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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
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
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