication Networks. Proc. 15th Allerton Conference on c3. Meditch, J. S. and J. C. Mandojana (1979). A Decentralized Algorithm for Optimal Routing in DataCommunication Networks. Proc. 18 th IEEE Decision and Control Conference, Fort Lauderdale, Florida. Sauri, J. P., J. W. Wong and J. A. Field (1980). On Fairness in Packet Switching Networks. Proceedings Compcon. Segall, A. (1977). The Modeling of Adaptive Routing in DataCommunication Networks. IEEE Trans. on Communications, COM25, 8595. Singh, M. G. and A. Titli (1978). Systems: DeCOmposition, Optimization and Control. Pergamon Press, Oxford.