The Delay-Capacity Product for Store-and-Forward Communication Networks: Tree Networks by IZHAK RUBIN Department of System Science School of Engineering and Applied Science University of California Los Angeles, California 90024 Communicated by A. V. Balakrishnan ABSTRACT

A store-and-forward communication network under a maximal message delay criterion is considered. It is shown that the overall channel capacity C and the associated minimal maximal delay T, as well as the maximal delay T and the associated minimal overall capacity C, are characterized by a unique Delay-Capacity (yC) product number. The latter is related to a Delay-Capacity product (TC)* number, uniquely determined solely by the topological structure of the communication network. Basic characteristics of the optimal delay and capacity assignment, a useful algoritm for the calculation of (TC)* and simple upper and lower bounds on (TC)*, are derived for store-and-forward tree networks. Synthesis considerations and applications to hierarchical communication networks are noted.

1. Introduction S t o r e - a n d - f o r w a r d c o m m u n i c a t i o n n e t w o r k s are of p a r t i c u l a r r e c e n t interest. D a t a is t r a n s m i t t e d b e t w e e n the n e t w o r k ' s terminals (being c o m p u t e r s , teleprinters, v o c o d e r s etc.) t h r o u g h the switching centers a s s o c i a t e d with each station. A t e a c h switching center, a message (or a submessage, p a c k e t ) is s t o r e d in a buffer, j o i n s a queue, a n d w h e n its turn comes (following a n a p p r o p r i a t e service discipline) is b e i n g t r a n s m i t t e d a l o n g the o u t g o i n g c h a n n e l within the determ i n e d r o u t e t o w a r d s its destination. A m a j o r c o n s i d e r a t i o n in e v a l u a t i n g the p e r f o r m a n c e of a c o m m u n i c a t i o n n e t w o r k is the time d e l a y e x p e r i e n c e d b y a m e s s a g e b e t w e e n its source a n d d e s t i n a t i o n terminals. I n particular, one g e n e r a l l y wishes to use as a p r a c t i c a l d e l a y i n d e x of p e r f o r m a n c e the m a x i m a l d e l a y of a n y m e s s a g e t r a n s m i t t e d t h r o u g h the n e t w o r k (or within a n h i e r a r c h i c a l subnetwork). H o w e v e r , studies

This work was supported by the Office of Naval Research under Grant N00014-69-A-0200-4041 and N00014-75-C-0609 and by the National Science Foundation under Grant ENG 75-03224. 197 APPLIED MATHEMATICS~ OPTIMIZATION,VOI. 2, No. 3 © 1976 by Springer-Verlag New York Inc.

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up-to-date have generally considered as a network message delay measure a delay T averaged with-respect-to the message terminals' class (see, for example, [5] and the references therein). Thus, if )to and 7~/denote the traffic intensity and the (steady-state) average message delay, respectively, for message flow between the i-th and the j-th terminals (i~:j,)tii=O), the latter averaged message delay measure T is given by T = 52i,j )t~)t- ~¥u, where )t = 52i,j )to., since Xu)t- ~ yields the probability that an arbitrary message will belong to an i-j traffic flow. Measure T thus weights each terminal delay by its proportional traffic flow. This weighting is generally undesirable since one wishes the network (or an appropriate subnetwork, see Section 6) to offer its terminals, for each class of messages, maximal delay independent of the source-destination message flow rate. Hence, we consider in this paper a maximal delay measure 7 given by y = Max~j Y0" Utilizing delay measure 7 (clearly, T~< 7), we consider here the capacity assignment problem. For that purpose, we assume the message delay along each channel to be approximated by the delay resulting when the arrival process at the channel is Poisson with the message lengths being exponentially distributed independently chosen at each channel. See [5] for the common use of this approximation in computer-communication networks, and [4] for simulation studies regarding such approximations in packet-switching communication networks. For packet-switching store-and-forward communication networks with non-simultaneous multi-terminal flows, exact queueing results are obtained in [1], [2] and analysis and synthesis results are given in [3]. The capacity assignment problem is presented in Section 2. Consideringthe minimal overall channel capacity C*(y) for a communication network with maximal delay 7, and the minimal possible maximal delay ,if(C) for a network with overall capacity C, it is shown in Section 3 that ( , / C ) ~ C*(y)y =,/C0+ (yC)*, where C o depends on (~.) and (yC)* is a unique Delay-Capacity number associated solely with the topological structure of the network. A useful algorithm for the calculation of the Delay-Capacity product for a tree network is derived in Section 4. Simple bounds on the Delay-Capacity number of a tree network are obtained in Section 5, and the notion of a delay-center is introduced to characterize the situations at which the latter upper and lower bounds are tight. In Section 6, synthesis considerations are presented, and the applications of the results to hierarchical networks, with different maximal delay requirements for the different subnetwork classes, are noted.

2. The Capacity Assignment Problem Preliminaries. We consider a communication network which topologically is described as a tree graph, T=(V,F), with set of vertices V, V= (vi), and set of edges F, F - (bi). Graph T is connected, has no cycles, and is assumed to have n vertices and subsequently n - 1 edges. Each vertex is considered to be a storeand-forward switch, serving the arriving messages on a first-come first-served basis. Each branch bi, i = 1, 2 .... , n - 1 , is a communication channel whose

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capacity is Ci [bits/sec.]. Messages are sent between vertex v; and vj at random times following Poisson statistics with rates X0 [mess./sec.], i , j = 1, 2 , . . . , n . For notational simplicity, assume X0=Xsi, for each i, j, and let each branch b i represent a full-duplex communication channel whose capacity is C i in either direction. Since in a tree network there is a unique (shortest) path between any two vertices, 'we choose this path to route the messages. The overall message flow intensity, in any one direction, along edge bi, subsequently follows from the tree topological structure T and the terminal message-flow rates (X0}, and is denoted as h i [mess./sec.]. Clearly, 2~;= E

E

Xk,

(1)

where V0 (Vii) denotes the set of vertices connected to v i (vj) when edge b i = [% vj] is removed.

