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A Dynamic Flow Assignment Problem for a Telephone Trunk Network
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A DYNAMIC FLOW ASSIGNMENT PROBLEM FOR A
TELEPHONE TRUNK NETWORK J. Filipiak 1 1/~lIt/ltl
11/
(. 1,"11111.
\,,11111, 1-." 11;:'111111111 .::'
IIlId
"1"1 ·11 '/1111111111111((1 11111/\,
")().f}=i<) l\' U!l;f/;L
,·\ ((u/nll.'
fI/
,\l l tIIl/g
([1If/
,\l dol/Uf,!..,}·,
Po/and
Abstract. The dynamic flow approach to analysis and synthesis of control in a telephone trunk network is presented. The individual circuit trunk is modelled as a nonlinear dynamic system. The trun k description serves as a basis for the construction of t he net work model. The class of homeostatic network mod els i s defined in whi c h the deadlock periods do not occur. For the homeostat.ic network with congestion control and adaptive routing the dynamiC flow assignment algorithm maximizing the revenue production rate, defined by the steady-state costate solution of optimal control problem is given. The algorithm belongs to the class of shortest route algorithms. Keywords. Telecommunication network, queueing models, nonlinear control systems, flow control, routing, homeostatic behavior, steady-state costate solution.
INTHODUCTI ON
proposed. At the first stage the optimal flow pattern is found and then, at the second stage, this flow pattern is used to define the procedures of call processing at nodes. In this paper we discuss in detail the first stage optimization problem.
The past decade has witnessed a s i g-
nificant development of different switching techniques. The circuit switching is still dominating the design of communication networks, however. The general problem concerning the exploitation of circuit switching network is how to operate it so as to 6~arantee the efficient utilization of network channels under a ll conditions of traffic load. Various partial tasks related to that basic question are formulated which have been recognized as forming a multi-layer hierarchy presented in Table 1. Its hi ghes t layer concerns the network eng ineering . At present, there is a vast and well established literature on the synthesis of the optimal network structure. Thus, we assume that the corres ponding procedure has been performed and in consequence we have at our disposal the network of d efinite structure. The question which we tend to examine in this paper is how to manage the traffic within the existing network. It is intuitively clear that in the case of multinode networks the construction of th e model comprising all events in the system,i.e.,the model describing in detail processing of all individual calls is difficult. Instead a two stage procedure is usually 681
Recently, a relatively new chapter of the telephone network traffic analysis has been devoted to that class problems. Camoin (1976), Hennet and Titli (197 9) , Garcia and Hennet (1980), Garcia, Hennet and Titli (1 981) consider the flow a ss ignment in a stationary as well as a nonstationary
TABLE 1
I
Hierarchy of Network Problems
Time horizon
Network problem
Long-term planning Average-term Im anagement Short-term loperation
Network dimensioning Flow assignment Processing of individual calls
J. Filipiak
682
environment. A similar approach is presented also by Demidovic and Malasenko (1980). This paper is a sequel to those stud ies. Using a balance equation and statistics for a stationary state we approximate the dynamic trunk behavior. Then we construct the network model and investigate its global stability conditions. Finally, the model is used to find the traffic assignment algorithm. The general methodology present ed in this paper has been applied previously to the optimization of dynamic flows in s-f networks ( Filipiak, 1982 ) •
dependence fout=#(X). Intuitively, it is clear that it is increasing and has the form depicted in Fig.1. However, for heavy loads switches can be kept
).l(X)
1
I
"f"i-l v I I
DESCRIPTION OF A CURCUIT TRUNK
In the simplest case of an infinite channel group with Poisson arrivals and exponential service times the trunk averages are described by the equation ( Feller, 196 1 ) : i(t) = -x(t) + f.In (t) (1) where x(t) is the number of calls processed at time t (also, it is the intensity of traffic leaving the system), and f . (t) is the intensity of offered traff!B in Erlangs. If we assume the finite server, e.g., the r·l/M/m s ys tem without queue, an arr iving entity is accepted unless all c hannels are taken. Denote by fc(t) and fret ) flow intensities of carried and r e jected traffic, respectively. S inc e the increase of input traffic causes the nonlinear increase of the overflow we de scribe the sys tem with t he nonlinear differential equation: x(t) = -p.(x(t)] + fin(t )
(3a)
x
c
m Fig.1
x
Typical trunk characteristic m=number of channels
busy with call attempts which do not complete and do not result in connection. That phenomenon degrades the throughput as shown in Fig.2, and significantly changes qualitative behavior of the overall network. In such cases, before the optimization procedure can be applied the network stability must be analysed.
