Network Planning VITMM215 Markosz Maliosz PhD 10/05/2016
Outline Telephone network dimensioning – Traffic modeling – Erlang formulas – Exercises 2
Telephone Network Circuit switching Each voice channel is identical For each call one channel is allocated A call is accepted if at least one channel is idle Goal: network dimensioning Question to answer: How many circuits are required to satisfy subscribers’ needs ? Input: traffic statistics – subscribers’ behavior: when, how often are calls arriving? how long are the call durations? 3 3
Arrival Process In our case: telephone calls arriving to a switching system described as stochastic point process we consider simple point processes, i.e. we exclude multiple arrivals the i th call arrives at time T i N(t) : the cumulative number of calls in the half- open interval [0; t[ N(t) is a random variable with continuous time parameter and discrete space N(t) t 4 4
Arrivals and Departures N(t) D(t) N ( t ) to be the cumulative number of arrivals up to time t D ( t ) to be the cumulative number of departures up to time t L ( t ) = N ( t ) - D ( t ) is the number of calls at time t 5
N(t) N(t) Equations D(t) D(t) Average arrival rate: λ (t) = N(t)/t F(t) = area of shaded region from 0 to t in the figure = total service time for all customers = carried traffic volume Average holding time: W(t) = F(t)/N(t) Average number of calls: L(t) = F(t)/t = W(t)N(t)/t = W(t) λ (t) 6
Traffic Volume Volume of the traffic: the amount of traffic carried during a given period of time Traffic volume in a period divided by the length of the period is the average traffic intensity in that period = average number of calls 7
Traffic Variations Traffic fluctuates over several time scales – Trend (>year) Overall traffic growth: number of users, changes in usage Predictions as a basis for planning – Seasonal variations (months) – Weekly variations (day) – Daily profile (hours) – Random fluctuations (seconds – minutes) In the number of independent active users: stochastic process Except the last one, the variations follow a given profile, around which the traffic randomly fluctuates 8
Traffic Variations Source: A. Myskja, An introduction to teletraffic, 1995 . 9
Busy Hour It is not practical to dimension a network for the largest traffic peak describe the peak load, where singular peaks are averaged out Busy Hour = The period of duration of one hour where the volume of traffic is the greatest. Operator’s intention: spreading the traffic – By service tariffs busy hour period is the most expensive less important calls are started outside of the busy hour, and typically last longer Recommendations define how to measure the busy hour traffic – There are several definitions (ITU E.600, E.500) – An operator may choose the most appropriate one 10
Busy Hour Measurements ADPH (Average Daily Peak Hour) – one determines the busiest hour separately for each day (different time for different days), and then averages over e.g. 10 days TCBH (Time Consistent Busy Hour) – a period of one hour, the same for each day, which gives the greatest average traffic over e.g. 10 days FDMH (Fixed Daily Measurement Hour) – a predetermined, fixed measurement hour (e.g. 9.30-10.30); the measured traffic is averaged over e.g. 10 days 11
Traffic Model Average arrival rate: λ (t) – depends on time, however it has a very strong deterministic component according to the profiles In the busy hour period the average arrival time is considered stationary: λ , and the arrival process is considered as a Poisson process with intensity λ – Time homogenity – Independence The future evolution of the process only depends upon the actual state. Independent of the user(!) – modeling all users in the same way The average holding time ( W(t) ) is also considered to be stationary, and exponentially distributed with intensity μ 12
Traffic Model N(t) – Poisson process: Poisson – in time interval ( t , t + τ] the number of calls follows a Poisson distribution with parameter λτ – Expected number of calls = λ τ – λ = arrival intensity [1/hour] Exp. W(t) = W – exp. distribution – Expected value = 1/ μ = h – h – average holding time [min] (!) f(x; μ ) = μ e - μ x 13 13
