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Introduction to Queuing Theory

Queuing 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. - (Wolff) The primary tool for studying these

problems of congestions is known as queueing

theory. - (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.

http//www2.uwindsor.ca/hlynka/queue.html

Applications of Queuing Theory

- Telecommunications
- Traffic control
- Determining the sequence of computer operations
- Predicting computer performance
- Health services (eg. control of hospital bed

assignments) - Airport traffic, airline ticket sales
- Layout of manufacturing systems.

http//www2.uwindsor.ca/hlynka/queue.html

Example application of queuing theory

- In many retail stores and banks
- multiple line/multiple checkout system ? a

queuing system where customers wait for the next

available cashier - We can prove using queuing theory that

throughput improves increases when queues are

used instead of separate lines

http//www.andrews.edu/calkins/math/webtexts/prod

10.htmQT

Example application of queuing theory

http//www.bsbpa.umkc.edu/classes/ashley/Chaptr14/

sld006.htm

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
- Average waiting time
- Probability queue is at a certain length
- Probability a packet will be lost

Model Queuing System

- Use Queuing models to
- Describe the behavior of queuing systems
- Evaluate system performance

Characteristics of queuing systems

- Arrival Process
- The distribution that determines how the tasks

arrives in the system. - Service Process
- The distribution that determines the task

processing time - Number of Servers
- Total number of servers available to process the

tasks

Kendall Notation 1/2/3(/4/5/6)

- Six parameters in shorthand
- First three typically used, unless specified
- Arrival Distribution
- Service Distribution
- Number of servers
- Total Capacity (infinite if not specified)
- Population Size (infinite)
- Service Discipline (FCFS/FIFO)

Distributions

- M stands for "Markovian", 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)

Kendall Notation Examples

- M/M/1
- Poisson arrivals and exponential service, 1

server, infinite capacity and population, FCFS

(FIFO) - the simplest realistic queue
- M/M/m
- Same, but M servers
- G/G/3/20/1500/SPF
- General arrival and service distributions, 3

servers, 17 queue slots (20-3), 1500 total jobs,

Shortest Packet First

Analysis of M/M/1 queue

- Given
- l Arrival rate of jobs (packets on input link)
- m Service rate of the server (output link)
- Solve
- L average number in queuing system
- Lq average number in the queue
- W average waiting time in whole system
- Wq average waiting time in the queue

M/M/1 queue model

L

Lq

l

m

Wq

W

Littles Law

System

Arrivals

Departures

- Littles Law Mean number tasks in system mean

arrival rate x mean response time - Observed before, Little was first to prove
- Applies to any system in equilibrium, as long as

nothing in black box is creating or destroying

tasks

Proving Littles Law

Arrivals

Packet

Departures

1 2 3 4 5 6 7 8

Time

J Shaded area 9 Same in all cases!

Definitions

- J Area from previous slide
- N Number of jobs (packets)
- T Total time
- l Average arrival rate
- N/T
- W Average time job is in the system
- J/N
- L Average number of jobs in the system
- J/T

Proof Method 1 Definition

in System (L)

1 2 3 4 5 6 7 8

Time (T)

Proof Method 2 Substitution

Tautology

M/M/1 queue model

L

Lq

l

m

Wq

W

L?W Lq?Wq

Poisson Process

- For a poisson process with average arrival rate

, the probability of seeing n arrivals in time

interval delta t

Poisson process exponential distribution

- Inter-arrival time t (time between arrivals) in a

Poisson process follows exponential distribution

with parameter

M/M/1 queue model

L?W Lq?Wq W Wq (1/µ)

Solving queuing systems

- 4 unknowns L, Lq W, Wq
- Relationships
- LlW
- LqlWq
- W Wq (1/m)
- If we know any 1, can find the others

Analysis of M/M/1 queue

- Goal A closed form expression of the probability

of the number of jobs in the queue (Pi) given

only l and m

Equilibrium conditions

Define to be the probability of having

n tasks in the system at time t

Equilibrium conditions

l

l

l

l

n1

n

n-1

m

m

m

m

Solving for P0 and Pn

- Step 1
- Step 2

Solving for P0 and Pn

- Step 3
- Step 4

Solving for L

Solving W, Wq and Lq

Online M/M/1 animation

- http//www.dcs.ed.ac.uk/home/jeh/Simjava/queueing/

mm1_q/mm1_q.html

Response Time vs. Arrivals

Stable Region

linear region

Example

- On a network gateway, measurements show that the

packets arrive at a mean rate of 125 packets per

second (pps) and the gateway takes about 2

millisecs to forward them. Assuming an M/M/1

model, what is the probability of buffer overflow

if the gateway had only 13 buffers. How many

buffers are needed to keep packet loss below one

packet per million?

Example

- Measurement of a network gateway
- mean arrival rate (l) 125 Packets/s
- mean response time (m) 2 ms
- Assuming exponential arrivals
- What is the gateways utilization?
- What is the probability of n packets in the

gateway? - mean number of packets in the gateway?
- The number of buffers so P(overflow) is lt10-6?

Example

- Arrival rate ?
- Service rate µ
- Gateway utilization ? ?/µ
- Prob. of n packets in gateway
- Mean number of packets in gateway

Example

- Arrival rate ? 125 pps
- Service rate µ 1/0.002 500 pps
- Gateway utilization ? ?/µ 0.25
- Prob. of n packets in gateway
- Mean number of packets in gateway

Example

- Probability of buffer overflow
- To limit the probability of loss to less than

10-6

Example

- Probability of buffer overflow P(more than

13 packets in gateway) - To limit the probability of loss to less than

10-6

Example

- Probability of buffer overflow P(more than

13 packets in gateway) ?13 0.2513

1.49x10-8 15 packets per billion packets - To limit the probability of loss to less than

10-6

Example

- Probability of buffer overflow P(more than

13 packets in gateway) ?13 0.2513

1.49x10-8 15 packets per billion packets - To limit the probability of loss to less than

10-6

Example

- To limit the probability of loss to less than

10-6 - or

Example

- To limit the probability of loss to less than

10-6 - or 9.96

Empirical Example

M/M/m system