OPTIMAL PLANNING OF TELECOMMUNICATION NETWORKS

WITH LOGICAL AND PHYSICAL ASPECTS

G. R. Mateus and H. P. L. Luna (1) L. C. M. Lage

Departamento C. Computqiio UFMG Telecomdcqks M. Gerais Belo Horizonte - MG - Brasil

ABSTRACT

Digital technology applied to switching and transmission equipment for telecommunication networks has introduced a deep change on their design and dimensioning principles, in particular, with respect to the planning of junction net- works. The problem can be established as an economic optimization problem of alternative routing network with minimum total junction cost for specified end to end con- gestion values between each pair of exchanges in the tele- phone and computer communication networks. The ob- jective function takes into account the economies of scale associated with the modularity of resources, and the model considers the existing network, as well as reallocation of installed switching and transmission equipment. The main aspects that we consider in an emphatic way is the opti- mization strategy, composed of logical and physical net- work design. The problem is formulated as a non-linear integer programming model that can be solved by mixed algorithms.

Keywords. Telecommunication; large scale systems; opti- mization; communication netw0rks;junction network plan- ning.

INTRODUCTION

Due to the complexity and dimension of a Telecommuni- cation Network, a planning methodology is necessary to minimize the investment costs to supply the requirements of traffic demand with a satisfactory grade of service. This. aim is achieved by an efficient and integrated framework for computer assisted planning which requires the appli- cation of optimization techniques. The overall problem is decomposed in smaller subproblems, with the use of math- ematical and computational methodologies that adopt a divide-and-conquer strategy. It results Mathematical Pro- gramming Models solved by optimization techniques that

(1) The authors have research fellowships from Con- selho Nacional de Desenvolvimento Cientifico e Tecnolbgi- CO (CNPq) and computational resources from the Funda- + de Amparo B Pesquisa do Estado de Minas Gerais (FAPEMIG).

takes advantage of particular properties of each system (Minoux (1989)). In this sense, we can elaborate a plan by steps that do not cause appreciable errors.

Digital technology applied to transmission and switching equipment for telecommunication networks has introduced a deep change on their design and dimensioning principles. In particular, with respect to planning junction networks, there are some new aspects such as channel modularity and multiplexing hierarchy of transmission systems, mod- ularity of switching systems, common channel signalling, non-hierarchical instead of hierarchical structure, flexibil- ity of routing strategies, etc, that modify the conventional parameters on which the economic dimensioning of the network is based.

One of the ways of dealing with this large scale problem is the decomposition in time and space. The time decompo- sition analyses the short, medium and long term planning. The space one analyses the interurban and urban networks both being studied at different levels of description. We can physically identify the main subsystems. The plan- ning process can be seen in a top-down manner, beginning with the urban and interurban network until the telephone subscribers, and it becomes engineering projects at lower levels.

The interurban networks are characterized by long routes without derivations in the means of communication; the common device of support is the atmosphere. The ur- ban network satisfies the communication necessities of a restricted community and the devices of support are phys- ical. In this context, the primary network (that links an exchange and its subscribers) sometimes predominates taking into account as dimension as economical aspects. But the importance of the junction network (that links the exchanges) increases with the size of the city and the digitalization. The main question, concerning both the junction and the primary networks, refers to the location of exchanges. The interdependency among the locations of exchanges suggests the use of quadratic formulations for this problem (see Mateus and Luna, 1989b). Focus- ing the junction network planning, the grade of service requirements introduce nonlinear (stochastic) constraints in the integer problem, but convexity properties of a con- tinuous relaxed version can be explored to overcome some difficulties.

