One moment model for telephone traffic

One moment model for telephone traffic F. Le Gall Laboratoire d'Automatique et d'Analyse des Syst&nes 7, avenue du Colonel Roche, 31400 Toulouse, France [Received January 1982)

This paper deals with the derivation of an analytical model for telephone networks in order to evaluate the performance of a routing policy and to determine optimal dynamic policies. For that purpose, the model must be accurate but sufficiently tractable. A one moment model is proposed, which is an analytical model based on the calculation of input rates and output rates (functions of the control parameters of the system) and on the determination of the blocking probabilities of the trunk groups through the Erlang loss formula introducing, for that purpose, fictitious offered traffics. Numerical experiments are given to show the model's performance. Key words: telecommunication networks, mathematical modelling, birthdeath process, one moment model Introduction

Probabilistic m o d e l l i n g

So far, telephone network behaviour was somewhat 'fixed': the call progression in the network was determined a priori. But thanks to the introduction of micro-processors and information transmission, call routing can now be made much more flexible. We can think of dynamic routing improving the efficiency and reliability of telephone networks. Then, the problem consists of determining an optimal dynamic routing strategy. This is still an open question in the area of circuit-switching networks due to its relative youth. Some results have been obtained by simulation of routing algorithms 1° or from the theory of learning automata. a'9 But to the best of our knowledge there is no analytical model which permits the determination of an optimal routing policy for large networks. Indeed, many investigations have been done concerning the evaluation of the grade of service in a network. 1,2,4-7,11-13 But these algorithms are not available for the determination of a dynamic optimal policy because of their procedure (step by step or iterative) or of the burden of the computation. The aim of this paper is to contribute to the derivation of an analytical model in order to test the performances of a network (end-to-end service quality ...) and to determine an optimal routing policy. We describe the most accurate (but unrealistic) method of characterizing the network under classical assumptions (Poisson distributed arrivals, exponential negative service time). Then we show how to derive an aggregated model of the mean from the results of the previous section. This is illustrated with some examples and results are given to show the performance of the model.

A telephone network can be represented by a transport graph with source nodes 1, sink nodes J, and transit nodes K. These nodes are linked by trunk groups with finite capacities. Calls are routed through the network according to a routing policy: when a free connection (.path or sequence of trunk groups) can be established between an origin I and a destination J, the call remains in the system for a random conversation time. Otherwise, the call is lost. The following hypotheses are made:

0307-904X/82/060415-09/$03.00 © 1982 Butterworth & Co. (Publishers) Ltd.

H1 : the inputs/J of the network (calls originated at I and destinated to J ) are Poisson distributions with parameter kz.i H2: the service time is exponential negative (mean r) H3: only one event (call arrival or call termination) can occur in a small interval of time. Under these assumptions, the telephone process is a discrete birth-death process, s The possible events, birth or death of a call IJ, affect only one path I, . . . , J in the network, namely the path which has been chosen by the routing policy to connect the call IJ: the discrete state space of the system is then the set of integers representing the instantaneous number of calls holding on the different paths connecting the input nodes I to the output nodes J. This set of integers completely describes the system. The considerations lead us to a probabilistic model of the system, the random variables representing the instantaneous number of calls holding on the different paths I . . . . ,J.

Appl. Math. Modelling, 1982, Vol. 6, December 415

Mode/for telephone traffic: F. Le Gall Suppose there are L origin-destination paths in the graph. Let us denote:

The resolution of system (1) provides:

iv(t) (p = 1, . . . , L): the instantaneous state of path p at

aggregating the system, we can compute the marginal probabilities of the instantaneous states of the different paths and trunk groups; in particular the end-to-end blocking probabilities and the blocking probabilities of each trunk group

the probability of instantaneous state I(t) E a¢

time t (number of calls holding on path p)

iv(t ) depends of course on the routing policy (choice of the connections between I and J ) and on the structure of the network (capacities of the trunk groups of the path)

we can also compute the average occupation of the paths and trunk groups and the grade of service of the network.

iv(t ) can take a f'mite number of values: iv(t ) E {0, 1,2, . . . , i~nax}

Unfortunately, this modelling is time consuming (even impossible for large networks) because the combinatory of the system becomes tremendous as the size of the graph (values of the capacities and number of nodes) grows. This is why we have to look for a more realistic model.

l(t) = (il(t) . . . . . it(t))': the instantaneous state vector of the network at t (where A' denotes the transposed vector of A)

I(t) E a¢" = (to . . . . , Ii . . . . . /max} Jdescribes the combinatory of the system, i.e. the set of possible values of I(t), taking the constraints of capacities in the graph into account. So, ,kt depends on the structure of the network (number of nodes and capacities of the trunk groups) and can have a tremendous number of elements as the size of the graph grows.

