1. Queueing Theory 2008

2. Queueing theory definitions
(Bose) “the basic phenomenon of queueing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.”
(Kleinrock) “We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory." "Any system in which arrivals place demands upon a finite capacity resource may be termed a queueing system.”
(Mathworld) “The study of the waiting times, lengths, and other properties of queues.”
排队论是专门研究带有随机因素，产生拥挤现象的优化理论。也称为随机服务系统。

3. Applications of Queueing Theory
Telecommunications
Determining the sequence of computer operations
Predicting computer performance
One of the key modeling techniques for computer systems / networks in general
Vast literature on queuing theory
Nicely suited for network analysis
Traffic control
Airport traffic, airline ticket sales
Layout of manufacturing systems
Health services (eg. control of hospital bed assignments)

4. Queuing theory for studying networks
View network as collections of queues
FIFO data-structures
Queuing theory provides probabilistic analysis of these queues
Examples:
Average length (buffer)
Average waiting time
Probability queue is at a certain length
Probability a packet will be lost

5. Model Queuing System Use Queuing models to
Describe the behavior of queuing systems
Evaluate system performance
Customers
Queue
Server
Queuing System

6. Customer n Customer n+1 Arrival event Begin service End service Delay
Activity
Time
Interarrival
Arrival event
Begin service
End service
Delay
Activity
Time
Customer n+1

7. Characteristics of queuing systems Kendall Notation 1/2/3(/4/5/6)
Arrival Distribution
Service Distribution
Number of servers
Total storage (including servers) (infinite if not specified)
Population Size (infinite if not specified)
Service Discipline (FCFS/FIFO)

8. Distributions
M: stands for "Markovian / Poisson" , implying exponential distribution for service times or inter-arrival times.
D: Deterministic (e.g. fixed constant)
Ek: Erlang with parameter k
Hk: Hyperexponential with param. k
G: General (anything)

9. Poisson process & exponential distribution
Inter-arrival time t (time between arrivals) in a Poisson process follows exponential distribution with parameter (mean)
无后效性 － 不管多长时间(t)已经过去，逗留时间的概率分布与下一个事件的相同 
fT(t) t

10. Examples
M/M/1:
Poisson arrivals and exponential service, 1 server, infinite capacity and population, FCFS (FIFO)
the simplest ‘realistic’ queue
M/M/m/m
Same, but
m servers,
m storage (including servers)
Ex: telephone

11. Analysis of M/M/1 queue l: Arrival rate (mean) of customers (jobs)
Given:
l: Arrival rate (mean) of customers (jobs)
(packets on input link)
m: Service rate (mean) of the server
(output link)
Solve:
L: average number in queuing system
Lq average number in the queue ~ “1”
W: average waiting time in whole system
Wq average waiting time in the queue ~ “1/m”

12. M/M/1 queue model l m Wq W L Lq

13. Derivation
since all probability sum to one

14. Solving W, Wq and Lq
For stability, mean arrival rate must be less than mean service rate Utility factor < 1

15. Response Time vs. Arrivals

16. Example On a network router, measurements show
the packets arrive at a mean rate of 125 packets per second (pps)
the router takes about 2 millisecs to forward a packet
Assuming an M/M/1 model
What is the probability of buffer overflow if the router had only 13 buffers
How many buffers are needed to keep packet loss below one packet per million?

17. Example Arrival rate λ = 125 pps Service rate μ = 1/0.002 = 500 pps
Router utilization ρ = λ/μ = 0.25
Prob. of n packets in router =
Mean number of packets in router =

18. Example
Probability of buffer overflow: = P(more than 13 packets in router) = ρ13 = = 1.49x = 15 packets per billion packets
To limit the probability of loss to less than 10-6:
= 9.96