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The junction network planning can be partitioned i l l t 1 i 1 c ~ - phases: the routing strategy definition, trunk iletwork t l i -

mensioning and physical network organization. Tlie rout- ing plan must be more flexible in order to take advan- tage of the features provided by the new technology; self- adjusting capability may become an important economic factor, requiring the use of software for network opera- tions management. In this. sense, at the planning level the routing rules can be simplified, thus enabling empha- sis on the trunk network optimization, composed of logical and physical aspects. The logical network dimensioning is based on congestion limitslimposed by the traffic in the exchanges and by the communication channels capacities. Physical network aspects are concerned with the selection of transmission media, their physical lay-out and the num- ber of digital carrier systems for each PCM hierarchical lwel (Luna, Mateus and Lage, 19SS).

Given the offered traffic between each origin-destination pair and the routing plan, the purpose of this paper is to establish a mathematical model to treat the most impor- tant aspects related to planning a mixed junction network with a gowing digital component. The problem can be seen as the economic optimization of an alternative rout- ing network with minimum total junction cost for spec- ified end to end congestion values between each pair of exchanges in the telephone network. The objective func- tion takes into account the economies of scale associated with the modularity of resources, and the model considers the existing network.

The problem can be solved by mixed integer-continuous algorithms, based on traffic carrying efficiency formulae and on the consideration of multiple module sizes. Chance constraints reflect the congestion limits, in such a way that the problem is formulated as a rather difficult non-linear integer programming model, although some heuristics can be used in order to take advantage of certain particular aspects. In this sense, good approximate formulae can be used to compute end-to-end blocking probabilities to avoid extra computational effort in integer programming. The use of large scale mathematical programming techniques must also be considered: decomposition, linearization and relaxation are the most useful in this case.

Other important aspects that we analyse are some gen- eral planning assumptions that will become the problem simpler. The crucial impact of non-voice services is sup- posed to be included in the traffic demand. We also sup- pose that the network topology, the grade of service, and network costs, such as costs of transmission systems in physical links and of multiplexing facilities in transmission nodes are known. Another important aspect that will be discussed concerns convexity properties of the Erlang B formula and its consequences.

At last, the conclusion section include the future implica- tions of this paper.

NETWORK PLANNING MODEL

The trunk network planning problem in multi-exchange areas consists of determining the optimal numb& of com- munication channels between exchanges, minimizing the total cost and considering the quality of service required. The communication network capacities are located at the arcs and nodes. The arcs receive the communication chan- nels while the nodes receive the structure of the exchanges. The aim is optimize the number of channels to attend the traffic demand originating at each source node and destinated for each siik node, subject to specified end to end congestion values and the technical restrictions of the equipments. In addition, we can route the traffic from a node to another node directly by a high-usage route or by one or more alternative routes. Firstly, the traffic is offered to the high-wage route. If this route has not suf- ficient capacity to handle this traffic, then the overflow trafic is offered to an alternative route, and so on until it is routed via a final route. After then, the traffic is lost or blocked. A network has a grade of aewice that represents the capability to handle all the traffic offered.

The grade of service constraint can be defined point-to- point in the route, but the widely accepted approach is a node-to-node blocking probability that do not exceed a limit b. The traffic is also assumed to be Poisson such that the mean of the traffic offered is equal to its variance. Besides, the number of calls are independent Poisson pro- cesses while the duration of calls are usually assumed to be an exponential distribution independent of the arrival times. In this case, a poissonian traffic with mean U offered to an integer number n of channels satisfies the Erlang B loss function: B(n,u) = b (e.g. see Kubasik, 1985).

The particular aspects presented by this optimization pro- blem suggest to decompose the system in three phases: ( 1 ) the routing strategy definition, (2) trunk network di- mensioning, and (3) physical route identification. If a sub- scriber, connected to exchange i, wants to establish a com- munication with a subscriber connected to exchange j , the first step is to define a path in the junction network. This call can be routed via a final route, or a high-usage route with alternative via tandem, or by a final route via tan- dem. Therefore, we need to defme a routing plan. On the other hand, the “network intelligence” is rising with the digitalization process, controlled by the software imbeded in digital switching exchanges. In this sense, the com- mon channel signalling networks also permit to introduce dynamic routing strategies, exploring the noncoincidence of the t r a c peaks. These aspects must be treated by the network management system, that controls the digi- tal network aciency in real time. Here, we consider that the routing plan is given. The trunk network dimension- ing consists. in determining the optimal number of high- usage and final channels that minimizes the total junc-

