CPE 619 Introduction to Queuing Theory

1
CPE 619Introduction to Queuing Theory

• Aleksandar Milenkovic
• The LaCASA Laboratory
• Electrical and Computer Engineering Department
• The University of Alabama in Huntsville
• http//www.ece.uah.edu/milenka
• http//www.ece.uah.edu/lacasa

2
Overview

• Queueing Notation
• Rules for All Queues
• Little's Law
• Types of Stochastic Processes

3
Queueing Models What Will You Learn?

• What are various types of queues?
• What is meant by an M/M/m/B/K queue?
• How to obtain response time, queue lengths, and
  server utilizations?
• How to represent a system using a network of
  several queues?
• How to analyze simple queueing networks?
• How to obtain bounds on the system performance
  using queueing models?
• How to obtain variance and other statistics on
  system performance?
• How to subdivide a large queueing network model
  and solve it?

4
Basic Components of a Queue
2. Service time distribution
1. Arrival process
6. Service discipline
4. Waiting positions
5. Customer Population
3. Number of servers
Example students at a typical computer terminal
roomwith a number of terminals. If all terminals
are busy,the arriving students wait in a queue.
5
Kendall Notation A/S/m/B/K/SD

• A Arrival process
• S Service time distribution
• m Number of servers
• B Number of buffers (system capacity)
• K Population size, and
• SD Service discipline

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Arrival Process

• Arrival times
• Interarrival times
• tj form a sequence of Independent and Identically
  Distributed (IID) random variables
• The most common arrival process Poisson arrivals
• Inter-arrival times are exponential IID ?
  Poisson arrivals
• Notation
• M Memoryless Poisson
• E Erlang
• H Hyper-exponential
• G General ? Results valid for all distributions

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Service Time Distribution

• Time each student spends at the terminal
• Service times are IID
• Distribution M, E, H, or G
• Device Service center Queue
• Buffer Waiting positions

8
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 ? Processor Sharing (PS)
• Infinite Server (IS) fixed delay
• 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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Common Distributions

• M Exponential
• Ek Erlang with parameter k
• Hk Hyper-exponential with parameter k
• D Deterministic ? constant
• G General ? All
• Memoryless
• Expected time to the next arrival is always 1/l
  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 arrivals 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 discipline is first-come-first-served
• Defaults
• (Only the first 3 parameters are sufficient to
  indicate the type)
• Infinite buffer capacity
• Infinite population size
• FCFS service discipline
• G/G/1 G/G/1/1/1/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 inter-group 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 servers

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Key Variables
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Key Variables (contd)

• t Inter-arrival time time between two
  successive arrivals
• l Mean arrival rate 1/EtMay be a function
  of the state of the system, e.g., number of jobs
  already in the system
• s Service time per job
• m Mean service rate per server 1/Es
• Total service rate for m servers is mm
• n Number of jobs in the system. This is also
  called queue length
• Note Queue length includes jobs currently
  receiving service as well as those waiting in the
  queue

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Key Variables (contd)

• nq Number of jobs waiting
• ns Number of jobs receiving service
• r Response time or the time in the system
  time waiting time receiving service
• w Waiting time Time between arrival and
  beginning of service

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Rules for All Queues

• Rules The following apply to G/G/m queues
• 1. Stability Condition l lt
  mmFinite-population and the finite-buffer
  systems are always stable
• 2. Number in System versus Number in Queuen
  nq nsNotice that n, nq, and ns are random
  variables. EnEnqEnsIf the service rate
  is independent of the number in the queue,
• Cov(nq,ns) 0

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Rules for All Queues (contd)

• 3. Number versus Time If jobs are not lost due
  to insufficient buffers, Mean number of jobs in
  the system Arrival rate ? Mean response time
• 4. Similarly, Mean number of jobs in the queue
  Arrival rate ? Mean waiting time
• This is known as Little's law.
• 5. Time in System versus Time in Queue
  r w sr, w, and s are random
  variables
• Er Ew Es

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Rules for All Queues (contd)

• 6. If the service rate is independent
• of the number of jobs in the queue,
  Cov(w,s)0

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Little's Law

• Mean number in the system Arrival rate ? 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)
• Based on a black-box view of the system
• In systems 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