P(t) denotes the probability vector of the instantaneous state of the system: P(t) = (Po(t), . . . , Pi(t) . . . . . pmax(t))' with Pi(t) l~ probability (l(t) = Ii), i = O , . . . , max. /i(t ) ~ dP(t) dt The dynamic of the system is then given in terms of probabilities: /i(t) = Q(X, r, Jff, u ) P

(1)

The elements qq (i = 0, . . . , m a x ; / = 0 . . . . . max) of matrix Q are the rates of transition from state li E ~ to state

D y n a m i c m o d e l o f the m e a n The mean occupancies of the different trunk groups in the network are adequate variables to describe the network; indeed, these variables have a physical meaning since trunk groups do exist. Moreover, the number of trunk groups in a network is smaller than that of the paths (e.g. for a twolevel graph (Figure 1 ) with S source nodes, T transit nodes and P sink nodes, there are (S + P ) . T trunk groups and S.P. T paths between the sources and the sinks). Of course this model is an aggregate one and we lose some exact information concerning the system (end-toend blocking of a path) but we see that, introducing realistic assumptions, we can, from the knowledge of the characteristics of the trunk groups (mean occupancy, blocking) completely describe the system (end-to-end grade of service, efficiency) with fair accuracy. First we will detail the aggregation procedure in the case of a single trunk group and then generalize it to the case of a whole network.

1iEJ.

Ifli and 1/are not adjacent states then according to assumption H3, qii = 0. Otherwise, qq depends on:

A g g r e g a t e d m o d e l f o r a single t r u n k - g r o u p

the inputs of the system X = ( . . . , Xzs. . . . )' the mean service time r the characteristics of the network .A'(finite capacities of the trunks between the nodes in the network)

Let us consider a single trunk group with Poisson arrival (parameter X) and capacity N (Figure 2). The control variable u can be omitted here because of the unicity of the path (one trunk group) between the origin I

the routing policy u (choice of the sequence of trunks groups connecting I to J in the graph).

1 1

If •

Ii=li+

,

1

(0 . . . . , 0 1 0 , . . . , 0 ) ' '1' in column k, k = 1 . . . . ,L

then

•

qq = ~ij(~, JCc,u)

•

(2)

If /j = I i - - (0 . . . . . 010 . . . . )' '1' in column k, k = 1 . . . . . L then

•(t) qii--

T

S k= l

T

.1

416 Appl. Math. Modelling, 1982, Vol. 6, December

Figure 1

P

Model f o r telephone traffic: F. Le Gall k

I

N

©

J

define at t i m e t, a fictitious offered traffic y ( t ) such t h a t :

©

X(t) = y(t). (1 -- £(y(t), N))

Figure 2

X(t) mean carried traffic in the transient state and the destination J. If there is an idle circuit in the trunk group (i.e. if the instantaneous state of the trunk group is equal to 0, 1,2 . . . . . N - - 1), the call enters the system. Otherwise, it is rejected. System (1) is written: /i = Q(X, 1", N ) e

P~(t)

E(y(t),

f((t)

= --

X(t)

+ X. (1

--P~r(t))

T

(3)

e~(t)

E(y(t), N )

=

(6)

X(t) = y(t). (1 -- E(y(t), N))

(/~o (t) . . . . . PN(t))' = Q(X, 1",N).(po(t) . . . . . PN(t))' with Pi(t): probability of i circuits occupied at t. Writing down the transition rates qq of Q is straightforward:

can be solved by integration from the initial conditions. P~(t) and PN(t) have been plotted for different values of k and N (one example is given in Figure 3). P~v(t) is an accurate estimate Of PN(t ). For t ~ ~ :

. . . . ,N}

x(~)

= 0

Y(~) = xr

e~(oo) = pN(oO) = e ( X r ,

qq = 0 if i q~{.]-- 1,],/+ 1}

N)

So a single trunk group can be accurately characterized in the steady and in transient states by its mean occupancy and its blocking probability.

