Teletraffic Theory AGH Eng

[ Pobierz całość w formacie PDF ]
//-->Instantaneous traffic intensity xt123calls•••Basic Concepts ofTeletraffic TheoryCxttMeasure of telephone trafficcalledinstantaneous traffic intensityin moment „t” isnumber of simultaneous calls „xt”.Stanisław StochSwitching Systems1Stanisław StochSwitching Systems2Traffic volume – an example5minQ = 1call • 5minQ = 5calls • 1minTraffic volume Q123calls•••1minQ = 5 PM4minxt21[PM] = callminuteQ =∑xi•∆ti∆i=1LCxt∆t1∆t2…∆ti…∆tL1min3minttQ = 2calls •1min + 1call •3minLQ = x1•∆t1+ x2•∆t2+ ... + xL•∆tL=∑xi•∆ti∆∆∆∆i=1Stanisław StochSwitching Systems3Stanisław StochSwitching Systems4Traffic volume Q123Traffic volume Q123callscalls••••••Q =∑xi•∆ti∆i=1MLCxtQ =∑xi•∆ti∆i=1MLCxtQ =∑xj•∆tj=1∆t ∆t ∆t ∆t ∆t ∆t ∆t ∆t ∆ttQ =∑xj•∆tj=1TQdt dt dt dt dt dt dt dtdttTQ =∫xtdtRepresentation of traffic volume is the area on thegraph of instantaneous traffic intensity „xt”.Stanisław StochSwitching Systems5Stanisław StochSwitching Systems61Average traffic intensity A – the definitionQ=A•TA=1TQTT123Units of traffic intensityThe unit of traffic intensity „A” is 1 erlang.Interpretation of big values:A = 10 erl means – in every moment there are(averagely) 10 simultaneous calls (10 circuits busy)Interpretation of small (fractional) values:A = 0,1 erl means – probability of existence of onecall (of occupation of some circuit in a group)in every moment equals 0,11erl = 1call– for simultaneous calls onlycallsA•••CxtA=A=∫xtdtQdt dt dt dt dt dt dt dtdttT1Lx1•∆t1+ x2•∆t2+ ... + xL•∆tL=∑xi•∆tiTTi=1x1+ x2+ x3+ ... + xLA=for equal∆tonly, where:T = L •∆tLStanisław StochSwitching Systems8In USA:CCS(hundredcall seconds per hour)1 erl = 36 CCSStanisław StochSwitching Systems9Units of traffic volume QDaily variations of trafficUnits of traffic volume Q:Example typical for Poland.number ofoccupieddevicesPM = callminuteQ [PM] = A [erl] • T [min]callhour1 erl • 1 h = 1 erl • 60 min = 60 PMIn USA:pcm (paidcall minutes)= 1PM (conversation)246810121416182022hours of dayStanisław StochSwitching Systems10Stanisław StochSwitching Systems11Busy HourWeakly variations of trafficTime period 10:00 – 11:00Busy hour is the uninterrupted period of60 min during which the traffic is a maximum.The busy hour is defined asthat four consecutive quarter hourswhose traffic intensity is the greatest.Mon Tu WeStanisław StochSwitching Systems12ThFrSatSu13Stanisław StochSwitching Systems2Seasonal variations of trafficLong-term variations of trafficThe traffic shows generally a consistenttendency to increase. This increase is notuniform. Decline is also possible.Therefore, high and low values may havedifferent relative increase. It is generally010203040506070809 10 1112Monthrather difficult to distinguish betweengrowth and seasonal variations.Stanisław StochSwitching Systems14Stanisław StochSwitching Systems15Average holding time h (call duration)połączenia(eg.billing)q1=1• h1q2=1• h2•••qC=1• hCQ123123123Calls overstepping the T-period boundarycallscallsQ=C•hA•T=C•hUnits are important:10 erl • 1 h = 10 erl • 60 min = 600 PM = 200 calls • 3 minStanisław StochSwitching Systems17Mean calling rate (mean call intensity)A•T=C•hλ=CTA=C•h =λ•hT[ calls / h ]•λis basic measure for control devices /processors (setting time of a call isindependent on call duration)• BHCA (BusyHour Call Attempts)– maximalallowed vallue ofλ• A is measure of traffic intensityStanisław StochSwitching Systems19•••C•••C•••ChxtAdt dt dt