,.ue x) degraded curve
(2)
where ,u, [x(t )J=fout(t ) is the total intenSity of traffic leaving the system. We have ,u,[x (t )]=fc(t )+fr(t ) . Since the tra ffic intensity is measured in Erlang s t hus fc (t )=x (t ) and fr(t)= ,u.[x(t )]-x(t ) . Consequently, the complet e model is:
x
f =X
I
The basic element of a telephone trunk network is a multichannel link with an access through the line-switching system. Its model is called here a line switched concentrator.
}lex) -
Y ! r =)l(x)-x
( 3b)
( 3c)
Now the trouble consist s in finding
m
Fi g .2
x
Degradation of throughput due to the overload of the switch
Example To illustrate the general considerations of this s ection we shall find the detaile:i form of model (3) for the simplest possible case of the J·1/M/m system without queue. The stationary s tate of the ~ /M/m system is described by the Erlang B formula:
Dynamic Flow Assignment Problem
m!
p (m, A)
(4)
where: p=loss probability, A=intensity of offered traffic, m=number of channels. For stationar~ state we have f.l.n =f ou t=A, and x= l1-p(m,A)]A. Thus X=[1-p(m,fout)]fout • Unfortunatelv. we need a reverse dependence and it can be obtained merely by an approximation.\n example of approximation function is x for x )lex) =
~
683
pting and routing calls. Based on those new possibilities different control actions are proposed. In this paper we assume that at nodes: a) the adaptive routing may be implemented, i.e., a decision on where the traffic is to be forwarded may be taken according to the actual network state; b) the congestion control may be executed, i.e., it may be determined at nodes how much traffic shall be carried in relation to the offered traffic. The model of the network with congestion control and adaptive routing is developed as follows.
m/2
Network configuration
) m/2-aln[2( m-x) /mJ
for x ) m/2 ( 5)
with a as a parameter. In order to confirm the validity of model (3) with function )..L(') specified by (5) simulation experiments has been carried for a twelve channel primary group ( fin (t)= A·1( tl, a=7 ) • The results are shown in Fig.3. The mean has been calculated from 50 simulation runs. NETWORK MODEL Present-day telephone systems equipped with electromechanical switches offer limited possibilities as regards traffic control. The computerized switching centers of the future, however, allow one a great flexibility of acce-
It is defined that: N={i,j, .•• } is the collection of nodes, L= {(i, j), (j,k), ••• } is the collection of unidirectional links, O(j)= {(j,k): (j,k) E:L} , l(j)={(i,j):(i,j)EL} • 'Ehe destination node shall be denoted by q. Route R~={j=j1, •.• ,jk=l} from node j to 1 is a sequence of k nodes so that (jn' jn+1 )(L. Loop is a route from j to j. Link (m,n) is downstream from (i,j) if there exists a loop-free route R~ passing through nodes j and m. We denote (m,n) ( (i, j). Dynamic flow pattern Since in our approach we consider averages, the congestion control and
o o
o o
CD
o o
A=6.
W~ I-
ITo 1-0 (f)~
o
o
- APPROXIMATION .SIMULATION
N
o
o
o-r----f---~----,---_,r----r----~--~----_.----r_--~
-0.50
0.50
1.50
TIME Fig.3.
LS S-W
2.50
3.50
Dynamic flow model of the twelve channel primary group
J. Filipiak
684
routing are defined by: ~j(t)= congestion control variable defining which portion of traffic r. generated in node J j and destinedto q may be admitted to the network; ~jk(t)= routing variable which defines the portion of total traffic incoming to j routed to trunk (j,k) • According to these definitions:
o o
~CPj(t) i
i
d
jk
(6a)
j EN
1
(t) .s. 1
(Gc) jEN ~ ~'k(t) = o (j) J Consider now the traffic rejected at links (j,k)EO(j). We shall assume that fixed portion ?j of that traffic is reiterated with some reiteration delay D., measured in units of call duration. J Denote by x. a number of calls wal. t.lng J for the re-entry. Thus, the intensity of retries equals x./D . • Consequently, J J the balance equation for reiterated traffic is given by:
J
J
L
J
(7)
Lx.. +X . /D . J 1 J J J
Substituting that expression to Eq.3a written separately for each trunk we get J
J
J J
1o {LN
LX .. +X J. /D J. J
I(j) lJ
(j,k)eL with constraints (6).
b.[
f . k = cl . k [ Cf· r . + J J J J I( j)
,u. k + 0:: . k [ cp . r . +
J =
(1-l(!')
Now, we are in a position to complete the model. The amount of traffic entering node j and routed to trunk (j,k) totals
=-
In order to describe the performance of the telephone network various criteria have been taken into account. The typical examples are (Gimpelson, 1974) : a) traffic handling capacity or completion rate; b) grade of service; c) trunk occupancy; d) revenue production rate. To include all these indices the objective functional is defined as follows
J
jEN
xJ.k
DYNAMIC FLOW PATTERN
(,u'k-x'k)'
J O( j)
J
Obviously, in the homeostatic network the deadlock periods do not occur. The following are the simple conditions sufficient for homeostasis: a) the network is loop-free; b) functions ,ujk(') are single-valued and increasing.