Assuming independent messages, exponentially distributed with average length/~- x [bits/mess.], the message average delay 7; along edge b; is assumed to be given by (see Introduction and [1]-[2], [4]-[5]) /.t- l ~{i -- Ci _ ~ k i ~ _ 1 '

(2)

when Ci>~i~ -1, and 7;= oo when C i <~i~1-1. With no loss in generality, we assume henceforth/~-1 = 1. The overall network delay criterion 7 is chosen to be the maximal average message delay for any terminal flow. Thus, 7 = M ax 70, 1,1

(3)

where 7o.= 7(%.) denotes the average delay of a message directed from t)i to vj, along the unique shortest v i - v j path ~ro. (Clearly, 7o = 7si, for each i, j.) A Reduced Presentation of the Capacity Assignment Problem. Assume the topological structure of the tree graph T and the message-flow rates {?~0} are given. (Assume X0 > 0, V i, j.) Our problem then reduces to assigning capacities to the edges, such that an appropriate performance criterion is optimized. We choose the objective measure to include the network delay measure 7, (7 < ~ ) and the overall capacity assigned C = X ~ 2 ~ C r Thus, for a specified overall capacity cost C, we wish to assign capacities (C;} such that the delay measure 7 associated with the resulting graph is minimal. We denote the latter delay as 7"(C), where 7*(C)=Min (Ci)

n--1 } 7: ~, C ; = C . ;=1

(4)

Equivalently, if a delay measure 7 is prescribed, and thus requiring 7j~ < 7 for each terminal pairj, k, we wish to choose { C;} so that C is minimized. The latter

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minimal overall capacity is then denoted as C*(T), where C*(7) = Min ( C: 7jk < 7, Vj, k}. {c,} -

(5)

Using formula (2) we observe that for Yi< c~ we must require channel i capacity to be larger then Ci°, Ci > Ci 0 __ ~ki~ - 1.

(6)

Hence, we require an overall capacity of at least Co=Y~=~C~° for achieving y < m. We can thus decompose each capacity value as follows i = 1,2,...,n- 1.

Ci=di-J~Ci 0,

(7)

Furthermore, using (7) in (2) we obtain (setting t~-l= 1) Yi=l/Ci.

(8)

Thus, setting C = C - Co, we have ~,*(C) = ~*(C ),

(9a)

C*(T) = Co+ C*(y),

(9b)

where ~*(C)=Min "

{e,)

n-1 } y: ~] C ) = C , T i = I / C ' i ,

(10)

i=l

C*(y) = Min ( C:yjk < "l, Vj, k, vi = 1/ Ci )"

(4)

(11)

Thus, the capacity assignment problem then reduces to equivalent problems (10)-(11), which replace formula (2) for the edge delay by Yi-1/Ci. The latter edge delay formula is independent of the edge message flow rate (X;), so that the capacity assignment problem is reduced to a pure topological problem of weighted graphs with weight characteristics (8). Equivalently, one can regard the reduced problem as the capacity assignment problem for a network which incorporates only edge transmission delays, ~'i---~-l/Ci for each branch bi whose capacity is assumed to be Ci (no waiting-time delays are involved). We will present in this paper the solution to problems (10)-(11). 3. The Delay-Capacity (TC)* Number of a Communication Network We wish to establish the relationship between ~*(C) and C*(7), show that one is the inverse function of the other, and prove that with each network we can associate a unique Delay-Capac~y Number, denoted ('yC)*, characterizing the minimal value of the product 7" C (when considering the reduced problem).

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201

For that purpose, we first obtain the functional form of ~*(C). Lemma 1. For each C > O, ~/*( d ) = a / C ,

(12)

where a is a positive constant. Proof. Let "~*(C)=3'p Then, a capacity assignment exists which sets Ci = C; 0), i = 1..... n - 1, and yields network delay "/1.Using assignment C; = C;(2) = xC; 0), where x is an arbitrarypositive number, we obtain a network whose overall capacity is C = ((2)= xC. Since, 3,i(2)= 1/ (i(2)= 1/xCi (1)= x-13'i°), the network under assignment ((i(2)} has a delay measure "~(2)=x-13' v Hence, ~*(xC) < x - ~ * ( C ) . We wish furthermore to show that we also have ~*(xC) /> x - 1~,((). For that purpose, assume ~*(xC)< x - 1 ~ , ( ( ) . Consider the assignr.lent (~i(3)), satisfying d(3)='~,di(3)~'Xd, which achieves ~*(xC). Choosing Ci(4)= x-lCi (3), we have ~(4)= (~, 3'}4_)=x3'/(3), 3'(4)= x.~,(x~)" Therefore, the latter assignment yields ,~*(C) < x q * ( x C ) < x.x-L~.*(C)=,7*(C), a contradiction. Consequently, Vx > 0, we have ~,*(xC) = x - l~*(C), or

x d ~ * ( x d ) = d,~*(d ) = a = const.

Q.E.D.

To obtain the functional form of C*(3'), we prove that the latter is just the inverse function of ~*(C), denoted (,~*)-1(-). Lemma 2. For each 3'E(O, o~),

d*(3') = (~*)- 1(3').

(13)

Thus, C*(3'1) = C 1 if and only if ~*(CO=3' 1, VC1,3' 1. Proof. Let ~*(C1)= 3'1" Then, clearly, d*(3'l ) < C 1. We wish to show that we also have C*~3'1)~> C 1, and subsequently conclude that C*(3'l)= C r Hence, assume that C*(3'1)= C2< C I. Subsequently, ~*(C2)< 3'1. Also, since ,~*(C)= a~ C is a strictly monotone decreasing function in C, we obtain 3', = ? * ( c , ) <

~?*(c2) < 3',,

a contradiction. Lemmas 1-2 imply that

C*(3') = a/3', 9*(d)= a/d,

Q.E.D.

(14)

so that we can define a unique number a--(3'C)* to denote the minimal Delay-Capacity product for the reduced network.

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Theorem 1. For the reduced network, a unique Delay-Capacity number (yC)* exists, yielding

vd*(v)=

I

(15)

One readily shows that the Delay-Capacity characteristics indicated by Theorem 1 are valid iff the appropriate edge delay function is of the form 7i--1/Ci. Also, following the proofs of Lemmas 1-2, we observe that the conclusions of Theorem 1 remain valid if an arbitrary graph structure, under fixed path routing, is considered.

Proposition 1. The conclusions of Theorem 1 remain valid if an arbitrary graph structure, under fixed path routing, is considered. I The detailed analysis for arbitrary graph structure will be presented elsewhere. We will present here a procedure for calculating the Delay-Capacity number for a tree network. We note that the Delay-Capacity product for the primary problem, denoted as (7C), is related to (7C)* according to (16a)

( v c ) = vc0 + (vc)*,

so that the minimal delay under overall capacity C and minimal capacity under delay 7 are given, respectively, by

v*(c) = ( v c ) * ( c - Co)-'

(16b)

c*(v)= Co+(vc)*y-',

(16c)

when C > C0.

4. The Delay-Capacity Number of a Tree Network

We wish to determine the (7C)* number of a tree network. To observe the characteristics of the evaluation procedure, we study first two simple tree structures. Only the reduced problem need be considered. Consider a path network P,, shown in Fig. 1. Assume the i-th edge to have capacity C i and to cause delay 7i = Ci -l, C=]~i=~Ci, with the optimal values of C i and 7i (assumed by a network achieving the (7C)* value) denoted as C* and -- I ~n- lc -1 n-- I .c~n--1 - 1 Y*, respectively. We have y = ~ n i=17i= i=1 i , C - __ ~,i=lCi~-,~i=lYi . By symmetry, to minimize C (given 7) or minimize 7 (given C), we obtain for

C1

C2

C3

Cn_1

0

0

O

0

v1

v2

v3

v4

°,°

Figure 1. A path network.