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Proof of Little's Law
4
NumberinSystem
3
2
1
1
2
3
4
5
6
7
8
Time
Time

• 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
• Mean time in the system J/N
• Mean Number in the system J/T (N/T) ?(J/N)
  Arrival rate ? Mean time in the system

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Application of Little's Law
Arrivals
Departures

• Applying to just the waiting facility of a
  service center
• Mean number in the queue Arrival rate ? Mean
  waiting time
• Similarly, for those currently receiving the
  service, we have
• Mean number in service Arrival rate ? Mean
  service time

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Example 30.3

• A monitor on a disk server showed that the
  average time to satisfy an I/O request was 100
  milliseconds. The I/O rate was about 100
  requests per second. What was the mean number of
  requests at the disk server?
• Using Little's law
• Mean number in the disk server
• Arrival rate ? Response time
• 100 (requests/second) ?(0.1 seconds)
• 10 requests

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Stochastic Processes

• Process Function of time
• Stochastic Process Random variables, which are
  functions of time
• Example 1
• n(t) number of jobs at the CPU of a computer
  system
• Take several identical systems and observe n(t)
• The number n(t) is a random variable
• Can find the probability distribution functions
  for n(t) at each possible value of t
• Example 2
• w(t) waiting time in a queue

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Types of Stochastic Processes

• 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 w(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. A. Markov who defined and analyzed
  them in 1907
• Markov Chain discrete state Markov process
• Markov ? It is not necessary to know how long the
  process has been in the current state ? 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 Processes

• 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 server and individual arrivals (not bulk
  arrivals)

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Poisson Processes

• Interarrival time s IID and exponential ?
  number of arrivals n over a given interval (t,
  tx) has a Poisson distribution ? arrival
  Poisson process or Poisson stream
• Properties
• 1.Merging
• 2.Splitting If the probability of a job going to
  ith substream is pi, each substream is also
  Poisson with a mean rate of pi l

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Poisson Processes (contd)

• 3.If the arrivals to a single server with
  exponential service time are Poisson with mean
  rate l, the departures are also Poisson with the
  same rate l provided l lt m

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Poisson Process (contd)

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

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Relationship Among Stochastic Processes
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Summary

• Kendall Notation A/S/m/B/k/SD, M/M/1
• Number in system, queue, waiting, serviceService
  rate, arrival rate, response time, waiting time,
  service time
• Littles Law Mean number in system Arrival
  rate X Mean time spent in the system
• Processes Markov ? Memoryless, Birth-death ?
  Adjacent states Poisson ? IID and exponential
  inter-arrival

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Analysis of A Single Queue
33
Overview

• Birth Death Processes
• M/M/1 Queue
• M/M/m Queue
• M/M/m/B Queue with Finite Buffers
• Results for other Queueing systems

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Birth-Death Processes

• Jobs arrive one at a time (and not as a batch)
• State Number of jobs n in the system
• Arrival of a new job changes the state to n1 ?
  birth
• Departure of a job changes the system state to
  n-1 ? death
• State-transition diagram

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Birth-Death Processes (contd)

• When the system is in state n, it has n jobs in
  it.
• The new arrivals take place at a rate ln
• The service rate is mn
• We assume that both the inter-arrival times and
  service times are exponentially distributed

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Theorem State Probability

• The steady-state probability pn of a birth-death
  process being in state n is given by
• Here, p0 is the probability of being in the zero
  state

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Proof

• Suppose the system is in state j at time t. There
  are j jobs in the system. In the next time
  interval of a very small duration Dt, the system
  can move to state j-1 or j1 with the following
  probabilities

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Proof (contd)

• If there are no arrivals or departures, the
  system will stay in state j and, thus
• Dt small ? zero probability of two events (two
  arrivals, two departure, or a arrival and a
  departure) occurring during this interval
• pj(t) probability of being in state j at time t

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Proof (contd)

• The jth equation above can be written as follows
• Under steady state, pj(t) approaches a fixed
  value pj, that is

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Proof (contd)

• Substituting these in the jth equation, we
  get
• The solution to this set of equations is

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Proof (contd)