(6b)
(j,k)EL
X.= -x./D. + VZ·
Network ~ =. y(~,~,~,t) is homeostatic if it has only one equilibrium which is asymptotically stable.
(8)
Network Homeostatic Behavior It has been observed that due to the fluctuations in nonlinear systems the jumps between equilibria occur which manifest themselves as deadlocks. That phenomenon has been investigated extensively for random access systems. In a telephone network setting a similar problem has been discussed by Nakagoma and Mori (1973). In the presented approach it is formalized as follows. Denote by ~jk the right-hand side of Eq.8, and by ~. the right-hand side of J Eq.7, respectively.
-
~
J J
L
O(j)
(,lL ' k-X'k)J J
J
ajkx jk } dt
(9)
where: 'l'jr j is the volume of accepted traffic, (1-l2j) ('ujk-Xjk) describes the grade of service, and x jk corresponds to a trunk occupancy. If the coefficients b j and a jk are interpreted as economical factors (e.g., b . J is treated as a charge at node j, and a jk is the unit cost of exploiting trunk (j,k)) the overall integral is proportional to a revenue production rate. Formally, the dynamic optimization problem is to find control variables t
~
-
L
- (1-V;j) -
~
LN
b.[rD .r . J ~J
J
~j)()ljk-Xjk)J- ~
ajkx jk
Pjk{-fljk+djk[Cfjr j + frj
t
ij
Dynami c Flow Assignment Probl em
~ Pj (
+ x/Djl}-
L
+
-X/Dj ( 10)
"l°()J.jk-Xok)] J
O( j)
H(~*,;e",.!f·,S!.,t) =
H( ~~ ,;e'l: ,~,!£., t )
( 11 )
!!,!£. and Pjk=-~H/OXjk' (j,k)E L,
( 12a)
p o =-~H/dx o , J J
( 12b)
jEN
The maximization of the Hamiltonian may be decomposed into nodes as follows: H(~·,;e.,~,!£.,t)
max
L
(13)
J
where ~j={djk:(j,k)EO(j)}, and H(~~,:2·,~,!£.,t)=
max Cfj
:L
max [bo-
O(j )
J
cp o
c:ljk >0 and Pjk > P j Thus: )
J
Pjk=P j
(15a)
implies c:(jk=O.
( 15b)
only i f
= (1-1'1O(m )(dl1 rmn /dx mn -1)
Property. The dynamic flow pattern induced by the steady-state costate solution Pjk=O' Pj=O,is such that at each node j all used routes to destination have the same length P j and are shorter than the unused routes:
J
Q(R3k) = Pj
(19)
where R3k denotes the used route from j to q, whose first trunk is (j,k) • Proof. From (12 ) for PJok=O and Pj =0 we have b j (1-?j )( dJlj k/ dx jk- 1 ) + a jk - P okd}.J.jk/dx ok +
c:(JokPJok = PJo
J
( 16) +
Consequently, from (14) it follows that:
erJ0=1
for
Pj < b j
( 17a)
Cf'j=O
for
Pj ) b j p o= b o J J
( 17b)
OSCfji1
n
P Okd ok]Cfjrj
(14) Define Pj=inf {Pjk: (j,k)e:L} • Taking into account constraint (6c) we get from (13) that
L
The length Q of route Rqj is given by Q(R q )= L (18) wmn (m,n)e:R~ j J where b 0 +a 1 w = m mn mn mn 1+ <5mn (k, 1) > (m, n) 1+S kl
J
A
o( j
The steady-state costate solution of optimal control problem is defined to be one in which both Pjk and Pj equals zero. Before we consider the implications of that solution the route length is defined first.
omn
p °kdjk
O(j)
ifj
Steady State Costate Solution
and
=
~j
= max
considered problem can hardly be found. Instead, in the remainder of this paper we consider the steady state solution of costate equations.
J
where Pjk and Pj are costate variables. According to the Pontryagin's maximum principle on the optimal time path, we have:
= max
685
for
A
( 17c)
The optimal control consists in the integration of state and costate equations (15 ) and (17 ) . Since for the infinite time horizon the Pontryagin,s principle does not state additional constraint s on costate variables the general solution of the
J
L rf..klPkl O(k )
Pj I'2j ( dfJjk/ dX jk- 1 ) = 0 ( 20a)
and
( 20b ) For the used trunk (j,k) we have Pjk=Pj • Moreover
O~ )
c:(klPkl = Pk
J. Filipiak
686
Now, if trunk (k,l) is used then Pkl =~k . That allow us to repeat the same reasoning as for (j,k ) • Movin g down the route along the used trunks we &et (1 8) •
.
proved in the previous section. If m trunks ( j,k)EO(j) are to be used, and moreover 0< 'Pj < 1, then at j:
Finally, since Pjk=Pj and the procedure applies to any route from j to q we get the result.
