Introduction to Queuing Theory

1
Chapter 30

• Introduction to Queuing Theory

2
Contents

• Queuing Notation
• Rules for all queues
• Littles Law
• Types of Stochastic Processes

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Queuing Notation
4
Arrival Process

• Arrival times t1, t2, tj
• Interarrival time tj tj -tj-1
• tj form a sequence of Independent and
  Identically Distributed (IID) random variables
• Exponential IID gt Poisson
• Erlang
• Hyper-exponential
• General Results valid for all distributions

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Service Time Distribution

• Time each students spends at the terminal
• Service times are IID
• Distribution Exponential, Erlang,
  Hyper-exponential, General.
• General gt Results apply to all service time
  distribution
• Note jobs customers
• device service center queue
• Buffer waiting positions

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Service Disciplines

• First-Come-First-Served (FCFS)
• Last-Come-First-Served (LCFS)
• Last-Come-First-Served with Preempt and Resume
  (LCFS-PR)
• Round-Robin (RR) with a fixed quantum. Small
  Quantum gt Processor Sharing (PS)
• Infinite Server (IS) fixed delay

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Service Discipline (2)

• Shortest Processing Time First (SPT)
• Shortest Remaining Processing Time First (SRPT)
• Shortest Expected Processing Time First (SEPT)
• Shortest Expected Remaining Processing Time First
  (SERPT)
• Biggest-In-First-Served (BIFS)
• Loudest-Voice-First-Served (LVFS)

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Kendall notation

• A/S/m/B/K/SD
• A arrival process
• S service time distribution
• m number of serves
• B number of buffers (system capacity)
• K population size
• SD service disciple

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Common Distribution

• M Exponential
• Ek Erlang with parameter k
• Hk Hyper-exponential with parameter k
• D Deterministic gt constant
• G General gt All
• Note Exponential M memoryless
• Expected time to the next arrival is always 1/?
  regardless of the time since the last arrival.
• Remembering the past history does not help.

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Example

• M/M/3/20/1500/FCFS
• Time between successive arrival is exponentially
  distributed.
• Service times are exponentially distributed.
• Three servers.
• 20 buffers 3 service 17 waiting
• After 20, all arriving jobs are lost.
• Total of 1500 jobs that can be serviced.
• Service disciple is fist-come-first served.

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Defaults

• Infinite buffer capacity
• Infinite population size
• FCFS service disciple
• gt The first three of the six parameter are
  sufficient
• G/G/1G/G/1/8/8/FCFS

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Group Arrivals Service

• Bulk arrivals/service.
• Mx x represents the group size.
• Gx a bulk arrival or service process with
  general intergroup times.
• Examples.
• Mx/M/1 single server queue with bulk Poisson
  arrivals and exponential service times.
• M/Gx/m Poisson arrival process, bulk service
  with general service time distribution, and m
  server.

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Common Random Variables
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Key Variables.

• t Interarrival time time between two
  successive arrivals.
• ? Mean arrival rate 1/E(t)
• May be a function of the state of the system,
  e.g., number of jobs already in the system.
• s Service time per job
• µ Mean service rate per server 1/E(?)
• Total service rate for m servers in mµ

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Key Variables

• n Number of jobs in the system
• This is also called queue length
• Note queue length includes jobs currently
  receiving services as well as those waiting in
  the queue
• nq Number of jobs waiting
• ns Number of jobs receiving service
• r Response time or the time in the system.
• time waiting time receiving services
• ? waiting time
• time between arrival and beginning of service.
• All of the above are random variables except ?
  and µ.

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Rules for All Queues

• The following apply to G/G/m queues
• Stability condition
• ?lt mµ
• Finite-population and the finite-buffer systems
  are always stable.

