Network Design and Planning (sq16) Analysis of offered, carried and lost traffic in circuit-switched systems Massimo Tornatore Dept. of Electronics, Information and Bioengineering Politecnico di Milano Dept. Computer Science University of California, Davis tornator@elet.polimi.it
Summary General considerations Statistical traffic characterization Analysis of server groups Dimensioning server groups 2 Traffic theory
Summary General considerations Definitions Parameters Traffic characterization Analysis of server groups Dimensioning server groups 3 Traffic theory
Network Basic concepts We are dealing with circuit-switched networks with given resources/capacity System that we analyse servers A B m offered traffic/ users/sources 4 Traffic theory
Network Basic concepts 3+(1) fundamental parameters A : offered load m : Service system with certain capacity P : quality of service (e.g. delay or blocking probability) F: functional characteristics (e.g. queueing discipline, routing technique, etc.) Problems Dimensioning (synthesis, network planning) • Given A, P (and F), find m at minimum cost/capacity Performance evaluation (analysis) • Given A , m (and F), find P Management (traffic engineering) • Given A and m , find F optimizing P 5 Traffic theory
Network Basic concepts For each model a statistical characterization needed for Traffic sources Server systems Traffic Service sources system 6 Traffic theory
Network Basic concepts Sources S traffic sources Generate connection requests (calls) • Busy source: source engaged in a service request Otherwise the user is not busy or free • Average number of busy sources = Average amount of offered traffic Servers m system servers Satisfy requests issued by sources • Busy server: server engaged in a service to a source for a time duration requested by the source (holding time of the connection) Average number of busy servers = Average amount of carried traffic Congestion: a connection request is not accepted ⇒ Blocked request Denied request (loss systems) Delayed request (waiting systems) 7 Traffic theory
Network Basic concepts E[ θ ]: average holding time of a connection Offered traffic Λ o : average rate of connection requests A o : average number of connection requests issued in a time interval equal to the average holding time ⇒ A o = Λ o E[ θ ] = Λ o / µ Carried traffic Λ s : average acceptance rate of connection requests (statistical equilibrium) A s : average number of connection requests accepted in a time interval equal to the average holding time ⇒ A s = Λ s E[ θ ] = Λ s / µ Lost traffic Λ p : average refusal rate of connection requests A p : average number of connection requests denied in a time interval equal to the average holding time ⇒ A p = Λ p E[ θ ] = Λ p / µ A o , A s , A p adimensional ⇒ Erlang 8 Traffic theory
How do we use queueing theory for traffic characterization? A p 9 Traffic theory
Summary General considerations Traffic characterization Statistical behaviour Modeling of offered traffic Analysis of server groups Dimensioning server groups 10 Traffic theory
Traffic description Statistical behaviour H H ´ ´ 1 6 Relevant time instants 1 H H ´ ´ 3 7 Time of service request 2 H H ´ ´ 4 8 Time of service completion 3 X(t, ω ) = Number of servers H H ´ ´ 2 5 4 busy at time t of realization X(t) ω of the process 4 H H 7 8 3 Assumptions H H H 6 7 8 Stationarity 2 E [to,to+ τ ] [X(t, ω )] = A τ (t 0 , ω ) H H H H H H 2 3 4 5 6 7 • 1 = A τ ( ω )= A( ω ) H H H H 1 3 4 5 Ergodicity 0 t + τ t t A( ω ) = A 0 0 • 11 Traffic theory
Traffic description Two main parameters Holding time θ (duration of the call/request) • It is the inverse of the service rate: E( θ )=1/ µ • We will stick to the traditional assumption of negative exponential distribution of the holding time – Simple and practical Interarrival time T (time between the arrival of two calls) • It is the inverse of the arrival rate E( Τ )=1/ λ • We will consider the traditional assumption (Poisson), as well as two other cases (Bernoulli and Pascal) 12 Traffic theory