Q(R
) = b j ,(j,k)€O(j) (21) jk Differentiating Eqs.21 with respect to time we get
Conge s tion Control
( u,v)ER jk (k,l»(u,v)
Let us impose now the additional requirements on coefficients b o and ao k • J J Namely, we s hall require that at the steady state the exploitation of each trunk is profitable, i.e., that for each fOk and xok=U , xo= O, the trunk J J J profit is positive:
(dWuv/dxkl)xkl = 0
~
- ajkx jk >0 where fjk is the flow intensity on trunk (j,k). It is easy to check that this requirement implies bj-b k > a jk • Assume further that none node is shut-down. Under those additional constraints we shall prove the following theorem. Theorem. I f bj-b k > a jk ' (j,k)E:L, and Cfo> 0, jEN, then 'P 0=1 for each load, J J i.e., the congestion control variables are inactive. Proof. Assume that 0 < 'Pj < 1 . Accoruing to (17) cpo is different than 1 .. J only if Po 2 b 0 • However, from (20), J J taking into account (19) , we get 1
..
+ - - ( a 0k+Pk) 1+0 1+0 J jk jk Moreover, since 'Pk > 0 then Pk ~ bk • Consequently
~ + -1- ( a 0k +b ) P 0< b 0 k J J 1+ 0 1+d J jk jk Finally, substituting ajk+b k < b j we get po < b 0 and that yields the J J contradiction. A
ALGORITHM Now we demonstrate how to calculate the routing and congestion control variables uSing property of the steady-state costate flow pattern
(22)
Substituting from state equations for x we get m linear algebraic kl equations for m+1 unknown variables. The additional equation is the constraint O( j )
Ojk
L
I::
dJok= 1
(23)
That reasoning results in the following algorithm. 1•
2.
3.
4. 5.
Calculate the length wO k of each trunk. J Moving up the routes from the destination label each node j with distance ~o • J At each node calculate Q(R ) jk for all output trunks; if Q(R jk ) =Pj ~ b j then trunk (j, k) is to be used. For each node solve equations (22) and (23) • Integrate state equations. CONCLU::lION
In this paper the dynamic flow model of the telephone trunk network has been presented. Using that model the nonstationary flow pattern induced by the steady state solution of costate equations has been found. The properties of the steady-state costate flow pattern have been exa~ined, and the corresponding algorithm of calculating the routing and congestion control variables has been presented. The algorithm belongs to the class of shortest route algorithms.
Concluding the results of this paper it is to be stressed that since for the implementation of the presented procedure the current estimates of traffic intensities and network state are required it may be difficult to do on-line calculations. However, during the normal exploitation the
Dynami c Flow Ass i gnment Problem
daily traffic demand patterns are fixed and thus th e application of the solution for preplanned rerouting is very t entative. Es pecially, the benefits should be gained in such classical applications as the noncoincidenc e of busy hours in different countri e s or in business and re s idential districts of a city. REF EIlliNCES Ca moin, B . ( 1976) . f.lodeHe analytique de traffic dan s un re s eau telephonique. Ann. Teleco mmunic., 11, 23 9- 267 . Demidovic, 0 .1., and J . E. Malasenko ( 1980 ) . A model f or n e twork flow c ontrol. Tekh.Kibern., No. 6 , 86 - 90 ( Engli s h tran s lat i on in: Eng . Cybe rn., USA ) . Fe ller, W. ( 196 1 ) . An Int r oduction to Probability Theory and Its Applications, Vol.1, 2nd ed. Wiley, New York . Chap.17. Fili piak , J . (1 982) . Optimal control of s t or e- a nd-fo r ward n etwork s. Optim. Control Appl.Methods, L, 155 -17 6 .
687
Garcia, J.M., and J.C. Hennet (1980). Models and control structure for a large telephone network. Proc. 2nd IFAC Symp.Large Scale S~ Pe r gamon Press, pp.4 87-495. Garcia, J.M., J.C. Hennet, and A. Titli (1981). Optimization of routing in interurban telephone n e tworks. Large ~cale Sfstems, 1, 257-267. Gimpelson, L.A. (974). Network manag ement: Design and control of communication networks. El.Comm., 49, 4-22. Hennet,J.C., and A.Titli (1979). Decentralized control of telecommunications networks. In M.Singh, A.Titli Eds. , Large Scale Systems Engineering Applications, North-Holland, pp. 353-371. Nakagoma, Y., and H. Mori (1973) • Flexible routin g in the global communication networks. Proc. ITC-7, Sto c kholm, 1973. -----
V
I F .\(
\, ',11', 1\\ . P ~d;llld
CO\I\ll·:\IL\TIO:\ .\:\[) TR .\:\SJ>()RT .\TI():\ S\SI E\IS
i..11".gC ."'Lilt· S\ ,It' n l,
I ~I.'''': : \
A DYNAMIC FLOW ASSIGNMENT PROBLEM FOR A
TELEPHONE TRUNK NETWORK J. Filipiak 1 1/~lIt/ltl
11/
(. 1,"11111.
