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On the capacity assignment problem in packet-switching computer networks
On the capacity assignment problem in packet-switching computer networks E. D. Sykas Department of Electrical Engineering, National Technical Universi O, of Athens, Scopelou 42, Athens 113 63, Greece (Received September 1984; revised Februat T 1986)
The capacity assignment problem in a packet-switching communication network is examined with a new look and under general assumptions about the form of the network cost function and a general class of delay measures, which includes as special cases all the previously proposed delay criteria. Four optimal functions: the optimal delay (cost) versus the maximum permissible cost (delay), and the optimal capacities versus the maximum permissible cost or delay, are defined. Several propositions describing the relations between these functions are given and their form is found if a separability property holds. An interesting application of these results to the flow and capacity assignment problem is also presented. The relations between different versions of the Capacity Assignment Problem (CAP) are given together with sufficient conditions for the uniqueness of the solution. Finally, algorithms for the solution of the CAP are proposed. Emphasis is given to the analysis of an algorithm appropriate for minimax delay criteria. Numerical results and comparisons with other delay measures are also included. Keywords: computer networks, packet switching, capacity assignment, delay measures, flow and capacity assignment The general topological design problem of the packetswitching computer communication networks can be defined as follows: given the node locations and the traffic requirements during the peak-hour, find the appropriate topology, routing policy and communication line capacities so that the total communication network cost is minimized, under reliability and line capacity constraints and so that the packet delay is less than a prescribed value. The Capacity Assignment Problem (CAP) arises as a subproblem of the previous general design problem, if the network topology and the routing policy are known. The CAP has been extensively studied, not only because it is the simplest topological design subproblem, but because it readily leads to well-defined mathematical optimization problems as well and, its efficient solution is critical for the successful development of more general design algorithms. The
346 AppI. Math. Modelling, 1986, Vol. 10, October
method to find the solution of the CAP mainly depends on the form of the network cost function and the "measure' of the packet delay. Analytic solutions of the CAP are possible, for few special choices of the network cost function and the packet delay measure. Note that the cost of leasing a communication line is a more or less known function of the line capacity and length. On the contrary, the designer is free to choose the appropriate packet delay measure. Several delay criteria have been proposed, ~--~which are usually characterized either as 'system optimizing" criteria] or as 'user optimizing', if they are appropriate for the needs of the individual user. 2-5 The basic aim of this paper is twofold. First, to give general results on the CAP as independent as possible of the specific form of the network cost function, and second, to present a new general packet delay measure.
0307-904X/86/115346-11/$03.1111 © 1986 Bunerworth & Co. (Publishers) Ltd
CAP in packet-switching networks: E. D. Sykas Thus, the network model is described. The new general packet delay measure is then presented. Its generality lies in the fact that not only all the previously proposed (as far as one knows) delay criteria and other new ones result from it as special cases, but also it is flexible enough to satisfy particular design objectives. A mathematical formulation of the CAP and a dual form of it, denoted by DCAP, are given. Several general results and propositions relative to the optimal solutions of the CAP and D C A P as well as sufficient conditions of the uniqueness of their solution are also presented. Two techniques for the solution of the CAP are presented. The first is the most general and is based on the method of Lagrange multipliers. The second is appropriate for the case where our delay criterion reduces to a minimax one. Finally, numerical results for the case of a wellknown network example are given.
The c o m m u n i c a t i o n n e t w o r k model A computer communication network using the packetswitching technique, consisting of N nodes v,,, n = 1, 2 . . . . . N, and L lines, b~, i = 1, 2 . . . . . L, and topologically described as a connected and undirected graph G = ( V , B ) , where V is the set of nodes and B is the set of lines, is considered. Each node v, is considered to act as a store-and-forward switching station with infinite storage capacity, and each line b~ is considered to be an error-free communication channel the capacity of which is Ci (bits/s). Packets originating at the node v,, and destined for the node v,, arrive at random times following Poisson statistics with rate r,,,, (packets/s), n, m = 1, 2 . . . . . N. The 'requirement matrix', Jr,,,,], is assumed to be time invariable and all packets, without distinction of origin and destination, are assumed to have lengths that are independent random variables exponentially distributed with mean 1//x(bits). Packets originating at the node v,, and terminating at the node v,, can travel through the network following any path *r,,mconnecting their source-destination nodes. The routing policy, which is assumed to be fixed, specifies the time invariable probabilities pk,, with which the path 7r,,mk is used. So p~mr, m packets per second travel along the path 7r,~,, and a total packet flow Ai or equivalently a bit flow f~ = Ai//z is created across the line by The propagation time p~, that is, the time required for a packet to propagate down the length of the channel b~ and the nodal processing times k,, are assumed to be so small that they can be neglected. The previous assumptions together with the celebrated 'independence assumption '~ allows one to model the packet switching computer communication network as a stochastic network of queues, in which each queue behaves as an independent M / M / 1 queue. 6,7Therefore, the average packet delay can be easily computed by the expression: L
T = Z (A,/1,)T,
(1)
T, - - -
1
G-A;
p-i - - -
L
D(C) = Z
d,(C,)
(4)
i=1
In most applications both C~ and di(C3 assume discrete values. It is, however, customary to approximate the real costs to continuous functions. For example, the following function: d (G) = (kt + k2dl)C
(2)
(5)
was found to be sufficiently general to represent a large variety of specific service offerings) In the present study of the CAP the results presented as independent as possible of the specific form of the line cost functions di(Ci). The presentation of the results given later in the paper proceeds from the most general assumptions about cost functions to the most specific ones.
A new packet delay measure Let us assume that packets travelling in the network are distinguished into R classes. The definition of a packet class is arbitrary and a packet can change class after visiting a node while travelling in the network. Let Tr be the average delay of class r packets. Then, Tr is given as a linear combination of the line delays Ti, that is: L
T, = ~" ariTi
(6)
i=1
where ar~ are non-negative constants specified by the definition of the packet classes. The proposed new class of packet delay measures 4.9 is given by:
T(p)= [ RrZiWrTPr]
(7)
where p is a real number and Wrare non-negative weight constants which can be defined in accordance with the particular needs of the problem. It is the usual choice for communication networks, although not a restrictive one, to assume that w~ is the ratio of the rate of class r packet arrivals, Tr(packets/s), to the total rate of packet arrivals in the network, T, that is: w~= y r / y , r =
1,2 . . . . . g
(8)
Before describing how one can obtain all the previously proposed optimization criteria and other new ones, the behaviour of TO,) as a function of p should be considered. According to the definitions given elsewhere, I° TO,) is the mean of order p for the positive numbers T~. Its definition can be extended for p = + oo, 0, if .
T(_®) ~m®TO,) = min{Tr}
where y(packets/s) is the total rate of external packet arrivals in the network, that is:
(3)
The network communication cost D(C) is the sum of the line leasing costs di(Ci), that is:
A
i=1
y = ~ " r,m
and T i is the average queueing plus transmission delay on the line bi given by the well-known M / M / 1 formula:
(9)
R
7"(o) =~ lim TO,) p--*0 A
=
]-~rrW r
(10)
r= 1
T+®)=( p-,+®limTro~,= max{T,}
(11)
n,m
Appl. Math. Modelling, 1986, Vol. 10, October
347
CAP in packet-switching networks: E. D. Sykas It is proved that TO,) is a strictly increasing and continuous function with respect to p. For p = - 1 , 0, 1, To,) reduces to the weighted harmonic, geometric, and arithmetic mean for the Tr, respectively. If the weights wrare given by equation (8), then: R
L
(Tr/T)Tr =-- Z
TII) = Z r=l
(Ai/y)T/= ?
(12)
i= 1
Therefore, for p = 1 our criterion is equivalent to the corresponding one proposed by Kleinrock j and is a system optimizing criterion. However, the most interesting case arises when p = + ~ . Then, Tct,~ reduces to a user optimizing criterion, that is: 7"(+=) = max{T,}
(13)
The specific form of To,I for three different definitions of packet classes is now considered. (a) Let class r include packets transmitted across the line bi. Then 7", = T i and if the weights Wr are given by equation (8), then:
ro,~ =
(,~i/~,) r f
(14)
This criterion has been proposed by Meister et al. -~For p = 1 equation (14) reduces to equations (1) or (12) and f o r p = ~ one obtains:
T(=) = max{ Ti}
(15)
If the weights w~ are given by w, = yP, here Yr = ai, then To,) reduces to:
LO,) =
Lf
(16)
where L~ is the average queue length in the channel b~, which is the criterion used by Komatsu et al.'- In general, this choice for the weights w,, instead of the one in equation (8), results in queue length measures, because of Little's theorem,7 instead of delay measures. (b) Let us assume that packets are distinguished into classes on the basis of their source-destination nodes. There are at most N ( N - 1) such classes and Tip) is given by equation (7) where: L
7", = Z a'iTi
(17)
i=1
arl = ~ - ' p ~ , k:bierr~r k W r =
(18)
(19)
r,/y
and p k, rr~, r, are the corresponding to the class r, p k...... rr k..... r ...... respectively. (c) Let us assume that packets are distinguished into classes according to the path they follow travelling from their source node to their destination node. Obviously, the number of these paths, and therefore the number of the classes, is specified by the routing policy. In this case, To,) is given by equation (7) where: Yr = Z
Ti' i:bierrr i
ldl r ~-
rrPr/7
(20)
(21)
and p,, 7J'r, r~ are corresponding to the class r, p,,,,,,k ~r,,,,,,k r,,,,, respectively. For p = oo this criterion reduces to that proposed by Rubin. -~
348
Appl. Math. Modelling, 1986, Vol. 10, October
Mathematical formulation The CAP can be mathematically formulated as follows: t~ (CAP)
Given: The network topology, the traffic matrix and the routing policy, and therefore, the flow vector f = (f~ ,f2 . . . . . . fL)' Minimize: The network cost D(C) with respect to: the capacities C = (C1, C2 . . . . .
Q)'
subject to: f~< C and T(C) <, Tm,x The Dual CAP (DCAP) can similarly be defined as follows: (DCAP) Given: The topology, the traffic matrix and the routing policy Minimize: The pack delay measure T(C) with respect to: C = (C~, C2 . . . . . CL)' subject to: f ~ C and D(C) <<.Dr,,,~ The packet delay measure T(C) can assume any one of the forms presented in the previous section. Four functions are defined below related to the optimal solution of the previous problems: (1) 3-he optimal cost function D*(T): It gives the minimal value of the network cost as a function of the maximal admissible delay T. (2) The optimal delay function T*(D): It gives the minimal value of the packet delay as a function of the maximal admissible network cost D. (3) and (4) The optimal capacities functions C*(D) and C*(T): They give the optimal capacities as functions of the maximal admissible network cost D or the maximal admissible packet delay T, respectively.