Xif0~
qi, i+l

N)

The system:

or more explicitly:

i,/e{O

=

i(t) qi, i - I = qu

1"

Aggregated model for a network

i(t))

X* + ~

=-

Consider now a general network. Its probabilistic representation per path is given by:

X+ = 0 i f i = N

I"

=Xif0~
P(t)

Let us denote:

=

Q(X, r, Jf~,u).P(t)

Let us define xk(t ) the mean number of calls holding on path k:

N X ( t ) = ~. ipi(t ) i=o

k -- 1, . . . , L ;x(t) -- (Xl(t) . . . . . XL(t))' The dynamic o f x k ( t ) is given by:

the mean occupancy of the trunk group. Aggregating the dynamic system of the probabilities (P = QP) we obtain the dynamic of the mean:

2k(t) =

xk(O

+ ek(x, k, % u~, u)

(7)

T

x(o

ek(x, k, r, .~, U) = X~-(1 --ek(t)) in which:

)((t) = e(X, X, r, N) -- - -

T

k~ depends on the inputs k and the command u

That is, the variation of the mean is equal to the input rate minus the output rate. Let us explicitly state the input rate: In the steady state: e(X, X, r , N ) = X-(1 --PN(=)) PN(O°): blocking probability in the steady state of the trunk group with Poisson input and exponential negative service time.

Pk(t) is the end-to-end blocking probability of path k and has no general expression. It depends on the capacities of the different trunk groups of path k, of k and u. Suppose path k is the sequence of trunk groupslkTx, T~T2 . . . . , TkJk (Figure 4). A call can connect IJ if there is at least one idle circuit on each trunk group of path k, i.e.:

PN(OO): is given by the well-known Erlang loss formula: 10-

(X1-)N/N!

PN(OO) = - -

N (xry

Y' i=0

-- E(X1-, N )

(5)

wl

0 *

8

P.

i!

So X('~) = h r - ( 1 --E(Xr, N)): mean carried traffic in the steady state

e

£

X~=18 N= 20

¢~ e ¢-

4

X1- is the mean offered traffic in the steady state. In the transient state: e(X, X, r, N ) = X. (1 - - P N ( t ) ) There is no simple analytical expression of PN(t) but we can provide an accurate estimate P~(t)ofPN(t) extending the use of the Erlang loss formula in the transient state as follows:

0

N

2 o 200

400

I

I

CoO0

800

Time, (s) Figure 3

A p p l . Math. Modelling, 1982, Vol. 6, December 417

Mode/for telephone traffic: F. Le Gall T1

IK

T2

0

0

TK

O-

<3

JK

0

Figure 4

see (Figure 5). When a call (l,J) arrives he is sent to route k i with probability a~j (/J). If there is at least one busy trunk-group on this path the call is rejected. So that :

(10)

~k~(l,J) = ~Ij'~k(IS) • )kij

(1 --Pk(t)) = Q(lkTl and T1T2 . . . . and TkJk)

kob(t) =

X~(t) + - -

the right term of the equation designing the probability that each trunk group has at least one idle circuit. In a large network, we can assume the ~ndependence of the trunk groups blocking on a path because of the dispatching of the traffic through the network. So:

X

with:

1"[

(l --PAB(t))

To determine PAS(t), we introduce a fictitious offered traffic YAB(t) such that:

Z

E

Yl~(t)~

k(2,J)e~/b

PAB blocking probability of trunk group AB.

1"I (1--PAB(t)) ABEk

and + ~'~ ]"I (I --PAB(t))

(8)

ABEk

OlfJ(I'J) xIJT

Overflow (Figure 7) Overflow consists of: Testing the occupancy of the first trunk of a first path

Let us denote:

kt(l,J ) (first choice route). If this trunk has at least one

An the set of paths including trunk group AB

idle circuit then the call is sent to this path.

XAB(t ) the mean occupancy of trunk group AB: XAB(t ) = ~

Otherwise (i.e. if the first trunk of the first choice route is busy), the call is routed by a second choice path k2(I, J)

Xk(t)

(Figure 6).

k E ~FA B

x ( o = (...,XAB(t) . . . .