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tion costs, and satisfies the grade of service and equipment constraints. The grade of service constraint is expressed by end-to-end blocking probability in each route. And, with respect to each equipment we take into account the type, capacity, alternative routing, availability, etc. Given the number of channels between switching centres, we can identify the physical routes to carry the traffic, analysing the transmission system costs (cables: type, attenuation, loop resistence; or multiplex systems: type, capacities, distance) and security aspects. The transmission system costs depend on the PCM system evolution in high mul- tiplexing hierarchy. And, the security aspects analyse the possibility of assuring at least two physical paths to carry the traffic between exchanges.

Given the routing plan and the traffic offered, we introduce a model for optimizing the dimensioning of alternative routing network with regard to existing means and con- sidering logical and physical aspects, like channel modu- larity, multiplexing hierarchy of transmission systems, and one-way and both-way circuits. Before we introduce the model, we discuss other models and use the opportunity to present some theoretical assumptions.

The routing phase is treated in Girard and C6tC (1983). Berry (1983), Berry and Harris (1985), Combes, Horn and Kerneis (1983), Kleinewillinghofer-Kopp (1985), Roosma (1985) and hsemberg (1987) present the dimensioning phase (2). While the phases (2) and (3.) ,are discussed in Hedberg (1983) and Yoshida and Okazaki (1985). The phase (3) is treated by Garcia (1987) and Okano, Makino and Miki (1984), Minoux (1989).

Berry (1983) presents an optimization problem, based on a non-linear integer programming model, to solve the junc- tion network dimensioning considering full availability, al- ternative routing, grade of service constraints, channel mo- dularity and t e l e t r a c properties.

Another mixed non-liear integer programming model is proposed by Hedberg (1983) to solve the dimensioning problem for a given alternative routing, considering log- ical and physical aspects, the-channel modularity and ex- isting equipment. The n routes are divided in high-usage, j = 1,. . . , k, and final, j = k+ 1,. . . , n. The problem con- sists in finding the optimal number of channels z) in each. route j. A final route j is dimensioned to find the grade of service conditions, depending on high-usage routes 2,:

~j >_ Dj(z1 , . . . , ~ k ) , j = IC + 1 , . . . , T I (1)

where Dj is the dimensioning function. Let be given the Erlang loss function B(n, U ) and a continuously differen- tiable extension B ( z , U ) with a continuous variable z. The new function B can be write by B ( z j , Mj,V,) = b, where x j is the continuous number of channels, Mj is the mean of the traffic offered to route j, 5 is the variance of the traffic offered to route j and b the blocking probability. Hence, the function Dj is the inverse of the continuous function B , or Dj = B-’(Mj,Vj,b). Assuming that both

functions B and Dj are convex we can assure convergence to a minimal cost number of circuits for a specified grade of service b.

The overflow traffic in route j, with mean mj and variance w j , can be obtained by Wilkinson’s or Hayward’s methods (see Jagerman, 1984):

mj = MjB(n j / z j , M , / z j ) and

where zj = Vj /Mj . And, the mean Mi and variance route i , that receives the overflow traffic, are given by:

in

M i = a i + ) m j and T I , = u i + C v j ( 3 ) j - + i j - i

where a; is the “fresh” or “background” poissonian traffic offered to route i.

Hedberg implemented an heuristic based on a branch-and- bound algorithm. This technique generates a sequence of quadratic subproblems, that require exterior penalty functions, and they are solved by the gradient method.

