ITC 15 / V. Ramaswami and P.E. Wirth (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
179
Performance Model and Multi-Hour-Optimization of Circuit Switched N e t w o r k s Driven by Optimized Dynamic Routing Rainer Stademann Siemens AG, Hofmannstr. 51, D-81359 Muenchen, Germany 1. Abstract The subject of this paper is a highly precise and numerically efficient Erlang fixed-point model of nonhierarchical networks that are driven by Siemens' event-dependent Optimized Dynamic Routing (ODR). Based on extensive simulations it is shown that the calls alternatively routed by the dynamic routing automata arrive at links approximately as Poisson processes with the calls on average having access to an effective number M* of statistically independent two-link paths. The phenomenological parameter M* can easily be determined by simulation. M* is generally non-integer and differs considerably from the nominal M currently used by many models. When incorporated into the Erlang fixed-point model it leads to excellent precision. Simulations also indicate applicability to least-busy-alternative (LBA) routing. Explicit numerical solution of the full set of link-state equations is avoided making the model very efficient. This efficiency allows for direct use of general purpose programming methods on top of the performance model to solve the multi-hour problem. Design results for asymmetric networks are presented and checked by simulation. An extension to IPP (Interrupted Poisson Process) external offers is described and first analysis results are also shown for the IPP case. 2. Introduction and Overview
Operators of nonhierarchical, strongly meshed networks driven by dynamic routing gain large economic advantages which result from an increase in network performance, simplified network management and a higher network robustness compared to conventional hierarchically structured networks with fixed alternate routing. Therefore performance modeling and the design problem of dynamically driven networks are of considerable interest (e.g. [ 1-9]). A performance model that provides high precision while being numerically efficient allows for the application of general purpose programming methods to tackle the network dimensioning problem. Such a model could also be a base for an online network management feedback loop operating with a time constant in-between those of real-time dynamic routing (seconds) and the long term capacity management (years), and providing additional congestion control or multiple grade of service (GOS) classes for example. Fixed-point models that approximate adaptive routing by stationary load sharing [3, 4, 5] ignore the effects of dynamic path selection in utilizing transient spare network capacity and systematically overestimate blocking probabilities [4]. Therefore several fixed-point models of state-dependent dynamic routing networks being recently published [6, 7, 9] take care of
180
dynamic path selection by a link-state dependent arrival rate. However this is on one hand difficult to apply to event-dependent routing like ODR. On the other hand it is numerically expensive for networks and links of realistic size because it requires an explicit numerical solution of the birth-death link-state equations. Things get even worse in the IPP case when a complete state transition matrix must be used within the link model. The current paper follows a more phenomenological approach and is based on the results of extensive simulations. The work was more driven by the objective of numerical efficiency, correct reproduction of simulation results, and by the applicability to real world networks than by the ambition to find a closed stochastic description without the use of any phenomenological parameters. The contents of the paper is organized as follows. First a comprehensive description of Optimized Dynamic Routing is given. Chapter 4 introduces the central definition of this paper entitled link associated call loss probability. Typical results of extensive simulations are presented which demonstrate that the link associated call loss probability can be approximated by a function of the time congestion of the link only. This function has a simple interpretation. It leads to an end-to-end blocking of a one moment model with overflowing calls having access to an effective number M* of alternate two-link paths. M* is incorporated into a performance model as the single phenomenological parameter. M* mainly depends on the network size and on the routing automata used. Simulations of least-busy-alternative (LBA) routing indicate the applicability of this approach also for state-dependent dynamic routing. Chapter 5 presents analysis results for the case of external Poisson offers and demonstrates the feasibility of using a general purpose reduced-gradient programming method on top of the performance model. Single- as well as multi-hour design results are checked by simulation. Finally it is shown in chapter 6 that even in the case of external IPP offers, the performance model reproduces simulation results with good accuracy when using an approximate link-state model for IPP offers that efficiently accounts for state protection.