\,,11111, 1-." 11;:'111111111 .::'
IIlId
"1"1 ·11 '/1111111111111((1 11111/\,
")().f}=i<) l\' U!l;f/;L
,·\ ((u/nll.'
fI/
,\l l tIIl/g
([1If/
,\l dol/Uf,!..,}·,
Po/and
Abstract. The dynamic flow approach to analysis and synthesis of control in a telephone trunk network is presented. The individual circuit trunk is modelled as a nonlinear dynamic system. The trun k description serves as a basis for the construction of t he net work model. The class of homeostatic network mod els i s defined in whi c h the deadlock periods do not occur. For the homeostat.ic network with congestion control and adaptive routing the dynamiC flow assignment algorithm maximizing the revenue production rate, defined by the steady-state costate solution of optimal control problem is given. The algorithm belongs to the class of shortest route algorithms. Keywords. Telecommunication network, queueing models, nonlinear control systems, flow control, routing, homeostatic behavior, steady-state costate solution.
INTHODUCTI ON
proposed. At the first stage the optimal flow pattern is found and then, at the second stage, this flow pattern is used to define the procedures of call processing at nodes. In this paper we discuss in detail the first stage optimization problem.
The past decade has witnessed a s i g-
nificant development of different switching techniques. The circuit switching is still dominating the design of communication networks, however. The general problem concerning the exploitation of circuit switching network is how to operate it so as to 6~arantee the efficient utilization of network channels under a ll conditions of traffic load. Various partial tasks related to that basic question are formulated which have been recognized as forming a multi-layer hierarchy presented in Table 1. Its hi ghes t layer concerns the network eng ineering . At present, there is a vast and well established literature on the synthesis of the optimal network structure. Thus, we assume that the corres ponding procedure has been performed and in consequence we have at our disposal the network of d efinite structure. The question which we tend to examine in this paper is how to manage the traffic within the existing network. It is intuitively clear that in the case of multinode networks the construction of th e model comprising all events in the system,i.e.,the model describing in detail processing of all individual calls is difficult. Instead a two stage procedure is usually 681
Recently, a relatively new chapter of the telephone network traffic analysis has been devoted to that class problems. Camoin (1976), Hennet and Titli (197 9) , Garcia and Hennet (1980), Garcia, Hennet and Titli (1 981) consider the flow a ss ignment in a stationary as well as a nonstationary
TABLE 1
I
Hierarchy of Network Problems
Time horizon
Network problem
Long-term planning Average-term Im anagement Short-term loperation
Network dimensioning Flow assignment Processing of individual calls
J. Filipiak
682
environment. A similar approach is presented also by Demidovic and Malasenko (1980). This paper is a sequel to those stud ies. Using a balance equation and statistics for a stationary state we approximate the dynamic trunk behavior. Then we construct the network model and investigate its global stability conditions. Finally, the model is used to find the traffic assignment algorithm. The general methodology present ed in this paper has been applied previously to the optimization of dynamic flows in s-f networks ( Filipiak, 1982 ) •
dependence fout=#(X). Intuitively, it is clear that it is increasing and has the form depicted in Fig.1. However, for heavy loads switches can be kept
).l(X)
1
I
"f"i-l v I I
DESCRIPTION OF A CURCUIT TRUNK
In the simplest case of an infinite channel group with Poisson arrivals and exponential service times the trunk averages are described by the equation ( Feller, 196 1 ) : i(t) = -x(t) + f.In (t) (1) where x(t) is the number of calls processed at time t (also, it is the intensity of traffic leaving the system), and f . (t) is the intensity of offered traff!B in Erlangs. If we assume the finite server, e.g., the r·l/M/m s ys tem without queue, an arr iving entity is accepted unless all c hannels are taken. Denote by fc(t) and fret ) flow intensities of carried and r e jected traffic, respectively. S inc e the increase of input traffic causes the nonlinear increase of the overflow we de scribe the sys tem with t he nonlinear differential equation: x(t) = -p.(x(t)] + fin(t )
(3a)
x
c
m Fig.1
x
Typical trunk characteristic m=number of channels
busy with call attempts which do not complete and do not result in connection. That phenomenon degrades the throughput as shown in Fig.2, and significantly changes qualitative behavior of the overall network. In such cases, before the optimization procedure can be applied the network stability must be analysed.