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Rules for All Queues (2)

• Number in Systems versus Number in Queue
• n nq ns
• Notice that n, nq, ns are random variables.
• En Enq Ens
• If the service rate is independent of the number
  of jobs in the queue,
• Cov(nq, ns)0
• Var(n)Var(nq)Var(ns)

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Rules for All Queues (3)

• Number verses Time If jobs are not lost due to
  insufficient buffers.
• Mean number of jobs in the system (Arrival
  rate) x (Mean response time)
• Mean number of the jobs in the queue
• (Arrival rate) x (Mean waiting time)
• This is Littles law

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Rules for All Queues (4)

• 4. Time in System versus Time in Queue
• r ? s
• r, ?, s are random variables.
• Er E? Ens
• If the service rate is independent of the number
  of jobs in the queue,
• Cov(?, s)0
• Var(r)Var(?)Var(s)

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Littles Law

• Mean number in the system Arrival rate x mean
  response time
• This relationship applies to all systems or parts
  of systems in which the number of jobs entering
  the system is equal to those completing service.
• Named after Little (1961)

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Littles Law (2)

• Based on a black-box view of the system
• Arrival Departures
• In system in which some jobs are lost due to
  finite buffers, the law can be applied to the
  part of the system consisting of the waiting and
  serving positions.

Black Box
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Arrival and Departure Time Date
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Proof of Littles Law

• Monitor the system for
• a time interval T. If T is large,
• Arrivals departures N
• Arrival Rate Total arrivals/Total time N/T
• Hatched areas total time spent inside the
  system by all jobs J

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Proof of Littles Law (2)

• From (c)
• Mea time in the system J/N

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Proof of Littles Law (3)

• From (b)
• Mea Number in the system J/T
• N/T x J/N
• Arrival rate x Mean time in the system

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Application of Littles Law

• Applying to just the waiting facility of a
  service center

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Application of Littles Law

• Mean number in the queue Arrival rate x Mean
  waiting time
• Similarity, for those currently receiving the
  service, we have
• Mean number in service Arrival rate x Mean
  service time

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Examples of Littles Law

• A monitor on a disk server showed that the
  average time to satisfy an I/O request was 100
  ms. The I/O rate was about 100 request/s. What
  was the mean number of request at the disk
  server?
• Mean number in the disk server
• Arrival rate x Response time
• (100 request/s) x (0.1 s)
• 10 request

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Stochastic Processes

• Process Function of time.
• Stochastic Process Random variable, which are
  function of time
• Examples 1 n(t) number of job at the CPU of a
  computer system.
• Take several identical system and observe n(t),
  it is random variable.
• Example 2 w(t) waiting time in a queue

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Type of Stochastic Process

• Discrete or continuous state Processes
• Markov Processes
• Birth-death Processes
• Poisson Processes

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Discrete/Continuous State Processes

• Discrete finite or countable
• Number of jobs in a system n(t) 0, 1, 2, ..
• n(t) is a discrete state process
• The waiting time ?(t) is a continuous state
  process.
• Stochastic Chain discrete state stochastic
  process.

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Markov Processes

• Future states are independent of the past and
  depend only on the present.
• Named after A. Markov who defined and analyzed
  them in 1907.
• Markov Chain discrete state Markov process
• Markov gt it is not necessary to know how long
  the process has been in the current state gt
  state time has a memoryless (exponential)
  distribution.
• M/M/m queues can be modeled using Markov
  processes.
• The time spent by a job in such a queue is a
  Markov process and the number of jobs in the
  queue is a Markov chain.

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Birth-Death Process

• The discrete space Markov processes in which the
  transitions are restricted to neighboring states.
• Process in state n can change only to state n1
  or n-1.
• Example the number of jobs in a queue with a
  single serve and individual arrivals (not bulk
  arrivals)

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Poisson Processes

• Interarrival time sIID exponential
• gtnumber of arrivals n over a given interval (t,
  tx) has a Poisson distribution.
• gt arrival Poisson process or Poisson stream

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Property 1

• Merging

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Property 2

• Splitting If the probability of a job going to
  i-th substream is pi, each substream is also
  Poisson with a mean rate of pi ?.

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Property 3

• If the arrivals to a single server with
  exponential service time are Poisson with mean
  rate ?, the departures are also Poisson with the
  same rate ?, provided ?ltµ.

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Property 4

• If the arrivals to a service facility with m
  service centers are Poisson with a mean rate ?,
  the departures also constitute a Poisson stream
  with the same rate ?, provided ?lt?µi. Here the
  servers are assumed to have exponentially
  distributed service times.

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Relationship Among Stochastic Processes