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Internet Queuing Delay Introduction

Internet Queuing Delay Introduction. How many packets in the queue? How long a packet takes to go through?. The M/M/1 Queue. An M/M/1 queue has Poisson arrivals (with rate λ) Exponential service times (with mean 1/μ, so μ is the “service rate”). One (1) server An infinite length buffer

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Internet Queuing Delay Introduction

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  1. Internet Queuing Delay Introduction • How many packets in the queue? • How long a packet takes to go through?

  2. The M/M/1 Queue • An M/M/1 queue has • Poisson arrivals (with rate λ) • Exponential service times (with mean 1/μ, so μ is the “service rate”). • One (1) server • An infinite length buffer • The M/M/1 queue is the most basic and important queuing model for network analysis • Terminology: A/B/c/K • A: interarrival time distr. B: service time distr. • c: number of servers K: size of buffer • M: expo. distr. G: general distr. D: deterministic

  3. The Poisson Arrival Model • Examples • Customers arriving to a bank • Packets arriving to a buffer • The rateλ of a Poisson process is the average number of events per unit time • The time interval between arrivals follows exponential distribution:

  4. Exponential Distribution Prob. time

  5. State Analysis of M/M/1 Queue • n : number of customers in the system (including queue + server) • pn : steady state probability of finding n customers in the system • =/: Traffic rate (traffic intensity) State transition diagram

  6. State Analysis of M/M/1 Queue • # of transitions  = # of transitions  piare probabilities: : prob. the server is working ( is called “server utilization”)

  7. State Analysis of M/M/1 Queue • N: # of customers in the system

  8. : remaining service time of the customer in service Exponential r.v. with mean 1/ due to memoryless property of expo. Distr. : service time of i-th customer State Analysis of M/M/1 Queue • W: waiting time for a new arrival • T: sojourn (response) time

  9. M/M/1 Queue Example • A router’s outgoing bandwidth is 100 kbps • Arrival packet’s number of bits has expo. distr. with mean number of 1 kbits • Poisson arrival process: 80 packets/sec • How many packets in router expected by a new arrival? • What is the expected waiting time for a new arrival? • What is the expected access delay (response time)? • What is the prob. that the server is idle? • What is P( N > 5 )? • Suppose you can increase router bandwidth, what is the minimum bandwidth to support avg. access delay of 20ms?

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