General results A series of propositions concerning the above defined optimal functions are given with the only assumption that the functions di(C~) are continuous and strictly increasing with respect to the capacities C~. The full proofs are not given since they are relatively easy to obtain, but a brief outline of them is presented when necessary. (a) The inequality constraints in the definitions of the C A P and D C A P hold as equalities, that is, if C* is the optimal solution, then:
T(C*) = Tm,x or D(C*) = Din, x
(22)
The proof is given by contradiction, because if the constraints hold as strict inequalities, one can always find another better solution satisfying the constraints as equalities. (b) The optimal cost function D*(T) is the inverse function of the optimal delay function T* (D), that is:
D*(T*(D)) = D and T*(D*(T)) = T
(23)
The proof is based on the comparison of the solutions of CAP and D C A P when Tm,,x = T and Dm~,x = D*(T) and mainly depends on the fact that the constraints hold as equalities. (c) The optimal cost and delays D*(T) and T*(D) are decreasing functions. It follows readily from the results (a) and (b). (d) The optimal solutions of the C A P and D C A P are
the same if Tmax and Dm,,~ are related as follows:
CAP in packet-switching networks: E. D. Sykas Tm,,x =
T*(Dmax) o r Dm,x =
D*(Tm.x)
(24)
that is, for the optimal capacities C*(D), C*(T) one has C*(D*(T)) = C*(T) and C * ( T * ( D ) ) = C*(D) (25)
Convexity and uniqueness A second series of propositions follows if one supposes that, in addition to the previous assumptions, the functions di(Ci) are convex. Of course, the delay measure T(C) given by equation (2) is strictly convex forp/> 1. (e) The optimal delay and cost functions D*(T) and T*(D) are convex. This can be found by investigating the relationships between the optimal solutions of the CAP(DCAP) when Tm,~ is Ti, 7",_,or wT l + (1 - w ) ~ , O~ 1).
Separabifity results Since C t> f, instead of minimizing with respect to the capacities, one could use the excess capacities C, where: 0i = Ci - fi
(26)
Then note that for the delay measure one has:
T(C) = - -
and 7"(w0) -
?(0)
/x
(SCAP)
Minimize:/5(0)^ with respectto: C subject to: C>~ 0 and 7~(0) ~< Tmax
(SDCAP) Minimize: 7"(0) with respectto: C subject to: C>~ 0 a n d / 5 ( 0 ) ~< Dmax It should be noticed that identical results to those ones described previously hold for the SCAP and the SDCAP. (g) The solutions of the CAP (DCAP) and the SCAP (SDCA P) are closely related as shown below: C*(T) = f + O*(/xr)
(30)
C*(D) = / + 0*(D - Do(f))
(31)
D*(T) = D0(f) +/5*(/xT)
(32)
T*(D) -
?(O - Do(f)) /.z
(33)
where the functions 0*(T), 0*(D), /5*(T) and 7~(D) correspond to the C*(T), C*(D), D*(T) and T*(D) optimal functions respectively, but are defined for the SCAP and the SDCAP. (h) The optimal functions O*(T), O*(D), /5*(T) and T*(D), are given by:
7~,/5*( T) = D('F*( D) ),, = d *
(34)
TC*(T) =
(35)
C*(D) = c*
where d* is a constant number and c* a constant vector. Moreover, the constant c* is the optimal solution of the following problem: (MP) Minimize: b ( 0 ) , with respectto: C subject to: C > 0 and 7~(0) ~< 1 and d* is the corresponding minimum cost, that is, d* = D(c*). The proof is based on the application of propositions (b), (d) and (f) to a set of SCAP and SDCAP problems with specifically defined Tmax and Dmax. (i) The optimal functions C*(T), C*(D), D*(T) and T*(D) are given by: C*
C*(T) = / + / . t T
(36)
[o - D,,(r)l','",
w> 0
(27)
C*(D) = / + [
c77
.] c
(37)
w
If one expands the previous property to the cost functions, namely, if for the network cost function D(C) one has:
D(C) = D,,(f) + / 5 ( 0 )
(28)
where the excess cost function/5(0) satisfies the following relationship: /5(w0) = w"/5(0)
problems, the Simplified CAP (SCAP) and the Simplified DCAP (SDCAP) as follows:
a> 0
(29)
some simplifications of the problem are possible. First we define two simplified versions of the CAP and DCAP
d*
D*(T) = D,(f) + p. T" tO - O,,(f)J
(38)
(39)
Finally, note that the most general form of cost functions satisfying the assumption in equation (29) is: L
D(C) = D0(f) + Z d~O/
(40)
i=1
Appl. Math. Modelling, 1986, Vol. 10, October
349
CAP in packet-switching networks: E. D. Sykas In brief, the importance of the previous results is that one not only obtains explicit forms for the optimal functions, but also only a single minimization problem has to be solved in order to find them. That is, the solution of the MP is sufficient in order to find the optimal tradeoff curves for the CAP or DCAP. The fact that one can find the form of the optimal trade-off curve may facilitate the solution of other more complex design problems. In order to illustrate this, consider the Flow and Capacity Assignment Problem (FCAP), defined as follows: (FCAP) Given: the network topology and the traffic matrix Minimize: the network cost D(C) with respect to: the capacities C = (Cl, C2 . . . . . CL)' and the flows
f = ( f , , f 2 . . . . . fL)'
subject to: f<~ C, T(C) <~Tm,x a n d f i s a multicommodity flow satisfying the traffic requirement matrix. Assume that a minimax criterion for the delay measure holds, that is, p = oo and T(C) = max{T~} and the cost function D(C) is linear with respect to the capacities Ci, that is: L
Do(f) = Z
difi
i=l
Then, d* does not depend on the flows f/as it is readily obtained from MP, therefore, from equation (38) the minimization of the network cost D(C) reduces to the minimization of the function Do(f) with respect to f. Obviously, the optimal solution is a shortest-path flow. Thus, it has been proved that the optimal solution of minimax FCAP, with linear costs is a shortest-path flow, where the link costs are the coefficients d i. In practice, d~ is proportional to the physical length of a link, in this case, the optimal routing scheme follows the shortest, geographically, route connecting every pair of end nodes.
where 4,i(C) are positive scalar functions of the vector C, C I> f and are defined by the network cost function D(C) and the elimination procedure. For example, if: di( Ci) = 4C~'
then:
[ [ Dm.,x\ (ai(C)=~)(~)Q
_,,) i/2
where: L
(45)
Sifi
Q = Zi=1 (Ci-fi) 2 Employing any standard algorithm for the solution of systems of nonlinear equations, one can find the solution of the system of equation (43) and therefore the optimal solution of the DCAP. In the same way, a similar system of equations can be found for the solution of the CAP. Note that the form of equation (43) suggests that simple iterative schemes are also possible. For the special case when p = ~ in equation (7) one obtains a very useful design criterion, that is: T = max{T,}, r = 1,2 . . . . . R
xi -
1
(47)
TmaxP,(G - f/)
Then, the following problem is equivalent to the CAP: L (,, (NLMP) Minimize: d(x) = ~ d i + i=
1
with respect to: x
The uniqueness of the solution of the CAP (DCAP) and the fact that for the optimal solution the inequality delay (cost) constraint holds as equality allows one to use the classical method of Lagrange multipliers to find the optimal solution of the CAP (DCAP). For instance, after the application of the Lagrange multipliers method to the D C A P , one obtains the following system of L + 1 nonlinear equations:
subject to: x >t O, Ax <~1
OD /3"ff"~-/3di(Ci)
(C,
/J~i Tmax'
where A is a R × L matrix with the constants ari given in equation (6) as elements. The proposed algorithm is based on three theorems, proved in the Appendix, which are stated below and describe some useful properties which can be applied to every convex minimization problem with linear constraints. Theorem 1
Z di(Ci) = Omax i= l
/3 > 0
R
Si = ~ , w, a~iTp-1
(42)
r=l
Eliminating/3 from the previous equations one can obtain a new system, of L nonlinear equations, of the form: Ci =fi + cki(C), i= 1, 2 . . . . . L (43)
350
1)
i = 1, 2 . . . . . L (41)
L
where:
(46)
Then the previous procedure cannot be applied and special design algorithms are essential, since the approach by finding an approximate solution allowing p to assume large values is bound to fail because of overflow errors. An algorithm appropriate for this special case will be proposed later. First, one defines a new minimization problem equivalent to the CAP. Let us introduce a new variable x, which is the normalized link delay scaled by Trnax,that is:
Algorithms
Si
(44)
Appl. Math. Modelling, 1986,Vol. 10, October
Let f(x) be a continuous convex function defined on an open and convex set C c R L with continuous first partial derivatives on C. Let X be a convex subject of C and g(x) = x'Vf(x*) where x*eX. If g(x) >I g(x*) for every xeX, then the point x* is an optimal solution of the problem min f(x), with respect to x, xeX. Theorem 2 Let f(x) be a continuous convex function defined on an open and convex set C c R L with continuous first
CAP in packet-switching networks: E. D. Sykas partial derivatives on C. Let X be a C and g(x) = x ' V f ( x * ) where x * e X . point x * * e X such that g(x**) < g(x*) a w, we[0,1] such that f ( x ) < f(x*), + (1 -
convex subset of If there exists a then there exists where x = wx**
w)x*.
Theorem 3 Let f ( x ) be a continuous convex function on an o p e n and convex set C with continuous first derivatives. Let x* be an optimal solution of the p r o b l e m min f ( x ) with respect to x, x e X , where X is a convex subset of C. If g(x) = x ' V f ( x * ) "then g(x) >- g(x*) for every x e X . T h e a b o v e t h e o r e m s can be applied to the N L M P and then the following algorithm results. It is tacitly assumed that the objective function d(x) in the N L M P is convex. Obviously, this is the case if di(Ci) belongs to the p o w e r law or logarithmic line cost function families described previously. T h e algorithm given below can always be applied, but its c o n v e r g e n c e is g u a r a n t e e d only for convex objective functions d(x). Step 0: Initialization Set k = 0
x" -
whered=max
1
b' = Lg
J
(50)
The proposed algorithm always converges, because in accordance with Theorem 2, d k > d T M for every k. In order to prove that the sequence d k converges to d* and not to another point, first observe that if one applies the property that every strictly convex function lies above its tangent to the function f(w) = d((1 - w)xk + wx*) one obtains: d* >1 d k + gk(x*) -- gk(x~)
(51)
But gk(x *) 1> gk(zk), SO one has: 0 <. d k - d* <~gk(xk) -- gk(zk)
(52)
T h e r e f o r e , one need only show that g~(x k) - gk(zk) converges to 0. This is easy to p r o v e because: d ( w z k + (1 - w)x k) <~ d k + w(gk(z k) - gk(xk)) + 4 w2M
(53)
where M is an u p p e r b o u n d for the of the function f(w) = d ( w z k + (1 T h e n , by substituting the expression side of equation (53) by its m a x i m u m to w, one obtains:
second derivative - w)xk), we[0,1]. in the right-hand value with respect
d k - d T M I> [gk(zk) -- gk(xk)]2 > 0 (2M)
(54)
d _ a,
Obviously x" is a feasible solution, that is. x % X { x : x e R L , A x <~ l . x >! O}
=
Step 1 : Linear s u b p r o b l e m Find the optimal solution z k of the p r o b l e m : rain gk(x) = x ' V d ( x k ) , with respect to x, .reX
but, the sequence d k - d T M converges to 0, and therefore, so does gk(zk) -- gk(xk) and this c o m p l e t e s the p r o o f that the p r o p o s e d algorithm converges to the optimal solution. Finally, a simple e x a m p l e is given of the application of the p r o p o s e d algorithm to a n e t w o r k with L = 3 links, where: D ( C ) = d l C I + d2C 2 + d3C 3
d t ~ d2 ~ d3
Step 2: Stopping rule If ]gk(xk) -- gk(zk)l < e, where e is a positive tolerance, then Stop (x k is the optimal solution with such a tolerance), otherwise go to step 3.
1
(1) x °=(½,'~_,~)' '
Find the optimal solution w* of the p r o b l e m : m i n f k ( w ) = d ( w z k + (1 - w)xk), we[0.1].
(2) g°(x) = - 4 ( d . x , + dzx2 + d',,x3)
S e t x k+l= w*z k + (I - w*).r k k = k + 1, and go to step 1
if dl + d, > d3 then z" = (4., 4., ~)'
As a b y - p r o d u c t of the previous algorithm a lower bound for the o p t i m u m cost d* can be o b t a i n e d if d(x) is convex. T h a t is: k = 0, 1,2 . . . .