• The sum of all probabilities must be equal to
  one

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M/M/1 Queue

• M/M/1 queue is the most commonly used type of
  queue
• Used to model single processor systems or to
  model individual devices in a computer system
• Assumes that the interarrival times and the
  service times are exponentially distributed and
  there is only one server
• No buffer or population size limitations and the
  service discipline is FCFS
• Need to know only the mean arrival rate l and the
  mean service rate m
• State number of jobs in the system

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Results for M/M/1 Queue

• Birth-death processes with
• Probability of n jobs in the system

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Results for M/M/1 Queue (contd)

• The quantity l/m is called traffic intensity and
  is usually denoted by symbol r. Thus
• n is geometrically distributed. Utilization of
  the server Probability of having one or more
  jobs in the system

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Results for M/M/1 Queue (contd)

• Mean number of jobs in the system
• Variance of the number of jobs in the system

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Results for M/M/1 Queue (contd)

• Probability of n or more jobs in the system
• Mean response time (using the Little's law)Mean
  number in the system Arrival rate Mean
  response timeThat is

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Results for M/M/1 Queue (contd)

• Cumulative distribution function of the response
  time
• The response time is exponentially distributed.
  ? q-percentile of the response time

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Results for M/M/1 Queue (contd)

• Cumulative distribution function of the waiting
  time
• This is a truncated exponential distribution. Its
  q-percentile is given by
• The above formula applies only if q is greater
  than 100(1-r). All lower percentiles are zero.

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Results for M/M/1 Queue (contd)

• Mean number of jobs in the queue
• Idle ? there are no jobs in the system
• The time interval between two successive idle
  intervals
• All results for M/M/1 queues including some for
  the busy period are summarized in Box 31.1 in the
  book.

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Example 31.2

• 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 two
  milliseconds to forward them. Using an M/M/1
  model, analyze the gateway. What is the
  probability of buffer overflow if the gateway
  had only 13 buffers? How many buffers do we need
  to keep packet loss below one packet per million?
  
• Arrival rate l 125 pps
• Service rate m 1/.002 500 pps
• Gateway Utilization r l/m 0.25
• Probability of n packets in the gateway
  (1-r)rn 0.75(0.25)n)

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Example 31.2 (contd)

• Mean Number of packets in the gateway (r/(1-r)
  0.25/0.75 0.33)
• Mean time spent in the gateway ((1/m)/(1-r)
  (1/500)/(1-0.25) 2.66 milliseconds
• Probability of buffer overflow
• To limit the probability of loss to less than
  10-6 We need about ten buffers.

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Example 31.2 (contd)

• The last two results about buffer overflow are
  approximate. Strictly speaking, the gateway
  should actually be modeled as a finite buffer
  M/M/1/B queue. However, since the utilization is
  low and the number of buffers is far above the
  mean queue length, the results obtained are a
  close approximation.

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Summary

• Birth-death processes Compute probability of
  having n jobs in the system
• M/M/1 Queue Load-independent gt Arrivals and
  service do not depend upon the number in the
  system ln l, mnm
• Traffic Intensity r l/ m
• Mean Number of Jobs in the system r/(1-r)
• Mean Response Time (1/m)/(1-r)

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M/M/1 Queue

• Parameters
• Traffic intensity
• Stability Condition
• Traffic intensity r must be less than one
• Probability of zero jobs in the system p0 1- r
• Probability of n jobs in the system
• Mean number of jobs in the system

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M/M/1 Queue (contd)

• Variance of number of jobs in the system
• Probability of k jobs in the queue

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M/M/1 Queue (contd)

• Mean number of jobs in the queue
• Variance of number of jobs in the queue
• Cumulative distribution function of the response
  time
• Mean response time
• Variance of the response time
• q-percentile of the response time Er
  ln100/(100-q)
• 90-percentile of the response time 2.3Er

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M/M/1 Queue (contd)

• Cumulative distribution function of waiting time
• Mean waiting time
• Variance of the waiting time
• q-percentile of the waiting time
• 90-percentile of the waiting time
• Probability of finding n or more jobs in the
  system rn

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M/M/1 Queue (contd)

• Probability of serving n jobs in one busy period
  
• Mean number of jobs served in one busy period
  Kleinrock, p. 218, eq. 5.155, with
  
• Variance of number of jobs served in one busy
  period
• Mean busy period duration
• Variance of the busy period

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M/M/m Queue