Rosemberg (1987) presents a non-linear programming heu- ristic for computing optimal link capacities in a multi-hour alternative routing communication network. There are n high-usage routes A + i , l < i 5 n, with z , channels in route i and ( n + 1) final routes, A --t B -t i , l < i 5 n, with yo channels in final route A + B and yi channels in’final routes B -t i , l 5 i 5 n. Each high- usage route i, 1 < i < n, carries a load of (lib erlangs in period h, h E H. And, each final route i, 0 < i < n, carries a load of z i h erlangs in hour h E H, that is the “fresh” or “background” traffic, plus the overflow traffic, the part of a i h not carried on high-usage route z. Let $,, be the number of channels to carry the “background” traffic on final route i in hour h at the required grade of service. Once this traffic is poissonian, the yyh satisfies the Erlang loss function: B(yf,,, Ciih) = b.

The problem consists in determining the number of chan- nels x = (11,. . . , x,) and y = (yo,, . . , yn) that carry the traffic a i h and 6 i h at lowest cost, subject to the grade of service constraints:

Supposing that the “background” traffic is poissonian and sufficiently large, an overflow traffic of e erlangs to a final route requires Xe additional channels to attend the proba- bility in b. The constant X (approximately 1.15) is seldom

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applied despite of errors that can occur in its application. Hence, in (4) and (5 ) , when a traffic U l h , in hour h, is of- fered to z, channels, the overflow traffic is a,hB(x , ,a ,h) , requiring Xa,hB(I,, a s h ) channels in final route, that is added to the &, circuits.

The Rosemberg's model is convex if the function B is con- vex in 5, and a > 0. We note that B(n,a) has not been proved to be convex on the positive integer values of n (Messerli, 1972). The proposed algorithm is a convergent iterative process that also makes use of quadratic subprob- lems, penalty functions and heuristics.

Garcia (1987) treats the physical network organization taking into account the security aspects. The mathemati- cal model corresponds to a linear minimal cost multicom- modity flow problem that considers an arc-route matrix instead of the traditional node-arc formulation.

Our purpose is to introduce a new model composed of logical and physical network design, considering new and existing exchanges, minimizing the fixed costs of installa- tion and expansion, plus communication fixed costs, and plus variable costs. The model can be viewed as a merge between Rosemberg's (1987) framework, for logical dimen- sioning, with Garcia's (1987) approach, for the physical design.

The premisses, constants and variables are: (1) An exchange can carry different products. (2) A product can correspond to an origin-destination

group of traffic between two exchanges, p E Pod; or else it can correspond to the traffic between a re- mot unit and its exchange, p E Pelr. The product is indexed by p , and it is supposed given the traffic demand of d p erlangs in the busy hour, that is the

'60-minute period during the day when the traffic is the highest. The product is associated with an ori- gin destination direction, but it can also be related to both-way traffic. In this case, the d p value is the sum of the traffic in both directions. Any link between a remot unit and its exchange is both-way.

(3) The traffic associated to a product is carried through one or more physical paths.

(4) A path is composed by one or more arcs indexed by 2;

each arc i links two or more transshipment points of the urban network, and it is characterized by a unique transmission facility. The transmission media can be conventional cables (i E I,,), PCM cable (i E I p c m )

or fiber optic system (i E I f o ) . Thus, each arc has a modularity m,, e.g., 1,30, 120,480 and 1920 channels. All the arcs belong to set I = I c c ~ I p c m ~ I ~ o . For each arc it is also known the number of existing channels b,, and let y, be the number of additional modules in arc i with unit cost c,.

( 5 ) Each route is characterized by a product and a path and the cost to maintain the route r is I,.

(6) For each product p E Pod there is only one alterna- tive route. The calls are offered to a set of first choice routes, r E Rp, that make use of different paths. But, ,the overfiow traffic can only be carried through one specified route rp . The traffic between a remote unit and its exchange is only carried by first choice routes. It is not considpd any alternative routing for this case. In this way, the necessary number of channels ep, to flow a product p E Per,, can be previously es- tablished, such that the blocking probability B(e,, d p ) attends the required grade of service.

