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Unformatted text preview: Lectures 5 & 6 6.263/16.37 Introduction to Queueing Theory Eytan Modiano MIT, LIDS Eytan Modiano Slide 1 Packet Switched Networks Packet Network PS PS PS PS PS PS PS Buffer Packet Switch Messages broken into Packets that are routed To their destination Eytan Modiano Slide 2 Queueing Systems • Used for analyzing network performance • In packet networks, events are random – Random packet arrivals – Random packet lengths • While at the physical layer we were concerned with biterrorrate, at the network layer we care about delays – How long does a packet spend waiting in buffers ? – How large are the buffers ? • In circuit switched networks want to know call blocking probability – How many circuits do we need to limit the blocking probability? Eytan Modiano Slide 3 Random events • Arrival process – Packets arrive according to a random process – Typically the arrival process is modeled as Poisson • The Poisson process – Arrival rate of λ packets per second – Over a small interval δ , P(exactly one arrival) = λδ + ο(δ) P(0 arrivals) = 1  λδ + ο(δ) P(more than one arrival) = 0(δ) Where 0(δ)/ δ −> 0 δ −> 0. – It can be shown that: P(n arrivalsininterval T) = ( λ T ) n e − λ T n ! Eytan Modiano Slide 4 The Poisson Process P(n arrivalsininterval T) = ( λ T ) n e − λ T n ! n = number of arrivals in T It can be shown that, E[n] = λ T E[n 2 ] = λ T +( λ T) 2 σ 2 = E[(n E[n]) 2 ] = E[n 2 ]E[n] 2 = λ T Eytan Modiano Slide 5 Interarrival times • Time that elapses between arrivals (IA) P(IA <= t) = 1  P(IA > t) = 1  P(0 arrivals in time t) = 1  e λ t • This is known as the exponential distribution – Interarrival CDF = F IA (t) = 1  e λ t – Interarrival PDF = d/dt F IA (t) = λ e λ t • The exponential distribution is often used to model the service times (I.e., the packet length distribution) Eytan Modiano Slide 6 Markov property (Memoryless) P ( T ≤ t 0 + t  T > t ) = P ( T ≤ t ) Pr oof : P ( T ≤ t 0 + t  T > t ) = P ( t 0 < T ≤ t 0 + t ) P ( T > t ) t 0 + t ∫ λ e − λ t dt − e − λ t  t 0 t 0 + t − e − λ ( t + t ) + e − λ ( t ) t = = = ∞ ∞ e − λ ( t ) ∫ λ e − λ t dt − e − λ t  t 0 t 0 = 1 − e − λ t = P ( T ≤ t ) • Previous history does not help in predicting the future! • Distribution of the time until the next arrival is independent of when the last arrival occurred! Eytan Modiano Slide 7 Example • Suppose a train arrives at a station according to a Poisson process with average interarrival time of 20 minutes • When a customer arrives at the station the average amount of time until the next arrival is 20 minutes – Regardless of when the previous train arrived • The average amount of time since the last departure is 20 minutes!...
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 Spring '09
 Queueing theory, Eytan Modiano, Eytan Modiano Slide

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