3. Synopsis of Optimized Dynamic Routing Optimized Dynamic Routing (ODR) is an event-dependent dynamic routing strategy for strongly meshed nonhierarchical networks which is implemented within the Siemens' EWSD switching system and operates principally as follows. The incoming node of the nonhierarchical network layer offers a call to the direct one-link path first, if existing. If the direct one-link path is blocked, the call is offered to alternate twolink paths which are selected from a dynamically changing set of active paths managed per commodity (o-d-requirement pair). The maximum number M of adjacent links that are hunted by the incoming node for a free channel is fixed by administration to prevent inefficient searching. State protection is used to guarantee network stability [10]. An alternate two-link path is transiently deactivated by the ODR automata if a blocking event occurs either on the first link (i.e. during huntifig) or on the second link (crank-back). Since the path is not replaced, the set of active paths reduces in size. If the set becomes empty the deactivated paths are reactivated again. The reactivation is done transiently while a call is processed. When an alternate path is not blocked, only a small number of succeeding overflowing calls are routed via this path before the selection changes cyclically to the next
181 path of the active set. This allows for exploitation of transiently available network capacity, while the load is distributed adaptively to all available alternate two-link paths. ODR allows for crank-back and re-routing, i.e. the call is re-routed by the incoming node over a second two-link path if the first selected two-link path was blocked on the second link. To prevent inefficient re-routing attempts, re-routing is restricted to once per call. Additionally the re-routing feature can be switched on/off by administration. The difference from other event-dependent routing strategies like Dynamic Alternative Routing (DAR) [3] becomes clear for unbalanced load situations when some of the alternate paths provide spare capacity while others are almost blocked by directly routed traffic. ODR will quickly and autonomously reduce the set of active paths to those paths providing available capacity. The overflow traffic is adaptively distributed to these paths until reactivation occurs. Using this simple scheme, ODR is able to "learn" an unbalanced load situation which makes it especially suitable for networks with multiple busy hours. ODR is going to be introduced into the Deutsche Telekom long-distance network. 4. A Performance Model Based on Link Associated Call Loss Probability Event driven simulations and the performance model are based on the following approximations concerning the stochastic behavior of the network. External traffic requirements from node i to node j with mean Aq are stationary and either described by a Poisson process (Poisson offers) or Interrupted Poisson Process (IPP offers). Call holding times are independent and exponentially distributed. Call setup times are negligible. Blocked calls do not return and have zero holding times. Calls are not blocked by switching nodes. The analytical model is additionally based on the assumption of statistical independence of the states for the two links lik and lkj within one path ~ikj" However, it is not assumed that the states of the twolink paths with fixed commodity (i, j) and different transit nodes k are independent. Each routing attempt will change the state of an event-dependent, decentralized dynamic routing automata like ODR to one which is more correlated with the network state, helping to minimize the number of future call loss events, i.e. the correlation leads to a reduced network call loss probability. Additionally multiple overflow and crank-back features further reduce the network call loss probability. On the other hand these features tend to increase the correlation between the states of different alternate paths of the same commodity leading to a higher blocking probability. Based on the example of ODR it is the objective of this chapter to show that all these effects can be quantitatively taken into account within the framework of a link decomposition model by replacing the call congestion Puj for alternate traffic on link Iij by a smaller link associated call loss probability Pij *. Note, call congestion denotes the probability that a link is blocked when a call arrives at the link, but the call is not necessarily lost because it may overflow to another path. ODR offers the same direct link overflowing call sequentially to different alternate twolink paths until a path is found which is available, or the call is lost because either re-routing failed or the maximum number M of alternate paths allowed to be searched per call is exceeded. However, under the above assumptions unsuccessful call offers do not change the network state (although the state of the routing automata changes), and the sequence of call offers can be replaced by a single call offer to the last path which was selected by the ODR
182 automata. In the following this path is called the 'finally selected path' for a call. With the notation of
P[s]
being the probability that statement s is true and
P[slls2]
being the
conditional probability that s1 is true when s2 is true, the mean carried alternate traffic
hlij
on
link l~/is given by
hlij = Z Aik PoikP[gijk is finally selected] P[gijk is availablelrcijk k~i,j + ~_~AkjPokjP[rtkij is finally selected]P[rckij is availablelrckij k~i,j
is finally selected] (1) is finally selected]
with P0 denoting the call congestion of links for directly routed calls. Since the links within a nonhierarchical network usually carry mixtures of direct traffic and many different alternate traffic parcels, the states of the two links within one alternate path are assumed to be statistically independent. This leads to the following approximation: P[path
rtijk is available[path rcijk is finally selected]
= P[ lij is available]rcijk is finally selected] P[ ljk is availablel~ijk is finally selected]
(2)
= P[lij is avaJlable[lij is on a finally selected path]P[ljk is availablelljk is on a finally selected path] =: (1- Pij*)(1- Pjk*) The second approximation in equation (2) assumes that the conditional link availability is independent from the particular commodity (i,k). The corresponding conditional link blocking probability defines the link associated call loss probability p*. It is the call congestion observed for alternate calls on links of paths finally selected by the routing automata. With approximations (2) equation (1) becomes:
hlij = Z {AikPOikPijk (1- Pij *)(1- Pjk*)+ Akj POkjPkij (1- Pki *X 1 -- Pij *)} k~:i,j
r
Here the abbreviation 9 ijk: P[path
] rr,ijk is finally selected]
(3)
was used. The path selection
probability p depends on the dynamic routing method. Following a similar argument as in [3] the selection probability of a path is for ODR proportional to the mean time the path remains in the set of active paths. This is because ODR distributes the overflow traffic equally to all active alternate paths for each commodity. Since a path is deactivated if a blocking event occurs the mean time a path remains active is inversely proportional to the call congestion of
Pijk ~ 1/ [(1- Plij XI-- Pljk )] 9 The selection function p ijk together with the link associated call loss probabilities Pij *
the path, i.e.
