# Introduction - PowerPoint PPT Presentation

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Introduction. Definition M/M queues M/M/1 M/M/S M/M/infinity M/M/S/K. Queuing system. A queuing system is a place where customers arrive According to an “arrival process” To receive service from a service facility Can be broken down into three major components The input process

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Introduction

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### Introduction

• Definition

• M/M queues

• M/M/1

• M/M/S

• M/M/infinity

• M/M/S/K

### Queuing system

• A queuing system

• is a place where customers arrive

• According to an “arrival process”

• To receive service from a service facility

• Can be broken down into three major components

• The input process

• The system structure

• The output process

Customer

Population

Service

facility

Waiting

queue

### Characteristics of the system structure

λ: arrival rate

μ: service rate

• Queue

• Infinite or finite

• Service mechanism

• 1 server or S servers

• Queuing discipline

• FIFO, LIFO, priority-aware, or random

λ

μ

### Queuing systems: examples

• Multi queue/multi servers

• Example:

• Supermarket

• orchestrator

• Multi-server/single queue

• Bank

• immigration

.

.

.

M: Markovian

D: constant

G: general

Cx: coxian

### Kendall notation

• David Kendall

• A British statistician, developed a shorthand notation

• To describe a queuing system

• A/B/X/Y/Z

• A: Customer arriving pattern

• B: Service pattern

• X: Number of parallel servers

• Y: System capacity

• Z: Queuing discipline

### Kendall notation: example

• M/M/1/infinity

• A queuing system having one server where

• Customers arrive according to a Poisson process

• Exponentially distributed service times

• M/M/S/K

• M/M/S/K=0

• Erlang loss queue

K

### Special queuing systems

• Infinite server queue

• Machine interference (finite population)

μ

λ

.

.

S repairmen

N

machines

### M/M/1 queue

• λn = λ, (n >=0); μn = μ (n>=1)

λ

μ

λ: arrival rate

μ: service rate

### Traffic intensity

• rho = λ/μ

• It is a measure of the total arrival traffic to the system

• Also known as offered load

• Example: λ = 3/hour; 1/μ=15 min = 0.25 h

• Represents the fraction of time a server is busy

• In which case it is called the utilization factor

• Example: rho = 0.75 = % busy

3

3

2

2

1

1

busy

idle

### Queuing systems: stability

N(t)

• λ<μ

• => stable system

• λ>μ

• Steady build up of customers => unstable

1 2 3 4 5 6 7 8 9 10 11

Time

N(t)

1 2 3 4 5 6 7 8 9 10 11

Time

### Example#1

• A communication channel operating at 9600 bps

• Receives two type of packet streams from a gateway

• Type A packets have a fixed length format of 48 bits

• Type B packets have an exponentially distribution length

• With a mean of 480 bits

• If on the average there are

• 20% type A packets and 80% type B packets

• Calculate the utilization of this channel

• Assuming the combined arrival rate is 15 packets/s

### Performance measures

• L

• Mean # customers in the whole system

• Lq

• Mean queue length in the queue space

• W

• Mean waiting time in the system

• Wq

• Mean waiting time in the queue