Traffic Intensity A – traffic intensity – A = λ * h – A [1], often written as Erl (Erlang) Example: individual subscriber – λ = 3 [1/hour] – h = 3 [min] = 0.05 [hour] – A = 3 [1/hour]* 0.05 [hour] = 0.15 [Erl] Example: 10 000 line switch – λ = 20 000 [1/hour] – h = 3 [min] = 0.05 [hour] – A = 20 000 [1/hour]* 0.05 [hour] = 1000 [Erl] 14 14
Typical Traffic Intensities Typical traffic intensities per a single source are (fraction of time they are being used) – private subscriber 0.01 - 0.04 Erlang – business subscriber 0.03 - 0.06 Erlang – mobile phone 0.03 Erlang – PBX (Private Branch Exchange) 0.1 - 0.6 Erlang – coin operated phone 0.07 Erlang 15
Traffic Modeling Agner Krarup Erlang (1878 – 1929) – Danish mathematician, statistician and engineer Conditions: – n identical channels – Blocked Calls are Cleared (BCC) – The arrival process is a Poisson process with intensity λ – The holding times are exponentially distributed with intensity μ (corresponding to a mean value 1/ μ ) The traffic process then becomes a pure birth and death process, a simple Markov process – A= λ / μ 16
Infinite number of channels State diagram – nr. of busy channels: If the system is in statistical equilibrium, then the system will be in state [i] the proportion of time p(i), where p(i) is the probability of observing the system in state [i] at a random point of time, i.e. a time average When the process is in state [i] it will jump to state [i+1] λ times per time unit and to state [i-1] i μ times per time unit 17
Infinite number of channels In equilibrium state – Node equations: – Cut equations: Normalization restriction: 18
Infinite number of channels Derivation of cut equations: A= λ / μ 19
Infinite number of channels Using the normalization constraint: State probabilities: Carried traffic = offered traffic = A No congestion, no traffic loss 20
Limited number of channels State diagram: Normalization condition becomes: State probabilities: 21
Erlang B formula Time congestion: – The probability that all n channels are busy at a random point of time is equal to the portion of time all channels are busy (time average) Call congestion: – The probability that a random call attempt will be lost is equal to the proportion of call attempts blocked. If we consider one time unit, we find by summation over all possible states: Carried traffic = Lost traffic = 22
Erlang B formula Conditions for applicability: – Gives good results if number of subscribers is much greater, than the number of channels (around 10x) – Subscribers initiate calls independently from each other (not applicable e.g. if a TV advertisement presents a phone number and many people call it) – The only reason for blocking is if all channels are busy – Blocked Calls are Cleared, no waiting queue – Subscribers do not repeat call attempt, if call was blocked – The channel is occupied only by the particular subscribers, no resource sharing 23 23
Erlang B formula 24 24
Erlang B formula Example: 3 employees in an office, each of them calls 3 times in an hour with 3 minutes talking time. Question: How many channels are needed for max. 5% blocking? (1? 2? 3??) Answer: – λ = 3*3 [1/hour] – h = 3 [min] = 0.05 [hour] – A = 3*3 [1/hour]* 0.05 [hour] = 0.45 [Erl] E(1) =31% E(2) =6.5% (not enough!) ( E (3) =1%, in reality: E (3 )=0) – 3 channels are needed 25 25
Erlang B formula E.g. 1000 subscriber and n channels: – λ = 1000*3 [1/hour] – h = 3 [min] – A = 1000*3 [1/hour]* 0.05 [hour] = 150 [Erl] – E(n): n 100 150 155 160 200 E(n) 34% 6,2% 4,3% 2,8% 0,0015% If the number of subscribers are large, the required number of channels (n) for a satisfactory blocking ratio converges to A 26 26
Erlang B formula If A and achievable blocking is given, how to calculate n? n A ! n By probing E ( A ) n i n A Recursive method: i ! i 0 – Def.: I n (A) = 1 / E n (A) – I 0 (A) = 1, that is with 0 channel the blocking = 1 – I n (A) = I n-1 (A) * n / A + 1 – E.g. if the goal is: E n (A) = 1 / I n (A) < 0.05 – I n (A) > 1/0.05 = 20 27 27
Extended Erlang B Extended Erlang B: a certain percentage of blocked calls are reattempted – Iterative calculation with extra parameter, the Recall Factor: R f – A 0 :initial traffic intensity 1. Calculate E n (A 0 ) with Erlang B 2. Nr. of blocked calls: B = A 0 E n (A 0 ) 3. Nr. of recalls: R = R f B 4. New offered traffic: A 1 = A 0 + R 5. Return to step 1 and iterate until value of A is stabilized 28
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