(7) Let A be an arc-route matrix where each element is defined by:

if arc i is used by route r ; 6: otherwise. air =

(8) Let A be an alternative routing matrix where each element is defined by:

if arc i is used by route rp ; otherwise. 6 i p = { 1

The variables are: x r - the number of channels in route r ; y i - the number of additional modules in arc i; and z p - the number of channels in first choice routes for product P. The mathematical programming formulation is:

subject to

The model considers the logical network dimensioning and the physical network optimization, with the modularities of routes and arcs mi. The alternative routing is pre- viously defined and takes into account a unique alterna- tive route; the remot units and both-way routes are also treated.

We can emphasize some aspects of this model. The con- stant X can vary into a great interval. The results can be compromised by errors that generally occur. But, it can be applied to reduce the computational effort, and after then, we can use a more precise technique like Wilkin- son's or Hayward's methods, (2) and (3). Moreover, we can change the constant X taking into account the route efficiency to obtain different values for A.

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The alternative routing is previously defined. We justify this condition since we are interested in the structurd as- pects, thus simplifying the operational aspects. The rout- ing optimization would be treated in red time by a net- work management software.

The alternative routing established permits only second choice routes. This means that the network designed will have a better practical performance if other hierarchical levels are considered. The precision of the theoretical re- sults can thus be compromised for a lot of hierarchical levels; in such cases, the model complexity rises sensibly. In other words, the netwoik dimensioning can be super- estimated, since the complete network flexibility has not been explored.

Our model is based on Rosemberg problem, but we empha- size the physical network and modularities, as proposed by Garcia (1987). The Hedberg model also considers phys- ical network and modularities, but our physical network is more fle+ble and the use of the constant X in the first steps becomes the problem simpler.

The proposed model is a very complex non-linear inte- ger problem. The non-linear constraints and the integer variables to attend the modularity, are the complicated aspects. The Erlang loss function B is non-explicit and non-continuously differentiable. But, considering a real number representing the number of channels, this exten- sion of the B formula results in a continuously differen- tiable and convex function (Messerli, 1972). And, under convexity assumption, an iterative process guarantees con- vergence to a global optimum. The other complication is the integer variables. In order to attend this condition, the problem can be treated by decomposition or enumeration algorithms. We are investigating both strategies jointly with some heuristics.

The decomposition algorithms generally follow Benders partitioning method and need an enumeration scheme, since the master program reduces to an integer program- ming problem. Their computational success to solve large- scale multi-product models have been consolidated by Ge- offrion and Graves (1974); and their capability to solve stochastic versions has been presented by Franca and Luna (1982). The enumeration algorithms are generally from the branch-and-bound type, as suggested by Geoffrion and Marsten (1972) and Mateus and Luna (1989a). Although the heuristic methods can not assure a global optimum, they have been quite succesful in getting good solutions, Lage (1989). The algorithms generally decompose the problem in two independent subproblems in each step. In a higher level is solved the logical dimensioning and a slave program treats the physical network. The subproblems feedback on each other.

CONCLUSIONS

We introduced a non-linear integer problem to optimize the planning of telecommunication networks considering logical and physical aspects. For this problem we empha- sized the use of the constant X to evaluate the number of additional channels to carry the overflow traffic; the alternative routing is previously defined, and the model permits one or more first choice routes and the network modularities. The function B properties were discussed and we suggested &me kind of algorithms to treat the model. Now, we are investigating the network manage- ment and another model that takes into account a lot of second choice routes and module size on exchanges and routes. For both models, the decomposition and enumer- ation algorithms have been tested.

REFERENCES

Berry, L. (1983). Optimal dimensioning of circuit switched digital networks. Tenth International Teletmfic Congress, Ses~ion~2.1, paper n. 06 (Australia], Montreal. Berry, L., and R. Harris (1985). Modular design of a large metropolitan telephone network: a case study. Eleventh Intemational Teletrafic Congress, Session 2.3A-5 [Aus- tralia], Kyoto.