describe the properties of the dynamic routing automata within the network equations ( 3 ).
183
100% o
90% - - - - p * ( b l ) , 9 .9
=~
80%
-
/
I l I single hour engineered asymmetric seven node network driven by ODR M* = 2.5
symmetric seven node network under uniform loads driven by ODR p*(bl), M ' = 2.8
t ~,,..o~:" 12 0.%o~./~"
r
(D
E <
9
70%
symmetric seven node network under ~ ' " uniform loads driven by LBA routing "~::~ 120%
. . . . . . p*(bl), M* = 3.2
~.'f
"~. 60%
.-_~
for asymmetric network
50~ n
~0~~
o
_"
40%
l FI ;
0
.--.
8
i02~.,/"
30%
Z',':~0~,
<
-~ _J
ioo~
20%
I Z,'.'-
~~ioo~
I
To examine the link associated call loss probabilities p* in detail, extensive simulations of dynamically driven nonhierarchical networks were carried out. Figure 1 shows a characteristic example of the results, p* is plotted dependent on the time congestion bl for alternate traffic. Results for two totally different seven node networks with traffic requirements varying from 95% to 120% of the nominal network loads are shown. The strongly asymmetric first network was engineered for 1% grade of service (GOS) under Poisson requirement A (1) of Table 1. Link sizes nij vary between 79 and 657
channels. Results are shown as open dots with each dot representing a (bl, p*)-pair of a single link. 9~&" 95% The second network is symmetric 0% 0% 20% 40% 60% 80% 100% with 90 channels per link. Results are Time Congestion bl of Links for Alternate Traffic shown as filled squares. Each square represents the average over all links Figure 1" Link associated call loss probability p* in of the symmetric network under a dependence upon time congestion b] uniform load. For both networks ODR was simulated with M=5 and at most one re-routing attempt per call. Each point represents a sample of at least 15000 final selection events. The 95% confidence interval is about the order of the point sizes. (The meaning of the filled triangles for LBA routing is discussed later.) The results clearly indicate that the link associated call loss probability p* is mainly determined by the time congestion b1 of the link. The additional influence of other parameters as link size, state protection, mean and variance of the overflow traffic parcels turns out to be minor. Despite the strong difference between the symmetric network and the strongly asymmetric network the interdependence between time congestion and link associated call loss probability is very similar. The full and dashed lines in Figure 1 each show a function of the form /,Ty
10%
-
9
~
~ o~ nomin~1 load I ~ GC 1
98%- for symmetric network
p * (bl)= 1 - ~ 1 - (1- ( 1 - b l ) 2 ) M*
(4)
with M* being determined by a least squares fit based on the results obtained for the asymmetric and symmetric networks driven by ODR to be M*=2.5 and M*=2.8, respectively.
184
Equation (4) has a simple interpretation for symmetric networks under uniform load. The endto-end blocking (EEB) Bsym in symmetric networks is related to p* and b 1via
nsym : P0(l_ (1_ p,)2).._ Po(l_(l_bl)2) M*.