,.ue x) degraded curve
(2)
where ,u, [x(t )J=fout(t ) is the total intenSity of traffic leaving the system. We have ,u,[x (t )]=fc(t )+fr(t ) . Since the tra ffic intensity is measured in Erlang s t hus fc (t )=x (t ) and fr(t)= ,u.[x(t )]-x(t ) . Consequently, the complet e model is:
x
f =X
I
The basic element of a telephone trunk network is a multichannel link with an access through the line-switching system. Its model is called here a line switched concentrator.
}lex) -
Y ! r =)l(x)-x
( 3b)
( 3c)
Now the trouble consist s in finding
m
Fi g .2
x
Degradation of throughput due to the overload of the switch
Example To illustrate the general considerations of this s ection we shall find the detaile:i form of model (3) for the simplest possible case of the J·1/M/m system without queue. The stationary s tate of the ~ /M/m system is described by the Erlang B formula:
Dynamic Flow Assignment Problem
m!
p (m, A)
(4)
where: p=loss probability, A=intensity of offered traffic, m=number of channels. For stationar~ state we have f.l.n =f ou t=A, and x= l1-p(m,A)]A. Thus X=[1-p(m,fout)]fout • Unfortunatelv. we need a reverse dependence and it can be obtained merely by an approximation.\n example of approximation function is x for x )lex) =
~
683
pting and routing calls. Based on those new possibilities different control actions are proposed. In this paper we assume that at nodes: a) the adaptive routing may be implemented, i.e., a decision on where the traffic is to be forwarded may be taken according to the actual network state; b) the congestion control may be executed, i.e., it may be determined at nodes how much traffic shall be carried in relation to the offered traffic. The model of the network with congestion control and adaptive routing is developed as follows.
m/2
Network configuration
) m/2-aln[2( m-x) /mJ
for x ) m/2 ( 5)
with a as a parameter. In order to confirm the validity of model (3) with function )..L(') specified by (5) simulation experiments has been carried for a twelve channel primary group ( fin (t)= A·1( tl, a=7 ) • The results are shown in Fig.3. The mean has been calculated from 50 simulation runs. NETWORK MODEL Present-day telephone systems equipped with electromechanical switches offer limited possibilities as regards traffic control. The computerized switching centers of the future, however, allow one a great flexibility of acce-
It is defined that: N={i,j, .•• } is the collection of nodes, L= {(i, j), (j,k), ••• } is the collection of unidirectional links, O(j)= {(j,k): (j,k) E:L} , l(j)={(i,j):(i,j)EL} • 'Ehe destination node shall be denoted by q. Route R~={j=j1, •.• ,jk=l} from node j to 1 is a sequence of k nodes so that (jn' jn+1 )(L. Loop is a route from j to j. Link (m,n) is downstream from (i,j) if there exists a loop-free route R~ passing through nodes j and m. We denote (m,n) ( (i, j). Dynamic flow pattern Since in our approach we consider averages, the congestion control and
o o
o o
CD
o o
A=6.
W~ I-
ITo 1-0 (f)~
o
o
- APPROXIMATION .SIMULATION
N
o
o
o-r----f---~----,---_,r----r----~--~----_.----r_--~
-0.50
0.50
1.50
TIME Fig.3.
LS S-W
2.50
3.50
Dynamic flow model of the twelve channel primary group
J. Filipiak
684
routing are defined by: ~j(t)= congestion control variable defining which portion of traffic r. generated in node J j and destinedto q may be admitted to the network; ~jk(t)= routing variable which defines the portion of total traffic incoming to j routed to trunk (j,k) • According to these definitions:
o o
~CPj(t) i
i
d
jk
(6a)
j EN
1
(t) .s. 1
(Gc) jEN ~ ~'k(t) = o (j) J Consider now the traffic rejected at links (j,k)EO(j). We shall assume that fixed portion ?j of that traffic is reiterated with some reiteration delay D., measured in units of call duration. J Denote by x. a number of calls wal. t.lng J for the re-entry. Thus, the intensity of retries equals x./D . • Consequently, J J the balance equation for reiterated traffic is given by:
J
J
L
J
(7)
Lx.. +X . /D . J 1 J J J
Substituting that expression to Eq.3a written separately for each trunk we get J
J
J J
1o {LN
LX .. +X J. /D J. J
I(j) lJ
(j,k)eL with constraints (6).
b.[
f . k = cl . k [ Cf· r . + J J J J I( j)
,u. k + 0:: . k [ cp . r . +
J =
(1-l(!')
Now, we are in a position to complete the model. The amount of traffic entering node j and routed to trunk (j,k) totals
=-
In order to describe the performance of the telephone network various criteria have been taken into account. The typical examples are (Gimpelson, 1974) : a) traffic handling capacity or completion rate; b) grade of service; c) trunk occupancy; d) revenue production rate. To include all these indices the objective functional is defined as follows
J
jEN
xJ.k
DYNAMIC FLOW PATTERN
(,u'k-x'k)'
J O( j)
J
Obviously, in the homeostatic network the deadlock periods do not occur. The following are the simple conditions sufficient for homeostasis: a) the network is loop-free; b) functions ,ujk(') are single-valued and increasing.