Combes, M., F. Horn, and J. Kerneis (1983). Simple mod- ular algorithn for dimensioning digital links. Tenth Tele- trafic Congress, Session 4.3B1 paper n. 04 [Fiance], Mon- treal.

Francs, P.M., and H.P.L. Luna (1982). Solving stochastic transportation location problems by generalized Benders decomposition. !hnportation Science, 16, n.2, 113-126. Garcia, A. S. (1987). Planejamento do entroncamento em redes telefBnicas urbanas em process0 de digitalizqk. Tese de Doutorado em Engenharia Elktrica, FEE, UNI- CAMP, Campinas. Geoffrion, A.M., and G.W. Graves (1974). Multicommod- ity distribution system design by Benders decomposition. Management Science, 20, 822-844.

Geoffrion, A.M., and R.E. Marsten (1972). Integer pro- gramming algorithms: a framework and state-of-the-art survey. Management Science, 18, n. 9,465-489. Girard, A., and Y. CBtd (1983). Sequential routing op- timization for circuit switched networks. Tenth Interna- tional Teletrafic Congress, Session 4.3B1 paper n. 08 [Ca- nada], Montreal.

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Hedberg, I. (1983). Optimization of alternative routing networks with regard to existing equipment and modu- larities. Tenth Intemational Teleirafic Congress, Session 4.3A, paper n. 03 [Sweden], Montreal. Jagerman, D.L. (1984). Methods in traffic calculations. ATtYT Bell Lab. Tech. J . , 63, 1283-1310. Kleinewillinghofer-Kopp, R. ( 1985). An heuristic proce- dure for optimizing a switching network. Eleventh Inter- national Teletrafic Congress, Session 2.4A-1 [Germany], Kyoto. Kubasik, J. (1985). On some numerical methods for the computation of Erlang and Rngset functions. Eleventh In- ternational Teletrafic Congress,Session 5.2B-3-1 [Poland], Kyoto. Lage, L.C.M. (1989). Planejamento da rede de entronca- mentos de keas urbanas multi-centrais em an~biente de transiqb tecnolbgica. Dissertq% de Mestrado, DCC- UFMG, Belo Horizonte. Luna, H.P.L., G.R. Mateus, and L.C.M. Lage ( 1988). Mo- delos de plkejamento de redes telefhnicas em areas niulti- Centrais. RT 011/88, DCC-UFMG, Belo Horizonte. Mateus, G.R., and H.P.L. Luna (1989a). Avan~os nos projetos de 0timizq-b de redes telefhnicas: localizaqk, cronograma de implants+ de centrais e entroncamento. RT 013/89, DCC-UFMG, Belo Horizonte.

Mateus, G.R. and H.P.1, Luna (1989b). Combinatorial optimization in telephonic network planning.ALI0-EURO Workshop on Practical Combinatorial Optimization, Rio de Janeiro. Messerli, E.J. (1972). Prwd of a convexity property of the Erlang B formula. Bell Sy tt. Tech. J., 51, 951-953. Minoux, M. (1989). Netwo: k Synthesis and Optimum Net- work Design Problems: b 'odels, Solution Methods and Applications. Network, 19, 913-360.

Okano, Y., M. Makino, and '' Miki (1984). Transmission network optimization in junct ion networks. Proceedings of ICC'84, Amsterdam, 185-188 Roosma, A. (1985). Optimizatbn of digital network struc- tures. Eleventh International Teletrafic Congress, Session 2.4A-4 [The Netherlands], Kyoto. Rosemberg, E. (1987). A nonlinear programming heuristic for computing optimal link capacii ies in a multi-hour al- ternate routing communications ne \work. Operations Re- search, 35, n.3, 354-363. Yoshida, M., and H. Okazaki (1985). Digital telephone junction network planning system considering logical and physical network aspects. Eleventh Iniemational Teletruf- fic Congress, Session 2.3A-2 [Japan], Kyoto.

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