For the first choice route:

)'

X~,(I, J ) = a~ (l'J) • ~.H

Summing equations (8) gives the dynamic of the trunk groups: f(ab(t) = --Xab(t------)+ E 1"

h~ I"[ (1 --PAB(t))

kebab

For the second choice route:

(9)

•k,(l,J)*

---- e~J(I'~OX/JPIT ,,

(12)

AB~k

For each trunk group, the variations of the mean occupancy is equal to the input rate minus the output rate. The problem is to determine the input rate, i.e. ;k~ and the blocking probabilities o f the different trunk-groups

(PAB(t)). These quantities depend mainly on the routing policy. We examine two elementary cases, namely load sharing and overflow, the superposition of these two policies then being straightforward.

Ct K1

XIJJ•

( K~ )

~ K2 IJ

(K2)

•

I

Figure 5 of the input rate

T,2

Load sharing

(K2)

Load sharing consists in sharing the input f l o w s / J on different routes k~ (I, J), k2(I, J) . . . . with parameters ak,(/,J) ,~k~(Y,J) IJ

( I--PAB(O)

A B e { k(l, J) - Ib }

The traffic offered to trunk group Ib is smooth because of the blocking of the trunk groups all along the path: the blocking of the smooth traffic is computed through the Erlang loss formula introducing a fictitious Poissonian offered traffic the mean of which is smaller than the mean of the actual smooth traffic.

So:

Determination

(ll)

~'[

olkj(IJ) )klJ T

k(l,J)e .2;Fib

= 1--PAB

T

'

ABCk(/,J)

Yn,(t) =

Q(AB) = probability of at least one idle circuit on trunk group AB

ek(x,X,r , .¢g',u)=~.~

5". ~k¢~J).~IJ k(l,J)e ~rab ~IJ

T

CAB(0 = E(YAB(0, NAB) NAB capacity of trunk group AB. In the steady state:

= Q ( I k r , ) ' . . . " a(TkJk)

xk(t)

-

XAB(t ) = YAB(t)(1 -- PAB(t))

Q(IkT1 and TIT2 . . . . and TkJ,.,)

YCk(t) =

-

(K~)

. such that:

' ~IJ ' "" E o ki(l'J) -iJ ki(1,J)

1

418 Appl. Math. Modelling, 1982, Vol. 6, December

Figure 6

Mode/for telephone traffic: F. Le Gall

IT] : first trunk o f the first choice route. We must be careful at the computation of the call blocking of the first trunk IT~ of the second choice route: this blocking is in fact a conditional blocking, a call being offered to IT~ if and only ifITl is busy. The call blocking of trunk group IT~ is then: N1

PlTtzandlT~ P ~ r ~/rr ', =

Figure 7

Prr~

In the case of overflow:

gab(t) gab(t) = -- - -

I

+

E

4Jl.J)x,j

K

E[ (1 --PAB(t)) AB~k(Z,J)

+

j

•

tx/j I (I'])XH (PtT',--PIT', and ITS)

•

•

,

•

l(1,J)/2(/,J)~ ~ab

x (1--PT~T])...(1--PT~J)

(13)

The first summation concerns the first choice routes and the second one the second choice routes. Blocking probabilities PAn are computed as described in the case o f load sharing. To compute PAB and ¢0, we consider links AB and CD together and introduce a fictitious offered traffic )rAn, CD(t) such that:

gAB(t) -Jr XCD(t ) = YAB, CD(t)( 1--PAB a n d CD(t))

Figure 8

I

PAB and CO(t) = E(YAB, CD(t),NAB + NCD) (14) So, we can compute the input rate in the case of load sharing and of overflow, the case o f mixed policy being then straight forward.

K

Some applications of the method We now illustrate the procedure with two examples: an unidirectional network with a mixed policy and then a bidirectional network with a mixed policy.

UnMirectional network with a mixed policy The network is represented in Figure 8. It is composed o f source nodes I connected to transit nodes K by upstream trunk groups IK. The transit nodes K are connected to sink nodes J by downstream trunk groups KJ. Calls are originated at I and terminated at J. The mixed policy is defined as follows: First: for each flow XIj making a load sharing on routes

IKJ with parameter ~J'Ka' such that: ~. OIlKJ = 1 K c~IKjXIjr is rejected by trunk IK with probability PIK. Second: for alKSXtJrPtK (overflow from trunk group IK destinated to J ) making a load sharing on routes IkJ, k :/:K, with parameter On(ks such that:

Figure 9

Appl. Math. Modelling, 1982, Vol. 6, December 419

Model for telephone traffic: F. Le Gall

The dynamic of the system is given by:

~, ~IKkJ = 1 kqK

XIK ) ( / K ---- - - - -

(see Figure 10). Writing down the dynamic of the system is now straightforward:

"(

+ E

+ E (OtIKi;kli + OtirI;kil)(l - - P r o ) ( 1 - - P / x ) i E

{OIkKiOtlki;kli(PIk--PIKandlk)(1--PiK)

i kCK XIK = -- XIK + E OlIKJ;klJ(1-- PIK) (1 -- PKJ) ~" j Jr- Z

E

+~ikKlOtikI;kiI(Pik--PiK and ik)( 1 --PIK)}

(18)

PIK = E(YIK,NIK)

fJIkxJOIIkS;klS(elk "LPIK a . d I k )

Plrandlk = E(YIKk, NIK + Nlk)

J kV=K

x (1 -- PKJ)

XIK = YIK( I -- PIK)

(15)

XKJ = -- XK'--~J+ ~. OqKg;kly(1 -- PIK)(1 -- PKJ) 1" I + ~. ]E OIkXJOtIkJ;kld(Plk--PIx and Ik) I k~eK

X m + X/k = Ymk(1 -- P m a.d Ik)

The first summation concerns first choice traffic (from I to i or from i to I ) and the second one second choice traffic (from I to i or from i to I). The efficiency of flow Ii is given by:

X (1 -- PKj ) eli = E OtlKi(1--PIK)(1--PiK) K

The first summation concerns the first choice traffic and the second summation the second choice traffic; with:

+ E fllkKiOtlki(Plk--eIkandlK)(1--PiK) k--kK

PIK = E(YlK,NIK)

Prj= E(yKj, Nm)

The global loss of the system is given by:

PIK and Ik =E(YlKk, NIK + Nlk) Xzr

=

1

Loss

YIK(1--PIK) since

XIK + Xlk = YIKk( 1 --PIK and Ik)

XIK = ~. (Iki holding calls) + T. (iKI holding calls)

The efficiency of flow H is given by:

i

els = T. a l K j ( I - - P I K ) - ( I - - P K j )

i

~'. XIK = 2 • (IKi holding calls) LK, i

K

1,K

+ ]E ~IkKJalkJ(Ptk --PIk and IKX 1 -- PKJ) k#=K (16) The global loss of the system is given by:

Loss = E ; k / j r - - E XIK = E ;kzjr-- E XKJ (17) l,J I,K I,J K,J Bidirectional network with a mixed policy The network is represented in Figure 9. It is composed of nodes K, connected by bidirectional trunk groups IK. Calls are originated at a node 1, transited by K, and are terminated at I'. Calls can progress from I to K or from K to 1: the trunk groups are bidirectionals. The routing policy is the same as the one proposed in the case of a unidirectional network.

(K1) J

f [3IKr K1j

•

So, one can then characterize the system by: mean occupancy of trunk groups blocking probabilities of trunk groups end-to-end blocking global loss These two examples are of course nonexhaustive. The model can easily be extended to networks with several transit-levels or to more general policies (e.g. sequence of overflows, i.e. testing at least three routes in sequence for a given call). Knowing the structure of the graph and the routing policy, writing down the equations of the dynamic of the trunk groups of the network is straightforward, in the light of remarks earlier in this paper. In all the cases, the model can be solved by integration from initial conditions (X(0), P(0)) or using relaxation methods in the steady state. If we are interested only in the ' steady sti~te, the latter method is faster than the former and quite adequate: it consists of an iterative procedure on y(t) (fictitious offered traffic) which converges quickly (less than ten iterations for a precision of 10 -3 for network with more than 10 nodes, i.e. less than 5 s CPU on IMB 370/168). Numerical experiments

I~IKF

KJ • •

I o.75 ~

= F XHr - 2 IXKXzr LJ

XKj = YKj(1--PKJ)

0-

(19)

~ ~

( K F)

Figure 10

420 Appl. Math. Modelling, 1982, Vol. 6, December

The accuracy of the method was tested for different policies on various networks. The results were compared to those obtained by the Markov model in the case of very small networks (resolution of system,6 = QP) and to eventby-event simulations for larger networks. In Tables 1-8

Model for telephone traffic: F. Le Gall Table 1 Network of Figure 11. Capacity of the trunk group IK