0

O

Vn_l

Vn

Delay-Capacity Product for Store-and-Forward Communication Networks

203

i=1,2 ..... n--l, 7* = 7 / ( n

-- 1) = ( n -- 1 ) / C,

(17a)

1) = ( n - 1 ) / 7 ,

(17b)

C* = C/(n-

so that the D e l a y - C a p a c i t y p r o d u c t is equal to (yC)*

= (n -

1) 2 = ( I P , I),2

(18)

being the square of the p a t h length (]P.I). A m o r e general tree structure to be considered now is shown in Fig. 2. In this n e t w o r k p a t h ~r1 between v a n d v 1 is of length x 1 and the delay across it is 71 = 7(~rl) • Similarly, v - v 2 path ~r2 and v - v 3 path ~r3 are of length x 2 and x 3 with delays V(~r2)=72 and 7(~r3)=73, respectively. By the above indicated solution for the optimal assignment in a path network, each edge in 7ri will be

"X~v C1 o~ .,JD

C1

0

Cl

0

C1 Vl

C1

0

0

-..

0

0

..o/, ~° <..,,,:, >J

~ /,'~_?@"

"K""*" x23

x1

©

©

©

v23

v

v1

(b)

Figure 2. A specific tree network. (a) the network. (b) an equivalent weighted network.

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|ZHAK RUBIN

assigned the same capacity value C~, i = 1,2,3. We then have Ci=x~7~ -1, i = 1, 2, 3, and subsequently obtain C ~- X2"}tl 1 "Jr"X272 1 -Jr" X32'~3 1

(19a)

7 = Max(71 + 72, 71 + "/3,72 + 73).

(19b)

Minimizing C, given 7, one readily obtains by variational analysis the resulting delay-capacity values to be given as follows. Assume x 1= MaX(Xl,X2,X3) and set x23=(x2+ x2) 1/2, x = x I +x23. Then, if x23 < Xl, we obtain the optimal assignment to be 72 -- 73 -- X23X- 17,

7~ = X l X - 17,

(20a)

C*=xi(7")-1,i=1,2,3, and subsequently ( T C ) * = x 2.

(20b)

If X23 ~ Xl, we obtain 7~ -- "* -- 7" ---I2-3-7/2

C~' = 2 x i / 7 ,

(21a)

i = 1,2,3,

and subsequently (7C)* = 2(x 2 + x 2 + x2).

(21b)

In particular, we note that if x23 ~ x l, assignment (20a) will yield 7~' - 7"3 > 7 / 2 resulting with a path (~r2 U ~r3) whose delay is larger than 7. In the latter case, a stationary solution to the optimization problem does not exist, and the solution lies on the boundary of the constraintsset (19b), as indicated by (21a). We further note that once the optimal delay distribution for the equivalent weighted network has been determined, the delay and capacity values for the channels (edges of the original graph) are readily obtained by a uniform assignment over each tandem path. We wish next to formulate corresponding procedures for evaluating the (7C)* number and the optimal assignment for an arbitrary tree network. Consider an arbitrary vertex v in graph G, such that degree (v)= deg(v)= K/> 3. (Clearly, if no such vertex exists the network assumes a path structure, for which the (7C)* is given by (18).) Decompose graph G into the union -

a = C,(v) u aa(V) u . . .

u

(22)

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205

where Gi(v ) is the subgraph containing the i-th edge li=(v,vi) incident with v and the edges contained in the subgraph consisting of all vertices connected to vi, when edge (v, vi) is removed (see Fig. 3(a)). The paths in Gi(v), starting from v, are denoted as (~r~/,j = 1,2,...,Li}. An optimal assignment will induce the latter paths to assume the delays (7"(~r,7)}, which follow the characteristics indicated by the following Lemma. L e m m a 3. For each i, 1 < i < K, such that iv ~ k, some k, a vertex v and its decomposition (22), we have

7*(~r,j)=yv
each j = l , 2 ..... L i,

iv~k.

(23)

\

GK(V) 0

O

VK

1

/

"~

GI(V)

0

G2(v)

(a)

Gi(v)

Figure 3. Graph decomposition around vertex v. (a) decomposition into Gi(v), i = 1,2..... K. (b) subgraph Gi(v ).

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IZHAK RUBIN

For G~:(v), we have Max 7*(%j.)= 7 - 7v. J

(24)

Proof Assume graph G to assume an optimal capacity assignment, yielding delay and capacity values of 7 and C, respectively. Let (with no loss of generality), considering vertex v and decomposition (22), MaxjT*(%) =

7*('h'il) ----A7/],

for each i = 1,2,.. .,K, and let MaxiT/] = 7*kl. Furthermore, assume Maxi:i=~k 7/] = 7~'1. Clearly, we must have 7~1 "<<,//2, and subsequently 7/] < 7/2, each ig=k. For if 7 ~ ' 1 = 7 " ( r q l ) > 7 / 2 , we have 7~'1=7*(%1)> 7~'1>7/2, and subsequently 7*(%1to %1) > 7 (i.e., the path %1to %1 consisting of a tandem connection at v of rrll and %1 assumes a delay higher than 7), a contradiction. Hence, 7/] < Y~'l< 7/2, each i=/=k. Furthermore, we must have 7"el =7-7~'1. To prove this equality, we observe that edge ee = (v, vk) (as well as any other edge) must be contained in at least one path assuming maximal delay 7. For if not, the capacity of ek can be decreased until the delay of the latter path increases to 7, subsequently reducing (7C)*, a contradiction. The path of delay 7 containing G must clearly be %1 to %1, since the delay of any path ~r incident with v satisfies 7*(~r)< y * 0 r n U %0" Hence, 7 = 7"(7/'11 to 'a'kl)--'~ 7~1 q- 7*('a'kl), and therefore 7 * ( % 0 = 7 - 7T1 and Eq. (24) holds. For any i, i=/=k, we also have 7,.]=7~'v For if for s o m e / , i4=k, we have * 7~] < 7~'1, one obtains 7*(%1 tO %1) -- 7il# + 7kl < 7]~1+ (7 - 7~'1)= 7. Subsequently, the capacity of edge ei=(v,v~) can be decreased to yield a lower (7C)* value, a contradiction. To finally verify Eq. (23), we must also show that for each i, i=~ k, we have 7~ = 7~, each j = 1 , 2 , . . . , L r For that purpose, considering subgraph Gi(v), assume 7* < 7/] = 7~'1< 7 / 2 , s o m e j . The end edge (incident with the end vertex of degree one) of path ~rU, say 1o, is not contained in any other path (for then a cycle would be generated, contradicting the tree structure). Hence, the capacity of lg/ can be reduced to increase 7~ to 71~1, resulting therefore with a lower (7C)* value, a contradiction. Eq. (23) subsequently follows, letting 771 = 7v. Q.E.D. To obtain the optimal assignment within each sub-tree Gi(v ), i4=k, we consider an arbitrary directed tree rooted at vertex v (see Fig. 4) and introduce a vertex distance measure l~ at each vertex u. The latter measure l, is associated with the subtree G (u) rooted at u. Decomposition (22) can be applied to G (u) to obtain G ( u ) = tO N= 1 Gi (u) where o d ( u ) = N > 1 (od(u) denoting the out-degree of vertex u). Definition 1. For a tree rooted at v, the distance measure Iu at each vertex u is defined as follows: 1. lu = 0 for each end-vertex u (i.e., such that o d ( u ) = 0). 2. If o d ( u ) = N > 1, and u 1, u2..... uN are the vertices adjacent from u, we set