(6b)
(j,k)EL
X.= -x./D. + VZ·
Network ~ =. y(~,~,~,t) is homeostatic if it has only one equilibrium which is asymptotically stable.
(8)
Network Homeostatic Behavior It has been observed that due to the fluctuations in nonlinear systems the jumps between equilibria occur which manifest themselves as deadlocks. That phenomenon has been investigated extensively for random access systems. In a telephone network setting a similar problem has been discussed by Nakagoma and Mori (1973). In the presented approach it is formalized as follows. Denote by ~jk the right-hand side of Eq.8, and by ~. the right-hand side of J Eq.7, respectively.
-
~
J J
L
O(j)
(,lL ' k-X'k)J J
J
ajkx jk } dt
(9)
where: 'l'jr j is the volume of accepted traffic, (1-l2j) ('ujk-Xjk) describes the grade of service, and x jk corresponds to a trunk occupancy. If the coefficients b j and a jk are interpreted as economical factors (e.g., b . J is treated as a charge at node j, and a jk is the unit cost of exploiting trunk (j,k)) the overall integral is proportional to a revenue production rate. Formally, the dynamic optimization problem is to find control variables t
~
-
L
- (1-V;j) -
~
LN
b.[rD .r . J ~J
J
~j)()ljk-Xjk)J- ~
ajkx jk
Pjk{-fljk+djk[Cfjr j + frj
t
ij
Dynami c Flow Assignment Probl em
~ Pj (
+ x/Djl}-
L
+
-X/Dj ( 10)
"l°()J.jk-Xok)] J
O( j)
H(~*,;e",.!f·,S!.,t) =
H( ~~ ,;e'l: ,~,!£., t )
( 11 )
!!,!£. and Pjk=-~H/OXjk' (j,k)E L,
( 12a)
p o =-~H/dx o , J J
( 12b)
jEN
The maximization of the Hamiltonian may be decomposed into nodes as follows: H(~·,;e.,~,!£.,t)
max
L
(13)
J
where ~j={djk:(j,k)EO(j)}, and H(~~,:2·,~,!£.,t)=
max Cfj
:L
max [bo-
O(j )
J
cp o
c:ljk >0 and Pjk > P j Thus: )
J
Pjk=P j
(15a)
implies c:(jk=O.
( 15b)
only i f
= (1-1'1O(m )(dl1 rmn /dx mn -1)
Property. The dynamic flow pattern induced by the steady-state costate solution Pjk=O' Pj=O,is such that at each node j all used routes to destination have the same length P j and are shorter than the unused routes:
J
Q(R3k) = Pj
(19)
where R3k denotes the used route from j to q, whose first trunk is (j,k) • Proof. From (12 ) for PJok=O and Pj =0 we have b j (1-?j )( dJlj k/ dx jk- 1 ) + a jk - P okd}.J.jk/dx ok +
c:(JokPJok = PJo
J
( 16) +
Consequently, from (14) it follows that:
erJ0=1
for
Pj < b j
( 17a)
Cf'j=O
for
Pj ) b j p o= b o J J
( 17b)
OSCfji1
n
P Okd ok]Cfjrj
(14) Define Pj=inf {Pjk: (j,k)e:L} • Taking into account constraint (6c) we get from (13) that
L
The length Q of route Rqj is given by Q(R q )= L (18) wmn (m,n)e:R~ j J where b 0 +a 1 w = m mn mn mn 1+ <5mn (k, 1) > (m, n) 1+S kl
J
A
o( j
The steady-state costate solution of optimal control problem is defined to be one in which both Pjk and Pj equals zero. Before we consider the implications of that solution the route length is defined first.
omn
p °kdjk
O(j)
ifj
Steady State Costate Solution
and
=
~j
= max
considered problem can hardly be found. Instead, in the remainder of this paper we consider the steady state solution of costate equations.
J
where Pjk and Pj are costate variables. According to the Pontryagin's maximum principle on the optimal time path, we have:
= max
685
for
A
( 17c)
The optimal control consists in the integration of state and costate equations (15 ) and (17 ) . Since for the infinite time horizon the Pontryagin,s principle does not state additional constraint s on costate variables the general solution of the
J
L rf..klPkl O(k )
Pj I'2j ( dfJjk/ dX jk- 1 ) = 0 ( 20a)
and
( 20b ) For the used trunk (j,k) we have Pjk=Pj • Moreover
O~ )
c:(klPkl = Pk
J. Filipiak
686
Now, if trunk (k,l) is used then Pkl =~k . That allow us to repeat the same reasoning as for (j,k ) • Movin g down the route along the used trunks we &et (1 8) •
.
proved in the previous section. If m trunks ( j,k)EO(j) are to be used, and moreover 0< 'Pj < 1, then at j:
Finally, since Pjk=Pj and the procedure applies to any route from j to q we get the result.