(5)
Therefore M* can be interpreted as the effective number of statistically independent alternate paths which are available to overflow calls under the assumption of Poisson arrivals. While (5) is applicable only to the special case of symmetric networks under uniform load, (4) can be applied to all links of an asymmetric nonhierarchical network. Although equation (2) makes no sense for state-dependent routing, a p* can be defined in the case of symmetric networks from the EEB-value B sym and the direct traffic blocking P0 using (5). Figure 1 shows as filled triangles the results obtained from simulations of LBA routing. It gives M* = 3.2 for the example of the symmetric seven-node network even though under LBA routing each call has full access to 5 alternate paths. The simulation results shown in Figure 1 demonstrate the accuracy of equation (4) for seven-node engineered nonhierarchical networks under single-hour Poisson offers. Additional simulation results strongly support that this functional dependence holds with similar accuracy for any number of network nodes N with M* mainly depending on N and on the routing strategy.
5. Single- and Multi-Hour Optimization with External Poisson Offers In the case of extemal Poisson offers call congestion P0 equals time congestion b0 . Above simulation results indicate that the link arrival process of alternatively routed calls is in good approximation a Poisson process also (i.e. Pl = b~). Therefore the Erlang B formula extended for protected and unprotected traffic offers may serve as a sufficiently accurate link-state model. Together with equations (3), (4) a set of fixed-point equations (FPE) results which can be solved by relaxation once the phenomenological parameter M* is determined by simulation. The solution of the FPE needs less than a second on a 133 MHz PC for the example sevennode networks of Table 1, since no birth-death link-state equations must be solved explicitly. This makes it feasible to use a standard reduced-gradient optimization method on top of the FPE. It takes about 3 minutes to dimension a seven-node network for a single-hour of Table 1. The single-hour design of a 14-node network requires about one hour. A more efficient numerical evaluation of Eflang-B than the well-known recursion formula can further reduce the execution times. Table 1 shows two example busy hours A (1) and A (2) (as both-way requirement) of a seven-node network with bi-directional links together with the corresponding single-hour design r e s u l t s n (1) , n (2) and the two-hour design result n ( l & 2 ) . All results are based on ODR with M=2. The corresponding M* was determined by simulation to be 1.7. As objective function the total number of network channels was used. For both single-hour designs the number of channels reserved for direct traffic was fixed to~n / 2.
185
i,j
1,2
1,3
1,4
1,5
1,6
1,7
2,3
2,4
2,5
A (1) A(2) n (1) n(2) n(l&2)
E(I&2)
400.7 500.0 415 509 443 5
262.1 262.5 279 277 277 7
214.2 137.5 233 159 208 3
64.0 37.5 79 52 78 3
181.4 87.5 201 109 194 3
159.9 100.0 178 121 176 3
353.5 200.0 370 219 347 15
452.1 375.0 469 394 529 5
138.3 112.5 157 130 141 3
365.0 337.6 225.0 275.0 383 356 246 293 351 334 4 5
2,6
2,7
i,j A(1) A(2) n(1) n (2) n(1~2) E(I&2)
3,4 525.2 637.5 539 647 585 6
3,5 117.8 100.0 135 117 122 3
3,6 326.0 237.5 345 258 335 4
3,7 238.1 137.5 257 158 241 4
4,5 203.3 275.0 222 292 274 9
4,6 554.0 700.0 569 713 641 8
4,7 394.1 450.0 412 467 445 6
5,6 169.6 212.5 188 231 206 6
5,7 121.2 137.5 139 156 139 6
6,7 647.0 1025 657 1023 895 12
total 6225 6225 6583 6572 6961 120
Table 1" Both-way Poisson requirements A (1), A (2) [Erlang] and for 1% GOS engineered single-hour networks n (1) , n (2) and multi-hour network n (1&2) with state protection E (l&2) . 1
I
I
Single-Hour Engineered Network n(2)at Busy Hour A(2)
I
[
I
1.0% 0.5% 0.0% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t3b "=-o~1.5% I
........
Multi-Hour Engineered Network n(l&2)at Busy Hour A(1)
1.0%
_
51__
.i'l .
.
0.s%
.
i[
tu 0.0% . . . . . . . . . . . . . . . . . . . . . .
.
.
