Internet Queuing Delay Introduction

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Presentation transcript:

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 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

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

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

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

State Analysis of M/M/1 Queue W: waiting time for a new arrival : service time of i-th customer : remaining service time of the customer in service Exponential r.v. with mean 1/\mu due to memoryless property of expo. Distr. T: sojourn (response) time This T is the access delay at edge router in the problem 9 of Chapt. 2

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?