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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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   2 1 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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