(1+

lu= i=1

=

I2[@(u)] i=1

,

(25)

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207

where l[Gi(u)]=l ~ +1 denotes the distance associated with subtree Gi(u ). The distance measure of the tree rooted at v is lv = l [G (v)]. We note that the evaluation of distance l, at each vertex of a tree rooted at v, proceeds backwards from the end-vertices towards the root vertex v (see Fig. 4). We can associate a distance measure with an edge e = (v i, vj) by setting

l(e) = Iv, - l9.

(26)

By (25), for each edge e in the tree rooted at v we have l(e)= 1. However, for the purpose of calculating lv, we can reduce the tree by replacing, for any vertex u, subtree G (u) with an edge e(u)= (u, ul) assuming the distance l[e(u)] = l~. This is equivalent to replacing G (u) by a tandem path containing l, channels. This replacement is continued (with distances calculated each time for the present tree) until a single edge incident with v remains. The distance associated with the latter edge is lv. v5

0

v4

1

v3

v6

/

0

~5

v9 0

+.,/2

0

2+% i

;~v2= [(1+-,,/5)2 + (2 +.,/2) 2] 1/2

Figure 4. A directed tree rooted at v and the corresponding vertex distances. The evaluation of distances (l,} for subtree Gi(v ), i ~ k , of Lemma 3 yields the optimal assignment for this tree, as shown by the following Theorem.

Theorem 2. For the subtree Gi(v ) of Lemma 3, for each i= 1..... K, such that i:~k, the optimal delay and capacity values assigned to any edge em.=(Vm,V.), y*(e) and C*(e), respectively, are given by y*(e,,,.) = [ 1 +/,~ ]

-1

3'v,.,

(27a)

q/v. = 7v,. -- T*(emn)

(27b)

7v = 7~'1,

(27c)

C*(e) = [y*(e)]-1,

(27d)

where and, for each edge e,

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IZHAK RUBIN

where (lv} are the distance measures evaluated for Gi(v ) rooted at v. The (minimal) Delay-Capacity product for Gi(v), rooted at v, (3'C)*(@(v)), is subsequently given by (7C)*( Gi (v)) = { 1[ @ (v)] }2 = 12 [ G, (v)] = (1 + lv,)2.

(28)

Proof. Consider the directed tree Gi(v ), rooted at v. Let {u) 0) be the set of vertices whose out degrees are higher than one and so that each such vertex u~0 is not positively (towards) adjacent to any other vertex of degree higher than one (i.e., e = (uSO, u ) ~ od(u)< 1. For example, in Fig. 4, {u) 0} = {v3,Vg}). Applying Lemma 3 to vertex uS0 and the decomposition around it, we conclude that each path with root u) 1), in the directed tree G (uj°)) rooted at u(1) assumes the same (optimal) delay value 7u)l~< 7 / 2 (since 7v ~<3,/2). Consequently, the overall edge capacity value for G (u)l)), denoted C (u)l)), is related to 3'~mby

C(U~I)) =12(',['I-~uJ/~Jml--'~.

(29)

Subsequently, we reduce G,.(v) by replacing each subtree G (u.0)) with a weighted directed edge e)l)=(u)l),t~ 0)) with distance measure l(e)0~=#,,, so that the delay-capacity values across it satisfy (29). For the resulting G/°)(v) graph, we similarly obtain the set of vertices {u) 2)} (defined as (u) 0) for Gi(v)), and conclude that each path in G(u) 2)) assumes the same delay, y~2), so that the overall capacity in G (uj(2)) (that value in G/°)(v) readily observed to be equal to the capacity value in G,.(v)) satisfies relation (29) with u) 0 replaced by u) 2). The graph is subsequently reduced by replacing G (u) 2)) by a weighted edge rooted at uj(2) with weight l(e)= luj=). We note that each reduction results with a simpler tree which assumes the same minimal delay and capacity values as Gi(v ). We continue reducing the trees until a directed path with root v, and containing edge (v, vO, results. Since the latter path assumes the same minimal delay and capacity values as G,(v), the delay-capacity product for @(v) is equal to that of this path. Hence, following (18), we conclude (28). Furthermore, since edge (v, Vl) is in both G/(v) and the path, its optimal capacity and delay values are the same in both, setting the path delay equal to 3'v. Following Eq. (17), the path assignment is uniform, so that the delay assigned to (V, Vl) is proportional to its distance measure (being 1) vs. the distance of the overall path (being 1 + lv,). Hence, 2/([v, vi]) = 7 v - 7v, = [1 + Iv,]- 17v. The same assignment procedure follows for G (vi) with overall delay value 3%. In particular, if od(vi)> 2, each subtree G~(v~) assumes maximal delay 3'v,, and the assignment proceeds recursively for each Gi(v ). The optimal delay and capacity values thus follow Eqs. (27). Q.E.D. Following Lemma 3 and Theorem 2, we observe that, considering vertex v, we can remove subtree U~:~4:~ G~(v) and replace it with a weighted edge ev incident with v with distance value 12(ev)= Y,~:i~k 12(ai(v)) • The optimal capacity and delay values along this edge satisfy 3,(%)= 12(ev)[C(ev)] - i. The minimal capacity and delay values of G remain subsequently unchanged under this reduction.

Delay-Capacity Product for Store-and-Forward Communication Networks

209

Corollary 1. The Delay-Capacity product of G remains unchanged if at vertex v, subtree U i:i=/=k Gi(v ) is replaced by a weighted edge ev incident at v, with distance measure l ( ev) given by 12[@(v)].

12(e~)= ~

[

(30)

i:iv~k

We note that the property stated in Corollary 1 has been used repeatedly in the proof of Theorem 2 for vertices within the rooted tree Gi(v ). For each such vertex u we observe that Ui:i~k Gi(u)= G(u) being the subtree rooted at u of directed tree Gi(v). Considering a vertex v and subtrees (G~(v)}, as in Lemma 3, we wish to identify G~(v). The following Theorem yields a characterization of Gk(v ) in terms of the distance measures of the subtrees. For that purpose, we let lv[Gi(v)]=li(v ) be the distance measure associated with directed tree Gi(v ) A

rooted at v. We also set G ( v ) = t_Ji:i~ k Gi(v ), with distance measure l~[G(v)] given by (30), and G ( v ) = G - G(v) with distance lv[G(v)].

