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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

Introduction

  • Definition

  • M/M queues

    • M/M/1

    • M/M/S

    • M/M/infinity

    • M/M/S/K


Queuing system

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

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

Queuing systems: examples

  • Multi queue/multi servers

    • Example:

      • Supermarket

      • Blade centers

        • orchestrator

  • Multi-server/single queue

    • Bank

    • immigration

.

.

.


Kendall notation

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

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

Special queuing systems

  • Infinite server queue

  • Machine interference (finite population)

μ

λ

.

.

S repairmen

N

machines


M m 1 queue

M/M/1 queue

  • λn = λ, (n >=0); μn = μ (n>=1)

λ

μ

λ: arrival rate

μ: service rate


Traffic intensity

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


Queuing systems stability

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

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

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


Mean queue length m m 1

Mean queue length (M/M/1)


Mean queue length m m 1 cont d

Mean queue length (M/M/1) (cont’d)


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