Lectures5_6 - Queueing

Lectures5_6 - Queueing - Lectures 5 & 6...

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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 bit-error-rate, 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 packets per second Over a small interval , P(exactly one arrival) = + () P(0 arrivals) = 1 - + () P(more than one arrival) = 0() Where 0()/ > 0 > 0. It can be shown that: P(n arrivalsininterval T) = ( T ) n e T n ! Eytan Modiano Slide 4 The Poisson Process P(n arrivalsininterval T) = ( T ) n e T n ! n = number of arrivals in T It can be shown that, E[n] = T E[n 2 ] = T +( T) 2 2 = E[(n -E[n]) 2 ] = E[n 2 ]-E[n] 2 = T Eytan Modiano Slide 5 Inter-arrival times Time that elapses between arrivals (IA) P(IA <= t) = 1 - P(IA > t) = 1 - P(0 arrivals in time t) = 1 - e- t This is known as the exponential distribution Inter-arrival CDF = F IA (t) = 1 - e- t Inter-arrival PDF = d/dt F IA (t) = e- 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 e t dt e t | t 0 t 0 + t e ( t + t ) + e ( t ) t = = = e ( t ) e t dt e t | t 0 t 0 = 1 e 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 inter-arrival 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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Lectures5_6 - Queueing - Lectures 5 & 6...

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