Traffic description Modelling service duration 0.16 0.14 Avg. holding time (min) Valor medio 1.6 0.12 Frequency of occurence 0.10 Frequenza di presentazione 0.08 Oltre 10 min. 0.06 − t { } = e θ ˜ Pr θ > t 0.04 0.02 0 1 2 3 4 5 6 7 8 9 10 11 Holding time (min) Tempi di tenuta 0 (min) Possible histogram of holding times and corresponding approximation though exponential distribution 13 Traffic theory
Traffic description Interarrival time distributions As for the interarrival time we will see three distributions: Pascal, Bernoulli, Poisson Why are they interesting? See next slides 14 Traffic theory
Traffic characterization Poisson 50.00 45.00 40.00 35.00 Parameters 30.00 A o = Λ o = 30 25.00 m = 50 20.00 15.00 10.00 50 100 150 200 250 300 350 15 Traffic theory
Traffic characterization Bernoulli 50.00 45.00 40.00 35.00 Parameters A o = 30 30.00 m = 50 25.00 S = 40 20.00 15.00 10.00 50 100 150 200 250 300 350 16 Traffic theory
Traffic characterization Pascal 50.00 45.00 40.00 35.00 Parameters A o = 30 30.00 m = 50 25.00 c = 10 20.00 15.00 10.00 50 100 150 200 250 300 17 Traffic theory
Traffic description How do we model the three previous traffic behaviors? We use a birth & death process [X(t)] to represent the offered traffic Births: arrivals of service requests Deaths: service completions In general b&d processes are characterized by two parameters [ ] • E X ( t ) [ ] [ ] Var X ( t ) = • Var ( ) VMR (peakednes s factor) X t or [ ] E X ( t ) same characterization for all traffic types (offered, carried, lost) Typically, modelling simplicity suggests t [ ] [ ] − − µ θ θ > = E = t • D eaths : exponentia l - Pr t e e ] ( ) [ k λ t − λ = = t • B irths : Poisson - Pr X ( t ) k e k ! In this lecture we go beyond Poisson (VMR = 1) and we also consider Smoothed traffic (VMR < 1) - Bernoulli Peaked traffic (VMR >1) - Pascal 18 Traffic theory
Offered traffic model Assumptions Arrival and service processes Indipendent identically distributed (IID) interarrival times IID service times Arrival and service process mutually independent Ergodicity Stationarity 19 Traffic theory
Offered traffic model Single source Source model Two states: idle (0) or busy (1) { } ′ → ∆ = λ ∆ Pr 0 1 in ( t , t + t ) | 0 t { } → ∆ = µ ∆ Pr 1 0 in ( t , t + t ) | 1 t ⇒ interarrival and service times with exponential distribution and λ ' = conditioned average interarrival rate (idle source) • µ = conditioned average rate of service completion (busy source) • Steady-state limiting probabilities ′ µ λ = = = q q A 0 1 o ′ ′ λ + µ λ + µ ′ λ µ 1 1 1 = + → λ = µ = A individual average interrival rate o ′ ′ λ µ λ λ + µ ′ λ λ α = = = = a q offered traffic by a source 1 ′ µ λ + µ + α 1 ′ λ λ q a α = = = = 1 offered traffic by an idle source µ − µ − λ − 1 q 1 a 1 20 Traffic theory
Offered traffic model Multiple sources Single source model ensures that the occupancy process of a source groups is markovian continuous-time and time-homogeneous with discrete states of birth & death type In formulas, this mean that the transition probabilities can be written as { } ′ → ∆ = λ ∆ Pr 0 1 for a source in ( t , t + t ) | n busy sources, 0 t n { } → ∆ = µ ∆ Pr 1 0 for a source in ( t , t + t ) | n busy sources, 1 t n IID service times (also called source occupancy times) ⇒ µ n = n µ Interbirth times described by three models ( ) ′ ′ λ = λ − • Bernoulli S n [ S sources] n ′ λ = Λ = λ ∞ • Poisson [ sources] ( ) n o ′ ′ λ = λ + ∞ • Pascal c n [ sources] [ c integer] n 21 Traffic theory
Offered traffic model Steady state characterization State probabilities in steady- state conditions derived by queues M/M/∞ X = Number of sources busy at the same time A o = A s = E[ X ] = Λ o / µ ( ) ′ λ λ S − S n = n − = = = = Bernoulli p a 1 a n 0 ,..., S a p n 1 ′ µ λ + µ n n λ a − = a = Poisson p e a n µ n ! ( ) + − ′ λ c n 1 n c = α − α α = Pascal p 1 n µ n 22 Traffic theory
Recommend
More recommend
