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Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Basic Definitions and Terminology Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Course – assigned reading M. Stasiak, M. Głąbowski, P. Zwierzykowski: Modeling and Dimensioning of Mobile Networks: from GSM to LTE, John Wiley and sons Ltd., January 2011. Iversen V.B., ed., Teletraffic Engineering, Handbook, ITU, Study Group 2,Question 16/2 Geneva, January 2005, published on-line. 2Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Arrival stream 3Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Stochastic point process Possible realization of the stochastic point process 4Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Process parameters Λ o intensity of arriving calls P k (t) o probability of k calls arrival within time interval of length t f(t) o inter-arrival time distribution 5Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Arrival Poisson processes properties Stationarity o stream intensity is not time-dependable o λ(t)= λ =const Memorylessness (independence of all time instants) o number of arrivals occurring within the time interval t 1 is independent of the number of arrivals occurring within the time interval t 2 Singularity o in a given time point only one arrival can occur 6Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Lack of memory property Distribution function of time interval between consecutive calls (inter-arrival time) is exponential function: We assume that inter-arrival time interval is equal to t. Let us determine the conditional probability so that this interval lasts for at least time τ. So, we can 7 0 t+ tT Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Lack of memory property Taking into account the distribution function we have The conditional probability have to receive the following value: 8 )( )( tTTPee tt Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Singularity property Let us consider time interval Δt → 0. It results from the singularity property that probability of appearance of more then one arrivals within the time interval Δt is going towards 0: o where ( t) is infinitely small value if compared with Δt Elementary probabilities: 9Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Poisson stream parameter The flow parameter (t) at time point t is defined as the limit of quotient: Probability of appearing at least one arrival within time interval t+ t time interval length t 0: 10Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Poisson stream - characteristics Probability of appearance of k arrivals at time t: for k=0 i k=1 we receive: 11 k Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Poisson stream – characteristics Inter-arrival time distribution: Mean value and variance of inter-arrival time: Peakedness coefficient: 12Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Streams operations Superposition of Poisson streams Random decomposition of Poisson stream 13 Stream 1 Stream 2 Stream 3 Stream 1 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Streams operations Erlang k-decomposition o inter-arrival time distribution o mean value of inter-arrival time o variance of inter-arrival time o disorder coefficient 14 Stream 1 Stream 2 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Markov process definition A stochastic process is called the Markov process when the future trajectory of the process depends only on the present state S(t 0 ) at the time point t 0, but is independent of how this state has been obtained. 15 pastfuture Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Markov process in the M/M/2 system A service process in the M/M/2 system (trunk group with two channels) is the Markov process when: o Arrival process is the Poisson process, o Service time has exponential distribution. 16 Trajectory of the service process in M/M/2 system Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Service stream 17Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Trajectory of the Markov process 18Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Exponential service time Distribution function: Density function: Mean value and variance: 19Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Service stream At time point t there are k servers busy. The probability of service termination in i servers within Δt time-interval can be determined on the basis of Bernoulli distribution for i successful events, when total number of events is equal to k: Probability of service termination in one server within Δt time-interval: 20Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Service stream For i=0 we obtain the probability of the event that within, time interval Δt, there are no terminations among k busy servers: Termination probability by at least one server: P 1 (Δt ) decomposition (into series): Service stream parameter: 21 k t t kt 0t )( lim)( Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Markov proces 22Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Birth and death process in M/M/2 system state 0 all links are free state 1 one busy link state 2 two links are busy 23 blocking state 0 1 2 2 1 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Birth and death process in M/M/2 system Infinite number of traffic sources Finite number of busy servers 24 0 1 2 2 1 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Kolmogorov equations Determination of the probability P 0 (t + Δt) Events within time Δt: o Was in state "0" and transferred into state "1": λ Δt o Was in state "0" and remained in state "0": 1- λ Δt o Was in state "1" and transferred into state "0": μ Δt 25Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Kolmogorov equations 26 Was in state "0" and remained in state "0"Was in state "1" and transferred into state "0" Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Kolmogorov equations 27Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Kolmogorov equations Solution: 28 Solution of Koplmogorov equations in M/M/2/0 system for λ=μ=1, P0(0)=1 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Steady-state distribution 29 Probability calculations: Solution In the steady-state regime of the process, the state probabilities are not time-dependable Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Steady states Interpretation of the probability [Pi] V : The state probability is interpreted as the proportion of the time in which the system remains in state i: 30Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Stream value in state i 31 Stream value in the state i for : ii P Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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State equations in