Table 6 Mean occupancies of the trunk group XlK in the steady state

2

3 2

1 2 3 4

20 25 20 35

25 20 20 15

30 15 35 10

16.04

1

Table 2 Network of Figure 11. Capacity of the trunk group KJ

2

j ~ ~

1

2

3

3

1 2 3 4

20 20 35 25

20 15 25 20

10 35 15 30

21.24 I

+0.4%

16.13

20.64

20.47 I I --2.9% 19.88

15.66 15.36

16"46 I

17"111

--3.0%

15.96 29.25

4

3

--2.8%

--1.4%

--1.9%

"--2.9%

11.85 I I --2.7% 11.53

29-43I --1.7% 28.92

12.47[

8.49

11.83

8.37 --5.0%

--5.3%

28.83

--0.5%

25.16

16.62

I

25.29 I

Simulation model/Relative error

Table 3 Matrix of offered traffic ~lJ~" (Erlang) Table 7 Mean occupancies of the trunk grou p XKj in the steady

1 2 3 4

0 17 20 12

2

3

4

18 0 30 16

24 15 0 30

30 22 21 0

state

Table 4 CZlKJ parameters

j ~

1

2

17.26 I I --1.9% 16.94

15.97

1

16.14

2

I

--3.1% 15.47

1

12 13 14 21 23 24 31 32 34 41 42 43

1 0 0 0 0 1 1 0 0 0 0 1

2

3

28.35

3 0 1 0 1 0 0 0 0 1 0 1 0

0 0 1 0 1 0 0 1 0 1 0 0

4

--1.6%

12.05

J

28.04

20.40

20.46 I I -1.2% 20.20

16.80

2

3

0 0 1 0 1 0 0 1 0 1 0 0

1 0 0 0 0 1 1 0 0 0 0 1

0 1 0 1 0 0 0 0 1 0 1 0

r

--1.5%

I

--1.9%

12.26 26.36

--1.8%

16.50

25.86

Table 8 End-to-end grade of service

1

1

--2.9%

12.46

--5%

2

3

4

Table 5 ~IKLJ parameters L

I

28.61

Simulation model/Relative error

/~1

~

+0.7%

27.76

21.47

-1.1%

7.75 I 7.81

12.24

--3.2%

15.61

i j ~

3

0.874 I

--

--0.3%

0.871

0.872

0.887 --2.0%

0.869

--2.9%

0.846

,KJ

112 123 134 221 233 214 311 332 324 431 422 413

2

3

0.916 I I --3.6% 0.883 0.819

I

--2.4%

0.799 4

0.815 0.768

0.840

0.952]

0.914 --3.6%

--

0.810

0.882 --2.2%

--5.1%

0.931

[

--5.8%

0.826I 0.787

--2.5% 0.891

0.837 0.987 --4.7%

--1.5% 0.923

Global loss: . Model: 35E/Simulation: 30E Simulation model/Relative error

A p p l . Math. Modelling, 1982, V o l . 6, December

421

M o d e / f o r telephone traffic: F. Le Gall Table 9 Network of Figure 12. Capacities NIK of trunk groups IK

I

J

1

1

K

35 42 30 33

1

2

2

2 3 4

2

3

33 35 41 39

39 39 24 60

Table 10 Offered traffic (Erlang) hlJ'r

4

4

Figure 11

1

0

2 3 4

18 8 30

2

3

4

10 0 20 24

12 12 0 10

16 20 22 0

= 202E

I,J

Table 11 Percentage load sharing parameters (etltKi 2X 100)

1 1 1 2 2 2 3 3 3 4 4 4

3

3

2 3 4 1 3 4 1 2 4 1 2 3

~1, KI= = 3 k=l

1

2

3

32.71 34.48 31.43 32.71 33.71 30.84 34.48 33.71 32.26 31.43 30.84 32.26

30.84 37.93 31.43 30.84 39.32 32.71 37.93 39.32 41.93 31.43 32.71 41.93

36.45 27.59 37.14 36.45 26.97 36.45 27.59 26.97 25.81 37.14 36.45 25.81

min (NIIK, NI2 K) min (Nlik, NI2k)

Table 12 Mean occupancies of trunk groups XIK in steady state

4 2

Figure 12 126.05 + 1.7%

1

(9 ... 13 respectively) the results obtained for the network of Figure 11 (12 respectively) are given. Network of Figure 11 is an unidirectional network with one transit-level; the routing policy is overflow. State equations are those given in the section on a unidirectional network with a mixed policy: for each origin destination couple 1J there is only one ctSKJ parameter and one/3/~:LJ parameter different from zero (flow IJ has only one first choice route (IKJ) and one second choice route (ILJ)). The network of Figure 12 is a bidirectional network with one transit level; the routing policy is load sharing. State equations are those given in the section on a bidirec-