dt dt dt dt dtdtxtTtTTtQ=A•TTDon't split calls into parts.Count each call only once – even if it appears in two(or more) subsequent observation periods.Assign it to that observation period, where the callbegins (count „call attempts”).Stanisław StochSwitching Systems18Traffic intensity in subscriber lineA=λ ≈2 calls/hC•h =λ•hTh≈2,4 min for local callsλ ≈0,1 call/h h≈3,8 min for long distance callsλ ≈0,1 call/h h≈3,2 min for calls to specialservices (9xxx)A = A1+ A2+ A3= 2 • 2,4 + 0,1 • 3,8 + 0,1 • 3,2≈≈6 (calls/h) • min = 6 (calls / 60min) • min = 0,1 erlSwitching Systems20Stanisław Stoch3Exercise 1.Exercise 2.In the observation time of T = 2ha subscriber group made C = 400 callswith average call duration of h = 3 min.Calculate average traffic intensity A for thatsubscriber group.A•T=C•hA=A=C⋅h 400⋅3==600T2In the trunk group, average number of busytrunks equals A = 20.In the observation time of T = 1h thenumber of C = 600 calls has been counted.Calculate mean holding time h.WRONGh=A•T=C•hA⋅T 20erl⋅1h 20calls⋅60min===2minC600calls600callsSwitching Systems22C⋅h 400calls⋅3min 400calls⋅3min===10erlT2h120minSwitching Systems21Stanisław StochStanisław StochTotal occupation time of i:th circuitτicircuitscircuits12Average occupation timeτcircuitsq1=1•τ1q2=1•τ2•••1212•••Stanisław StochMean occupancy (load per device)Aa==NInterpretations of„a”:a [%] =a [%] =a[erl]=Stanisław Stoch•••N•••NNqN=1•τNτixtAdt dt dt dt dt dt dt dtdtQτQ=N•τxtAdt dt dt dt dt dt dt dtdtttQ=A•TTQ=A•TTA•T=N•τUnits are important:10 erl • 1 h = 10 erl • 60 min = 600 PM = 20 trunks • 30 minSwitching Systems24Stanisław StochSwitching Systems25τT(0≤a≤1)τ[min]T[min]A[erl]N[=Amax]- (average) fraction of observationtime when one circuit is occupiedblank- (average) fraction ofsimultaneously occupied circuitsin a group, same as probability ofoccupation of one circuit- (average) traffic intensity percircuit (also called: load per device)Switching Systems26A[erl]N [dim’less]4Queuing systemAofBasic Concepts ofTeletraffic Theory( part II )AcArcircuitgroupAcAof- offered trafficAc- carried trafficAr- rejected trafficAof=Ar+AcStanisław StochSwitching Systems28Stanisław StochSwitching Systems29B - call congestionThe measure of quality of service iscall congestion B.B=ArAof=number of rejected callsnumber of offered calls(0≤B≤1)E - time congestionCalls are rejected, when all circuits are occupied.Another measure of quality of service istime congestion.xtnumber of circuits NcongestioncongestiontBecause measuring of Aris in many casesdifficult, another measure has beenintroduced (follows on next slide).Stanisław StochSwitching Systems30observation timeE=Stanisław Stochcongestion timeobservation timeSwitching Systems(0≤E≤1)31Comparison of B and Extnumber of circuits NComparison of B and E (cont.)xtnumber of circuits NCALLS→congestioncongestiontCALLS→congestioncongestionobservation timetobservation timeIf calls are independent (Poisson distribution) their timedistribution is independent on congestion intervals.Fraction of rejected calls is proportional to congestiontime, so:B=EStanisław StochSwitching Systems32If calls are not independent (distribution different toPoisson), in the congestion interval may appear more(as on the picture above) or less calls.Ratio of rejected calls to all calls is than not equal toratio of congestion time to observation time, so:B≠E(on the picture B > E)Stanisław StochSwitching Systems335 [ Pobierz całość w formacie PDF ]