Theorem 3. Consider vertex v and the decomposition around it as characterized by (22)-(24). We then have

Max K lv[ak(v)]=i:l
}.

(31)

Also, we have y~
if and only if

lk(v)=l~[G(v)]>lv[G(v)] ,

(32)

so that 3'v = 7 / 2 if and only if lv[G(v)] <<.lv[G(v)]. Proof We consider the tree under the optimal assignment. Assume first that 3'~= 3'/2. Then, proceeding as above we can replace G (v) and G (v) by weighted edges e~ and ev, incident at v, with distances lv[ G (v)] and lv [G (v)], respectively. Subsequently, if lv[G(v)]0, the delay-capacity product decreases. This is achieved by setting c to be smaller that the delay across [Vk,V] edge in G~(v), so that G(vk) can be replaced by a weighted edge evk so that l (e~k)+ 1 = lv[G (v)], and considering the path with edges evk, [vk, vl] and %. Consequently, if 3'v = 3,/2 we conclude that lv[G(v)]<<.lv[G(v)]. The above mentioned argument also shows that if l~[G (v)] < lv[G (V)] we must have 3'~ < y/2. Statement (32) thus follows. Eq. (31) follows by (32), observing that if 7 v < 3 ' / 2 we have l~[Gk(v)]>l~[G(v)] >1lv[Gi(v)] for each i, i4:k. Q.E.D. The results obtained above directly yield the following Algorithm for the calculation of the Delay-Capacity product and the optimal capacity assignment for a tree network G.

210

IZHAKRUBIN Algorithm 1. (Tree Delay-Capacity Product Algorithm)

Initiation: Let j = 1, G l = G. Step 1. For @, let L be the set of vertices of degree higher than two. 1.1 If L = ~ , the optimal assignment follows that of a path. Thus, the optimal

delay across each edge e is `/*(e)=`/l(e)/l(@). If l(e)> 1, the optimal assignment for the subtree reduced to e is obtained by Theorem 2. We have (`/C)*

=[;(o:)y.

Stop. 1.2. If L={u}, consider decomposition @= U i Gi(1.l), and compute li(u)=l . [Gi(u)]. Let l~(u)=Max i li(u), lk(u)=[Y,i:i=/=k li2(u)]1/2. 1.2.1 If lk(u ) >>,l~(u), replace G(u)= U i:i~k Gi(u) by a weighted edge incident at u with weight l~(u). Go to Step 1. 1.2.2 If l~(u)< ~(u), the optimal delay assignment across each Gi(u ), each i, is `/~= ,//2. The optimal assignment within each G~(u) follows then according to Theorem 2. We have (`/C)* =23~i li2(u). Stop. 1.3 If ILl > 1, go to Step 2.

Step 2. For @, let vf 1), vf2) be two vertices of degree higher than two, with no other such vertex in between. Consider the decomposition of Gj, Gj = G (vj(2)) U ~rvj2)v¢,) U G (vJl!), where %¢%7,, is the vf2)- vff) path, and G (vff)) is the subtree connected to ~('), i = 1, 2, obtained when the latter path is removed. Compute the distances /j(i)=/e)o [G(vff))], i=1, 2, ~('~)=z(~g40=l%,=,4,>l. Say, /j(2)>/j(l). Replace subtree G (v 51)) by a weighted edge incident at v.J (0 with weight l.i / 0) " 2.1. If/(2) ~/j(12) +/j0), go to Step 3. Step 3. For Gj, consider the decomposition around v)2), G/= uiGi(42)). Compute /,(vf2)) =/v2,[G,.(vf2))l. Let /l(vfZ))=Max, [/,(v)2))l, and/l(V}2))=[S~,:,=~l

;,2(v?)],/:.

3.1 If ll(Vf23)< ~(vf2)), the optimal delay assignment across each G;(vf 2)) is `/v]2~=`//2, and the optimal assignment for G;(vf2)), each i, follows according to Theorem 2. We have (2(C)* =2E; l;2(~<2)). Stop. 3.2 If ;,(vf 2)) > ;l(Vf2~), go to Step 4.

Step 4. In @, replace Gl(v)2)) = Ui:i561 GI(D:2)) by a weighted edge incident at v)2) with weight ll(V)2)). Denote the resulting graph by Gj+ v Let j ~ j + 1. Go to Step 1. [ We note that Algorithm 1 could be shortened b'y replacing instruction "go to Step 3" in Step 2.2 by "go to Step 1". The present version reduces however the number of iterations. It follows from Theorem 2, Corollary 1 and Theorem 3

Delay-Capacity Productfor Store-and-ForwardCommunicationNetworks

211

that Algorithm 1 yields the optimal assignment and the (7C)* number for a tree network. At each iteration of the algorithm, the algorithm either ends or the graph is reduced by at least one edge. The algorithm consequently terminates after a finite number of steps. We have thus shown the following.

Theorem 4. Algorithm t terminates after a finite number of steps and Yields the optimal assignment and the tree (yC)* number. [ A flow diagram of Algorithm 1 is shown in Fig. 5. Figs. 6-8 illustrate the use of Algorithm 1, showing the (yC)* number and the optimal assignment for three different tree networks. One observes that hand calculations of the (yC)* number are readily performed even for moderately large tree networks.

] L<°,' I ILl>,

CHOOSE,~'), v~2' COMPUTE {li(u)/ , ik(u),~k(U )

COMPUTE I~1}. I~2) i I (3'C)* = [I(GI)] G(vjI1))~ e, I(e)= I~1)

YES

NO

r NO

kLk t

] NO

,lo) °T, iv~21) J~J+I.Gj~Gj+

1

L

Figure 5. Flow diagram of Algorithm 1.

212

IZHAKRuBIy

5. Bounds on the Delay-Capacity Number. The Delay-Center

We develop here upper and lower bounds on the (7C)* number of a tree network. These bounds can be calculated at each stage of Algorithm 1, at any vertex of the merged network there, and will improve as the execution of the Algorithm progresses. At the termination of the Algorithm, either the upper or lower bound will become equal to the (7C)* number. The vertex at which the latter equality occurs will be subsequently characterized. A simple upper bound on (7C)* is obtained by assigning each branch of tree network G equal capacity, Ci = C / ( n - 1), where n is the number of vertices of G. Letting d(G) denote the diameter (see [6]) of G, we obtain the maximal delay to be 7=-d7i = dCi-1; and therefore under this assignment we have 7 C = ( n - 1) d(G). Hence,

(33)

(yC)* <(n - 1)d(G).

Considering an arbitrary vertex v and its corresponding subgraphs G (v) and (v) rooted at v, with corresponding distance measures Iv and lv, respectively, we obtain a set of useful simple upper and lower bounds.

