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212Queue theory,queueing network theoryQueues occur because of a temporary shortageof resources. This applies in many every-dayoccurrences, in the community as well as intechnology. In the telephony field queueingproblems aroused great interest almost sincethe first telephone exchanges were built. Thedevelopment of telecommunication technologyhas given rise to a multitude of problems, whichhave often required sophisticated mathematicaltreatment. The article deals with a difficultqueueing problem in modern telephony.Some basic conceptsThe purpose of queueing theory is to provide ananswer to the question of how long the waitingtime will be under different conditions. The limitfor what is acceptable must be decided individuallyin each case, depending on the overallfunction of the telecommunication system andthe expected reactions of the subscribers.Configurations. The queueing systems are builtup of servers. A simple queueing system consistsof a number of servers (one or several)working in parallel and carrying out the sametype of service, for example the devices in telephoneexchanges that work in delay systems.This type of system is treated with the aid ofqueueing theory (in a restricted sense).More complicated systems contain differenttypes of servers that must interwork in order tobe able to carry out their tasks. Such systemswork in several stages, e.g. the processors insystem AXE 10. The crossbar systems of typesARM and ARE (transit exchanges) also containsuch service systems. Such systems are treatedwith the aid of queueing network theory.The distribution function for the interarrivaltimes to the queue. Most common (and mostrealistic) is to assume an exponential distribution.If there is only one server the mean interarrivaltime must be longer than the mean servicetime, otherwise the queue will eventually becomeinfinite. The service time is either constantor varies with different call categories.There are a great number of queue handlingmethods. The most common one is FIFO (FirstIn, First Out). The calls can also be fetched fromthe queue at random. Other alternatives canmean different priority categories for differenttypes of calls.The result of a theoretical or empirical solutionto a queue problem is often given as a series ofcurves showing the distribution of the waitingtime for the calls.There were several reasonsforstarting asimulation study of AXE 10. The simulationresult was needed for comparisonswith analytical results, and the constructionof a simulation programmeprovides further insight into the behaviourof the system. Simulation alsohas an intrinsic value as a study of methods.Rather extensive compromises had tobe made when preparing the program.The central and regional processors andthe communication between them aredescribed in a fairly detailed way, whereasthe hardware connected to the regionalprocessors has been left out ofthe model. The functional blocks for theprocessing of incoming calls are describedin detail for the sequences ofevents considered to be of the greatestinterest.Jobs that are generated internally withinthe exchange are usually handled at alower priority level than jobs which concernthe traffic handling. However, certaininternal jobs have a degree of prioritysuch that they compete with thetraffic handling, and hence these havebeen included in the model.The simulation program comprises approximately8000 SIMULA statements. Itwas developed using a UNIVAC 100/80,and both the compiler capacity and theavailable storage space were fully utilized.ANALYTICAL MODELSIt is not possible to create a mathematicalanalytical model of the whole AXE 10processor system containing all the elementsdescribed by the Queue FlowModel, the signal operating times in differentprocessors, the behaviour of thesubscribers etc. The reason for this isthat there are no general analyticalmethods for networks of queues. It istherefore necessary to create differentmodels, depending on what features areto be studied. For overall analyses themost suitable methods are those in thebranch of queueing theory known asqueueing network theory. For detailedstudies, on the other hand, many usefulmethods are to be found in the branch ofqueueing theory that deals with queueingsystems consisting of only onestage 56 .Queueing network modelsA so-called Markovian queueing networkmodel requires simplification ofthe central processor structure. Thus itis necessary to refrain from giving certaintypes of signals priority. Moreover,no consideration can be paid to the factthat certain functional blocks generatemore than one signal. Finally an approximationhas to be made of the distributionof operating time.The Markovian queueing networkmodel can provide:1 Information regarding the mean flowsin different parts of the system, andthe mean load on different parts of thesystem.2 Information regarding the length ofqueues, the waiting time in differentparts of the system and sometimesthe total time through the whole system.Much of the structure in the Queue FlowModel for traffic handling, THL, can beretained in a Markovian queueing networkmodel of system AXE 10, but parallelactivities cannot be described in thelatter model. This is one of the majordisadvantages of this type of model.Fig. 6Model of the central processor (CP)