0
T h e n d(x) = d J x l + d2/x2 + d3/x3 and the solution steps ale as follows:
Step 3: N o n l i n e a r m a s t e r p r o b l e m
d k 1> d* t> b k
A=
(48)
where:
(3) g°(z°) = - 2 ( d r + d2 + d3) g°(x") = - 2 ( d l + d2 + d3) d o = 2(d, + d, + d3) b ~ = 2(d I + d, + d3)
(from e q u a t i o n (50))
T h e r e f o r e x* = x °. Hence:
d k = d(x k)
x*
and
(4.,'
"
d* = 2(d t + d2 + d3) b k = d e + gk(zk) -- gk(xk)
(49)
If the form of the function d(x) is known, other lower bounds can be found. For e x a m p l e , if D ( C ) satisfies the separability p r o p e r t y in equations (28) and (29) then a n o t h e r lower b o u n d is:
I f d 1 + d e ~< d 3 then z ° = (0,0,1)' (1) g ° ( z ° ) = - 4 d 3 g°(x°) = - 2 ( d l + d2 + d3) d t ' = 2(dr + & + d3)
Appl. Math. Modelling, 1986, Vol. 10, October
351
CAP in packet-switching networks: E. D. Sykas b " = (dr + d, + d 3)-" d3 2(dr + d_,) (2) f l ' ( , , , ) ~1- w
2d3 1+ w
- V ( 6 + d, |1'*
~
v-a;, + v T , + d,
..
=
(
+
in the minimization of average queue lengths instead of delays. Therefore, the generality and flexibility of the proposed delay measure allows the designer to tailor it so as to satisfy different objectives and particular design needs. The well-known example of the telegraph network examined in Reference 1 (pp. 22-23) is used in the following discussion since it has appeared in the literature -~.t2 in relation to different types of delay measures used in capacity assignment problems. In this example the network has N = 5 nodes and L = 7 links bt, b2• b3, ba, bs, b6, and b 7 connecting the node pairs 4-5, 3-4, 3-5, 2-5, 2-3, 1-3 and 1-2, respectively. The traffic matrix is symmetric and given by (in messages/s): 9.340
0.935
2.940
0.610-
L340
--
0.820
2.400
0.628
0.935
0.820
--
0.608
0.131
2.940
2.400
0.608
--
0.753
0.610
0.628
0.131
0.753
--
m
~
+ x/d~ + d2
(3) g'(,) = - ( v ~ , + x ~ - "
R=
dtxl + d2x ,. +x3 ) \dr+d_, dl+d, .t = (0 0 1)'
(4) gt(z') = - ( ~ / ~ + ~ d 2 ) ~g'(x') = - ( V ~ 3 + V ~ l + de)-' d' = ( V T , + b ~ = (v'-d73 + ,,/d~ + d,_) ~
Therefore x* = x ~. Hence:
(
~,fd, + d~
(V'-N3 °r- V ' d l
Numerical
A shortest-path deterministic routing scheme ~s used. Traffic from node vi to node v/, i, j = 1, 2 . . . . . 5, is always routed through the link i - j except for the following cases: (1) Traffic from v] to v5 is routed through v2 (2) Traffic from v] to v4 is routed through v3 (3) Traffic from w to v4 is routed through v5 This routing scheme results in the following loading of the network (one way traffic): (At. A: . . . . . AT) = (3.153, 3.548, 0.131, 3.638, 0.820, 3.875• 9.950)
#dr+d,
d* =
(55)
+ d2) 2
results
This section examines the performance of networks designed with the help of the proposed delay measure and compares it with that of networks designed with other previously proposed delay criteria. One should r e m e m b e r that the proposed packet delay measure in equation (7) is very general. In fact, the designer has three sets of parameters to cope with: the definition of the packet classes and therefore of the delays T,, r = 1, 2 . . . . . R, the weights w r, r = 1, 2 . . . . . R, and the exponent p. As was shown in an earlier section of this p a p e r all the previously proposed delay criteria result as special cases of equation (7) by appropriate choices of these parameters. New ones can also be defined as in case(c) of the section on a new packet delay measure. Moreover the proposed criterion is very flexible. For example, settingp = 1 one obtains a system optimizing criterion in the sense that the average network delay is minimized, while settingp = ~ one obtains a user optimizing criterion since the function to be minimized becomes more sensitive to the needs of the individual users. Also, choices of the type w, = 7rp result
352 Appl. Math. Modelling, 1986, Vol. 10, October
while the total traffic is A = A] + A2 + . . . A7 = 25.115. R = 10 classes of packets are distinguished according to their source-destination nodes. Classes 1-2, 1-3 . . . . . 1-5, 2-3 . . . . . 4-5 are n u m b e r e d as 1, 2 . . . . . 5, 6 . . . . . 10, respectively. Then, T, is simply the end-to-end delay across the network as seen by class r packets. Since all class r packets follow the same path, the constants ari in equation (6) are either 1 or 0 depending on whether the link bi is part of the path for class r packets or not, respectively, and are given as the elements of a R × L matrix A, where:
A=
-0
0
0
0
0
0
1-
0
0
0
0
0
1
0
0
1 0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
1 0
0
0
0
0
0
0
l
0
0
0
0
1
0
0
0
0
0
0
(56)
The weights w, in equation (7) are given by equation (8) where: (7~, 72 . . . . . 7t0) = (9.340, 0.935, 2.940, 0.610, 0.820, 2.400, 0.628• 0.608, 0.131• 0.753)
CAP in packet-switching networks: E. D. Sykas
I
200
150
150r
~100
~'100
i
3,4,5,6,9
1,3,4,6,7,10
? 50
50
OF I I
~,o 2
4
8
16
32 P
64
128
256
2,5,8,9
I 512 1024
Figure 1 Average delay T a n d end-to-end delays (1) T1, (2) T2, .... (10) 7-10versus p. Network is designed with proposed delay measure, equation (7), total network capacity is/~C = 192.0. Shortest path routing scheme described in text is used. Resulting traffic is 3.153, 3.548, 0.131, 3.638, 0.820, 3.875, 9.950 (messages/s) for links b 1, b 2. . . . . b 7, respectively
and 3' = 19.165. Finally, the line cost factors di are all considered equal to the unity. In other words, the overall network capacity is the cost function. The values of the average delay T a n d the end-to-end delays T~ for every packet class r, r = 1, 2 . . . . . 10, which result by designing the network with the proposed delay measure, equation (7), and the Meister et al. criterion, equation (14), respectively, for different values of the parameter p are presented in Table 1 and Table 2. The cost constraint is to use a total capacity /xC = 100.46 which results in a network utilization factor p = A/IzC = 0.25. It is interesting to note that in both cases the spread of the end-to-end delays T, decreases as p increases while the average delay increases slightly. Under the criterion in equation (14) the link delays TI =~ are all equal and then in the limit p = = the end-to-end delays T, are multiples of this constant quantity. The multiplication factor is the number of hops needed for class r packets to travel across the network. Under the proposed delay measure one sees a similar tendency in the sense that the end-to-end delays Tr are grouped in distinct sets, all Tr in a set having the same value. Except for special cases, it is not possible to achieve equality for all end-to-end delays T , r = 1, 2 . . . . . R, the reason being the limitations imposed by the routing scheme. If packets of a class have to follow a path that is a subset of the path for another packet class, then the delay for packets of the latter class will always be larger than the delay for packets of the former class. All one should expect is to limit the differences in delay among the classes as much as possible. The grouping of the end-to-end delays Tr in distinct sets is exhibited in Figures 1-3 and their corresponding final values are given in Table 3. The same network is used but with different routing schemes as indicated in the captions. It is clearly shown that the proposed delay measure is rather sensitive to the routing scheme. This behaviour should be expected since the definition of the packet classes depends on the paths they follow travelling through the network. Another interpretation of the previous results is possible if one defines the weighted standard deviation of the end-to-end delays T, given by:
01
I
I
I
I
I
2
4
8
16
I
I
I
I
I
32 64 128 256 512 1024 P Figure 2 Average delay T a n d end-to-end delays (1) T1, (2) T2, .... (10) 7"1oversus p. Network is designed with proposed delay measure, equation (7), and total network capacity is/~C = 192.0. A shortest path routing as in Figure 1 is used except that traffic from node v2 to v4 and from node v 1 to v5 is routed through v 3. Resulting traffic is 0.753, 5.948, 0.131,1.238, 3.220, 3.875, 9.950 (messages/s) for links b 1, b 2. . . . . b7, respectively
200
150
E _~ I00 3,4,6,9,10
T
50
8
o
2
4
1,7 8
~
16
32 P
64
128
2,5 256
512 1024
Figure 3 Average delay T a n d end-to-end delays (1) T1, (2) T2, .... (10) T10 class versus p. Network is designed with proposed delay measure, equation (7), and total network capacity is /~C = 192.0. A shortest path routing as in Figure 1 is used except that traffic from node v 1 to v 5 is routed through v3. Resulting traffic is 0.753, 5.948, 0.741, 0.628, 3.220, 4.485, 9.340 (messages/s) for links b 1, b 2. . . . . b 7, respectively
s=[Rr~=lWr(Tr-T)211/2
(57)
as the spread of the delays Tr. Then, the spread S is a measure of the uniformity of service over the various packet classes. A large spread in a network with a reasonable average delay performance indicates that some packet classes may have unncessarily short average delays while others suffer intolerably long delays. Generally, the shorter is the spread the greater is the uniformity of service attained. A spread S = 0 means that all end-to-end delays T~ are equal, that is, all packet classes receive the same service. As was explained before this is not always possible, but using the proposed delay measure the designer aims directly at reducing the spread as much as possible. The values of the spread for the three examined cases and for different loading
Appl. Math. Modelling, 1986,Vol. 10, October
353
CAP in packet-switching networks: E. D. Sykas 40
15C
30
I00
2O
50
I0
2
E -
3
I
I
I
I
2
4
8
16
I
I
I
I
I
32 64 128 256 512 1024 P Figure4 Spread S o f e n d - t o - e n d delays Trfor n e t w o r k of Figure 1 v e r s u s p. Curves f o r several average n e t w o r k utilizations: (1) p = 0.10, (2) p = 0.20, (3) p = 0.30, (4) p = 0.40 and (5) p = 0.50
200
2
4
8
16
32 64 128 256 512 1024 P Figure 7 Percentage increase in cost, f a c t o r k in e q u a t i o n (58), v e r s u s p for the n e t w o r k s of: (1) Figure 1, (2) Figure 2 and (3)
Figure 3
delay. To obtain a feeling for the penalties imposed on the overall network performance, the additional cost required if the average delay for the Tit I design (a system optimizing criterion) is to be retained in a T(m design (a user optimizing criterion) can be computed. The answer is not difficult to find if the separability results of the section on the mathematical formulation hold. From equation (37) it follows that the necessary percentage increase in cost is:
150 E % -~ I00 m
50
k I
I
I
I
I
2
4
8
32 P
64
I
i
128 256
I
512 1024
Figure 5 S p r e a d S o f e n d - t o - e n d delays Trfor n e t w o r k o f Figure 2 versus p. Curves f o r several a v e r a g e n e t w o r k utilizations: (1) p = 0.10, (2) p = 0.20, (3) p = 0.30, (4) p = 0.40 and (5) p = 0.50
2°°L 150
v
I
0
E
~-I00
,g 50 2
I
I
I
I
I
I
I
I
I
2
4
8
16
32 P
64
128
256
512
1024
1
-
D,,(f)
D
(58)
where the factor k depends on the network parameters but is independent of the cost D and given by: [average delay for T~) design]" k = L~delayfor T , ~ J - 1
(59)
The factor k can be interpreted as the additional cost percentage for a lightly loaded network, that is, when D is very large. A plot of the factor k versus the exponent p for the three examined networks in Figures 1-3 is shown in Figure 7. It is interesting to note that k is bounded and becomes almost constant as p increases. The two numerical techniques referred to in the section on algorithms were used for all computations. A modification of the Powell hybrid method '3 was used for the solution of the system of nonlinear equations, equations (43)-(45), The Jacobian was calculated by forward-difference approximation. Also, the algorithm described for the case p = ~ was slightly modified to improve its speed of convergence. In particular, a PARTAN step ~3 was added from iteration 2 and onwards, that is, an attempt to minimize f(x) in the direction x - xk_j is performed after step 3 (Nonlinear Master Problem) for k > 2.
