Queueing Systems Part I

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Queueing SystemsPart I

• J. M. Akinpelu

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Queueing Systems

• Examples
• Customers in a grocery store
• Cars on a highway
• Packets in a router network
• Call requests in a telephone network

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Queueing Systems

• Components
• waiting lines (queues)
• servers
• System
• a single queue and server
• a single queue and multiple servers
• multiple queues and a single server
• a network of queues and servers

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Queueing Systems

• Quantities of interest
• Mean number of users in the system
• Mean delay in the system
• Probability that all the servers are busy
• Probability that the queue exceeds some threshold
• Probability that a customer is dropped because
  the queue is full
• Size of queue needed to keep the probability of
  dropping a customer below some threshold

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Queueing Systems

• Delay components
• Queueing delay (time waiting for service)
• Service delay (or service time)
• Total delay in system

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Queueing Systems

• Parameters of a Queueing System
• Number of customers in the system at time t N(t)
• N(t) is the state of the system
• Probability that there are n customers in the
  system at time t Pn(t) PN(t) n
• Steady state probability

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Queueing Systems

• Parameters of a Queueing System
• Mean number of customers in the system at time t
  
• Number of customers in the system in equilibrium
  N
• Mean number of customers in the system in
  equilibrium
• Time average number of customers in the system at
  time t

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Queueing Systems

• Parameters of a Queueing System
• Delay in the system for the nth customer W(n)
• Delay in the system in equilibrium W
• Mean delay in equilibrium

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Ergodic Queueing Systems

• A queueing system is ergodic if the proportion of
  the time interval (0, x) that the system spends
  in state j converges to Pj as x ? ?.
• If a queueing system has a statistical
  equilibrium distribution, then the system is
  ergodic, and

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Littles Theorem

• Theorem 1. Assume that a system in which
  customers enter with rate ? is in equilibrium.
  Then
• that is, the mean number of customers in the
  system equals the arrival rate times the mean
  delay.

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Littles Theorem

• Littles theorem applies to any arrival-departure
  system with appropriate interpretation of the
  mean delay, mean arrival rate, and mean number in
  the system.
• In particular, there are no assumptions on
  arrival and service distributions or service
  disciplines.

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Examples of Littles Theorem

• Infinite queue with arrival rate ? (system
  queue)
• mean number in queue
• mean delay in queue waiting for service
• Heuristic argument The number of customers in
  the queue when you start service are the number
  that arrived during your wait.

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Examples of Littles Theorem

• Single server with arrival rate ?, service rate ?
  gt ? and infinite queue (system server)
• mean number in system
• server utilization ( ?)
• mean service time ( 1/?)

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Examples of Littles Theorem

• Queueing Network
• The system could be the entire network, a
  subnetwork, a class of customers,

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Justification of Littles Theorem

• Let ? (t) be the number of arrivals in (0, t), ?
  (t) be the number of departures in (0, t), and
  ?(t) be the shaded area. Note that N(t) ? (t) -
  ? (t), so that

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Justification of Littles Theorem
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Queueing Systems Notation

• A/B/s/C
• A and B identify the interarrival time and
  service time distributions, respectively
• s is the number of servers
• C is the storage capacity of the system (servers
  plus queue)

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Queueing Systems

• We will consider
• M/M/ systems
• M/G/1 system
• M exponential distribution
• G general distribution

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M/M/1 Queue

• M/M/1/? (or M/M/1) queueing system
• exponential interarrival distribution
• exponential service distribution
• 1 server
• infinite queue
• FCFS service discipline
• Note that this system can be represented as a
  birth-death process. Lets find the steady-state
  solution for the number of customers in the
  system, N.

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M/M/1 Queue
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M/M/1 Queue

• Another view

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M/M/1 Queue

• If ? lt ?, then
• or letting ? ?/? lt 1,

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M/M/1 Queue

• Example ? 0.7

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M/M/1 Queue

• Since N is geometrically distributed,
• and from Littles Theorem,

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M/M/1 Queue

• Whats the probability that there are n or more
  customers in the system?
• So if r is the desired probability of less than n
  in the system then

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M/M/1 Queue

• Example n 5

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M/M/1 Queue

• Expected number in queue
• Expected delay in queue (using Littles Theorem)

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M/M/1 Queue
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Equality of Outside Observers Distribution and
the Arriving Customers Distribution

• Theorem 2. In a system with Poisson arrivals, if
  ?j(t) is the probability that an arriving
  customer at time t finds j customers in the
  system, then

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Counterexample with Non-Poisson Arrivals

• Consider a queueing system consisting of one
  queue and one server. All customers have service
  times equal to 1, and the times between
  successive arrivals is greater than 1. Then each
  arrival sees an empty system (?0 1), but an
  outside observer sees P0 lt 1.

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M/M/1 Queue

• Lemma. If X1, , Xn are independent, identically
  distributed exponential random variables with
  parameter ?, and Y X1 Xn , then Y has a
  gamma distribution with density

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M/M/1 Queue

• Theorem 3. In an M/M/1 queueing system, let NA be
  the number of customers present when a customer
  arrives, then
• The delay in the system W has the equilibrium
  density

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M/M/1 Queue

• Corollary. In an M/M/1 queueing system,

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M/M/1 Queue

• Example ? 0.5, t 10

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Homework

• Problems 5.20, 6.10, 8.1, 8.6, 8.11
• Read Sections 8.3, 8.5

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References

• Erhan Cinlar, Introduction to Stochastic
  Processes, Prentice-Hall, Inc., 1975.
• Robert B. Cooper, Introduction to Queueing
  Theory, Second Edition, North Holland, 1981.
• Leonard Kleinrock, Queueing Systems, Volume I
  Theory, John Wiley Sons, 1975.
• Sheldon M. Ross, Introduction to Probability
  Models, Ninth Edition, Elsevier Inc., 2007.