Theorem 5. For a tree network G, each vertex v, corresponding subgraphs G (v) and G (v), rooted at v, with distance measures lv and lv, respectively, the DelayCapacity product'(TC)* of G satisfies

(l v + ()2 <(',/C)* <2(l~ + 72).

(34)

Proof The upper bound is obtained by assigning a maximal delay value 7 / 2 in G(v) and G(v) (i.e.; 7 v = y / 2 ) . The overall capacity value is then C-(7/2)-l(12)+(7/2)-1(72), thus yielding an upper bound on the minimal 7C number. The lower bound is obtained as follows. The maximal delay in G (v) is by Theorem 3 equal to 7 v < y / 2 . In G(v), the maximal delay of any path is 7 - 7 v > 7/2. However, not all maximal paths in G(V), rooted at v, will generally assume delay 7 - 7v > 7/2, since a path inside G (v) w i t h d e l a y higher than 7 may result. Nevertheless, assuming uniform 7 - 7v delay in G (v) for each maximal path rooted at v (as is the case for G (v) with uniform value 7v), a lower bound on the overall capacity value is obtained. Thus, c

lg(ro)-'

>

+ 7g( -

=(lo+

Q.E.D.

We note that the bounds of Theorem 5 can be applied at each execution stage of Algorithm 1. For the example of Fig. 6, considering vertex v}2) we obtain, 49 = (3.8 + 3.2) 2 ~<(7C)* = 49 < 2(3.82 + 3.22)----49.4. For the example

Delay-Capacity Product for Store-and-Forward Communication Networks

213

of Fig. 7, considering vertex V~2), we obtain 27---(2.4 + 2.8) 2 ~<(7C)* = 27.5 < 2(2.42+ 3.22) = 27.5. For the example of Fig. 8, considering vertex v~l), we have 134.6 = (2.2 + 9.4) 2 < ('~C)* = 134.6 < 2(2.22 + 9.42)--- 186.4, while considering vertex v2(2) one obtains 134.6 = (7.3 + 4.3) 2 < (7C)* = 134.6 < 2(7.32 + 4.32) - 143.5. Particular simple b o u n d s result from T h e o r e m 5 if we consider G (v)=,/,, as indicated by the following corollary. Corollary 2. For an arbitrary vertex v of a tree network G, let lv denote the distance measure of G considered as a directed tree rooted at v. Then, we have

l2

/

t

<<('yC)* < 21~

\

(35a)

/

\

/

I ;

I I I

/ / /

"- . . . .

a(v~21),~2) ~ a.2

-"

G(~I)I. ~I) _-2.8 (a)

2.8 0

3.2 0

0

v~1)

vl 2)

0

(b)

0.157

"~'6">N,,U) (e) Figure 6. An example of a delay-capacity product calculation for a tree network using Algorithm 1. (a) the network. (b) resulting path. (c) optimal delay assignment. Values show optimal edge delays. (yC)* = (2.8 + 1 + 3.2) 2 = 49.

214

IZHAK RUBIN

and subsequently Max[l~]<(vC)*<2. Min[12].

[

(35b)

Considering the network of Fig. 6, Eq. (35a) yields at vertex v~2) 24.7 = 12(v}2~) < (yC)* = 49 < 212(v~z)) = 49.4, at vertex v~t) 25.5 = 12(v~l)) < (yC)* = 49 < 212(v~ ')) = 51, and at an end-vertex v, 44.2 = 12(v) < (yC)* = 49 < 212(v) = 88.4. Inequality (33) for the network of Fig. 6 becomes (yC)* < ( 1 0 - 1). 6 = 54. We show now that the bounds in Eq. (34) are tight in the sense that we can find, for each tree network G, a vertex v at which either the lower bound or the upper bound for (34) equals (yC)*. Such a vertex v will be shown to be a delay-central vertex of G, defined as follows. The delay-eccentricity of vertex v, eo(v ), is defined as the maximal delay of any path starting from v. The delay-radius of network G, Ro(G ), is defined as R D ( G ) = M i n v eo(v ). A vertex v* is said to be a delay-central point of G if eo(v*)=Ro(G ). The delay-center of G, CD(G ), is the set of all delay-central vertices of G. The delay-diameter of G is do(G)= Max v eo(v ). (These definitions correspond to the regular graph theoretical ones, replacing the distance of a path by its delay; see [6].) Clearly, the delay-diameter of G is equal to 7, its maximal path delay. A characterization of the delay-center of a tree network is given by the following Theorem. We assume n > 2 (if n = 2, Co(G ) = V) and consider the decomposition around a vertex v to be, G (v), G (v), as defined for Theorem 3.

Theorem 6. For a tree network G under an optimal capacity assignment, consider vertex v~, the decomposition around it G (v~), G (v~) and the corresponding distance measures lvr=l~ [G(v~')], Ivt=lv~ [G(v~)]. Let v~ be the vertex adjacent to v~ in G(v~). Then, the delay-center of G is C o ( G ) = {v~} if and only if ~ < l~r + 1,

(36a)

and Co(G)= ~.tq, ~" * v 2*~~ if and only if ~vr= lvt + 1.

(36b)

Proof. If ~vt < lvt, then by Theorem 3 we have Yvt = V/2. Hence, eD(V~) = V/2 and for any vertex v~=v~ we have eD(V)>y/2=eD(V~{). Consequently, RD(G ) =

and

Assume now

lvr>l~r. Then, eo(v~)=~'-'t~r> 7/2.

For v~cG(v~O we have

Delay-Capacity Product for Store-and-Forward Communication Networks

215

= Max [ V - % t - V l , 7 1 + %t]' where 2q is the delay across x =[v~',v~]. If 7 - %r-~71 < V/2, we obtain (optimal path assignment) eD(v~)= "(1+ %* = (1 + l~t)(lvt + lvt)- 17. But, then e D (v~') = 3' - 7vt = ?vr(?vr+ lot)-~'/. For any vertexv, v--/=v'{, v~, we clearly have eD(v)>eD(v~) if veG(v~), and eD(v)>eD(v'~) if veG(v~). Hence, v~' is a delay-central vertex iff eo(v'{) 7 / 2 , then e D ( v ~ ) = 3' - % r - "(1 = e D ( v ' O "~1< eD(Vt)" Hence, v~' is a delay-central vertex only if 7 - % r - 7 1 < 7 / 2 , and eD(V~)

// % "~xx\

/ ~... ~

/ ~//G~v(12)), --~(12)=x/8 -

G2Wl , (2),t,'~2Wl , , (2),i \ CL

~, (1), ,,(1) i, ~1 = x / 2 ~

3.2 'JWl

1.4

(a)

=2

"...