Figure 6 S p r e a d S o f e n d - t o - e n d delays T, f o r n e t w o r k o f Figure 3 v e r s u s p. Curves f o r several a v e r a g e n e t w o r k utilizations: (1) p = 0.10, (2) p = 0.20, (3) p = 0.30, (4) p = 0.40 and (5) p = 0.50
Conclusions
levels of the network are shown in Figures 4-6. It is clear that the spread of delays drops abruptly as p increases and then remains almost constant. This reduction in the spread of the end-to-end delays Tr is achieved at the expense of the average packet
The capacity assignment problem in packet-switching communication networks was examined under general assumptions. A new packet delay measure, which includes as special cases all previously proposed ones, was presented. Several properties of the optimal solutions were discussed and an interesting application to
354 Appl. Math. Modelling, 1986, Vol. 10, October
CAP in packet-switching networks: E. D. Sykas Table 1 Average delay Tand end-to-end delays Tr, r = 1, 2 . . . . . 10 versus p. Network is designed with proposed delay measure, equation (7). Network utilization is p = 0.25
p
T
T1
T2
/-3
T4
Ts (ms)
7-6
7"7
7"8
T9
7"1o
1 2 4 8 16 32 64 128 256 512 1024
99.01 101.95 109.00 115.09 115.58 114.52 113.96 113.68 118.55 113.48 113.45
50.31 66.62 85.73 99.20 98.99 95.65 93.83 92.91 92.44 92.20 92.08
80.62 77.08 75.65 75.65 76.69 77.53 78.00 78.24 78.37 78.43 78.46
164.87 155.97 151.78 151.32 153.38 155.07 156.00 156.49 156.74 156.86 156.93
133.51 144.21 158.64 167.23 164.56 160.94 159.01 158.01 157.50 157.25 157.12
175.25 156.65 149.45 149.53 152.40 154.56 155.74 156.35 156.67 156.83 156.91
172.58 161.27 154.32 151.44 152.82 154.66 155.76 156.36 156.67 156.83 156.91
83.20 77.58 72.92 68.04 65.57 65.30 65.17 65.10 65.07 65.05 65.04
84.25 78.89 76.13 75.68 76.69 77.53 78.00 78.24 78.37 78.43 78.46
438.46 288.71 215.68 183.33 169.76 163.39 160.19 158.59 157.79 157.39 157.19
89.37 83.69 81.40 83.40 87.26 89.37 90.59 91.25 91.60 91.78 91.87
=
113.41
91.96
78.50
156.99
156.99
156.99
156.99
65.03
78.50
156.99
91.96
Table 2 Average delay Tand end-to-end delays. Tr, r = 1,2 . . . . . 10 versus p. Network is designed with Meister et al. delay measure, equation (14). Network utilization is p = 0.25
p
T
T1
7"2
7"3
7"4
7"5
7-6
T7
/-8
T9
7"10
1 2 4 8 16 32 64 128 256 512 1024
99.01 101.17 106.33 111.84 116.03 118.66 120.14 120.93 121.34 12t.54 121.65
50.31 59.83 70.10 78.87 84.98 88.67 90.71 91.79 92.34 92.62 92.76
80.62 81.92 84.65 87.58 89.83 91.24 92.04 92.46 92.68 92.79 92.85
164.87 166.29 170.81 176.03 180.12 182.73 184.20 184.99 185.40 185.60 185.71
133.51 143.49 155.83 167.07 175.14 180.09 182.84 184.30 185.05 185.43 185.62
175.25 137.47 115.49 104.08 98.42 95.64 94.27 93.58 93.24 93.08 92.99
172.58 171.41 173.94 177.81 181.09 183.23 184.46 185.12 185.46 185.64 185.72
83.20 83.66 85.73 88.20 90.16 91.42 92.13 92.51 92.71 92.81 92.86
84.25 84.37 86.16 88.45 90.30 91.49 92.16 92.53 92.71 92.81 92.86
438.46 253.36 166.66 127.60 109.63 101.11 96.96 94.92 93.91 93.41 93.16
89.37 87.75 88.22 89.61 90.92 91.81 92.33 92.61 92.76 92.83 92.87
~-
121.75
92.91
92.91
185.81
185.81
92.91
185.81
92.91
92.91
92.91
92.91
(ms)
Table 3 Average delay ~,, end-to-end delays Tr, r = 1, 2 . . . . . and spread S w h e n p = :~. Network is designed with proposed delay measure, equation (13). Total network capacity is ktC= 192.0. Results are for routing schemes of (1)Figure I, (2)Figure 2 and (3) Fig u re 3.
Case
T
T1
T2
T3
7-4
(1) (2) (3)
51.20 61.62 48.44
41.52 65.91 35.44
35.44 32.96 41.52
70.88 65.91 70.88
70.88 65.91 70.88
7-5
T6
T7
T8
T9
Lo
S
70.88 32.96 41.52
70.88 65.91 70.88
29.36 65.91 35.44
35.44 32.96 29.36
70.88 32.96 70.88
41.52 65.91 70.88
14.98 11.09 16.83
(ms)
the flow a n d c a p a c i t y a s s i g n m e n t p r o b l e m was given. F i n a l l y , a l g o r i t h m s for the s o l u t i o n were p r o p o s e d and n u m e r i c a l results for the case of a w e l l - k n o w n n e t w o r k e x a m p l e w e r e given and c o m p a r e d .
d~(G) Tr H.,' r
Yr
D( C) T( C) Nomenclature N L R
= n u m b e r of n o d e s = n u m b e r of links = n u m b e r of p a c k e t classes
1
= m e a n p a c k e t length (bits)
/z
nth n o d e ith link c h a n n e l c a p a c i t y o f link bi ( b i t s / s ) p a c k e t flow on link bi ( p a c k e t s / s ) bit flow on link bi ( b i t s / s ) Ci = C, - . ~ = excess c a p a c i t y on link hi (bits~s) Ti = a v e r a g e d e l a y on link bi (s) Li = a v e r a g e q u e u e length on link b~ ( p a c k e t s )
Vn
bi Ci Ai
= = = =
Y h p D*(T) T*(D) C*(T) C*(D)
cost of link bi a v e r a g e d e l a y o f class r p a c k e t s (s) weight for class r p a c k e t s class r p a c k e t arrival rate ( p a c k e t s / s ) total n e t w o r k cost delay measure average network delay total e x t e r n a l p a c k e t rate ( p a c k e t s / s ) total n e t w o r k traffic n e t w o r k utilization f a c t o r o p t i m a l cost function o p t i m a l d e l a y function o p t i m a l c a p a c i t y as a f u n c t i o n o f maximum permissible delay T = o p t i m a l c a p a c i t y as a f u n c t i o n o f m a x i m u m p e r m i s s i b l e cost D
= = = = = = = = = = = = =
References 1 Kleinrock. L. Communication nets-stochastic message flow and delay. IVlcGraw-Hill. New York, 1964
Appl. Math. Modelling, 1986, Vol. 10, October
355
CAP in packet-switching networks: E. D. Sykas 2 Komatsu, M., Nakamishi, H., Sanada, H. and Tezuka, Y. 'Extended optimum channel capacity assignment problems for message-switching networks', in 'Evolutions in Computer Communications' (ed. lnose, H.), North-Holland, Amsterdam, 1978 3 Meister, B., Miller, H. R. and Rudin, H. R. 'New optimization criteria for message-switching networks', IEEE Trans. Comm. Techn., COM-19, (3,256---260) June 1971 4 Protonotarios, E. N., Sykas, E. D. and Kappos, E. I. "Capacity assignment in packet switching networks: new optimization criteria, general results and algorithms'. Proc. IEEE huern. Conf. Circuits and Computers, ICCC'80, pp. 406--409, New York, NY, Oct. 1980 5 Rubin, I. 'The delay-capacity product for store-and-forward communication networks: tree networks', Applied Math. and Optim., 1976, 2, (3), 197-222 6 Jackson, J. R. 'Networks of waiting lines', Oper. Res., 1957, 5,518--521 7 Kleinrock, L. 'Queueing Systems, vol. I: theory', Wiley-Interscience, Chichester, 1975 8 Gitman, I. and Frank, H. 'Economic analysis of integrated voice and data networks: a case study', Proc. IEEE, 1978, 66. 1549-1570 9 Sykas, E. D. and Protonotarios, E. N. 'General results on the
capacity assignment problem in computer communication net10 11 12 13
works', Proc. 1EEE hltern. Syrup. on circuits and systems, ISCAS'82, Rome, Italy, May 1982 Mirtinovic, "Analytic inequalities', Springer-Verlag, 1970 Kleinrock, L. 'QueueingSystems. vol. II: computerapplications', Wiley-lnterscience, Chichester, 1976 Schwartz. M. 'Computer communication network design and analysis'. Prentice-Hall, Englewood Cliffs, NJ. 1977 Avriel, M. 'Nonlinear programming: Analysis and methods'. Prentice-Hall, Englewood Cliffs, N J, 1976
q(w) = f(wx** + (1 - w)x*) - f ( x * )
defined on [z,1], where z < 0 and zx** + (1 - z)x*eC. Since C is an open set, there exists such a n u m b e r z. By differentiation of q(w) one obtains:
dq(w) dw
- (x** - x*)'Vf(wx** + (1 - w)x*)
dq(O) dw
- g(x**)
- g(x*) < 0
Reference 11 one has:
f(x) >~f(x*) + (x - x*)Wf(x*) for e v e r y x e X (A1) Since g(x) >! g(x*) for every xeX, it follows that:
Since q(w) is a continuous, differentiable function, and: q(0) = 0
dq(O) dw
- - <
(A7) 0
(AS)
there exists an e > 0 such that:
q(w) < 0 for 0 < w < E
(A9)
Consequently, one has:
f(wx** + (1 - w)X*) < f ( x * ) f o r 0 < w < e (A10) Since X is a convex set, wx** + (1 - w)x*eX, 0 < w < e. Thus there exists a point Y:eX C_ C such that: (All)
Suppose that there exists an x ' e X such that:
g(x I) < g(x*)
(A12)
From T h e o r e m 2 it follows that there exists an 2eX such that:
f(ic) < f(x*) (A2)
Consequently from (A1) and (A2) one obtains:
f(x) >>-f(x*) for every xeX
(A6)
Proof of Theorem 3
Proof of Theorem 1 Since f(x) is a convex function from Corollary 4.23 in
(x - x*)'Vf(x*) >t 0 for every xeX
(A5)
Thus:
f(2) < f(x) Appendix
(A4)
(A3)
Hence x* is an optimal solution of the problem: min f(x), x e X
Proof of Theorem 2 Let q(w) be the function:
356 Appl. Math. Modelling, 1986, Vol. 10, October
(A13)
contradicting that x* is an optimal solution of the problem: mi.n f(x)
xeX
Consequently, there exists no x e X such that:
g(x) < g(x*)
(A14)
Hence g(x) >>-g(x*) for every xeX. That is, x* is an optimal solution of the problem: min g(x)
xeX.