I
i
', ',, I ',-', ' , , ,
', ,~Rh,, . . . .
i ', ', i : , ,
~--,~
Multi-Hour Engineered Network n(l&2)at Busy Hour A(2)
1.5% t
1
1.0% l
234~0,
134~07,2450,
~ , ~ ~
~
I
I "Aoa'~'s Simulation I Figure 2:
~
~
~
,23~67
123407
1234~7
1234~0
444444
~
000000
~
~
~
Requirement Pair, Commodity (i,j)
EEB in comparison of analysis and simulation for single- and multi-hour networks n (2) and n (1&2) of Table 1
186
.-. 150 e-
lOO
For the multi-hour design the 1 same set of state protection para-
. Busy . . . . Hour (1)
-
l
meters E (]e~2) was used in both
~" 50 hours and was determined directly ~-~ IEl,.n~,,~,,ml~n,_,.r,~,~,,I],,.-...n,,.~,!rl,l[],/-l,,.n~...,,d] through optimization. Since the "' -, -, t 9e--t~ 0 fixed-point model cannot reproduce the dynamic effects of 150 -o Busy Hour (2) Ir IIAnalysis network instability a lower bound 9~100 50 , ~ ~ ~ Flsimulati~ Of O'25(max(A(1) 'A (2) )) l/2 was set ~
"~
J"~
0
Figure 3"
~
~
J.r-~BI-I~
~m..~,-~,,.rl,---.~,_~ J
234567345674567567677 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6
to ~(1~2)" The multi-hour design gain is 519 channels (=7%) c o m p a r e d to a n e t w o r k u s i n g the
Link(i,j)
maximum of n(1) and n(2) (7480
hlij in comparison of analysis and simulation
channels). The upper part of Figure 2 compares simulated to analyzed EEB-values Bij for the single-hour
for multi-hour network n~l~2)of Table 1.
engineered network n(2) and requirement A (2) from Table 1. As a result of optimization all single-hour 1% GOS constraints turned out to be tight. Since for comparison with simulation the calculated non-integer channel number was rounded to the nearest integer, small deviations from the 1% GOS target show up. The relative error of the calculated EEBvalues B6 is on average about 5%. The largest errors of up to 20% show up for those commodities with extreme low (e.g. (1.5)) or high requirements (e.g. (6.7)). The corresponding result for network n(1) is not shown here but of comparable quality. Figure 2 (lower parts) and Figure 3 compare simulation and analysis results with respect to EEB-values
Bij and link carried alternate traffic
h 1 for the multi-hour network n (l&2) of
Table 1. Multi-hour problems represent stringent tests to performance analysis since a much more substantial part of the traffic will be dynamically routed on alternate paths than in singlehour designs. For those commodities with a tight constraint the error in EEB is less than 25%. In addition, the distribution of alternate traffic is very well reproduced by the analytical model. Note. that due to the multi-hour design link /24 carries in hour (1) about 10% and in hour (2) about 20% alternate traffic. Table 2 compares simulation and analysis results for the unplanned 30% overload example scenario published in ref. [7] but here based on an ODR driven network (M=5 and M*=2.5). Although the performance model presented here does not require explicit solution of the birthdeath link-state equation, a comparable accuracy to that reported in ref. [7] can be observed. i,j Analysis Simulation
1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6 total 0.96 0.03 5.37 0.10 0.53 0.00 17.15 0.20 4.50 0.22 0.01 0.02 0.22 1.27 0.00 30.57 0.86 0.04 7.71 0.08 0.57 0.00 15.59 0.70 3.87 0.29 0.04 0.07 0.52 1.51 0.00 31.83
Table 2: Lost traffic Aij
* Bij [Erlang] in comparison of analysis and simulation for 30%
overload scenario of reference [7]:
187
6. Analysis Results for the Case of External IPP Offers A nonhierarchical | long-distance network ~ I IAnal~is OSimulation 1 usually connects lower = layer regional sub- ~1% o networks. Nevertheless -~ high usage routes to ~ o and/or from regional ~, enodes which bypass the "' 0% 234567134567124567123567123467123457123456 long-distance network 111111222222333333444444555555666666777777 layer may still be Requi~entP~r,Commod~(i,j) economical for those traffic requirements Figure 4: Comparison of simulated and analyzed EEB for the which are substantial network n ~]) with external IPP traffic offers (90% ofA (~), through all busy hours, peakedness Zij = 5). For these requirements the nonhierarchical long-distance network is used as a final route for the peaked overflow traffic of the high usage routes only. However, for peaked external traffic offers with mean A and variance to mean ratio Z > 1 the Erlang B link-state model breaks down and must be replaced by a more appropriate model. The results of renewal theory were used as a base of a link-state model for IPP traffic offers (see e.g. [11]). To avoid in a first approach the heterogeneous link