422

A p p l . Math. M o d e l l i n g , 1982, V o l . 6, D e c e m b e r

26.49

+2.0% 29.74

--0.3% 24.12

30.30

30.39

0.0% 29.96

Simulation I Relative error

35.17 34.74

30.16 r

1.8%

30.71

[

--1.1%

32.57 r

--1.8%

31.97

I

--1.6%

29.90

29.36 4

0.0%

29.97

24.19 3

26.44 I 26.43

29.14 2

3

19.64 I

--0.1%

19.61

I

--1.2%

38.61 [ 39.18

+ 1.5%

Model for telephone traffic: F. Le Gall Table 13 End-to-end grade of service

0.934 0.916

2

3

4

0.912 I +0.4% 0.916 0.931 I 0.911 0.855 I

--2.1%

+ 1.0%

0.864

0.873

--1.9% 0.931 1_2.1% 0.911 0.864 0.903 0.896

-0.913

I -0.7% 0.839

-1.0%

+ 1.2%

0.849 0.828

--1.9%

+0.7%

0.896

0.834

0.854

--0.6% 0.831 i +0.4%

0.849

0.834

Global loss: . Model: 25.6Eo Simulation: 25.7E Simulation model [ Relative error

tional network with a mixed policy but with (JIKLJ = 0 (there are no second choice routes). Note then the given examples are worst cases since the percentage of loss (more than 10%) is much more important than actual cases (less than 3% o f loss). The accuracy of the model for real networks must, then, be better since for such networks the independence assumption of the blocking of trunk groups is much better verified.

Conclusion We have derived a one moment analytical model for telephone networks: under classical assumptions (Poisson offered traffic and negative exponential), we can write down the dynamic of the mean occupancies of the trunk groups according to the control policy ( X ( t ) = ~,~'(X, X, T, M/', U)). From various numerical experiments we can assert the accuracy of the model and its good performance for the resolution (time-saving). The interesting features of this model are as follows:

It is a one moment model, quickly solved and applicable to large networks. The introduction of fictitious traffic makes the model more accurate since we can take into account the characteristic o f the traffic offered (Poisson, smooth (variance lower than mean), peaky (overflow traffic, i.e. variance greater than mean).) The formulation is global (from origin to destination in one step) so that the computation o f the end-to-end grade o f service is straightforward, by contrast with well known algorithms like equivalent random theory 12 and Interrupt Poisson Process Method 4's which involve at each node aggregation and disaggregation o f flows. Finally, thanks to its relative simplicity (compared to second order method 1's'7'12) and since the control parameters are directly involved in the equations, we can think of evaluating the performance of dynamic routing algorithms and even, determine optimal routing policies for a network. This is beyond the scope o f this paper, but the first results, obtained in this area, confirms the applicability o f the model for that purpose.

References 1 Delbrouck, L. E. N. IEEE Trans. Commun. 1981, COM-29 (2), 85 2 Deschamps, P. J. 1EEE Trans. Commun. 1979, COM-27 (3), 603 3 Kleinrock, L. 'Queuing systems', John Wiley, 1975. 4 Kuczura, A. BSTJ 1973, 52 (3), 437 5 Kuczura, A. and Bajaj, D. IEEE Trans. Commun. 1977, COM-25 (2), 185 6 Manfield, D. and Downs, T. 1EEE Trans. Comrnun. 1979, COM-27 (8), 1169 7 Manfield, D. and Downs, T. 1EEE Trans. Commun. 1979, COM-27 (1), 44 8 Narendra, K. S. et al. IEEE Trans. Syst. Man. Cybern. 1977, SMC7 (11), 785 9 Narendra, K. S. and Thathachar, M. A. L. 1EEE Trans. Syst. Man. Cybern. 1980, SMC 10 (5), 262 10 Szybicki, E. and Bean, A. 'Advanced traffic routing in local telephone networks; performance of proposed call routing algorithms', ITC-9, 1979, Torremolinos, Spain 11 Chan, W. S. IEEE Trans. Commun. 1980, COM28(2),153 12 Wilkinson, R. I. BSTJ 1956, 35,421 13 Lin, P. M. et al. IEEE Trans. Commun. 1978, COM-26 (6), 754

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