X.~ ~

;>~=~

o

1.4

~ "~ -,~

o)

, 12))=2.4 Gl(V 121 1 ),~1(Vl ~'l(V~2)) =,,/8 ~ 3.2

/'/"/ J //"/ //~'~I 11~'" G3(v~2)), k3(vl2)1 = 2 (b)

% 0.353' ~O.z

Y

(c)

Figure 7. An example of (7C)* calculation using Algorithm 1. (a) the network. (b) distance measures for decomposition around v 1. Obtain %(2)=,//2, (7C*)=2(22+22+ 2.42) ~ 25.8. (c) optimal delay assignment ( 7"( g)); C*( g) = [7 ~'(e )]- i.

216

IZHAKRUBIN

subsequently (r < Ivr + 1. If the latter inequality holds we have lv¢= ~ - 1 < l~ < [v~+ 1 = l~ so that 7 - 7vT- ~1 < ~//2, by Theorem 3 and v~ is a delay-central vertex. Q.E.D. Theorem 6 enables us to identify the delay-center of G by directly observing the graph resulting at the last stage of Algorithm 1. In particular, from the proof of Theorem 6 we conclude the following result concerning the tightness of bounds (34) at a delay-central vertex.

Corollary 3. I f v~ is a delay-central point of G, the Delay-Capacity product is given by if ~t < Iv';

(lvr + lo ) ,

if lvt >

(37)

lvr

As an example, consideringthe network of Fig. 6 one obtains R D( G ) = 0.55 7, CD(G)= {v/2)}, (TC)*=(lvi2~+ lv1~)2=(3.2+3.8)2=49. For the network of Fig. 7, CD(G)= {v~2)}, RD(G)=0.5T ' (TC),=2(lv~2)..{_l£ia~ -2 ) = 2(3.22 + 2.42) -------25.8= For the network of Fig. 8, CD(G)={v}2)}, RD(G)=0.53 7, (7C)*=(l~=,+l~=~) = (5.3 + 6.3)2~ 134.6, as indicated by Eq. (37).

6. Synthesis Considerations. Hierarchical Networks Consider the problem of connecting n stations (vertices) by a tree network, so that its resulting (7C) number assumes a prescribed value. We characterize here the topological structure of those networks achieving the maximum and minimum (7C) value of any tree network. Considering the (7C)* number, the following result is obtained, showing the minimum (7C)* value to be achieved by the star network S, (where n - 1 vertices are adjacent to the n-th vertex), and the maximum (7C)* value to be realized by the path network P,.

Lemma 4. The Delay-Capacity (7C)* number of any n-vertex tree network, n >/3, satisfies 2(n - 1) < (yC)* <(n - 1)2.

(38)

The lower bound is realized if and only if the network is a star network S,, while the upper bound is realized if and only if the network is a Path Pn. Proof For any n-vertex tree network G, the diameter d ( G ) satisfies d ( G ) < n - 1 (observe that the network has n - 1 lines). Hence, by Eq. (33), we have (yC)* < (n - 1)d(G) .<<(n - 1) 2. Clearly, d ( G ) = n - 1 if and only if the network is the path Pn, thus proving the statement regarding the upper bound of (38). To prove the statement regarding the lower bound, consider a vertex v~" to be a delay-central vertex of G. Using Corollary 3 and~observing that minimal values of/2. and 7~. are obtained by having G (v~') and G(v~) be star trees rooted

Delay-Capacity Product for Store-and-Forward C o m m u n i c a t i o n N e t w o r k s

I

"~'

I

..... ~~"~"" ~ I

i//

~, (1), , (1)

" x t~Wl h ~ l

217

--~3.9

,,m._j iI " ~!12) = 1 I

" ( , v ~ 2 )

I I |

\

o o//o



o

\

q/

/ G(v~2)), t~2) -~ 6.6

\\~.

(a)

G2(v~ 2)) (2) ~2(Vl )=2

/-'----.... ,///,'5~/ | '~\ ,(~',,,~ " , /v!l)/

fit' // !

} G3(v~2))£3(v~2)) ~_ 4.9 "

%.

i

\ )

\

I

O

0//

I

GI(V~2)), ~llV~ 2)) -~ 6.3, ~llV~ 2)1 ~ 122 + 4.92)v2 -~ 5.3 (b)

Figure 8. An example of (TC)* calculation using Algorithm 1. (a) the network. (b) decomposition around vt2). (c) resulting network G2. (d) decomposition around v~2). (e) resulting path, (yC)*---134.6. (f) optimal edge delay assignment (7*(e)};C*(e) =[~*(01 -~.

at v~', the lower bound in (38) is obtained. Alternatively, let Gi be a network With i vertices and Delay-Capacity. product (yC)*. In G,+ 1, choose an e n d e d g e x = Iv, u] with end-vertex u. Considering the decomposition at v , l v = 1 and Iv > 1, so that by Theorem 3 the delay across x , %, satisfies 7v < 7/2. Hence, (yC)*_ l > 7 / % + ( 7 C ) * > 2+(7C)*, considering the overall capacity to equal that of x and the remaining n-vertex tree. Since (7C)~'=4, we obtain, by recursively applying the latter equality, (yC)* > / 2 ( n - 1), verifying (38), where the lower bound is observed to be realized iff Gn = Sn. Q.E.D.

218

IZHAK RUBIN

I

)m

I I I

x

\

Lo

\x

)

\\ I II I

\

~(212)=2

\\

/-'~ ~ - - - - "~ x

/ ,'o

v(,2 )

\

~ )% ,

I ~\

)

2

,.z o /

"~_~*/ ) ~(22)= (7.32+ 1)1/2= 7.4

~(d2)), ~)=,/~ ~ 2.2 (c)

/

t

C~ x

l.~x

I I .t

2.2

/ \~.

© ,

~ L~

I

,k

"\ "/ " (2) (2) "~: 0 ! O2lv 2 ),t~2(v 2 )= 1 .,v(~2) ~ - - ~,,"

(2) (2) G3(v 2 ), ~3(v2 ) = 4.2 (d)

Figure 8 (continued)

The Delay-Capacity (7C) product for a network is given by Eq. (16a) as the sum of ('tC)* and 7C 0. We have that C0=E i Xi, where X,. is the traffic flow across the i-th edge. Thus while (7C)* depends only on the topology and the edge capacities, and is independent of the terminal traffic intensities, the value yC0 (fixed 7) depends only on the topology and the edge traffic intensities. Considering n-vertex tree networks, the following theorem yields the topological characterization of the networks of lowest and highest Co values. The traffic matrix h=(?to. } is assumed to be uniform, so that hij=h for each i, j, i-¢=j

(X. ~ 0). Lemma 5. The capacity C o value of any n-vertex tree network, under uniform terminal traffic intensities ?to = X, V i, j, satisfies X(n- 1)2 < Co < l X n ( n : -

1).

(39)

The lower bound is realized iff the network is a star network S., while the upper bound is realized iff the network is a Path P .