The capacity assignment problem in a packet-switching communication network is examined with a new look and under general assumptions about the form of the network cost function and a general class of delay measures, which includes as special cases all the previously proposed delay criteria. Four optimal functions: the optimal delay (cost) versus the maximum permissible cost (delay), and the optimal capacities versus the maximum permissible cost or delay, are defined. Several propositions describing the relations between these functions are given and their form is found if a separability property holds. An interesting application of these results to the flow and capacity assignment problem is also presented. The relations between different versions of the Capacity Assignment Problem (CAP) are given together with sufficient conditions for the uniqueness of the solution. Finally, algorithms for the solution of the CAP are proposed. Emphasis is given to the analysis of an algorithm appropriate for minimax delay criteria. Numerical results and comparisons with other delay measures are also included. Keywords: computer networks, packet switching, capacity assignment, delay measures, flow and capacity assignment The general topological design problem of the packetswitching computer communication networks can be defined as follows: given the node locations and the traffic requirements during the peak-hour, find the appropriate topology, routing policy and communication line capacities so that the total communication network cost is minimized, under reliability and line capacity constraints and so that the packet delay is less than a prescribed value. The Capacity Assignment Problem (CAP) arises as a subproblem of the previous general design problem, if the network topology and the routing policy are known. The CAP has been extensively studied, not only because it is the simplest topological design subproblem, but because it readily leads to well-defined mathematical optimization problems as well and, its efficient solution is critical for the successful development of more general design algorithms. The
346 AppI. Math. Modelling, 1986, Vol. 10, October
method to find the solution of the CAP mainly depends on the form of the network cost function and the "measure' of the packet delay. Analytic solutions of the CAP are possible, for few special choices of the network cost function and the packet delay measure. Note that the cost of leasing a communication line is a more or less known function of the line capacity and length. On the contrary, the designer is free to choose the appropriate packet delay measure. Several delay criteria have been proposed, ~--~which are usually characterized either as 'system optimizing" criteria] or as 'user optimizing', if they are appropriate for the needs of the individual user. 2-5 The basic aim of this paper is twofold. First, to give general results on the CAP as independent as possible of the specific form of the network cost function, and second, to present a new general packet delay measure.
0307-904X/86/115346-11/$03.1111 © 1986 Bunerworth & Co. (Publishers) Ltd
CAP in packet-switching networks: E. D. Sykas Thus, the network model is described. The new general packet delay measure is then presented. Its generality lies in the fact that not only all the previously proposed (as far as one knows) delay criteria and other new ones result from it as special cases, but also it is flexible enough to satisfy particular design objectives. A mathematical formulation of the CAP and a dual form of it, denoted by DCAP, are given. Several general results and propositions relative to the optimal solutions of the CAP and D C A P as well as sufficient conditions of the uniqueness of their solution are also presented. Two techniques for the solution of the CAP are presented. The first is the most general and is based on the method of Lagrange multipliers. The second is appropriate for the case where our delay criterion reduces to a minimax one. Finally, numerical results for the case of a wellknown network example are given.
The c o m m u n i c a t i o n n e t w o r k model A computer communication network using the packetswitching technique, consisting of N nodes v,,, n = 1, 2 . . . . . N, and L lines, b~, i = 1, 2 . . . . . L, and topologically described as a connected and undirected graph G = ( V , B ) , where V is the set of nodes and B is the set of lines, is considered. Each node v, is considered to act as a store-and-forward switching station with infinite storage capacity, and each line b~ is considered to be an error-free communication channel the capacity of which is Ci (bits/s). Packets originating at the node v,, and destined for the node v,, arrive at random times following Poisson statistics with rate r,,,, (packets/s), n, m = 1, 2 . . . . . N. The 'requirement matrix', Jr,,,,], is assumed to be time invariable and all packets, without distinction of origin and destination, are assumed to have lengths that are independent random variables exponentially distributed with mean 1//x(bits). Packets originating at the node v,, and terminating at the node v,, can travel through the network following any path *r,,mconnecting their source-destination nodes. The routing policy, which is assumed to be fixed, specifies the time invariable probabilities pk,, with which the path 7r,,mk is used. So p~mr, m packets per second travel along the path 7r,~,, and a total packet flow Ai or equivalently a bit flow f~ = Ai//z is created across the line by The propagation time p~, that is, the time required for a packet to propagate down the length of the channel b~ and the nodal processing times k,, are assumed to be so small that they can be neglected. The previous assumptions together with the celebrated 'independence assumption '~ allows one to model the packet switching computer communication network as a stochastic network of queues, in which each queue behaves as an independent M / M / 1 queue. 6,7Therefore, the average packet delay can be easily computed by the expression: L
T = Z (A,/1,)T,
(1)
T, - - -
1
G-A;
p-i - - -
L
D(C) = Z
d,(C,)
(4)
i=1
In most applications both C~ and di(C3 assume discrete values. It is, however, customary to approximate the real costs to continuous functions. For example, the following function: d (G) = (kt + k2dl)C
(2)
(5)
was found to be sufficiently general to represent a large variety of specific service offerings) In the present study of the CAP the results presented as independent as possible of the specific form of the line cost functions di(Ci). The presentation of the results given later in the paper proceeds from the most general assumptions about cost functions to the most specific ones.
A new packet delay measure Let us assume that packets travelling in the network are distinguished into R classes. The definition of a packet class is arbitrary and a packet can change class after visiting a node while travelling in the network. Let Tr be the average delay of class r packets. Then, Tr is given as a linear combination of the line delays Ti, that is: L
T, = ~" ariTi
(6)
i=1
where ar~ are non-negative constants specified by the definition of the packet classes. The proposed new class of packet delay measures 4.9 is given by:
T(p)= [ RrZiWrTPr]
(7)
where p is a real number and Wrare non-negative weight constants which can be defined in accordance with the particular needs of the problem. It is the usual choice for communication networks, although not a restrictive one, to assume that w~ is the ratio of the rate of class r packet arrivals, Tr(packets/s), to the total rate of packet arrivals in the network, T, that is: w~= y r / y , r =
1,2 . . . . . g
(8)
Before describing how one can obtain all the previously proposed optimization criteria and other new ones, the behaviour of TO,) as a function of p should be considered. According to the definitions given elsewhere, I° TO,) is the mean of order p for the positive numbers T~. Its definition can be extended for p = + oo, 0, if .
T(_®) ~m®TO,) = min{Tr}
where y(packets/s) is the total rate of external packet arrivals in the network, that is:
(3)
The network communication cost D(C) is the sum of the line leasing costs di(Ci), that is:
A
i=1
y = ~ " r,m
and T i is the average queueing plus transmission delay on the line bi given by the well-known M / M / 1 formula:
(9)
R
7"(o) =~ lim TO,) p--*0 A
=
]-~rrW r
(10)
r= 1
T+®)=( p-,+®limTro~,= max{T,}
(11)
n,m
Appl. Math. Modelling, 1986, Vol. 10, October
347
CAP in packet-switching networks: E. D. Sykas It is proved that TO,) is a strictly increasing and continuous function with respect to p. For p = - 1 , 0, 1, To,) reduces to the weighted harmonic, geometric, and arithmetic mean for the Tr, respectively. If the weights wrare given by equation (8), then: R
L
(Tr/T)Tr =-- Z
TII) = Z r=l
(Ai/y)T/= ?
(12)
i= 1
Therefore, for p = 1 our criterion is equivalent to the corresponding one proposed by Kleinrock j and is a system optimizing criterion. However, the most interesting case arises when p = + ~ . Then, Tct,~ reduces to a user optimizing criterion, that is: 7"(+=) = max{T,}
(13)
The specific form of To,I for three different definitions of packet classes is now considered. (a) Let class r include packets transmitted across the line bi. Then 7", = T i and if the weights Wr are given by equation (8), then:
ro,~ =
(,~i/~,) r f
(14)
This criterion has been proposed by Meister et al. -~For p = 1 equation (14) reduces to equations (1) or (12) and f o r p = ~ one obtains:
T(=) = max{ Ti}
(15)
If the weights w~ are given by w, = yP, here Yr = ai, then To,) reduces to:
LO,) =
Lf
(16)
where L~ is the average queue length in the channel b~, which is the criterion used by Komatsu et al.'- In general, this choice for the weights w,, instead of the one in equation (8), results in queue length measures, because of Little's theorem,7 instead of delay measures. (b) Let us assume that packets are distinguished into classes on the basis of their source-destination nodes. There are at most N ( N - 1) such classes and Tip) is given by equation (7) where: L
7", = Z a'iTi
(17)
i=1
arl = ~ - ' p ~ , k:bierr~r k W r =
(18)
(19)
r,/y
and p k, rr~, r, are the corresponding to the class r, p k...... rr k..... r ...... respectively. (c) Let us assume that packets are distinguished into classes according to the path they follow travelling from their source node to their destination node. Obviously, the number of these paths, and therefore the number of the classes, is specified by the routing policy. In this case, To,) is given by equation (7) where: Yr = Z
Ti' i:bierrr i
ldl r ~-
rrPr/7
(20)
(21)
and p,, 7J'r, r~ are corresponding to the class r, p,,,,,,k ~r,,,,,,k r,,,,, respectively. For p = oo this criterion reduces to that proposed by Rubin. -~
348
Appl. Math. Modelling, 1986, Vol. 10, October
Mathematical formulation The CAP can be mathematically formulated as follows: t~ (CAP)
Given: The network topology, the traffic matrix and the routing policy, and therefore, the flow vector f = (f~ ,f2 . . . . . . fL)' Minimize: The network cost D(C) with respect to: the capacities C = (C1, C2 . . . . .
Q)'
subject to: f~< C and T(C) <, Tm,x The Dual CAP (DCAP) can similarly be defined as follows: (DCAP) Given: The topology, the traffic matrix and the routing policy Minimize: The pack delay measure T(C) with respect to: C = (C~, C2 . . . . . CL)' subject to: f ~ C and D(C) <<.Dr,,,~ The packet delay measure T(C) can assume any one of the forms presented in the previous section. Four functions are defined below related to the optimal solution of the previous problems: (1) 3-he optimal cost function D*(T): It gives the minimal value of the network cost as a function of the maximal admissible delay T. (2) The optimal delay function T*(D): It gives the minimal value of the packet delay as a function of the maximal admissible network cost D. (3) and (4) The optimal capacities functions C*(D) and C*(T): They give the optimal capacities as functions of the maximal admissible network cost D or the maximal admissible packet delay T, respectively.