the system M/M/2 For the M/M/2/0 system state equations take the following form: 32 In state i: Sum of incoming streams = sum of outgoing streams Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Generalized birth and death process State transition diagram: 33 so: Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Local balance equation Streams between neighboring states are in equilibrium Process solution 34 where: Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Local balance equation in M/M/2 35Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Concept of Traffic 36Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Telecommunication traffic Traffic as a process of capacity units occupancy where n(t) – number of occupied units at time T Units: o 1 SM (speech-minutes) o 1 Eh (Erlang-hour) o 1 Eh = 60 SM 37Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Telecommunication traffic intensity Traffic intensity: o where n(t) – number of occupied units at time T Units: 1 Erlang ( 1 Erl.) o 1 Erlang = 1 call serviced during time t when observation time is equal to t 38Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Telecommunication traffic and traffic intensity Traffic volume: Traffic intensity: service time t1t1 t3t3 t2t2 T 39Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Traffic intensity 40 service time t1t1 t3t3 t2t2 T t1t1 t3t3 t2t2 T busy timeidle time T=100% =time unit % of idle time % of busy time Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Traffic intensity Parameters: o V=4 - number of channels, o N=5 - number of time periods, o t obs =5T - period under consideration, o t i,j - occupancy of the j-th channel during the i-th time period o - call intensity o h - mean service time. 41Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Traffic intensity Def. 1 Traffic intensity is equal to the average number of simultaneously occupied channels during a given period of time under considerations. 42Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Traffic intensity Def. 2 Traffic intensity is the ratio of the sum of channel occupancy time during a given period of time under considerations with respect to this period. 43Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Traffic intensity Def. 3 The product of the average number of o calls (offered traffic) o connections (carried traffic) per time unit and the average time of connection. 44Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Traffic intensity Def. 4 The mean number of calls (connections) per mean service time offered traffic carried traffic 45Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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System Offered traffic Carried traffic Rejected traffic ATTENTION! Conventionally, under the notion of traffic we understand traffic intensity Kinds of traffic 46Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Kinds of traffic Carried traffic o the traffic carried by the group of servers during the time interval T Offered traffic o the traffic which would be carried if no calls were rejected due to lack of the capacity, i.e. unlimited number of servers. The offered traffic is a theoretical value and it cannot be measured Lost (rejected) traffic o the difference between offered traffic and carried traffic 47Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Quality of service in telecommunication systems 48Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Call and packet level in networks 49Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Concept of blocking Call congestion (Call loss probability) B(t 1, t 2 ) in time interval (t 1, t 2 ) is the fraction of all calls which are rejected due to lack of capacity N lost (t 1, t 2 ) with respect to all calls which are offered in the system N offered (t 1, t 2 ) 50Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Concept of blocking Time congestion (Blocking probability) E(t 1, t 2 ) in time interval (t 1, t 2 ) is the fraction of the time T blocking (t 1, t 2 ) when all servers are busy with respect to the total time of observation T(t 1, t2) 51Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Basic notions and parameters Traffic load capacity o the value of the offered traffic (traffic intensity) which can be serviced with the adopted value of blocking probability (loss probability) Load o the value of the carried traffic (traffic intensity) in the system Blocking o the state of system in which a call arriving at the input of the system cannot be serviced due to occupancy of all servers in the system Throughput o the probability of event that the given call will be serviced in the system 52Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Quality of service in communication systems (QoS) Packet delay (cell delay) o A delay considered between the moment of sending and receiving the packet (in appropriate nodes) Delay parameters (for example of ATM network) o CDT mean - Mean Cell Transfer Delay - statistical average delay of packet o CTD max - Maximum Cell Transfer Delay – maximum delay of packet, guaranteed by network with probability 1-α o CDVpeak-peak - Peak to Peak Cell Delay Variation – maximum delay decreased by constant system delay (i.e. propagation time, processing time in node) 53Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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constant max delay variation max delay Delay distribution α= l oss ratio delay 1-α Quality of service in communication systems (QoS) Interpretation of delay parameters 54Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Reasons for networks delay Constant and independent of network load o propagation time in physical layer o processing time in network node o minimum time the node wait for packet acknowledgement o bit rate of outgoing link bigger than incoming link 55Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Reasons for networks delay Dependent on network load o queuing in the buffers o queuing discipline, o priorities for given packet classes o mechanisms for packet streams shaping o resources reservation for given packet classes 56 server outgoing stream buffer incoming stream Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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Kinds of traffic at the packet level In majority of packet networks kinds of traffic are associated with parameters of offered services We can always distinguish the following traffic streams o Constant bit rate traffic o Variable bit rate traffic stream traffic, constant parameters of transmission adaptive traffic elastic traffic 57Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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