Introduction to Queuing Theory

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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.”

(Wolff) “The primary tool for studying these problems [of congestions] is known as queuing theory.”

(Kleinrock) “We study the phenomena of standing, waiting, and serving, and we call this study Queuing 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.”

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Applications of Queuing Theory Telecommunications Traffic control Determining sequence of computer

operations Predicting computer performance Health services (e.g., control of hospital

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

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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

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Example application of queuing theory

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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

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Model Queuing System

Server System Queuing System

Queue Server

Queuing System

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

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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

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

Six parameters in shorthand• First three typically used, unless specified

1. Arrival Distribution2. Service Distribution3. Number of servers 4. Total Capacity (infinite if not specified) 5. Population Size (infinite) 6. Service Discipline (FCFS/FIFO)

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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)

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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

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Analysis of M/M/1 queue

Given: • : Arrival rate of jobs (packets on input link) • : 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

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M/M/1 queue model

Wq

W

L

Lq

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Little’s Law

Little’s 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

Arrivals Departures

System

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Proving Little’s Law

J = Shaded area = 9

Same in all cases!

1 2 3 4 5 6 7 8

Packet #

Time

123

1 2 3 4 5 6 7 8

# in System

123

Time

1 2 3

Time inSystem

Packet #

123

Arrivals

Departures

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Definitions

J: “Area” from previous slide N: Number of jobs (packets) T: Total time : 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

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1 2 3 4 5 6 7 8

# in System(L) 1

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Proof: Method 1: Definition

Time (T) 1 2 3

Time inSystem(W)

Packet # (N)

123

=

WL TN )(

NWTLJ

WL )(

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Proof: Method 2: Substitution

WL TN )(

WL )(

))(( NJ

TN

TJ

TJ

TJ Tautology

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M/M/1 queue model

Wq

W

L

Lq

L=λWLq=λWq

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Poisson Process

For a poisson process with average arrival rate , the probability of seeing n arrivals in time interval delta t

0...)2Pr(

)1Pr()(...]!2

)(1[)1Pr(

1)0Pr()(1...!2

)(1)0Pr(

)(!

)()Pr(

2

2

ttott

ttte

ttott

te

tnEn

ten

t

t

nt

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Poisson process & exponential distribution Inter-arrival time t (time between

arrivals) in a Poisson process follows exponential distribution with parameter

1)(

)Pr(

tE

et t

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M/M/1 queue model

1

Wq

W

L

Lq

L=λWLq=λWqW = Wq + (1/μ)

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Solving queuing systems

4 unknowns: L, Lq W, Wq

Relationships: L=W Lq=Wq W = Wq + (1/)

If we know any 1, can find the others

0

n

nnPL

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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 and

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Equilibrium conditions

n+1nn-1

)(tPnDefine to be the probability of having n tasks in the system at time t

0)()(

lim,)(lim, when Stablize

)()()()()()(

)()()()(

)]1)()[(()]1)()[((])1)(1)[(()(

)]1)()[((])1)(1)[(()(

11

1000

11

100

t

tPttPPtP

tPtPtPt

tPttP

tPtPt

tPttP

tttPtttPtttttPttP

tttPtttttPttP

nn

tnn

t

nnnnn

nnnn

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Equilibrium conditions

11

10

)(

nnn PPP

PP

n+1nn-1

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Solving for P0 and Pn

Step 1

Step 2

0,0

2

201 , PPPPPPn

n

0

0

000

1,1,1

n

n

n

n

nn PPthenP

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Solving for P0 and Pn

Step 3

Step 4

, then

n

n0

n 1

1 n0

1

1 1

P0 1

nn0

1 and Pn n 1

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Solving for L

0

n

nnPL )1(0

n

nn )1(1

1

n

nn

(1 ) dd

11

0

)1(n

ndd

(1 ) 1(1 )2

)1(

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Solving W, Wq and Lq

W L

1 1

Wq W 1

1

( )

Lq Wq ( )

2

( )

L

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Online M/M/1 animation

http://www.dcs.ed.ac.uk/home/jeh/Simjava/queueing/mm1_q/mm1_q.html

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Response Time vs. Arrivals

1W

Waiting vs. Utilization

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2

W(s

ec)

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Stable Region

Waiting vs. Utilization

0

0.005

0.01

0.015

0.02

0.025

0 0.2 0.4 0.6 0.8 1

W(s

ec)

linear region

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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?

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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 gateway’s 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 <10-6?

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Example

Arrival rate λ = Service rate μ = Gateway utilization ρ = λ/μ = Prob. of n packets in gateway =

Mean number of packets in gateway =

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Example Arrival rate λ = 125 pps Service rate μ = 1/0.002 = 500 pps Gateway utilization ρ = λ/μ = 0.25 Prob. of n packets in gateway =

Mean number of packets in gateway =

(1 )n 0.75(0.25)n

1

0.25

0.570.33

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Example

Probability of buffer overflow:

To limit the probability of loss to less than 10-6:

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Example Probability of buffer overflow:

= P(more than 13 packets in gateway)

To limit the probability of loss to less than 10-6:

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Example Probability of buffer overflow:

= P(more than 13 packets in gateway) = ρ13 = 0.2513 = 1.49x10-8

= 15 packets per billion packets To limit the probability of loss to

less than 10-6:

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Example Probability of buffer overflow:

= P(more than 13 packets in gateway) = ρ13 = 0.2513 = 1.49x10-8

= 15 packets per billion packets To limit the probability of loss to

less than 10-6:

n 10 6

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Example To limit the probability of loss to

less than 10-6:

or

25.0log/10log 6n

n 10 6

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Example To limit the probability of loss to

less than 10-6:

or

= 9.96 ≈10 buffers

25.0log/10log 6n

n 10 6

Example

One fast server vs. m slow servers? In terms of delay, what will happen?

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Example

Customers arrive at a fast-food restaurant at a rate of 5/minute and wait to receive their order for an average of 5 minutes. Customers eat in the restaurant with probability 0.5 and carry out their order without eating with probability 0.5. A meal requires an average of 20 minutes. What is the average number of customers in the restaurant?

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Example

Empty taxis pass by a street comer at a Poisson rate of 2 per minute and pick up a passenger if one is waiting there. Passengers arrive at the street corner at a Poisson rate of 1 per minute and wait for a taxi only if there are fewer than 4 persons waiting; otherwise, they leave and never return. Find the average waiting time of a passenger who joins the queue.

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Example

The average time,T, a car spends in a certain traffic system is related to the average number of cars N in the system by a relation of the form T = a + bN2, where a > 0, b > 0 are give in scalars. What is the maximal car arrival rate that the

system can sustain?

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Example

A person enters a bank and finds all of the four clerks busy serving customers. There are no other customers in the bank, so the person will start service as soon as one of the customers in service leaves. Customers have independent, identical, exponential distribution of service time. What is the probability that the person will be the

last to leave the bank assuming that no other customers arrive?

If the average service time is 1 minute, what is the average time the person will spend?

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