16.36: Communication Systems Engineering Lecture 17/18: Delay Models for Data Networks Eytan Modiano
Eytan Modiano Slide 1 16.36: Communication Systems Engineering Lecture 17/18: Delay Models for Data Networks Eytan Modiano
Packet Switched Networks Messages broken into Packets that are routed To their destination Packet Network Buffer Packet Switch
Eytan Modiano Slide 2 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
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
Eytan Modiano Slide 3 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 ?
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 n packets per second Over a small interval s P(exactly one arrival)=78 P(O arrivals)=1-78 P(more than one arrival)=0 It can be shown that P(narrivalsinintervalT) (T'e
Eytan Modiano Slide 4 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 P(0 arrivals) = 1 - λδ P(more than one arrival) = 0 – It can be shown that: P(narrivalsinintervalT)= − ( ) ! λ λ T e n n T
The poisson process P(narrivalsinintervalT)-(Te-7 n number of arrivals in t It can be shown that E[]=T En]=AT +T). 02=El(n-E(OJ)2 ]=En21-E[n]=AT
Eytan Modiano Slide 5 The Poisson Process P(narrivalsinintervalT) = − ( ) ! λ λ T e n n T n = number of arrivals in T It can be shown that, E[n] = T E[n ] = T + ( T) = E[(n -E[n]) ] = E[n ] -E[n] = T 2 2 2 2 22 λ λ λ σ λ