blocking problem of the simultaneous IPP and Poisson arrival processes, both processes were replaced by a single IPP process with the peakedness of the externally offered traffic to calculate the link-state distribution. Note however, that within the FPE the time congestion b 1 determines the call loss probability p* of a link via equation (4) and thus can implicitly take care of the heterogeneous link blocking. It was checked through simulation that also in the case of external IPP traffic offers the link associated call loss probabilities p* can still be approximated by equation (4) using the same M* as for external Poisson offers. However, further investigation is still necessary to test the limits of equation (4). The renewal theory does not account for state protection, and the use of the resulting time congestion b~ when ignoring the reduced arrival rate into protected states does not give sufficient accuracy. Therefore the transition matrix of the link-state model was modified to account for the reduced arrival rate. However, in order to achieve the required numerical efficiency even in the case of several hundreds of channels per link, it is essential to avoid the solution of the complete set of link-state equations. Therefore only the link-states strongly affected by state protection were recalculated while for the majority of the other states the results of renewal theory were directly used. Figure 4 compares analysis with simulation results for network n c]) under 90% of the nominal network load A(1) of Table 1 and peakedness Z = 5 for all commodities. The intensity ~, of the Poisson source was chosen to be the equivalent Poisson offer computed from Equivalent Random Theory. The number of channels reserved for direct traffic is fixed per
188 link according to~n, i.e. twice as high as for Poisson offers to guarantee network stability. The results are based on ODR with M=5 and not more than one re-routing per call (i.e. M*=2.5). Note that n (~ is not engineered for IPP offers but for Poisson offers and therefore end-to-end blocking is not homogeneous. For most commodities simulation results are well reproduced, however for links with low requirements the analysis is too conservative. The time required for analysis of this example is 8 seconds on the above mentioned machine. 7. Conclusion
The simple phenomenological approach of link associated call loss probability based on an effective number of statistically independent alternate two-link paths is a very powerful means to model dynamically driven nonhierarchical networks -especially those which are eventdriven, e.g. ODR driven networks- with high precision. First investigations seem to support that this approach is not only applicable to the case of Poisson offers but also to the case of external IPP offers. The numerical efficiency of the performance model makes it feasible to use directly a reduced-gradient programming method for solving single- and multi-hour problems. 8. References
1. G.R.Ash, R.H.Cardwell, R.P. Murray, "Design and Optimization of Networks With Dynamic Routing", The Bell Systems Technical Journal, Vol. 60. No.8, Oct. 1981 2. J.Regnier, P.Blondeau, W.H.Cameron, "Grade of Service of a Dynamic Call-Routing System", ITC- 10, 1983 3. R.J.Gibbens, F.P. Kelly, P.B.Key, "Dynamic Alternative Routing - Modelling and Behaviour", ITC- 12, 1988, pp. 1019. 4. A.Girard, M.-A. Bell, "Blocking Evaluation for Networks with Residual Capacity Adaptive Routing", IEEE Transaction on Communications, Vol. 37., No. 12, Dec. 1989 5. R.J.Gibbens, F.P.Kelly, "Dynamic Routing in Fully Connected Networks", IMA Journal of Mathematical Control and Information, Vol.7,1990 6. D. Mitra, R.J. Gibbens, B.D. Huang, "State-Dependent Routing on Symmetric Loss Networks with Trunk Reservation - I", IEEE Transaction on Communications, Vol. 41., No.2, 1993 7. G.R. Ash, B.D.Huang, "An Analytical Model for Adaptive Routing Networks", IEEE Transaction on Communications, Vol. 41., No. 11, 1993 8. H.L.Hartmann, H.He, "Capacity Assignment for Circuit-Switched Reference Networks with Nonhierarchical Routing", European Transactions on Telecommunications, Vol. 6. No. 3, 1995 9. E.W.M.Wong, T-S.P. Yum, K.-M. Chan, "Analysis of the M and M 2 Routings in CircuitSwitched Networks", European Transactions on Telecommunications, Vol. 6. No. 5, 1995 10. R. S. Krupp, "Stabilization of Alternate Routing Networks", IEEE International Communication Conference, Philadelphia, Jun 1982, pp. 31.2.1 11. Syski, R., "Introduction to congestion theory in telephone systems", North-Holland, 1986