General results A series of propositions concerning the above defined optimal functions are given with the only assumption that the functions di(C~) are continuous and strictly increasing with respect to the capacities C~. The full proofs are not given since they are relatively easy to obtain, but a brief outline of them is presented when necessary. (a) The inequality constraints in the definitions of the C A P and D C A P hold as equalities, that is, if C* is the optimal solution, then:
T(C*) = Tm,x or D(C*) = Din, x
(22)
The proof is given by contradiction, because if the constraints hold as strict inequalities, one can always find another better solution satisfying the constraints as equalities. (b) The optimal cost function D*(T) is the inverse function of the optimal delay function T* (D), that is:
D*(T*(D)) = D and T*(D*(T)) = T
(23)
The proof is based on the comparison of the solutions of CAP and D C A P when Tm,,x = T and Dm~,x = D*(T) and mainly depends on the fact that the constraints hold as equalities. (c) The optimal cost and delays D*(T) and T*(D) are decreasing functions. It follows readily from the results (a) and (b). (d) The optimal solutions of the C A P and D C A P are
the same if Tmax and Dm,,~ are related as follows:
CAP in packet-switching networks: E. D. Sykas Tm,,x =
T*(Dmax) o r Dm,x =
D*(Tm.x)
(24)
that is, for the optimal capacities C*(D), C*(T) one has C*(D*(T)) = C*(T) and C * ( T * ( D ) ) = C*(D) (25)
Convexity and uniqueness A second series of propositions follows if one supposes that, in addition to the previous assumptions, the functions di(Ci) are convex. Of course, the delay measure T(C) given by equation (2) is strictly convex forp/> 1. (e) The optimal delay and cost functions D*(T) and T*(D) are convex. This can be found by investigating the relationships between the optimal solutions of the CAP(DCAP) when Tm,~ is Ti, 7",_,or wT l + (1 - w ) ~ , O~
Separabifity results Since C t> f, instead of minimizing with respect to the capacities, one could use the excess capacities C, where: 0i = Ci - fi
(26)
Then note that for the delay measure one has:
T(C) = - -
and 7"(w0) -
?(0)
/x
(SCAP)
Minimize:/5(0)^ with respectto: C subject to: C>~ 0 and 7~(0) ~< Tmax
(SDCAP) Minimize: 7"(0) with respectto: C subject to: C>~ 0 a n d / 5 ( 0 ) ~< Dmax It should be noticed that identical results to those ones described previously hold for the SCAP and the SDCAP. (g) The solutions of the CAP (DCAP) and the SCAP (SDCA P) are closely related as shown below: C*(T) = f + O*(/xr)
(30)
C*(D) = / + 0*(D - Do(f))
(31)
D*(T) = D0(f) +/5*(/xT)
(32)
T*(D) -
?(O - Do(f)) /.z
(33)
where the functions 0*(T), 0*(D), /5*(T) and 7~(D) correspond to the C*(T), C*(D), D*(T) and T*(D) optimal functions respectively, but are defined for the SCAP and the SDCAP. (h) The optimal functions O*(T), O*(D), /5*(T) and T*(D), are given by:
7~,/5*( T) = D('F*( D) ),, = d *
(34)
TC*(T) =
(35)
C*(D) = c*
where d* is a constant number and c* a constant vector. Moreover, the constant c* is the optimal solution of the following problem: (MP) Minimize: b ( 0 ) , with respectto: C subject to: C > 0 and 7~(0) ~< 1 and d* is the corresponding minimum cost, that is, d* = D(c*). The proof is based on the application of propositions (b), (d) and (f) to a set of SCAP and SDCAP problems with specifically defined Tmax and Dmax. (i) The optimal functions C*(T), C*(D), D*(T) and T*(D) are given by: C*
C*(T) = / + / . t T
(36)
[o - D,,(r)l','",
w> 0
(27)
C*(D) = / + [
c77
.] c
(37)
w
If one expands the previous property to the cost functions, namely, if for the network cost function D(C) one has:
D(C) = D,,(f) + / 5 ( 0 )
(28)
where the excess cost function/5(0) satisfies the following relationship: /5(w0) = w"/5(0)
problems, the Simplified CAP (SCAP) and the Simplified DCAP (SDCAP) as follows:
a> 0
(29)
some simplifications of the problem are possible. First we define two simplified versions of the CAP and DCAP
d*
D*(T) = D,(f) + p. T" tO - O,,(f)J
(38)
(39)
Finally, note that the most general form of cost functions satisfying the assumption in equation (29) is: L
D(C) = D0(f) + Z d~O/
(40)
i=1
Appl. Math. Modelling, 1986, Vol. 10, October
349
CAP in packet-switching networks: E. D. Sykas In brief, the importance of the previous results is that one not only obtains explicit forms for the optimal functions, but also only a single minimization problem has to be solved in order to find them. That is, the solution of the MP is sufficient in order to find the optimal tradeoff curves for the CAP or DCAP. The fact that one can find the form of the optimal trade-off curve may facilitate the solution of other more complex design problems. In order to illustrate this, consider the Flow and Capacity Assignment Problem (FCAP), defined as follows: (FCAP) Given: the network topology and the traffic matrix Minimize: the network cost D(C) with respect to: the capacities C = (Cl, C2 . . . . . CL)' and the flows
f = ( f , , f 2 . . . . . fL)'
subject to: f<~ C, T(C) <~Tm,x a n d f i s a multicommodity flow satisfying the traffic requirement matrix. Assume that a minimax criterion for the delay measure holds, that is, p = oo and T(C) = max{T~} and the cost function D(C) is linear with respect to the capacities Ci, that is: L
Do(f) = Z
difi
i=l
Then, d* does not depend on the flows f/as it is readily obtained from MP, therefore, from equation (38) the minimization of the network cost D(C) reduces to the minimization of the function Do(f) with respect to f. Obviously, the optimal solution is a shortest-path flow. Thus, it has been proved that the optimal solution of minimax FCAP, with linear costs is a shortest-path flow, where the link costs are the coefficients d i. In practice, d~ is proportional to the physical length of a link, in this case, the optimal routing scheme follows the shortest, geographically, route connecting every pair of end nodes.
where 4,i(C) are positive scalar functions of the vector C, C I> f and are defined by the network cost function D(C) and the elimination procedure. For example, if: di( Ci) = 4C~'
then:
[ [ Dm.,x\ (ai(C)=~)(~)Q
_,,) i/2
where: L
(45)
Sifi
Q = Zi=1 (Ci-fi) 2 Employing any standard algorithm for the solution of systems of nonlinear equations, one can find the solution of the system of equation (43) and therefore the optimal solution of the DCAP. In the same way, a similar system of equations can be found for the solution of the CAP. Note that the form of equation (43) suggests that simple iterative schemes are also possible. For the special case when p = ~ in equation (7) one obtains a very useful design criterion, that is: T = max{T,}, r = 1,2 . . . . . R
xi -
1
(47)
TmaxP,(G - f/)
Then, the following problem is equivalent to the CAP: L (,, (NLMP) Minimize: d(x) = ~ d i + i=
1
with respect to: x
The uniqueness of the solution of the CAP (DCAP) and the fact that for the optimal solution the inequality delay (cost) constraint holds as equality allows one to use the classical method of Lagrange multipliers to find the optimal solution of the CAP (DCAP). For instance, after the application of the Lagrange multipliers method to the D C A P , one obtains the following system of L + 1 nonlinear equations:
subject to: x >t O, Ax <~1
OD /3"ff"~-/3di(Ci)
(C,
/J~i Tmax'
where A is a R × L matrix with the constants ari given in equation (6) as elements. The proposed algorithm is based on three theorems, proved in the Appendix, which are stated below and describe some useful properties which can be applied to every convex minimization problem with linear constraints. Theorem 1
Z di(Ci) = Omax i= l
/3 > 0
R
Si = ~ , w, a~iTp-1
(42)
r=l
Eliminating/3 from the previous equations one can obtain a new system, of L nonlinear equations, of the form: Ci =fi + cki(C), i= 1, 2 . . . . . L (43)
350
1)
i = 1, 2 . . . . . L (41)
L
where:
(46)
Then the previous procedure cannot be applied and special design algorithms are essential, since the approach by finding an approximate solution allowing p to assume large values is bound to fail because of overflow errors. An algorithm appropriate for this special case will be proposed later. First, one defines a new minimization problem equivalent to the CAP. Let us introduce a new variable x, which is the normalized link delay scaled by Trnax,that is:
Algorithms
Si
(44)
Appl. Math. Modelling, 1986,Vol. 10, October
Let f(x) be a continuous convex function defined on an open and convex set C c R L with continuous first partial derivatives on C. Let X be a convex subject of C and g(x) = x'Vf(x*) where x*eX. If g(x) >I g(x*) for every xeX, then the point x* is an optimal solution of the problem min f(x), with respect to x, xeX. Theorem 2 Let f(x) be a continuous convex function defined on an open and convex set C c R L with continuous first
CAP in packet-switching networks: E. D. Sykas partial derivatives on C. Let X be a C and g(x) = x ' V f ( x * ) where x * e X . point x * * e X such that g(x**) < g(x*) a w, we[0,1] such that f ( x ) < f(x*), + (1 -
convex subset of If there exists a then there exists where x = wx**
w)x*.
Theorem 3 Let f ( x ) be a continuous convex function on an o p e n and convex set C with continuous first derivatives. Let x* be an optimal solution of the p r o b l e m min f ( x ) with respect to x, x e X , where X is a convex subset of C. If g(x) = x ' V f ( x * ) "then g(x) >- g(x*) for every x e X . T h e a b o v e t h e o r e m s can be applied to the N L M P and then the following algorithm results. It is tacitly assumed that the objective function d(x) in the N L M P is convex. Obviously, this is the case if di(Ci) belongs to the p o w e r law or logarithmic line cost function families described previously. T h e algorithm given below can always be applied, but its c o n v e r g e n c e is g u a r a n t e e d only for convex objective functions d(x). Step 0: Initialization Set k = 0
x" -
whered=max
1
b' = Lg
J
(50)
The proposed algorithm always converges, because in accordance with Theorem 2, d k > d T M for every k. In order to prove that the sequence d k converges to d* and not to another point, first observe that if one applies the property that every strictly convex function lies above its tangent to the function f(w) = d((1 - w)xk + wx*) one obtains: d* >1 d k + gk(x*) -- gk(x~)
(51)
But gk(x *) 1> gk(zk), SO one has: 0 <. d k - d* <~gk(xk) -- gk(zk)
(52)
T h e r e f o r e , one need only show that g~(x k) - gk(zk) converges to 0. This is easy to p r o v e because: d ( w z k + (1 - w)x k) <~ d k + w(gk(z k) - gk(xk)) + 4 w2M
(53)
where M is an u p p e r b o u n d for the of the function f(w) = d ( w z k + (1 T h e n , by substituting the expression side of equation (53) by its m a x i m u m to w, one obtains:
second derivative - w)xk), we[0,1]. in the right-hand value with respect
d k - d T M I> [gk(zk) -- gk(xk)]2 > 0 (2M)
(54)
d _ a,
Obviously x" is a feasible solution, that is. x % X { x : x e R L , A x <~ l . x >! O}
=
Step 1 : Linear s u b p r o b l e m Find the optimal solution z k of the p r o b l e m : rain gk(x) = x ' V d ( x k ) , with respect to x, .reX
but, the sequence d k - d T M converges to 0, and therefore, so does gk(zk) -- gk(xk) and this c o m p l e t e s the p r o o f that the p r o p o s e d algorithm converges to the optimal solution. Finally, a simple e x a m p l e is given of the application of the p r o p o s e d algorithm to a n e t w o r k with L = 3 links, where: D ( C ) = d l C I + d2C 2 + d3C 3
d t ~ d2 ~ d3
Step 2: Stopping rule If ]gk(xk) -- gk(zk)l < e, where e is a positive tolerance, then Stop (x k is the optimal solution with such a tolerance), otherwise go to step 3.
1
(1) x °=(½,'~_,~)' '
Find the optimal solution w* of the p r o b l e m : m i n f k ( w ) = d ( w z k + (1 - w)xk), we[0.1].
(2) g°(x) = - 4 ( d . x , + dzx2 + d',,x3)
S e t x k+l= w*z k + (I - w*).r k k = k + 1, and go to step 1
if dl + d, > d3 then z" = (4., 4., ~)'
As a b y - p r o d u c t of the previous algorithm a lower bound for the o p t i m u m cost d* can be o b t a i n e d if d(x) is convex. T h a t is: k = 0, 1,2 . . . .