Q(R
) = b j ,(j,k)€O(j) (21) jk Differentiating Eqs.21 with respect to time we get
Conge s tion Control
( u,v)ER jk (k,l»(u,v)
Let us impose now the additional requirements on coefficients b o and ao k • J J Namely, we s hall require that at the steady state the exploitation of each trunk is profitable, i.e., that for each fOk and xok=U , xo= O, the trunk J J J profit is positive:
(dWuv/dxkl)xkl = 0
~
- ajkx jk >0 where fjk is the flow intensity on trunk (j,k). It is easy to check that this requirement implies bj-b k > a jk • Assume further that none node is shut-down. Under those additional constraints we shall prove the following theorem. Theorem. I f bj-b k > a jk ' (j,k)E:L, and Cfo> 0, jEN, then 'P 0=1 for each load, J J i.e., the congestion control variables are inactive. Proof. Assume that 0 < 'Pj < 1 . Accoruing to (17) cpo is different than 1 .. J only if Po 2 b 0 • However, from (20), J J taking into account (19) , we get 1
..
+ - - ( a 0k+Pk) 1+0 1+0 J jk jk Moreover, since 'Pk > 0 then Pk ~ bk • Consequently
~ + -1- ( a 0k +b ) P 0< b 0 k J J 1+ 0 1+d J jk jk Finally, substituting ajk+b k < b j we get po < b 0 and that yields the J J contradiction. A
ALGORITHM Now we demonstrate how to calculate the routing and congestion control variables uSing property of the steady-state costate flow pattern
(22)
Substituting from state equations for x we get m linear algebraic kl equations for m+1 unknown variables. The additional equation is the constraint O( j )
Ojk
L
I::
dJok= 1
(23)
That reasoning results in the following algorithm. 1•
2.
3.
4. 5.
Calculate the length wO k of each trunk. J Moving up the routes from the destination label each node j with distance ~o • J At each node calculate Q(R ) jk for all output trunks; if Q(R jk ) =Pj ~ b j then trunk (j, k) is to be used. For each node solve equations (22) and (23) • Integrate state equations. CONCLU::lION
In this paper the dynamic flow model of the telephone trunk network has been presented. Using that model the nonstationary flow pattern induced by the steady state solution of costate equations has been found. The properties of the steady-state costate flow pattern have been exa~ined, and the corresponding algorithm of calculating the routing and congestion control variables has been presented. The algorithm belongs to the class of shortest route algorithms.
Concluding the results of this paper it is to be stressed that since for the implementation of the presented procedure the current estimates of traffic intensities and network state are required it may be difficult to do on-line calculations. However, during the normal exploitation the
Dynami c Flow Ass i gnment Problem
daily traffic demand patterns are fixed and thus th e application of the solution for preplanned rerouting is very t entative. Es pecially, the benefits should be gained in such classical applications as the noncoincidenc e of busy hours in different countri e s or in business and re s idential districts of a city. REF EIlliNCES Ca moin, B . ( 1976) . f.lodeHe analytique de traffic dan s un re s eau telephonique. Ann. Teleco mmunic., 11, 23 9- 267 . Demidovic, 0 .1., and J . E. Malasenko ( 1980 ) . A model f or n e twork flow c ontrol. Tekh.Kibern., No. 6 , 86 - 90 ( Engli s h tran s lat i on in: Eng . Cybe rn., USA ) . Fe ller, W. ( 196 1 ) . An Int r oduction to Probability Theory and Its Applications, Vol.1, 2nd ed. Wiley, New York . Chap.17. Fili piak , J . (1 982) . Optimal control of s t or e- a nd-fo r ward n etwork s. Optim. Control Appl.Methods, L, 155 -17 6 .
687
Garcia, J.M., and J.C. Hennet (1980). Models and control structure for a large telephone network. Proc. 2nd IFAC Symp.Large Scale S~ Pe r gamon Press, pp.4 87-495. Garcia, J.M., J.C. Hennet, and A. Titli (1981). Optimization of routing in interurban telephone n e tworks. Large ~cale Sfstems, 1, 257-267. Gimpelson, L.A. (974). Network manag ement: Design and control of communication networks. El.Comm., 49, 4-22. Hennet,J.C., and A.Titli (1979). Decentralized control of telecommunications networks. In M.Singh, A.Titli Eds. , Large Scale Systems Engineering Applications, North-Holland, pp. 353-371. Nakagoma, Y., and H. Mori (1973) • Flexible routin g in the global communication networks. Proc. ITC-7, Sto c kholm, 1973. -----