Delay-Capacity Product for Store-and-Forward Communication Networks

219

(')'C)* ~ (5.3 + 2 + 4.3) 2 ~ 134.6

(e)

Q)

0 0157

.o

0.15-/

0.03%

/.~,

o

0.0357 0.373,

(f)

Figure 8 (continued)

Proof Considering the i-th edge in an n-vertex tree G., let n~ be the number of vertices connected to one end-vertex of this edge, when this edge is removed from G.. Then, n--I

n--1

n--l

Co= Y~ x;=x Z n,(n-ni)>~X Z (n-ll=X(n-1) 2, i=1

i=1

i=1

since ni(n-ni)>/n- 1, obtaining thus the lower bound in (39). The latter is realized if G.= Sn, for then ni(n-ni)=n-1. To obtain the upper bound, consider an n + 1-vertex tree G.+ 1, with capacity Co(n + 1). Removing an endedge of G.+ l, say (v.,v.+l), an n-vertex tree G. is obtained with capacity Co(n). Adding now edge (v.,v.+ 1) back to G., the maximal traffic flow across this edge is clearly nX. Similarly, the maximal additional (v.+ l directed) flow across an edge incident with (v.,v.+ 1) is ( n - 1)X. Recursively, thus the maximal value of a v.+ l oriented flow in Gn+1 is (Y'7=l i)X. Hence, C 0 ( n + l ) < C0(n)+X(Y~7= 1 i),

220

IZHAKRuBIN

which yields n--1

C0(n)
n--I

1

~, n , ( n - n i ) = - d k n ( n

2

-1),

i=l

where ni = i shows this upper bound is realized iff Gn = Pn. Applying the results of Lemmas 4-5, we conclude the following.

Q.E.D.

Theorem 7. The Delay-Capacity product of any n-vertex tree network, n >>-3, with uniform traffic flows )tij = )t, satisfies 2 ( n - 1 ) + X y ( n - 1)2< yC < ( n - 1)2+ 1)Wn(n2-1).

(40)

The lower bound is realized if and only if the network is a star network Sn, while the upper bound is realized if and only if the network is a path Pn. I Clearly, non-uniform terminal traffic intensities will yield different topological structures for minimizing C o. Other network structures will result also for minimizing (~,C)* if additional topological constraints (like maximal node degree, etc.) are to be imposed. Such problems will be studied elsewhere. Finally, we note that communication networks will generally be hierarchical. A maximal delay criterion will thus be applied appropriately separately for each subnetwork of a specific hierarchical class. As example, consider the tree network presented in Fig. 9. The network is composed of local subnetworks Gi, 1 < i < 5, GL= u iGi, and an intra-local subnetwork GH, (a "long distance" subnetwork) connecting the local subnetworks, containing vertices vi, 1 <~i < 5, and the edges interconnecting them. One can associate a maximal delay 7i for each local subnetwork Gi, 1 < i < 5, and a maximal delay "/n for the "longdistance" subnetwork GH. The terminal traffic matrix yields the overall capacity values Co(i ) and Co(H ) for Gi and GH, respectively. The Delay-Capacity products for G~ and GH follow from the topological structure and are denoted as (~,C)* and (7C)~, respectively. By Eq. (16c), the minimal overall capacity C*(Tn,'/i) is given by

C*(yH'Yi)=[ C°(H)+ ~i C°(i)]+YH1(YC)* H+ i

(41)

In particular, for the network of Fig. 9, assume XU-- k, each i ~ j , within each Gi, and the traffic flow between each Gi and @, ira j, to be X1. The optimal delay assignment (being independent of the traffic flows) obtained by Algorithm 1 is shown in Fig. 9. The minimal overall capacity of Eq. (41) is then given by C*('/m "/i) = [ 18)tl + 210~] + 11.6y/~ 1 + [ 14.4y x- 1+ 18,/~- 1q.. 11.6y 3-1 + 27.53,~ 1+ 25y 5- 1].

Delay-Capacity Productfor Store-and-ForwardCommunicationNetworks

221

0.27=

0.375

//_0.3771 0.27E

0.3771

0.26"71 r

,. ~ ~ ' . / o . u,,Z/

\ 0.57~~ ,

/'y-"

\

GI

~k

/J' ~ -

,~X~ ~'' / i ~)-~i/ (7C)~- 144

~,

G5 I

0.37H

V / \

0.375

(7C)~= 25 0.375

//0"2572 0"2572 0"57;x\ /o ~ o\

G2

I

0'47H

)

,~,~

\~, CC' nn%~,7.. ~3"

I

--_-I,yc)~. = 18

/ / I

G3 1,

I G4 I

%.

0.33'4 0.2'74 C ~ ; ,

I

%\ /

/

0.4"/3 ~..~.

~/"-.... \~'~ (-yc)~ = 11.6 i (7C)~ = 11.6

~

¢' I-

~,I/

(7C)4, = 27.5

0"473 0.373

0.373 Figure 9. An hierarchical network and its optimal delay assignment. 7. Conclusions

We have considered a store-and-forward communication network under a maximum delay criterion. A unique Delay-Capacity product (7C)*, depending only on the topological structure of the network, is shown to characterize the minimal (yC) product of each communication network (under any path routing). Thus we note that while the excess capacity under an optimal capacity assignment for delay criterion T is distributed proportionally to the square-root of the edge traffic flow [5], the assignment under maximal delay Y depends on the network's topology alone. Basic characteristics of the optimal delay and capacity assignments for a tree network are developed, utilizing the specific network topology. A useful algorithm is derived for obtaining the DelayCapacity (7C)* number for a tree network. Using an appropriate distance measure on the graph, upper and lower bounds on (¥C)* are obtained and are shown to be appropriately tight at a delay-central vertex. Tree topological structures achieving minimal and maximal Delay-Capacity products are noted. Applications to hierarchical networks where various embedded subnetworks are assigned different maximal delay values are indicated. Extensions of the results obtained in this paper for tree networks to other topological structures, are presently under investigation.

222

IZHAKRUBIN

References [1] I. RUBIN, Communication networks: Message path delays, IEEE Trans. On Information Theory, 20, No. 6, (1974), pp. 738-745. [2] I. RUBIN, Message path delays in packet-switching communication networks, IEEE Tram. on Communications, 23, No. 2, (1975) pp. 186-192. [3] I. RtmiN, Multiterminal path flows for packet-switching communication networks, Networks, to appear. [4] I. RUBIN, An approximate time-Delay analysis for packet-switching communication networks, IBM Report, RC5202, IBM Thomas J. Watson Research Center, Yorktown-Heights, New York, Jan. 1975. [5] L. KLEINROCK,Scheduling, queueing, and delays in time-shared systems and computer networks, in Computer-Communication Networks, Ed. by N. Abramson and F. F. Kuo, Prentice-Hall, Englewood Cliffs, New Jersey, 1973. [6] F. HARARY,Graph Theory, Addison-Wesley, Reading, Mass., 1971. (Received May 12, 1975)

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