0
T h e n d(x) = d J x l + d2/x2 + d3/x3 and the solution steps ale as follows:
Step 3: N o n l i n e a r m a s t e r p r o b l e m
d k 1> d* t> b k
A=
(48)
where:
(3) g°(z°) = - 2 ( d r + d2 + d3) g°(x") = - 2 ( d l + d2 + d3) d o = 2(d, + d, + d3) b ~ = 2(d I + d, + d3)
(from e q u a t i o n (50))
T h e r e f o r e x* = x °. Hence:
d k = d(x k)
x*
and
(4.,'
"
d* = 2(d t + d2 + d3) b k = d e + gk(zk) -- gk(xk)
(49)
If the form of the function d(x) is known, other lower bounds can be found. For e x a m p l e , if D ( C ) satisfies the separability p r o p e r t y in equations (28) and (29) then a n o t h e r lower b o u n d is:
I f d 1 + d e ~< d 3 then z ° = (0,0,1)' (1) g ° ( z ° ) = - 4 d 3 g°(x°) = - 2 ( d l + d2 + d3) d t ' = 2(dr + & + d3)
Appl. Math. Modelling, 1986, Vol. 10, October
351
CAP in packet-switching networks: E. D. Sykas b " = (dr + d, + d 3)-" d3 2(dr + d_,) (2) f l ' ( , , , ) ~1- w
2d3 1+ w
- V ( 6 + d, |1'*
~
v-a;, + v T , + d,
..
=
(
+
in the minimization of average queue lengths instead of delays. Therefore, the generality and flexibility of the proposed delay measure allows the designer to tailor it so as to satisfy different objectives and particular design needs. The well-known example of the telegraph network examined in Reference 1 (pp. 22-23) is used in the following discussion since it has appeared in the literature -~.t2 in relation to different types of delay measures used in capacity assignment problems. In this example the network has N = 5 nodes and L = 7 links bt, b2• b3, ba, bs, b6, and b 7 connecting the node pairs 4-5, 3-4, 3-5, 2-5, 2-3, 1-3 and 1-2, respectively. The traffic matrix is symmetric and given by (in messages/s): 9.340
0.935
2.940
0.610-
L340
--
0.820
2.400
0.628
0.935
0.820
--
0.608
0.131
2.940
2.400
0.608
--
0.753
0.610
0.628
0.131
0.753
--
m
~
+ x/d~ + d2
(3) g'(,) = - ( v ~ , + x ~ - "
R=
dtxl + d2x ,. +x3 ) \dr+d_, dl+d, .t = (0 0 1)'
(4) gt(z') = - ( ~ / ~ + ~ d 2 ) ~g'(x') = - ( V ~ 3 + V ~ l + de)-' d' = ( V T , + b ~ = (v'-d73 + ,,/d~ + d,_) ~
Therefore x* = x ~. Hence:
(
~,fd, + d~
(V'-N3 °r- V ' d l
Numerical
A shortest-path deterministic routing scheme ~s used. Traffic from node vi to node v/, i, j = 1, 2 . . . . . 5, is always routed through the link i - j except for the following cases: (1) Traffic from v] to v5 is routed through v2 (2) Traffic from v] to v4 is routed through v3 (3) Traffic from w to v4 is routed through v5 This routing scheme results in the following loading of the network (one way traffic): (At. A: . . . . . AT) = (3.153, 3.548, 0.131, 3.638, 0.820, 3.875• 9.950)
#dr+d,
d* =
(55)
+ d2) 2
results
This section examines the performance of networks designed with the help of the proposed delay measure and compares it with that of networks designed with other previously proposed delay criteria. One should r e m e m b e r that the proposed packet delay measure in equation (7) is very general. In fact, the designer has three sets of parameters to cope with: the definition of the packet classes and therefore of the delays T,, r = 1, 2 . . . . . R, the weights w r, r = 1, 2 . . . . . R, and the exponent p. As was shown in an earlier section of this p a p e r all the previously proposed delay criteria result as special cases of equation (7) by appropriate choices of these parameters. New ones can also be defined as in case(c) of the section on a new packet delay measure. Moreover the proposed criterion is very flexible. For example, settingp = 1 one obtains a system optimizing criterion in the sense that the average network delay is minimized, while settingp = ~ one obtains a user optimizing criterion since the function to be minimized becomes more sensitive to the needs of the individual users. Also, choices of the type w, = 7rp result
352 Appl. Math. Modelling, 1986, Vol. 10, October
while the total traffic is A = A] + A2 + . . . A7 = 25.115. R = 10 classes of packets are distinguished according to their source-destination nodes. Classes 1-2, 1-3 . . . . . 1-5, 2-3 . . . . . 4-5 are n u m b e r e d as 1, 2 . . . . . 5, 6 . . . . . 10, respectively. Then, T, is simply the end-to-end delay across the network as seen by class r packets. Since all class r packets follow the same path, the constants ari in equation (6) are either 1 or 0 depending on whether the link bi is part of the path for class r packets or not, respectively, and are given as the elements of a R × L matrix A, where:
A=
-0
0
0
0
0
0
1-
0
0
0
0
0
1
0
0
1 0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
1 0
0
0
0
0
0
0
l
0
0
0
0
1
0
0
0
0
0
0
(56)
The weights w, in equation (7) are given by equation (8) where: (7~, 72 . . . . . 7t0) = (9.340, 0.935, 2.940, 0.610, 0.820, 2.400, 0.628• 0.608, 0.131• 0.753)
CAP in packet-switching networks: E. D. Sykas
I
200
150
150r
~100
~'100
i
3,4,5,6,9
1,3,4,6,7,10
? 50
50
OF I I
~,o 2
4
8
16
32 P
64
128
256
2,5,8,9
I 512 1024
Figure 1 Average delay T a n d end-to-end delays (1) T1, (2) T2, .... (10) 7-10versus p. Network is designed with proposed delay measure, equation (7), total network capacity is/~C = 192.0. Shortest path routing scheme described in text is used. Resulting traffic is 3.153, 3.548, 0.131, 3.638, 0.820, 3.875, 9.950 (messages/s) for links b 1, b 2. . . . . b 7, respectively
and 3' = 19.165. Finally, the line cost factors di are all considered equal to the unity. In other words, the overall network capacity is the cost function. The values of the average delay T a n d the end-to-end delays T~ for every packet class r, r = 1, 2 . . . . . 10, which result by designing the network with the proposed delay measure, equation (7), and the Meister et al. criterion, equation (14), respectively, for different values of the parameter p are presented in Table 1 and Table 2. The cost constraint is to use a total capacity /xC = 100.46 which results in a network utilization factor p = A/IzC = 0.25. It is interesting to note that in both cases the spread of the end-to-end delays T, decreases as p increases while the average delay increases slightly. Under the criterion in equation (14) the link delays TI =~ are all equal and then in the limit p = = the end-to-end delays T, are multiples of this constant quantity. The multiplication factor is the number of hops needed for class r packets to travel across the network. Under the proposed delay measure one sees a similar tendency in the sense that the end-to-end delays Tr are grouped in distinct sets, all Tr in a set having the same value. Except for special cases, it is not possible to achieve equality for all end-to-end delays T , r = 1, 2 . . . . . R, the reason being the limitations imposed by the routing scheme. If packets of a class have to follow a path that is a subset of the path for another packet class, then the delay for packets of the latter class will always be larger than the delay for packets of the former class. All one should expect is to limit the differences in delay among the classes as much as possible. The grouping of the end-to-end delays Tr in distinct sets is exhibited in Figures 1-3 and their corresponding final values are given in Table 3. The same network is used but with different routing schemes as indicated in the captions. It is clearly shown that the proposed delay measure is rather sensitive to the routing scheme. This behaviour should be expected since the definition of the packet classes depends on the paths they follow travelling through the network. Another interpretation of the previous results is possible if one defines the weighted standard deviation of the end-to-end delays T, given by:
01
I
I
I
I
I
2
4
8
16
I
I
I
I
I
32 64 128 256 512 1024 P Figure 2 Average delay T a n d end-to-end delays (1) T1, (2) T2, .... (10) 7"1oversus p. Network is designed with proposed delay measure, equation (7), and total network capacity is/~C = 192.0. A shortest path routing as in Figure 1 is used except that traffic from node v2 to v4 and from node v 1 to v5 is routed through v 3. Resulting traffic is 0.753, 5.948, 0.131,1.238, 3.220, 3.875, 9.950 (messages/s) for links b 1, b 2. . . . . b7, respectively
200
150
E _~ I00 3,4,6,9,10
T
50
8
o
2
4
1,7 8
~
16
32 P
64
128
2,5 256
512 1024
Figure 3 Average delay T a n d end-to-end delays (1) T1, (2) T2, .... (10) T10 class versus p. Network is designed with proposed delay measure, equation (7), and total network capacity is /~C = 192.0. A shortest path routing as in Figure 1 is used except that traffic from node v 1 to v 5 is routed through v3. Resulting traffic is 0.753, 5.948, 0.741, 0.628, 3.220, 4.485, 9.340 (messages/s) for links b 1, b 2. . . . . b 7, respectively
s=[Rr~=lWr(Tr-T)211/2
(57)
as the spread of the delays Tr. Then, the spread S is a measure of the uniformity of service over the various packet classes. A large spread in a network with a reasonable average delay performance indicates that some packet classes may have unncessarily short average delays while others suffer intolerably long delays. Generally, the shorter is the spread the greater is the uniformity of service attained. A spread S = 0 means that all end-to-end delays T~ are equal, that is, all packet classes receive the same service. As was explained before this is not always possible, but using the proposed delay measure the designer aims directly at reducing the spread as much as possible. The values of the spread for the three examined cases and for different loading
Appl. Math. Modelling, 1986,Vol. 10, October
353
CAP in packet-switching networks: E. D. Sykas 40
15C
30
I00
2O
50
I0
2
E -
3
I
I
I
I
2
4
8
16
I
I
I
I
I
32 64 128 256 512 1024 P Figure4 Spread S o f e n d - t o - e n d delays Trfor n e t w o r k of Figure 1 v e r s u s p. Curves f o r several average n e t w o r k utilizations: (1) p = 0.10, (2) p = 0.20, (3) p = 0.30, (4) p = 0.40 and (5) p = 0.50
200
2
4
8
16
32 64 128 256 512 1024 P Figure 7 Percentage increase in cost, f a c t o r k in e q u a t i o n (58), v e r s u s p for the n e t w o r k s of: (1) Figure 1, (2) Figure 2 and (3)
Figure 3
delay. To obtain a feeling for the penalties imposed on the overall network performance, the additional cost required if the average delay for the Tit I design (a system optimizing criterion) is to be retained in a T(m design (a user optimizing criterion) can be computed. The answer is not difficult to find if the separability results of the section on the mathematical formulation hold. From equation (37) it follows that the necessary percentage increase in cost is:
150 E % -~ I00 m
50
k I
I
I
I
I
2
4
8
32 P
64
I
i
128 256
I
512 1024
Figure 5 S p r e a d S o f e n d - t o - e n d delays Trfor n e t w o r k o f Figure 2 versus p. Curves f o r several a v e r a g e n e t w o r k utilizations: (1) p = 0.10, (2) p = 0.20, (3) p = 0.30, (4) p = 0.40 and (5) p = 0.50
2°°L 150
v
I
0
E
~-I00
,g 50 2
I
I
I
I
I
I
I
I
I
2
4
8
16
32 P
64
128
256
512
1024
1
-
D,,(f)
D
(58)
where the factor k depends on the network parameters but is independent of the cost D and given by: [average delay for T~) design]" k = L~delayfor T , ~ J - 1
(59)
The factor k can be interpreted as the additional cost percentage for a lightly loaded network, that is, when D is very large. A plot of the factor k versus the exponent p for the three examined networks in Figures 1-3 is shown in Figure 7. It is interesting to note that k is bounded and becomes almost constant as p increases. The two numerical techniques referred to in the section on algorithms were used for all computations. A modification of the Powell hybrid method '3 was used for the solution of the system of nonlinear equations, equations (43)-(45), The Jacobian was calculated by forward-difference approximation. Also, the algorithm described for the case p = ~ was slightly modified to improve its speed of convergence. In particular, a PARTAN step ~3 was added from iteration 2 and onwards, that is, an attempt to minimize f(x) in the direction x - xk_j is performed after step 3 (Nonlinear Master Problem) for k > 2.
Figure 6 S p r e a d S o f e n d - t o - e n d delays T, f o r n e t w o r k o f Figure 3 v e r s u s p. Curves f o r several a v e r a g e n e t w o r k utilizations: (1) p = 0.10, (2) p = 0.20, (3) p = 0.30, (4) p = 0.40 and (5) p = 0.50
Conclusions
levels of the network are shown in Figures 4-6. It is clear that the spread of delays drops abruptly as p increases and then remains almost constant. This reduction in the spread of the end-to-end delays Tr is achieved at the expense of the average packet
The capacity assignment problem in packet-switching communication networks was examined under general assumptions. A new packet delay measure, which includes as special cases all previously proposed ones, was presented. Several properties of the optimal solutions were discussed and an interesting application to
354 Appl. Math. Modelling, 1986, Vol. 10, October
CAP in packet-switching networks: E. D. Sykas Table 1 Average delay Tand end-to-end delays Tr, r = 1, 2 . . . . . 10 versus p. Network is designed with proposed delay measure, equation (7). Network utilization is p = 0.25
p
T
T1
T2
/-3
T4
Ts (ms)
7-6
7"7
7"8
T9
7"1o
1 2 4 8 16 32 64 128 256 512 1024
99.01 101.95 109.00 115.09 115.58 114.52 113.96 113.68 118.55 113.48 113.45
50.31 66.62 85.73 99.20 98.99 95.65 93.83 92.91 92.44 92.20 92.08
80.62 77.08 75.65 75.65 76.69 77.53 78.00 78.24 78.37 78.43 78.46
164.87 155.97 151.78 151.32 153.38 155.07 156.00 156.49 156.74 156.86 156.93
133.51 144.21 158.64 167.23 164.56 160.94 159.01 158.01 157.50 157.25 157.12
175.25 156.65 149.45 149.53 152.40 154.56 155.74 156.35 156.67 156.83 156.91
172.58 161.27 154.32 151.44 152.82 154.66 155.76 156.36 156.67 156.83 156.91
83.20 77.58 72.92 68.04 65.57 65.30 65.17 65.10 65.07 65.05 65.04
84.25 78.89 76.13 75.68 76.69 77.53 78.00 78.24 78.37 78.43 78.46
438.46 288.71 215.68 183.33 169.76 163.39 160.19 158.59 157.79 157.39 157.19
89.37 83.69 81.40 83.40 87.26 89.37 90.59 91.25 91.60 91.78 91.87
=
113.41
91.96
78.50
156.99
156.99
156.99
156.99
65.03
78.50
156.99
91.96
Table 2 Average delay Tand end-to-end delays. Tr, r = 1,2 . . . . . 10 versus p. Network is designed with Meister et al. delay measure, equation (14). Network utilization is p = 0.25
p
T
T1
7"2
7"3
7"4
7"5
7-6
T7
/-8
T9
7"10
1 2 4 8 16 32 64 128 256 512 1024
99.01 101.17 106.33 111.84 116.03 118.66 120.14 120.93 121.34 12t.54 121.65
50.31 59.83 70.10 78.87 84.98 88.67 90.71 91.79 92.34 92.62 92.76
80.62 81.92 84.65 87.58 89.83 91.24 92.04 92.46 92.68 92.79 92.85
164.87 166.29 170.81 176.03 180.12 182.73 184.20 184.99 185.40 185.60 185.71
133.51 143.49 155.83 167.07 175.14 180.09 182.84 184.30 185.05 185.43 185.62
175.25 137.47 115.49 104.08 98.42 95.64 94.27 93.58 93.24 93.08 92.99
172.58 171.41 173.94 177.81 181.09 183.23 184.46 185.12 185.46 185.64 185.72
83.20 83.66 85.73 88.20 90.16 91.42 92.13 92.51 92.71 92.81 92.86
84.25 84.37 86.16 88.45 90.30 91.49 92.16 92.53 92.71 92.81 92.86
438.46 253.36 166.66 127.60 109.63 101.11 96.96 94.92 93.91 93.41 93.16
89.37 87.75 88.22 89.61 90.92 91.81 92.33 92.61 92.76 92.83 92.87
~-
121.75
92.91
92.91
185.81
185.81
92.91
185.81
92.91
92.91
92.91
92.91
(ms)
Table 3 Average delay ~,, end-to-end delays Tr, r = 1, 2 . . . . . and spread S w h e n p = :~. Network is designed with proposed delay measure, equation (13). Total network capacity is ktC= 192.0. Results are for routing schemes of (1)Figure I, (2)Figure 2 and (3) Fig u re 3.
Case
T
T1
T2
T3
7-4
(1) (2) (3)
51.20 61.62 48.44
41.52 65.91 35.44
35.44 32.96 41.52
70.88 65.91 70.88
70.88 65.91 70.88
7-5
T6
T7
T8
T9
Lo
S
70.88 32.96 41.52
70.88 65.91 70.88
29.36 65.91 35.44
35.44 32.96 29.36
70.88 32.96 70.88
41.52 65.91 70.88
14.98 11.09 16.83
(ms)
the flow a n d c a p a c i t y a s s i g n m e n t p r o b l e m was given. F i n a l l y , a l g o r i t h m s for the s o l u t i o n were p r o p o s e d and n u m e r i c a l results for the case of a w e l l - k n o w n n e t w o r k e x a m p l e w e r e given and c o m p a r e d .
d~(G) Tr H.,' r
Yr
D( C) T( C) Nomenclature N L R
= n u m b e r of n o d e s = n u m b e r of links = n u m b e r of p a c k e t classes
1
= m e a n p a c k e t length (bits)
/z
nth n o d e ith link c h a n n e l c a p a c i t y o f link bi ( b i t s / s ) p a c k e t flow on link bi ( p a c k e t s / s ) bit flow on link bi ( b i t s / s ) Ci = C, - . ~ = excess c a p a c i t y on link hi (bits~s) Ti = a v e r a g e d e l a y on link bi (s) Li = a v e r a g e q u e u e length on link b~ ( p a c k e t s )
Vn
bi Ci Ai
= = = =
Y h p D*(T) T*(D) C*(T) C*(D)
cost of link bi a v e r a g e d e l a y o f class r p a c k e t s (s) weight for class r p a c k e t s class r p a c k e t arrival rate ( p a c k e t s / s ) total n e t w o r k cost delay measure average network delay total e x t e r n a l p a c k e t rate ( p a c k e t s / s ) total n e t w o r k traffic n e t w o r k utilization f a c t o r o p t i m a l cost function o p t i m a l d e l a y function o p t i m a l c a p a c i t y as a f u n c t i o n o f maximum permissible delay T = o p t i m a l c a p a c i t y as a f u n c t i o n o f m a x i m u m p e r m i s s i b l e cost D
= = = = = = = = = = = = =
References 1 Kleinrock. L. Communication nets-stochastic message flow and delay. IVlcGraw-Hill. New York, 1964
Appl. Math. Modelling, 1986, Vol. 10, October
355
CAP in packet-switching networks: E. D. Sykas 2 Komatsu, M., Nakamishi, H., Sanada, H. and Tezuka, Y. 'Extended optimum channel capacity assignment problems for message-switching networks', in 'Evolutions in Computer Communications' (ed. lnose, H.), North-Holland, Amsterdam, 1978 3 Meister, B., Miller, H. R. and Rudin, H. R. 'New optimization criteria for message-switching networks', IEEE Trans. Comm. Techn., COM-19, (3,256---260) June 1971 4 Protonotarios, E. N., Sykas, E. D. and Kappos, E. I. "Capacity assignment in packet switching networks: new optimization criteria, general results and algorithms'. Proc. IEEE huern. Conf. Circuits and Computers, ICCC'80, pp. 406--409, New York, NY, Oct. 1980 5 Rubin, I. 'The delay-capacity product for store-and-forward communication networks: tree networks', Applied Math. and Optim., 1976, 2, (3), 197-222 6 Jackson, J. R. 'Networks of waiting lines', Oper. Res., 1957, 5,518--521 7 Kleinrock, L. 'Queueing Systems, vol. I: theory', Wiley-Interscience, Chichester, 1975 8 Gitman, I. and Frank, H. 'Economic analysis of integrated voice and data networks: a case study', Proc. IEEE, 1978, 66. 1549-1570 9 Sykas, E. D. and Protonotarios, E. N. 'General results on the
capacity assignment problem in computer communication net10 11 12 13
works', Proc. 1EEE hltern. Syrup. on circuits and systems, ISCAS'82, Rome, Italy, May 1982 Mirtinovic, "Analytic inequalities', Springer-Verlag, 1970 Kleinrock, L. 'QueueingSystems. vol. II: computerapplications', Wiley-lnterscience, Chichester, 1976 Schwartz. M. 'Computer communication network design and analysis'. Prentice-Hall, Englewood Cliffs, NJ. 1977 Avriel, M. 'Nonlinear programming: Analysis and methods'. Prentice-Hall, Englewood Cliffs, N J, 1976
q(w) = f(wx** + (1 - w)x*) - f ( x * )
defined on [z,1], where z < 0 and zx** + (1 - z)x*eC. Since C is an open set, there exists such a n u m b e r z. By differentiation of q(w) one obtains:
dq(w) dw
- (x** - x*)'Vf(wx** + (1 - w)x*)
dq(O) dw
- g(x**)
- g(x*) < 0
Reference 11 one has:
f(x) >~f(x*) + (x - x*)Wf(x*) for e v e r y x e X (A1) Since g(x) >! g(x*) for every xeX, it follows that:
Since q(w) is a continuous, differentiable function, and: q(0) = 0
dq(O) dw
- - <
(A7) 0
(AS)
there exists an e > 0 such that:
q(w) < 0 for 0 < w < E
(A9)
Consequently, one has:
f(wx** + (1 - w)X*) < f ( x * ) f o r 0 < w < e (A10) Since X is a convex set, wx** + (1 - w)x*eX, 0 < w < e. Thus there exists a point Y:eX C_ C such that: (All)
Suppose that there exists an x ' e X such that:
g(x I) < g(x*)
(A12)
From T h e o r e m 2 it follows that there exists an 2eX such that:
f(ic) < f(x*) (A2)
Consequently from (A1) and (A2) one obtains:
f(x) >>-f(x*) for every xeX
(A6)
Proof of Theorem 3
Proof of Theorem 1 Since f(x) is a convex function from Corollary 4.23 in
(x - x*)'Vf(x*) >t 0 for every xeX
(A5)
Thus:
f(2) < f(x) Appendix
(A4)
(A3)
Hence x* is an optimal solution of the problem: min f(x), x e X
Proof of Theorem 2 Let q(w) be the function:
356 Appl. Math. Modelling, 1986, Vol. 10, October
(A13)
contradicting that x* is an optimal solution of the problem: mi.n f(x)
xeX
Consequently, there exists no x e X such that:
g(x) < g(x*)
(A14)
Hence g(x) >>-g(x*) for every xeX. That is, x* is an optimal solution of the problem: min g(x)
xeX.